3.637 \(\int \frac {1}{(d+e x)^{5/2} (a+c x^2)^2} \, dx\)

Optimal. Leaf size=930 \[ \frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (c x^2+a\right )}+\frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}} \]

[Out]

1/6*e*(-7*a*e^2+3*c*d^2)/a/(a*e^2+c*d^2)^2/(e*x+d)^(3/2)+1/2*(c*d*x+a*e)/a/(a*e^2+c*d^2)/(e*x+d)^(3/2)/(c*x^2+
a)+1/2*c*d*e*(-19*a*e^2+c*d^2)/a/(a*e^2+c*d^2)^3/(e*x+d)^(1/2)+1/8*c^(3/4)*e*arctanh((-c^(1/4)*2^(1/2)*(e*x+d)
^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2))*(c^2*d^4+34*a*c*d^2*e^2-7
*a^2*e^4+d*(-19*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a/(a*e^2+c*d^2)^(7/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^
2)^(1/2))^(1/2)-1/8*c^(3/4)*e*arctanh((c^(1/4)*2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d
*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2))*(c^2*d^4+34*a*c*d^2*e^2-7*a^2*e^4+d*(-19*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^
2)^(1/2))/a/(a*e^2+c*d^2)^(7/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/16*c^(3/4)*e*ln((e*x+d)*c^(1/2
)+(a*e^2+c*d^2)^(1/2)-c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))*(c^2*d^4+34*a*c*d^2
*e^2-7*a^2*e^4-d*(-19*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a/(a*e^2+c*d^2)^(7/2)*2^(1/2)/(d*c^(1/2)+(a*e^
2+c*d^2)^(1/2))^(1/2)+1/16*c^(3/4)*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)+c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c
^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))*(c^2*d^4+34*a*c*d^2*e^2-7*a^2*e^4-d*(-19*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)
^(1/2))/a/(a*e^2+c*d^2)^(7/2)*2^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 6.49, antiderivative size = 930, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {741, 829, 827, 1169, 634, 618, 206, 628} \[ \frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (c x^2+a\right )}+\frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(5/2)*(a + c*x^2)^2),x]

[Out]

(e*(3*c*d^2 - 7*a*e^2))/(6*a*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2)) + (c*d*e*(c*d^2 - 19*a*e^2))/(2*a*(c*d^2 + a*e
^2)^3*Sqrt[d + e*x]) + (a*e + c*d*x)/(2*a*(c*d^2 + a*e^2)*(d + e*x)^(3/2)*(a + c*x^2)) + (c^(3/4)*e*(c^2*d^4 +
34*a*c*d^2*e^2 - 7*a^2*e^4 + Sqrt[c]*d*(c*d^2 - 19*a*e^2)*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt
[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*(c*d^2
+ a*e^2)^(7/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(3/4)*e*(c^2*d^4 + 34*a*c*d^2*e^2 - 7*a^2*e^4 + Sqr
t[c]*d*(c*d^2 - 19*a*e^2)*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4
)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*(c*d^2 + a*e^2)^(7/2)*Sqrt[Sqrt[c]*d - S
qrt[c*d^2 + a*e^2]]) - (c^(3/4)*e*(c^2*d^4 + 34*a*c*d^2*e^2 - 7*a^2*e^4 - Sqrt[c]*d*(c*d^2 - 19*a*e^2)*Sqrt[c*
d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] +
Sqrt[c]*(d + e*x)])/(8*Sqrt[2]*a*(c*d^2 + a*e^2)^(7/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (c^(3/4)*e*(c^
2*d^4 + 34*a*c*d^2*e^2 - 7*a^2*e^4 - Sqrt[c]*d*(c*d^2 - 19*a*e^2)*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2]
+ Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(8*Sqrt[2]*a*(c*d
^2 + a*e^2)^(7/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 741

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(
a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 829

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((e*f - d*g)*(d
+ e*x)^(m + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d*f + a*
e*g - c*(e*f - d*g)*x, x])/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&
FractionQ[m] && LtQ[m, -1]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx &=\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (-2 c d^2-7 a e^2\right )-\frac {5}{2} c d e x}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}-\frac {\int \frac {-c d \left (c d^2+6 a e^2\right )-\frac {1}{2} c e \left (3 c d^2-7 a e^2\right ) x}{(d+e x)^{3/2} \left (a+c x^2\right )} \, dx}{2 a \left (c d^2+a e^2\right )^2}\\ &=\frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}-\frac {\int \frac {-\frac {1}{2} c \left (2 c^2 d^4+15 a c d^2 e^2-7 a^2 e^4\right )-\frac {1}{2} c^2 d e \left (c d^2-19 a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{2 a \left (c d^2+a e^2\right )^3}\\ &=\frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} c^2 d^2 e \left (c d^2-19 a e^2\right )-\frac {1}{2} c e \left (2 c^2 d^4+15 a c d^2 e^2-7 a^2 e^4\right )-\frac {1}{2} c^2 d e \left (c d^2-19 a e^2\right ) x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{a \left (c d^2+a e^2\right )^3}\\ &=\frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\frac {1}{2} c^2 d^2 e \left (c d^2-19 a e^2\right )-\frac {1}{2} c e \left (2 c^2 d^4+15 a c d^2 e^2-7 a^2 e^4\right )\right )}{\sqrt [4]{c}}-\left (\frac {1}{2} c^2 d^2 e \left (c d^2-19 a e^2\right )+\frac {1}{2} c^{3/2} d e \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}-\frac {1}{2} c e \left (2 c^2 d^4+15 a c d^2 e^2-7 a^2 e^4\right )\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\frac {1}{2} c^2 d^2 e \left (c d^2-19 a e^2\right )-\frac {1}{2} c e \left (2 c^2 d^4+15 a c d^2 e^2-7 a^2 e^4\right )\right )}{\sqrt [4]{c}}+\left (\frac {1}{2} c^2 d^2 e \left (c d^2-19 a e^2\right )+\frac {1}{2} c^{3/2} d e \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}-\frac {1}{2} c e \left (2 c^2 d^4+15 a c d^2 e^2-7 a^2 e^4\right )\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}-\frac {\left (c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (\sqrt {c} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 a \left (c d^2+a e^2\right )^{7/2}}+\frac {\left (\sqrt {c} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 a \left (c d^2+a e^2\right )^{7/2}}\\ &=\frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}-\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (\sqrt {c} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{4 a \left (c d^2+a e^2\right )^{7/2}}-\frac {\left (\sqrt {c} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{4 a \left (c d^2+a e^2\right )^{7/2}}\\ &=\frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}+\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.45, size = 321, normalized size = 0.35 \[ \frac {-\frac {\left (3 c d^2 e-7 a e^3\right ) \left (\frac {\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\sqrt {-a} \sqrt {c} d-a e}-\frac {\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\sqrt {-a} \sqrt {c} d+a e}\right )}{e}+15 c d (d+e x) \left (\frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\sqrt {-a} \sqrt {c} d-a e}-\frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\sqrt {-a} \sqrt {c} d+a e}\right )+\frac {6 (a e+c d x)}{a+c x^2}}{12 a (d+e x)^{3/2} \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(5/2)*(a + c*x^2)^2),x]

[Out]

((6*(a*e + c*d*x))/(a + c*x^2) - ((3*c*d^2*e - 7*a*e^3)*(-(Hypergeometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x)
)/(Sqrt[c]*d - Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d + a*e)) + Hypergeometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))
/(Sqrt[c]*d + Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d - a*e)))/e + 15*c*d*(d + e*x)*(-(Hypergeometric2F1[-1/2, 1, 1/2
, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d + a*e)) + Hypergeometric2F1[-1/2, 1, 1/2,
(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d - a*e)))/(12*a*(c*d^2 + a*e^2)*(d + e*x)^(3/
2))

________________________________________________________________________________________

fricas [B]  time = 4.96, size = 8281, normalized size = 8.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+a)^2,x, algorithm="fricas")

[Out]

1/24*(3*(a^2*c^3*d^8 + 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 + a^5*d^2*e^6 + (a*c^4*d^6*e^2 + 3*a^2*c^3*d^4*e^4
+ 3*a^3*c^2*d^2*e^6 + a^4*c*e^8)*x^4 + 2*(a*c^4*d^7*e + 3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)*x
^3 + (a*c^4*d^8 + 4*a^2*c^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4 + 4*a^4*c*d^2*e^6 + a^5*e^8)*x^2 + 2*(a^2*c^3*d^7*e +
3*a^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5 + a^5*d*e^7)*x)*sqrt(-(4*c^6*d^9 + 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4
- 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 + (a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6
*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14)*sqrt(-(11025*c^9*d^12*
e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 8
2026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 + 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*
a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d
^14*e^14 + 3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22
+ 91*a^15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^1
0*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14))*log((42
0*c^6*d^8*e^3 + 8421*a*c^5*d^6*e^5 + 36783*a^2*c^4*d^4*e^7 - 40817*a^3*c^3*d^2*e^9 + 2401*a^4*c^2*e^11)*sqrt(e
*x + d) + (105*a^2*c^6*d^10*e^4 + 4389*a^3*c^5*d^8*e^6 + 26274*a^4*c^4*d^6*e^8 - 34142*a^5*c^3*d^4*e^10 + 7525
*a^6*c^2*d^2*e^12 - 343*a^7*c*e^14 + 2*(a^3*c^9*d^19 + 15*a^4*c^8*d^17*e^2 + 64*a^5*c^7*d^15*e^4 + 112*a^6*c^6
*d^13*e^6 + 42*a^7*c^5*d^11*e^8 - 154*a^8*c^4*d^9*e^10 - 280*a^9*c^3*d^7*e^12 - 216*a^10*c^2*d^5*e^14 - 83*a^1
1*c*d^3*e^16 - 13*a^12*d*e^18)*sqrt(-(11025*c^9*d^12*e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1
360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28
+ 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9
*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e
^18 + 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28))
)*sqrt(-(4*c^6*d^9 + 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 - 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 + (a^3*
c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4
*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14)*sqrt(-(11025*c^9*d^12*e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e
^10 - 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^
14*d^28 + 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*
a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5
*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17
*e^28)))/(a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 +
21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14))) - 3*(a^2*c^3*d^8 + 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 +
a^5*d^2*e^6 + (a*c^4*d^6*e^2 + 3*a^2*c^3*d^4*e^4 + 3*a^3*c^2*d^2*e^6 + a^4*c*e^8)*x^4 + 2*(a*c^4*d^7*e + 3*a^
2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)*x^3 + (a*c^4*d^8 + 4*a^2*c^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4 + 4*
a^4*c*d^2*e^6 + a^5*e^8)*x^2 + 2*(a^2*c^3*d^7*e + 3*a^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5 + a^5*d*e^7)*x)*sqrt(-(4
*c^6*d^9 + 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 - 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 + (a^3*c^7*d^14 +
7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*
a^9*c*d^2*e^12 + a^10*e^14)*sqrt(-(11025*c^9*d^12*e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360
716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 +
14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^
18*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18
+ 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))/(
a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2
*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14))*log((420*c^6*d^8*e^3 + 8421*a*c^5*d^6*e^5 + 36783*a^2*c^4*d^4*e^7 -
40817*a^3*c^3*d^2*e^9 + 2401*a^4*c^2*e^11)*sqrt(e*x + d) - (105*a^2*c^6*d^10*e^4 + 4389*a^3*c^5*d^8*e^6 + 262
74*a^4*c^4*d^6*e^8 - 34142*a^5*c^3*d^4*e^10 + 7525*a^6*c^2*d^2*e^12 - 343*a^7*c*e^14 + 2*(a^3*c^9*d^19 + 15*a^
4*c^8*d^17*e^2 + 64*a^5*c^7*d^15*e^4 + 112*a^6*c^6*d^13*e^6 + 42*a^7*c^5*d^11*e^8 - 154*a^8*c^4*d^9*e^10 - 280
*a^9*c^3*d^7*e^12 - 216*a^10*c^2*d^5*e^14 - 83*a^11*c*d^3*e^16 - 13*a^12*d*e^18)*sqrt(-(11025*c^9*d^12*e^6 + 1
71990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^
5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 + 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^1
1*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^1
4 + 3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^
15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))*sqrt(-(4*c^6*d^9 + 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4
- 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 + (a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6
*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14)*sqrt(-(11025*c^9*d^12*
e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 8
2026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 + 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*
a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d
^14*e^14 + 3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22
+ 91*a^15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^1
0*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14))) + 3*(a
^2*c^3*d^8 + 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 + a^5*d^2*e^6 + (a*c^4*d^6*e^2 + 3*a^2*c^3*d^4*e^4 + 3*a^3*c^
2*d^2*e^6 + a^4*c*e^8)*x^4 + 2*(a*c^4*d^7*e + 3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)*x^3 + (a*c^
4*d^8 + 4*a^2*c^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4 + 4*a^4*c*d^2*e^6 + a^5*e^8)*x^2 + 2*(a^2*c^3*d^7*e + 3*a^3*c^2*
d^5*e^3 + 3*a^4*c*d^3*e^5 + a^5*d*e^7)*x)*sqrt(-(4*c^6*d^9 + 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 - 1155*a^3
*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 - (a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e
^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14)*sqrt(-(11025*c^9*d^12*e^6 + 1719
90*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c
^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 + 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*d
^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14 +
3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*
c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35
*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14))*log((420*c^6*d^8*
e^3 + 8421*a*c^5*d^6*e^5 + 36783*a^2*c^4*d^4*e^7 - 40817*a^3*c^3*d^2*e^9 + 2401*a^4*c^2*e^11)*sqrt(e*x + d) +
(105*a^2*c^6*d^10*e^4 + 4389*a^3*c^5*d^8*e^6 + 26274*a^4*c^4*d^6*e^8 - 34142*a^5*c^3*d^4*e^10 + 7525*a^6*c^2*d
^2*e^12 - 343*a^7*c*e^14 - 2*(a^3*c^9*d^19 + 15*a^4*c^8*d^17*e^2 + 64*a^5*c^7*d^15*e^4 + 112*a^6*c^6*d^13*e^6
+ 42*a^7*c^5*d^11*e^8 - 154*a^8*c^4*d^9*e^10 - 280*a^9*c^3*d^7*e^12 - 216*a^10*c^2*d^5*e^14 - 83*a^11*c*d^3*e^
16 - 13*a^12*d*e^18)*sqrt(-(11025*c^9*d^12*e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360716*a^3
*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 + 14*a^4*
c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^10
+ 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18 + 1001
*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))*sqrt(-(4
*c^6*d^9 + 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 - 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 - (a^3*c^7*d^14 +
7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*
a^9*c*d^2*e^12 + a^10*e^14)*sqrt(-(11025*c^9*d^12*e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360
716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 +
14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^
18*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18
+ 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))/(
a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2
*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14))) - 3*(a^2*c^3*d^8 + 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 + a^5*d^2*e
^6 + (a*c^4*d^6*e^2 + 3*a^2*c^3*d^4*e^4 + 3*a^3*c^2*d^2*e^6 + a^4*c*e^8)*x^4 + 2*(a*c^4*d^7*e + 3*a^2*c^3*d^5*
e^3 + 3*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)*x^3 + (a*c^4*d^8 + 4*a^2*c^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4 + 4*a^4*c*d^2*
e^6 + a^5*e^8)*x^2 + 2*(a^2*c^3*d^7*e + 3*a^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5 + a^5*d*e^7)*x)*sqrt(-(4*c^6*d^9 +
63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 - 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 - (a^3*c^7*d^14 + 7*a^4*c^6
*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*
e^12 + a^10*e^14)*sqrt(-(11025*c^9*d^12*e^6 + 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360716*a^3*c^
6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 + 14*a^4*c^1
3*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^10 +
3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18 + 1001*a^
13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^
14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10
+ 7*a^9*c*d^2*e^12 + a^10*e^14))*log((420*c^6*d^8*e^3 + 8421*a*c^5*d^6*e^5 + 36783*a^2*c^4*d^4*e^7 - 40817*a^3
*c^3*d^2*e^9 + 2401*a^4*c^2*e^11)*sqrt(e*x + d) - (105*a^2*c^6*d^10*e^4 + 4389*a^3*c^5*d^8*e^6 + 26274*a^4*c^4
*d^6*e^8 - 34142*a^5*c^3*d^4*e^10 + 7525*a^6*c^2*d^2*e^12 - 343*a^7*c*e^14 - 2*(a^3*c^9*d^19 + 15*a^4*c^8*d^17
*e^2 + 64*a^5*c^7*d^15*e^4 + 112*a^6*c^6*d^13*e^6 + 42*a^7*c^5*d^11*e^8 - 154*a^8*c^4*d^9*e^10 - 280*a^9*c^3*d
^7*e^12 - 216*a^10*c^2*d^5*e^14 - 83*a^11*c*d^3*e^16 - 13*a^12*d*e^18)*sqrt(-(11025*c^9*d^12*e^6 + 171990*a*c^
8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c^4*d^2*
e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 + 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*d^22*e^6
+ 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14 + 3003*a
^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4
*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))*sqrt(-(4*c^6*d^9 + 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 - 1155*a^3
*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 - (a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35*a^6*c^4*d^8*e
^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14)*sqrt(-(11025*c^9*d^12*e^6 + 1719
90*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 - 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 - 82026*a^5*c
^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 + 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 + 364*a^6*c^11*d
^22*e^6 + 1001*a^7*c^10*d^20*e^8 + 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 + 3432*a^10*c^7*d^14*e^14 +
3003*a^11*c^6*d^12*e^16 + 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 + 364*a^14*c^3*d^6*e^22 + 91*a^15*
c^2*d^4*e^24 + 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 + 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 + 35
*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 + 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 + a^10*e^14))) + 4*(9*a*c^2*d^4
*e - 55*a^2*c*d^2*e^3 - 4*a^3*e^5 + 3*(c^3*d^3*e^2 - 19*a*c^2*d*e^4)*x^3 + (6*c^3*d^4*e - 61*a*c^2*d^2*e^3 - 7
*a^2*c*e^5)*x^2 + 3*(c^3*d^5 + 3*a*c^2*d^3*e^2 - 18*a^2*c*d*e^4)*x)*sqrt(e*x + d))/(a^2*c^3*d^8 + 3*a^3*c^2*d^
6*e^2 + 3*a^4*c*d^4*e^4 + a^5*d^2*e^6 + (a*c^4*d^6*e^2 + 3*a^2*c^3*d^4*e^4 + 3*a^3*c^2*d^2*e^6 + a^4*c*e^8)*x^
4 + 2*(a*c^4*d^7*e + 3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)*x^3 + (a*c^4*d^8 + 4*a^2*c^3*d^6*e^2
+ 6*a^3*c^2*d^4*e^4 + 4*a^4*c*d^2*e^6 + a^5*e^8)*x^2 + 2*(a^2*c^3*d^7*e + 3*a^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5
+ a^5*d*e^7)*x)

________________________________________________________________________________________

giac [B]  time = 1.42, size = 1928, normalized size = 2.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+a)^2,x, algorithm="giac")

[Out]

-1/4*((a*c^3*d^6*e + 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 + a^4*e^7)^2*(c^2*d^3*e - 19*a*c*d*e^3)*abs(c) - (sqr
t(-a*c)*c^5*d^10*e + 37*sqrt(-a*c)*a*c^4*d^8*e^3 + 98*sqrt(-a*c)*a^2*c^3*d^6*e^5 + 82*sqrt(-a*c)*a^3*c^2*d^4*e
^7 + 13*sqrt(-a*c)*a^4*c*d^2*e^9 - 7*sqrt(-a*c)*a^5*e^11)*abs(-a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 - 3*a^3*c*d^2*e
^5 - a^4*e^7)*abs(c) + (2*a*c^9*d^17*e + 27*a^2*c^8*d^15*e^3 + 113*a^3*c^7*d^13*e^5 + 223*a^4*c^6*d^11*e^7 + 2
25*a^5*c^5*d^9*e^9 + 97*a^6*c^4*d^7*e^11 - 13*a^7*c^3*d^5*e^13 - 27*a^8*c^2*d^3*e^15 - 7*a^9*c*d*e^17)*abs(c))
*arctan(sqrt(x*e + d)/sqrt(-(a*c^4*d^7 + 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6 + sqrt((a*c^4*d^7
+ 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6)^2 - (a*c^4*d^8 + 4*a^2*c^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4
+ 4*a^4*c*d^2*e^6 + a^5*e^8)*(a*c^4*d^6 + 3*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4 + a^4*c*e^6)))/(a*c^4*d^6 + 3
*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4 + a^4*c*e^6)))/((a^2*c^6*d^12*e - sqrt(-a*c)*a*c^6*d^13 - 6*sqrt(-a*c)*a^
2*c^5*d^11*e^2 + 6*a^3*c^5*d^10*e^3 - 15*sqrt(-a*c)*a^3*c^4*d^9*e^4 + 15*a^4*c^4*d^8*e^5 - 20*sqrt(-a*c)*a^4*c
^3*d^7*e^6 + 20*a^5*c^3*d^6*e^7 - 15*sqrt(-a*c)*a^5*c^2*d^5*e^8 + 15*a^6*c^2*d^4*e^9 - 6*sqrt(-a*c)*a^6*c*d^3*
e^10 + 6*a^7*c*d^2*e^11 - sqrt(-a*c)*a^7*d*e^12 + a^8*e^13)*sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(-a*c^3*d^6*e - 3
*a^2*c^2*d^4*e^3 - 3*a^3*c*d^2*e^5 - a^4*e^7)) - 1/4*((a*c^3*d^6*e + 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 + a^4
*e^7)^2*(c^2*d^3*e - 19*a*c*d*e^3)*abs(c) + (sqrt(-a*c)*c^5*d^10*e + 37*sqrt(-a*c)*a*c^4*d^8*e^3 + 98*sqrt(-a*
c)*a^2*c^3*d^6*e^5 + 82*sqrt(-a*c)*a^3*c^2*d^4*e^7 + 13*sqrt(-a*c)*a^4*c*d^2*e^9 - 7*sqrt(-a*c)*a^5*e^11)*abs(
-a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 - 3*a^3*c*d^2*e^5 - a^4*e^7)*abs(c) + (2*a*c^9*d^17*e + 27*a^2*c^8*d^15*e^3 +
113*a^3*c^7*d^13*e^5 + 223*a^4*c^6*d^11*e^7 + 225*a^5*c^5*d^9*e^9 + 97*a^6*c^4*d^7*e^11 - 13*a^7*c^3*d^5*e^13
- 27*a^8*c^2*d^3*e^15 - 7*a^9*c*d*e^17)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^4*d^7 + 3*a^2*c^3*d^5*e^2 + 3
*a^3*c^2*d^3*e^4 + a^4*c*d*e^6 - sqrt((a*c^4*d^7 + 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6)^2 - (a
*c^4*d^8 + 4*a^2*c^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4 + 4*a^4*c*d^2*e^6 + a^5*e^8)*(a*c^4*d^6 + 3*a^2*c^3*d^4*e^2 +
3*a^3*c^2*d^2*e^4 + a^4*c*e^6)))/(a*c^4*d^6 + 3*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4 + a^4*c*e^6)))/((a^2*c^6*
d^12*e + sqrt(-a*c)*a*c^6*d^13 + 6*sqrt(-a*c)*a^2*c^5*d^11*e^2 + 6*a^3*c^5*d^10*e^3 + 15*sqrt(-a*c)*a^3*c^4*d^
9*e^4 + 15*a^4*c^4*d^8*e^5 + 20*sqrt(-a*c)*a^4*c^3*d^7*e^6 + 20*a^5*c^3*d^6*e^7 + 15*sqrt(-a*c)*a^5*c^2*d^5*e^
8 + 15*a^6*c^2*d^4*e^9 + 6*sqrt(-a*c)*a^6*c*d^3*e^10 + 6*a^7*c*d^2*e^11 + sqrt(-a*c)*a^7*d*e^12 + a^8*e^13)*sq
rt(-c^2*d - sqrt(-a*c)*c*e)*abs(-a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 - 3*a^3*c*d^2*e^5 - a^4*e^7)) + 1/2*((x*e + d
)^(3/2)*c^3*d^3*e - sqrt(x*e + d)*c^3*d^4*e - 3*(x*e + d)^(3/2)*a*c^2*d*e^3 + 6*sqrt(x*e + d)*a*c^2*d^2*e^3 -
sqrt(x*e + d)*a^2*c*e^5)/((a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*((x*e + d)^2*c - 2*(x*e
+ d)*c*d + c*d^2 + a*e^2)) - 2/3*(12*(x*e + d)*c*d*e^3 + c*d^2*e^3 + a*e^5)/((c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^
2*c*d^2*e^4 + a^3*e^6)*(x*e + d)^(3/2))

________________________________________________________________________________________

maple [B]  time = 0.32, size = 10403, normalized size = 11.19 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(5/2)/(c*x^2+a)^2,x)

[Out]

result too large to display

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2} + a\right )}^{2} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+a)^2,x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)^2*(e*x + d)^(5/2)), x)

________________________________________________________________________________________

mupad [B]  time = 4.74, size = 12390, normalized size = 13.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + c*x^2)^2*(d + e*x)^(5/2)),x)

[Out]

atan((((d + e*x)^(1/2)*(1568*a^16*c^5*e^28 + 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 + 29120*a^5*c^16*d
^22*e^6 + 128128*a^6*c^15*d^20*e^8 + 282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 - 282240*a^9*c^12*d
^14*e^14 - 1059840*a^10*c^11*d^12*e^16 - 1403904*a^11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 - 456512*a^13
*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 - 4160*a^15*c^6*d^2*e^26) + (-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^
(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*
(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a
^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*
d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*
a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(
1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14
+ 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21
*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 + 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 + 215040*
a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d^25*e^8 + 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^12 + 102
50240*a^12*c^13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 + 13178880*a^14*c^11*d^15*e^18 + 10250240*a^15*c^10*d
^13*e^20 + 6150144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24 + 931840*a^18*c^7*d^7*e^26 + 215040*a^19*c^6
*d^5*e^28 + 30720*a^20*c^5*d^3*e^30) - 1792*a^19*c^4*e^31 + 256*a^5*c^18*d^28*e^3 + 11776*a^6*c^17*d^26*e^5 +
119552*a^7*c^16*d^24*e^7 + 609280*a^8*c^15*d^22*e^9 + 1923328*a^9*c^14*d^20*e^11 + 4116992*a^10*c^13*d^18*e^13
+ 6243072*a^11*c^12*d^16*e^15 + 6825984*a^12*c^11*d^14*e^17 + 5364480*a^13*c^10*d^12*e^19 + 2945536*a^14*c^9*
d^10*e^21 + 1044736*a^15*c^8*d^8*e^23 + 183296*a^16*c^7*d^6*e^25 - 13568*a^17*c^6*d^4*e^27 - 12800*a^18*c^5*d^
2*e^29))*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4
*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*
a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^
8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*1i + ((d + e*x)^(1/2
)*(1568*a^16*c^5*e^28 + 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 + 29120*a^5*c^16*d^22*e^6 + 128128*a^6*
c^15*d^20*e^8 + 282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 - 282240*a^9*c^12*d^14*e^14 - 1059840*a^
10*c^11*d^12*e^16 - 1403904*a^11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 - 456512*a^13*c^8*d^6*e^22 - 10048
0*a^14*c^7*d^4*e^24 - 4160*a^15*c^6*d^2*e^26) - (-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d
*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 81
9*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*
c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2
*d^4*e^10)))^(1/2)*(256*a^5*c^18*d^28*e^3 - 1792*a^19*c^4*e^31 - (d + e*x)^(1/2)*(-(4*a^3*c^6*d^9 - 49*a^3*e^9
*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*
c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a
^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 +
35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 + 2048*a^6*c^19*d^31*e^2 + 30720*a^7
*c^18*d^29*e^4 + 215040*a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d^25*e^8 + 2795520*a^10*c^15*d^23*e^10 + 6150144*a
^11*c^14*d^21*e^12 + 10250240*a^12*c^13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 + 13178880*a^14*c^11*d^15*e^1
8 + 10250240*a^15*c^10*d^13*e^20 + 6150144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24 + 931840*a^18*c^7*d^
7*e^26 + 215040*a^19*c^6*d^5*e^28 + 30720*a^20*c^5*d^3*e^30) + 11776*a^6*c^17*d^26*e^5 + 119552*a^7*c^16*d^24*
e^7 + 609280*a^8*c^15*d^22*e^9 + 1923328*a^9*c^14*d^20*e^11 + 4116992*a^10*c^13*d^18*e^13 + 6243072*a^11*c^12*
d^16*e^15 + 6825984*a^12*c^11*d^14*e^17 + 5364480*a^13*c^10*d^12*e^19 + 2945536*a^14*c^9*d^10*e^21 + 1044736*a
^15*c^8*d^8*e^23 + 183296*a^16*c^7*d^6*e^25 - 13568*a^17*c^6*d^4*e^27 - 12800*a^18*c^5*d^2*e^29))*(-(4*a^3*c^6
*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c
^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c
^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a
^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*1i)/(((d + e*x)^(1/2)*(1568*a^16*c^5*e^28
+ 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 + 29120*a^5*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 + 28256
0*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 - 282240*a^9*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^16 - 1
403904*a^11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 - 456512*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 -
4160*a^15*c^6*d^2*e^26) - (-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7
*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9
*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^
6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2
56*a^5*c^18*d^28*e^3 - 1792*a^19*c^4*e^31 - (d + e*x)^(1/2)*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 3
15*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3
)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^
14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 +
21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 + 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 + 2150
40*a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d^25*e^8 + 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^12 +
10250240*a^12*c^13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 + 13178880*a^14*c^11*d^15*e^18 + 10250240*a^15*c^1
0*d^13*e^20 + 6150144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24 + 931840*a^18*c^7*d^7*e^26 + 215040*a^19*
c^6*d^5*e^28 + 30720*a^20*c^5*d^3*e^30) + 11776*a^6*c^17*d^26*e^5 + 119552*a^7*c^16*d^24*e^7 + 609280*a^8*c^15
*d^22*e^9 + 1923328*a^9*c^14*d^20*e^11 + 4116992*a^10*c^13*d^18*e^13 + 6243072*a^11*c^12*d^16*e^15 + 6825984*a
^12*c^11*d^14*e^17 + 5364480*a^13*c^10*d^12*e^19 + 2945536*a^14*c^9*d^10*e^21 + 1044736*a^15*c^8*d^8*e^23 + 18
3296*a^16*c^7*d^6*e^25 - 13568*a^17*c^6*d^4*e^27 - 12800*a^18*c^5*d^2*e^29))*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a
^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*
d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*
e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a
^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2) - ((d + e*x)^(1/2)*(1568*a^16*c^5*e^28 + 128*a^3*c^18*d^26*e^2
+ 3040*a^4*c^17*d^24*e^4 + 29120*a^5*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 + 282560*a^7*c^14*d^18*e^10 + 2
42016*a^8*c^13*d^16*e^12 - 282240*a^9*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^16 - 1403904*a^11*c^10*d^10*e^
18 - 1049440*a^12*c^9*d^8*e^20 - 456512*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 - 4160*a^15*c^6*d^2*e^26)
+ (-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e
^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*
d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*
d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*
c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^
6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^
9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 3
5*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 + 2048*a^6*c^19*
d^31*e^2 + 30720*a^7*c^18*d^29*e^4 + 215040*a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d^25*e^8 + 2795520*a^10*c^15*d
^23*e^10 + 6150144*a^11*c^14*d^21*e^12 + 10250240*a^12*c^13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 + 1317888
0*a^14*c^11*d^15*e^18 + 10250240*a^15*c^10*d^13*e^20 + 6150144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24
+ 931840*a^18*c^7*d^7*e^26 + 215040*a^19*c^6*d^5*e^28 + 30720*a^20*c^5*d^3*e^30) - 1792*a^19*c^4*e^31 + 256*a^
5*c^18*d^28*e^3 + 11776*a^6*c^17*d^26*e^5 + 119552*a^7*c^16*d^24*e^7 + 609280*a^8*c^15*d^22*e^9 + 1923328*a^9*
c^14*d^20*e^11 + 4116992*a^10*c^13*d^18*e^13 + 6243072*a^11*c^12*d^16*e^15 + 6825984*a^12*c^11*d^14*e^17 + 536
4480*a^13*c^10*d^12*e^19 + 2945536*a^14*c^9*d^10*e^21 + 1044736*a^15*c^8*d^8*e^23 + 183296*a^16*c^7*d^6*e^25 -
13568*a^17*c^6*d^4*e^27 - 12800*a^18*c^5*d^2*e^29))*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*
c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2)
- 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*
a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^1
1*c^2*d^4*e^10)))^(1/2) + 7448*a^13*c^6*d*e^25 - 32*a^2*c^17*d^23*e^3 - 72*a^3*c^16*d^21*e^5 + 8240*a^4*c^15*d
^19*e^7 + 72120*a^5*c^14*d^17*e^9 + 282240*a^6*c^13*d^15*e^11 + 648816*a^7*c^12*d^13*e^13 + 962976*a^8*c^11*d^
11*e^15 + 955440*a^9*c^10*d^9*e^17 + 633120*a^10*c^9*d^7*e^19 + 270040*a^11*c^8*d^5*e^21 + 67248*a^12*c^7*d^3*
e^23))*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d
^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^
2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*
c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*2i - ((2*e^3)/(3*(a*e^
2 + c*d^2)) + (20*c*d*e^3*(d + e*x))/(3*(a*e^2 + c*d^2)^2) + (c*e*(d + e*x)^2*(7*a^2*e^4 + 3*c^2*d^4 - 110*a*c
*d^2*e^2))/(6*a*(a*e^2 + c*d^2)^3) + (c^2*d*e*(19*a*e^2 - c*d^2)*(d + e*x)^3)/(2*a*(a*e^2 + c*d^2)^3))/(c*(d +
e*x)^(7/2) + (a*e^2 + c*d^2)*(d + e*x)^(3/2) - 2*c*d*(d + e*x)^(5/2)) + atan((((d + e*x)^(1/2)*(1568*a^16*c^5
*e^28 + 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 + 29120*a^5*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 +
282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 - 282240*a^9*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^1
6 - 1403904*a^11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 - 456512*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e
^24 - 4160*a^15*c^6*d^2*e^26) + (-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^
5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*
(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a
^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/
2)*((d + e*x)^(1/2)*(-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 +
189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(
1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*
e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^2
1*c^4*d*e^32 + 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 + 215040*a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d
^25*e^8 + 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^12 + 10250240*a^12*c^13*d^19*e^14 + 13178880*
a^13*c^12*d^17*e^16 + 13178880*a^14*c^11*d^15*e^18 + 10250240*a^15*c^10*d^13*e^20 + 6150144*a^16*c^9*d^11*e^22
+ 2795520*a^17*c^8*d^9*e^24 + 931840*a^18*c^7*d^7*e^26 + 215040*a^19*c^6*d^5*e^28 + 30720*a^20*c^5*d^3*e^30)
- 1792*a^19*c^4*e^31 + 256*a^5*c^18*d^28*e^3 + 11776*a^6*c^17*d^26*e^5 + 119552*a^7*c^16*d^24*e^7 + 609280*a^8
*c^15*d^22*e^9 + 1923328*a^9*c^14*d^20*e^11 + 4116992*a^10*c^13*d^18*e^13 + 6243072*a^11*c^12*d^16*e^15 + 6825
984*a^12*c^11*d^14*e^17 + 5364480*a^13*c^10*d^12*e^19 + 2945536*a^14*c^9*d^10*e^21 + 1044736*a^15*c^8*d^8*e^23
+ 183296*a^16*c^7*d^6*e^25 - 13568*a^17*c^6*d^4*e^27 - 12800*a^18*c^5*d^2*e^29))*(-(49*a^3*e^9*(-a^9*c^3)^(1/
2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105
*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(
a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 +
35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*1i + ((d + e*x)^(1/2)*(1568*a^16*c^5*e^28 + 128*a^3*c^18*
d^26*e^2 + 3040*a^4*c^17*d^24*e^4 + 29120*a^5*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 + 282560*a^7*c^14*d^18*
e^10 + 242016*a^8*c^13*d^16*e^12 - 282240*a^9*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^16 - 1403904*a^11*c^10
*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 - 456512*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 - 4160*a^15*c^6*d
^2*e^26) - (-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c
^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 83
7*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*
a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(256*a^5*c^18*d^28
*e^3 - 1792*a^19*c^4*e^31 - (d + e*x)^(1/2)*(-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8
+ 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*
c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^
2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4
*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 + 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 + 215040*a^8*c^17*d^27
*e^6 + 931840*a^9*c^16*d^25*e^8 + 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^12 + 10250240*a^12*c^
13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 + 13178880*a^14*c^11*d^15*e^18 + 10250240*a^15*c^10*d^13*e^20 + 61
50144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24 + 931840*a^18*c^7*d^7*e^26 + 215040*a^19*c^6*d^5*e^28 + 3
0720*a^20*c^5*d^3*e^30) + 11776*a^6*c^17*d^26*e^5 + 119552*a^7*c^16*d^24*e^7 + 609280*a^8*c^15*d^22*e^9 + 1923
328*a^9*c^14*d^20*e^11 + 4116992*a^10*c^13*d^18*e^13 + 6243072*a^11*c^12*d^16*e^15 + 6825984*a^12*c^11*d^14*e^
17 + 5364480*a^13*c^10*d^12*e^19 + 2945536*a^14*c^9*d^10*e^21 + 1044736*a^15*c^8*d^8*e^23 + 183296*a^16*c^7*d^
6*e^25 - 13568*a^17*c^6*d^4*e^27 - 12800*a^18*c^5*d^2*e^29))*(-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 +
315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^
3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d
^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8
+ 21*a^11*c^2*d^4*e^10)))^(1/2)*1i)/(((d + e*x)^(1/2)*(1568*a^16*c^5*e^28 + 128*a^3*c^18*d^26*e^2 + 3040*a^4*c
^17*d^24*e^4 + 29120*a^5*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 + 282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^1
3*d^16*e^12 - 282240*a^9*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^16 - 1403904*a^11*c^10*d^10*e^18 - 1049440*
a^12*c^9*d^8*e^20 - 456512*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 - 4160*a^15*c^6*d^2*e^26) - (-(49*a^3*
e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6
*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9
*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35
*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(256*a^5*c^18*d^28*e^3 - 1792*a^19*c^4*
e^31 - (d + e*x)^(1/2)*(-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2
+ 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3
)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^
12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*
a^21*c^4*d*e^32 + 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 + 215040*a^8*c^17*d^27*e^6 + 931840*a^9*c^1
6*d^25*e^8 + 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^12 + 10250240*a^12*c^13*d^19*e^14 + 131788
80*a^13*c^12*d^17*e^16 + 13178880*a^14*c^11*d^15*e^18 + 10250240*a^15*c^10*d^13*e^20 + 6150144*a^16*c^9*d^11*e
^22 + 2795520*a^17*c^8*d^9*e^24 + 931840*a^18*c^7*d^7*e^26 + 215040*a^19*c^6*d^5*e^28 + 30720*a^20*c^5*d^3*e^3
0) + 11776*a^6*c^17*d^26*e^5 + 119552*a^7*c^16*d^24*e^7 + 609280*a^8*c^15*d^22*e^9 + 1923328*a^9*c^14*d^20*e^1
1 + 4116992*a^10*c^13*d^18*e^13 + 6243072*a^11*c^12*d^16*e^15 + 6825984*a^12*c^11*d^14*e^17 + 5364480*a^13*c^1
0*d^12*e^19 + 2945536*a^14*c^9*d^10*e^21 + 1044736*a^15*c^8*d^8*e^23 + 183296*a^16*c^7*d^6*e^25 - 13568*a^17*c
^6*d^4*e^27 - 12800*a^18*c^5*d^2*e^29))*(-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 6
3*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*
d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^
12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^1
0)))^(1/2) - ((d + e*x)^(1/2)*(1568*a^16*c^5*e^28 + 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 + 29120*a^5
*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 + 282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 - 282240*a^9
*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^16 - 1403904*a^11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 - 4565
12*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 - 4160*a^15*c^6*d^2*e^26) + (-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4
*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d
^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e
^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^
10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*((d + e*x)^(1/2)*(-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9
+ 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9
*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^
7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e
^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 + 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 +
215040*a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d^25*e^8 + 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^1
2 + 10250240*a^12*c^13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 + 13178880*a^14*c^11*d^15*e^18 + 10250240*a^15
*c^10*d^13*e^20 + 6150144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24 + 931840*a^18*c^7*d^7*e^26 + 215040*a
^19*c^6*d^5*e^28 + 30720*a^20*c^5*d^3*e^30) - 1792*a^19*c^4*e^31 + 256*a^5*c^18*d^28*e^3 + 11776*a^6*c^17*d^26
*e^5 + 119552*a^7*c^16*d^24*e^7 + 609280*a^8*c^15*d^22*e^9 + 1923328*a^9*c^14*d^20*e^11 + 4116992*a^10*c^13*d^
18*e^13 + 6243072*a^11*c^12*d^16*e^15 + 6825984*a^12*c^11*d^14*e^17 + 5364480*a^13*c^10*d^12*e^19 + 2945536*a^
14*c^9*d^10*e^21 + 1044736*a^15*c^8*d^8*e^23 + 183296*a^16*c^7*d^6*e^25 - 13568*a^17*c^6*d^4*e^27 - 12800*a^18
*c^5*d^2*e^29))*(-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*
a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2)
- 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2
+ 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2) + 7448*a^13*c
^6*d*e^25 - 32*a^2*c^17*d^23*e^3 - 72*a^3*c^16*d^21*e^5 + 8240*a^4*c^15*d^19*e^7 + 72120*a^5*c^14*d^17*e^9 + 2
82240*a^6*c^13*d^15*e^11 + 648816*a^7*c^12*d^13*e^13 + 962976*a^8*c^11*d^11*e^15 + 955440*a^9*c^10*d^9*e^17 +
633120*a^10*c^9*d^7*e^19 + 270040*a^11*c^8*d^5*e^21 + 67248*a^12*c^7*d^3*e^23))*(-(49*a^3*e^9*(-a^9*c^3)^(1/2)
+ 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c
^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^
13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 3
5*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*2i

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(5/2)/(c*x**2+a)**2,x)

[Out]

Timed out

________________________________________________________________________________________