3.7.36 \(\int \frac {1}{(d+e x)^{3/2} (a+c x^2)^2} \, dx\) [636]

Optimal. Leaf size=845 \[ \frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

[Out]

1/2*e*(-5*a*e^2+c*d^2)/a/(a*e^2+c*d^2)^2/(e*x+d)^(1/2)+1/2*(c*d*x+a*e)/a/(a*e^2+c*d^2)/(c*x^2+a)/(e*x+d)^(1/2)
+1/8*c^(1/4)*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^(3/2)*d^3+13*a*d*e^2*c^(1/2)+(-5*a*e^2+c*d^2)*(a*e^2+c*d^2)^(1/2))/a/(a*e^2+c*d^2)^(
5/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/8*c^(1/4)*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^(3/2)*d^3+13*a*d*e^2*c^(1/2)+(-5*a*
e^2+c*d^2)*(a*e^2+c*d^2)^(1/2))/a/(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/16*c^(1/
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^(3/2)*d^3+13*a*d*e^2*c^(1/2)-(-5*a*e^2+c*d^2)*(a*e^2+c*d^2)^(1/2))/a/(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1
/2)+(a*e^2+c*d^2)^(1/2))^(1/2)+1/16*c^(1/4)*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^(3/2)*d^3+13*a*d*e^2*c^(1/2)-(-5*a*e^2+c*d^2)*(a*e^2+c*d^2)^(1/
2))/a/(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 2.15, antiderivative size = 845, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {755, 843, 841, 1183, 648, 632, 212, 642} \begin {gather*} \frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\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 )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\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 )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\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 )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\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 )^{5/2} \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 ) \sqrt {d+e x} \left (c x^2+a\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(3/2)*(a + c*x^2)^2),x]

[Out]

(e*(c*d^2 - 5*a*e^2))/(2*a*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]) + (a*e + c*d*x)/(2*a*(c*d^2 + a*e^2)*Sqrt[d + e*x]
*(a + c*x^2)) + (c^(1/4)*e*(c^(3/2)*d^3 + 13*a*Sqrt[c]*d*e^2 + (c*d^2 - 5*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)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(1/4)*e*(c^(3/2)*d^3 + 13*a*S
qrt[c]*d*e^2 + (c*d^2 - 5*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)^(5/2)*Sqrt[Sqrt[c
]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(1/4)*e*(c^(3/2)*d^3 + 13*a*Sqrt[c]*d*e^2 - (c*d^2 - 5*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)^(5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (c^(1/4)*e*(c^(3/2)*d^
3 + 13*a*Sqrt[c]*d*e^2 - (c*d^2 - 5*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)^(5/2)*Sqrt[
Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

Rule 212

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

Rule 632

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 642

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 648

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 755

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 841

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 843

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 1183

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)^{3/2} \left (a+c x^2\right )^2} \, dx &=\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (-2 c d^2-5 a e^2\right )-\frac {3}{2} c d e x}{(d+e x)^{3/2} \left (a+c x^2\right )} \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\int \frac {-c d \left (c d^2+4 a e^2\right )-\frac {1}{2} c e \left (c d^2-5 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 )^2}\\ &=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} c d e \left (c d^2-5 a e^2\right )-c d e \left (c d^2+4 a e^2\right )-\frac {1}{2} c e \left (c d^2-5 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 )^2}\\ &=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\frac {1}{2} c d e \left (c d^2-5 a e^2\right )-c d e \left (c d^2+4 a e^2\right )\right )}{\sqrt [4]{c}}-\left (\frac {1}{2} c d e \left (c d^2-5 a e^2\right )+\frac {1}{2} \sqrt {c} e \left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-c d e \left (c d^2+4 a e^2\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 )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\frac {1}{2} c d e \left (c d^2-5 a e^2\right )-c d e \left (c d^2+4 a e^2\right )\right )}{\sqrt [4]{c}}+\left (\frac {1}{2} c d e \left (c d^2-5 a e^2\right )+\frac {1}{2} \sqrt {c} e \left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-c d e \left (c d^2+4 a e^2\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 )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\left (\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {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 )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {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 )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {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 )^{5/2}}+\frac {\left (e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {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 )^{5/2}}\\ &=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 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 )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 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 )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {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 )^{5/2}}-\frac {\left (e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {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 )^{5/2}}\\ &=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 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 )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 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 )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 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 )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 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 )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.17, size = 338, normalized size = 0.40 \begin {gather*} \frac {\frac {2 \sqrt {a} \left (-4 a^2 e^3+c^2 d^2 x (d+e x)+a c e \left (2 d^2+d e x-5 e^2 x^2\right )\right )}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x} \left (a+c x^2\right )}+\frac {i \left (2 c d+5 i \sqrt {a} \sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\left (\sqrt {c} d+i \sqrt {a} e\right )^2 \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {i \left (2 c d-5 i \sqrt {a} \sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\left (\sqrt {c} d-i \sqrt {a} e\right )^2 \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{4 a^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)^2),x]

[Out]

((2*Sqrt[a]*(-4*a^2*e^3 + c^2*d^2*x*(d + e*x) + a*c*e*(2*d^2 + d*e*x - 5*e^2*x^2)))/((c*d^2 + a*e^2)^2*Sqrt[d
+ e*x]*(a + c*x^2)) + (I*(2*c*d + (5*I)*Sqrt[a]*Sqrt[c]*e)*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d +
e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/((Sqrt[c]*d + I*Sqrt[a]*e)^2*Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]) - (I*(2*c*
d - (5*I)*Sqrt[a]*Sqrt[c]*e)*ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*
e)])/((Sqrt[c]*d - I*Sqrt[a]*e)^2*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]))/(4*a^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3079\) vs. \(2(689)=1378\).
time = 0.50, size = 3080, normalized size = 3.64

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3080\)
default	\(\text {Expression too large to display}\)	\(3080\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

2*e^3*(c/(a*e^2+c*d^2)^2*((-1/4*(a*e^2-c*d^2)/a/e^2*(e*x+d)^(3/2)+1/4*d*(3*a*e^2-c*d^2)/a/e^2*(e*x+d)^(1/2))/(
c*(e*x+d)^2-2*c*d*(e*x+d)+e^2*a+c*d^2)+1/4/a/e^2*(1/4/a/c/e^2/(a*e^2+c*d^2)^(3/2)*(1/2*(-(2*(a*c*e^2+c^2*d^2)^
(1/2)+2*c*d)^(1/2)*c^(7/2)*d^6-14*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(5/2)*a*d^4*e^2+(2*(a*c*e^2+c^2*d^
2)^(1/2)+2*c*d)^(1/2)*c^(5/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^5-13*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*a
^2*d^2*e^4+14*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*(a*c*e^2+c^2*d^2)^(1/2)*a*d^3*e^2+13*(2*(a*c*e^2
+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a^2*d*e^4-5*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/
2)*(a*e^2+c*d^2)^(1/2)*a^2*c*d*e^4-4*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*a*c^2*d^3*e^2
+(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*c^3*d^5+5*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)
*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a^2*e^4+4*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(
1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a*c*d^2*e^2-(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+
c^2*d^2)^(1/2)*c^2*d^4)/c^(1/2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)+(a*e^
2+c*d^2)^(1/2))+2*(26*d*e^6*c*a^3+28*d^3*e^4*a^2*c^2+2*d^5*e^2*c^3*a-1/2*(-(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(
1/2)*c^(7/2)*d^6-14*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(5/2)*a*d^4*e^2+(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d
)^(1/2)*c^(5/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^5-13*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*a^2*d^2*e^4+14*
(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*(a*c*e^2+c^2*d^2)^(1/2)*a*d^3*e^2+13*(2*(a*c*e^2+c^2*d^2)^(1/2
)+2*c*d)^(1/2)*c^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a^2*d*e^4-5*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^
2)^(1/2)*a^2*c*d*e^4-4*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*a*c^2*d^3*e^2+(2*(a*c*e^2+c
^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*c^3*d^5+5*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)
^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a^2*e^4+4*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+
c^2*d^2)^(1/2)*a*c*d^2*e^2-(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)
*c^2*d^4)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)/c^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^
(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(
1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)))+1/4/a/c/e^2/(a*e^2+c*d^2)^(3/2)*(-1/2*(-(2*(a*c*e^2+c^2*
d^2)^(1/2)+2*c*d)^(1/2)*c^(7/2)*d^6-14*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(5/2)*a*d^4*e^2+(2*(a*c*e^2+c
^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(5/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^5-13*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3
/2)*a^2*d^2*e^4+14*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*(a*c*e^2+c^2*d^2)^(1/2)*a*d^3*e^2+13*(2*(a*
c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a^2*d*e^4-5*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d
)^(1/2)*(a*e^2+c*d^2)^(1/2)*a^2*c*d*e^4-4*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*a*c^2*d^
3*e^2+(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*c^3*d^5+5*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^
(1/2)*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a^2*e^4+4*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d
^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a*c*d^2*e^2-(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(a*c
*e^2+c^2*d^2)^(1/2)*c^2*d^4)/c^(1/2)*ln(-(e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)
-(a*e^2+c*d^2)^(1/2))+2*(-26*d*e^6*c*a^3-28*d^3*e^4*a^2*c^2-2*d^5*e^2*c^3*a+1/2*(-(2*(a*c*e^2+c^2*d^2)^(1/2)+2
*c*d)^(1/2)*c^(7/2)*d^6-14*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(5/2)*a*d^4*e^2+(2*(a*c*e^2+c^2*d^2)^(1/2
)+2*c*d)^(1/2)*c^(5/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^5-13*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*a^2*d^2*
e^4+14*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*(a*c*e^2+c^2*d^2)^(1/2)*a*d^3*e^2+13*(2*(a*c*e^2+c^2*d^
2)^(1/2)+2*c*d)^(1/2)*c^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a^2*d*e^4-5*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e
^2+c*d^2)^(1/2)*a^2*c*d*e^4-4*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*a*c^2*d^3*e^2+(2*(a*
c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*c^3*d^5+5*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2
+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a^2*e^4+4*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(a
*c*e^2+c^2*d^2)^(1/2)*a*c*d^2*e^2-(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2
)^(1/2)*c^2*d^4)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)/c^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d
^2)*c)^(1/2)-2*c*d)^(1/2)*arctan((-2*c^(1/2)*(e*x+d)^(1/2)+(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+
c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)))))-1/(a*e^2+c*d^2)^2/(e*x+d)^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)^2*(x*e + d)^(3/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5359 vs. \(2 (674) = 1348\).
time = 2.78, size = 5359, normalized size = 6.34 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x, algorithm="fricas")

[Out]

-1/8*((a*c^3*d^5*x^2 + a^2*c^2*d^5 + (a^3*c*x^3 + a^4*x)*e^5 + (a^3*c*d*x^2 + a^4*d)*e^4 + 2*(a^2*c^2*d^2*x^3
+ a^3*c*d^2*x)*e^3 + 2*(a^2*c^2*d^3*x^2 + a^3*c*d^3)*e^2 + (a*c^3*d^4*x^3 + a^2*c^2*d^4*x)*e)*sqrt(-(4*c^4*d^7
+ 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 - 105*a^3*c*d*e^6 + (a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^
6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10)*sqrt(-(1225*c^5*d^8*e^6 + 10780*a*c^4*d^6*e^8 + 21966
*a^2*c^3*d^4*e^10 - 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 + 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*
d^16*e^4 + 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 + 120*a^
10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 + 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 +
10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10))*log(-(140*c^4*d^6*e^3 + 1491*a*c^3*d^4
*e^5 + 3750*a^2*c^2*d^2*e^7 - 625*a^3*c*e^9)*sqrt(x*e + d) + (35*a^2*c^4*d^7*e^4 + 609*a^3*c^3*d^5*e^6 + 1977*
a^4*c^2*d^3*e^8 - 325*a^5*c*d*e^10 + (2*a^3*c^7*d^14 + 19*a^4*c^6*d^12*e^2 + 60*a^5*c^5*d^10*e^4 + 85*a^6*c^4*
d^8*e^6 + 50*a^7*c^3*d^6*e^8 - 3*a^8*c^2*d^4*e^10 - 16*a^9*c*d^2*e^12 - 5*a^10*e^14)*sqrt(-(1225*c^5*d^8*e^6 +
10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 - 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 + 10*a^
4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 + 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^10 + 2
10*a^9*c^4*d^8*e^12 + 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 + 10*a^12*c*d^2*e^18 + a^13*e^20)))*sqrt(-(
4*c^4*d^7 + 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 - 105*a^3*c*d*e^6 + (a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a
^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10)*sqrt(-(1225*c^5*d^8*e^6 + 10780*a*c^4*d^6*e^
8 + 21966*a^2*c^3*d^4*e^10 - 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 + 10*a^4*c^9*d^18*e^2 + 45
*a^5*c^8*d^16*e^4 + 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12
+ 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 + 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 + 5*a^4*c^4*
d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10))) - (a*c^3*d^5*x^2 + a^2*c^2*d
^5 + (a^3*c*x^3 + a^4*x)*e^5 + (a^3*c*d*x^2 + a^4*d)*e^4 + 2*(a^2*c^2*d^2*x^3 + a^3*c*d^2*x)*e^3 + 2*(a^2*c^2*
d^3*x^2 + a^3*c*d^3)*e^2 + (a*c^3*d^4*x^3 + a^2*c^2*d^4*x)*e)*sqrt(-(4*c^4*d^7 + 35*a*c^3*d^5*e^2 + 70*a^2*c^2
*d^3*e^4 - 105*a^3*c*d*e^6 + (a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a
^7*c*d^2*e^8 + a^8*e^10)*sqrt(-(1225*c^5*d^8*e^6 + 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 - 7700*a^3*c^2
*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 + 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 + 120*a^6*c^7*d^14*e^6
+ 210*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 + 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^
4*e^16 + 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2
*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10))*log(-(140*c^4*d^6*e^3 + 1491*a*c^3*d^4*e^5 + 3750*a^2*c^2*d^2*e^7 - 62
5*a^3*c*e^9)*sqrt(x*e + d) - (35*a^2*c^4*d^7*e^4 + 609*a^3*c^3*d^5*e^6 + 1977*a^4*c^2*d^3*e^8 - 325*a^5*c*d*e^
10 + (2*a^3*c^7*d^14 + 19*a^4*c^6*d^12*e^2 + 60*a^5*c^5*d^10*e^4 + 85*a^6*c^4*d^8*e^6 + 50*a^7*c^3*d^6*e^8 - 3
*a^8*c^2*d^4*e^10 - 16*a^9*c*d^2*e^12 - 5*a^10*e^14)*sqrt(-(1225*c^5*d^8*e^6 + 10780*a*c^4*d^6*e^8 + 21966*a^2
*c^3*d^4*e^10 - 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 + 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16
*e^4 + 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 + 120*a^10*c
^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 + 10*a^12*c*d^2*e^18 + a^13*e^20)))*sqrt(-(4*c^4*d^7 + 35*a*c^3*d^5*e^2 + 7
0*a^2*c^2*d^3*e^4 - 105*a^3*c*d*e^6 + (a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*
e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10)*sqrt(-(1225*c^5*d^8*e^6 + 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 - 770
0*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 + 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 + 120*a^6*c^7*
d^14*e^6 + 210*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 + 120*a^10*c^3*d^6*e^14 + 45*a^
11*c^2*d^4*e^16 + 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 1
0*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10))) + (a*c^3*d^5*x^2 + a^2*c^2*d^5 + (a^3*c*x^3 + a^4*x)*e^5 + (
a^3*c*d*x^2 + a^4*d)*e^4 + 2*(a^2*c^2*d^2*x^3 + a^3*c*d^2*x)*e^3 + 2*(a^2*c^2*d^3*x^2 + a^3*c*d^3)*e^2 + (a*c^
3*d^4*x^3 + a^2*c^2*d^4*x)*e)*sqrt(-(4*c^4*d^7 + 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 - 105*a^3*c*d*e^6 - (a^
3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10)*sqrt(-(
1225*c^5*d^8*e^6 + 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 - 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3
*c^10*d^20 + 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 + 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 + 252*a^8
*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 + 120*a^1...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + c x^{2}\right )^{2} \left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(3/2)/(c*x**2+a)**2,x)

[Out]

Integral(1/((a + c*x**2)**2*(d + e*x)**(3/2)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1350 vs. \(2 (674) = 1348\).
time = 5.01, size = 1350, normalized size = 1.60 \begin {gather*} \frac {{\left ({\left (a c^{2} d^{4} e + 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )}^{2} {\left (c d^{2} e - 5 \, a e^{3}\right )} {\left | c \right |} - {\left (\sqrt {-a c} c^{3} d^{7} e + 15 \, \sqrt {-a c} a c^{2} d^{5} e^{3} + 27 \, \sqrt {-a c} a^{2} c d^{3} e^{5} + 13 \, \sqrt {-a c} a^{3} d e^{7}\right )} {\left | a c^{2} d^{4} e + 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} \right |} {\left | c \right |} + 2 \, {\left (a c^{6} d^{12} e + 8 \, a^{2} c^{5} d^{10} e^{3} + 22 \, a^{3} c^{4} d^{8} e^{5} + 28 \, a^{4} c^{3} d^{6} e^{7} + 17 \, a^{5} c^{2} d^{4} e^{9} + 4 \, a^{6} c d^{2} e^{11}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{3} d^{5} + 2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4} + \sqrt {{\left (a c^{3} d^{5} + 2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )}^{2} - {\left (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}\right )} {\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )}}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}}\right )}{4 \, {\left (a^{2} c^{4} d^{8} e + \sqrt {-a c} a c^{4} d^{9} + 4 \, \sqrt {-a c} a^{2} c^{3} d^{7} e^{2} + 4 \, a^{3} c^{3} d^{6} e^{3} + 6 \, \sqrt {-a c} a^{3} c^{2} d^{5} e^{4} + 6 \, a^{4} c^{2} d^{4} e^{5} + 4 \, \sqrt {-a c} a^{4} c d^{3} e^{6} + 4 \, a^{5} c d^{2} e^{7} + \sqrt {-a c} a^{5} d e^{8} + a^{6} e^{9}\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | a c^{2} d^{4} e + 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} \right |}} + \frac {{\left ({\left (a c^{2} d^{4} e + 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )}^{2} {\left (c d^{2} e - 5 \, a e^{3}\right )} {\left | c \right |} + {\left (\sqrt {-a c} c^{3} d^{7} e + 15 \, \sqrt {-a c} a c^{2} d^{5} e^{3} + 27 \, \sqrt {-a c} a^{2} c d^{3} e^{5} + 13 \, \sqrt {-a c} a^{3} d e^{7}\right )} {\left | a c^{2} d^{4} e + 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} \right |} {\left | c \right |} + 2 \, {\left (a c^{6} d^{12} e + 8 \, a^{2} c^{5} d^{10} e^{3} + 22 \, a^{3} c^{4} d^{8} e^{5} + 28 \, a^{4} c^{3} d^{6} e^{7} + 17 \, a^{5} c^{2} d^{4} e^{9} + 4 \, a^{6} c d^{2} e^{11}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{3} d^{5} + 2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4} - \sqrt {{\left (a c^{3} d^{5} + 2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )}^{2} - {\left (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}\right )} {\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )}}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}}}}\right )}{4 \, {\left (a^{2} c^{4} d^{8} e - \sqrt {-a c} a c^{4} d^{9} - 4 \, \sqrt {-a c} a^{2} c^{3} d^{7} e^{2} + 4 \, a^{3} c^{3} d^{6} e^{3} - 6 \, \sqrt {-a c} a^{3} c^{2} d^{5} e^{4} + 6 \, a^{4} c^{2} d^{4} e^{5} - 4 \, \sqrt {-a c} a^{4} c d^{3} e^{6} + 4 \, a^{5} c d^{2} e^{7} - \sqrt {-a c} a^{5} d e^{8} + a^{6} e^{9}\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | a c^{2} d^{4} e + 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} \right |}} + \frac {{\left (x e + d\right )}^{2} c^{2} d^{2} e - {\left (x e + d\right )} c^{2} d^{3} e - 5 \, {\left (x e + d\right )}^{2} a c e^{3} + 11 \, {\left (x e + d\right )} a c d e^{3} - 4 \, a c d^{2} e^{3} - 4 \, a^{2} e^{5}}{2 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} {\left ({\left (x e + d\right )}^{\frac {5}{2}} c - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} c d + \sqrt {x e + d} c d^{2} + \sqrt {x e + d} a e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x, algorithm="giac")

[Out]

1/4*((a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)^2*(c*d^2*e - 5*a*e^3)*abs(c) - (sqrt(-a*c)*c^3*d^7*e + 15*sqrt(
-a*c)*a*c^2*d^5*e^3 + 27*sqrt(-a*c)*a^2*c*d^3*e^5 + 13*sqrt(-a*c)*a^3*d*e^7)*abs(a*c^2*d^4*e + 2*a^2*c*d^2*e^3
+ a^3*e^5)*abs(c) + 2*(a*c^6*d^12*e + 8*a^2*c^5*d^10*e^3 + 22*a^3*c^4*d^8*e^5 + 28*a^4*c^3*d^6*e^7 + 17*a^5*c
^2*d^4*e^9 + 4*a^6*c*d^2*e^11)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4
+ sqrt((a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)^2 - (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)*(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/((a^2*c^4*
d^8*e + sqrt(-a*c)*a*c^4*d^9 + 4*sqrt(-a*c)*a^2*c^3*d^7*e^2 + 4*a^3*c^3*d^6*e^3 + 6*sqrt(-a*c)*a^3*c^2*d^5*e^4
+ 6*a^4*c^2*d^4*e^5 + 4*sqrt(-a*c)*a^4*c*d^3*e^6 + 4*a^5*c*d^2*e^7 + sqrt(-a*c)*a^5*d*e^8 + a^6*e^9)*sqrt(-c^
2*d - sqrt(-a*c)*c*e)*abs(a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)) + 1/4*((a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^
3*e^5)^2*(c*d^2*e - 5*a*e^3)*abs(c) + (sqrt(-a*c)*c^3*d^7*e + 15*sqrt(-a*c)*a*c^2*d^5*e^3 + 27*sqrt(-a*c)*a^2*
c*d^3*e^5 + 13*sqrt(-a*c)*a^3*d*e^7)*abs(a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)*abs(c) + 2*(a*c^6*d^12*e + 8
*a^2*c^5*d^10*e^3 + 22*a^3*c^4*d^8*e^5 + 28*a^4*c^3*d^6*e^7 + 17*a^5*c^2*d^4*e^9 + 4*a^6*c*d^2*e^11)*abs(c))*a
rctan(sqrt(x*e + d)/sqrt(-(a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4 - sqrt((a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 +
a^3*c*d*e^4)^2 - (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)*(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 +
a^3*c*e^4)))/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/((a^2*c^4*d^8*e - sqrt(-a*c)*a*c^4*d^9 - 4*sqrt(-a
*c)*a^2*c^3*d^7*e^2 + 4*a^3*c^3*d^6*e^3 - 6*sqrt(-a*c)*a^3*c^2*d^5*e^4 + 6*a^4*c^2*d^4*e^5 - 4*sqrt(-a*c)*a^4*
c*d^3*e^6 + 4*a^5*c*d^2*e^7 - sqrt(-a*c)*a^5*d*e^8 + a^6*e^9)*sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(a*c^2*d^4*e +
2*a^2*c*d^2*e^3 + a^3*e^5)) + 1/2*((x*e + d)^2*c^2*d^2*e - (x*e + d)*c^2*d^3*e - 5*(x*e + d)^2*a*c*e^3 + 11*(x
*e + d)*a*c*d*e^3 - 4*a*c*d^2*e^3 - 4*a^2*e^5)/((a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4)*((x*e + d)^(5/2)*c - 2
*(x*e + d)^(3/2)*c*d + sqrt(x*e + d)*c*d^2 + sqrt(x*e + d)*a*e^2))

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Mupad [B]
time = 3.21, size = 2500, normalized size = 2.96 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + c*x^2)^2*(d + e*x)^(3/2)),x)

[Out]

atan((((-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3
*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d
^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^
7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*
a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8
*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480
*a^7*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 + 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*
a^11*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 + 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^1
5*c^5*d^3*e^20) + 3328*a^14*c^4*d*e^21 + 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*
e^7 + 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^1
5 + 95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19) + (d + e*x)^(1/2)*(128*a^3*c^13*d^18*e^2 - 800*a^12*c^4
*e^20 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 + 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 +
51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 + 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*
c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2)
- 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c
^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*1i - ((-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/
2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e
^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 +
10*a^9*c^2*d^4*e^6)))^(1/2)*(3328*a^14*c^4*d*e^21 - (d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1
/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*
e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4
+ 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7*c^13*d^19*e^4 + 92160
*a^8*c^12*d^17*e^6 + 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11*c^9*d^11*e^12 + 43008
0*a^12*c^8*d^9*e^14 + 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^5*d^3*e^20) + 256*a^5*
c^13*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d
^11*e^11 + 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*
e^19) - (d + e*x)^(1/2)*(128*a^3*c^13*d^18*e^2 - 800*a^12*c^4*e^20 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d
^14*e^6 + 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 + 51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14
+ 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*
d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2
))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*
e^6)))^(1/2)*1i)/(1000*a^10*c^4*e^19 - ((-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70
*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11
*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)
*(3328*a^14*c^4*d*e^21 - (d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 7
0*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^1
1*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2
)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 + 245760*
a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 + 245760
*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^5*d^3*e^20) + 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^1
2*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d
^9*e^13 + 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19) - (d + e*x)^(1/2)*(128
*a^3*c^13*d^18*e^2 - 800*a^12*c^4*e^20 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 + 30848*a^6*c^10*d^1
2*e^8 + 52480*a^7*c^9*d^10*e^10 + 51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 + 3200*a^10*c^6*d^4*e^16 - 2
432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4
+ 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*...

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