3.7.37 \(\int \frac {1}{(d+e x)^{5/2} (a+c x^2)^2} \, dx\) [637]
Optimal. Leaf size=930 \[ \frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}+\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
[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 = 4.19, 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 = {755, 843, 841,
1183, 648, 632, 212, 642} \begin {gather*} \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}} \end {gather*}
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 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)^{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 {\text {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 {\text {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 {\text {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 ) \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 )^{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 ) \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 )^{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 ) \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 )^{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 ) \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 )^{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 ) \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 )^{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 ) \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 )^{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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.09, size = 379, normalized size = 0.41 \begin {gather*} \frac {-\frac {2 \sqrt {a} \left (4 a^3 e^5-3 c^3 d^3 x (d+e x)^2+a^2 c e^3 \left (55 d^2+54 d e x+7 e^2 x^2\right )+a c^2 d e \left (-9 d^3-9 d^2 e x+61 d e^2 x^2+57 e^3 x^3\right )\right )}{\left (c d^2+a e^2\right )^3 (d+e x)^{3/2} \left (a+c x^2\right )}+\frac {3 \sqrt {-c d-i \sqrt {a} \sqrt {c} e} \left (-2 i c d+7 \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 )^4}+\frac {3 \sqrt {-c d+i \sqrt {a} \sqrt {c} e} \left (2 i c d+7 \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 )^4}}{12 a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^(5/2)*(a + c*x^2)^2),x]
[Out]
((-2*Sqrt[a]*(4*a^3*e^5 - 3*c^3*d^3*x*(d + e*x)^2 + a^2*c*e^3*(55*d^2 + 54*d*e*x + 7*e^2*x^2) + a*c^2*d*e*(-9*
d^3 - 9*d^2*e*x + 61*d*e^2*x^2 + 57*e^3*x^3)))/((c*d^2 + a*e^2)^3*(d + e*x)^(3/2)*(a + c*x^2)) + (3*Sqrt[-(c*d
) - I*Sqrt[a]*Sqrt[c]*e]*((-2*I)*c*d + 7*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)^4 + (3*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*((2*I)
*c*d + 7*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)^4)/(12*a^(3/2))
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3540\) vs.
\(2(778)=1556\).
time = 0.57, size = 3541, normalized size = 3.81
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| method |
result |
size |
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| derivativedivides |
\(\text {Expression too large to display}\) |
\(3541\) |
| default |
\(\text {Expression too large to display}\) |
\(3541\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^(5/2)/(c*x^2+a)^2,x,method=_RETURNVERBOSE)
[Out]
2*e^3*(-1/(a*e^2+c*d^2)^3*c*((1/4*c*d*(3*a*e^2-c*d^2)/a/e^2*(e*x+d)^(3/2)+1/4*(a^2*e^4-6*a*c*d^2*e^2+c^2*d^4)/
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/e^2/c/(a*e^2+c*d^2)^(3/2)*(1/2*(
c^(9/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^7+35*c^(7/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*d^5*e
^2-c^(7/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^6+27*c^(5/2)*(2*(a*c*e^2+c^2*d^2)
^(1/2)+2*c*d)^(1/2)*a^2*d^3*e^4-35*c^(5/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*d
^4*e^2-7*c^(3/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a^3*d*e^6-27*c^(3/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*
e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a^2*d^2*e^4+7*c^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d
)^(1/2)*a^3*e^6+19*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a^2*c^2*d^2*e^4+18*(a*e^2+c*d^2
)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*c^3*d^4*e^2-(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2
*c*d)^(1/2)*c^4*d^6-19*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a^2
*c*d*e^4-18*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*c^2*d^3*e^2+
(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^3*d^5)/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*(14*a^4*c*e^8-54*a^3*c^
2*d^2*e^6-70*a^2*c^3*d^4*e^4-2*a*c^4*d^6*e^2-1/2*(c^(9/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^7+35*c^(7/
2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*d^5*e^2-c^(7/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2
)+2*c*d)^(1/2)*d^6+27*c^(5/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a^2*d^3*e^4-35*c^(5/2)*(a*c*e^2+c^2*d^2)
^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*d^4*e^2-7*c^(3/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a^3
*d*e^6-27*c^(3/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a^2*d^2*e^4+7*c^(1/2)*(a*c*e
^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a^3*e^6+19*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^
(1/2)+2*c*d)^(1/2)*a^2*c^2*d^2*e^4+18*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*c^3*d^4*e^
2-(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^4*d^6-19*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)
^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a^2*c*d*e^4-18*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(
a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*c^2*d^3*e^2+(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*
d^2)^(1/2)+2*c*d)^(1/2)*c^3*d^5)*(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/e^2/c/(a*e^2+c*d^2)^(3/2)*(
1/2*(-c^(9/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^7-35*c^(7/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a
*d^5*e^2+c^(7/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^6-27*c^(5/2)*(2*(a*c*e^2+c^
2*d^2)^(1/2)+2*c*d)^(1/2)*a^2*d^3*e^4+35*c^(5/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/
2)*a*d^4*e^2+7*c^(3/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a^3*d*e^6+27*c^(3/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2
*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a^2*d^2*e^4-7*c^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)
+2*c*d)^(1/2)*a^3*e^6-19*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a^2*c^2*d^2*e^4-18*(a*e^2
+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*c^3*d^4*e^2+(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(
1/2)+2*c*d)^(1/2)*c^4*d^6+19*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/
2)*a^2*c*d*e^4+18*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*c^2*d^
3*e^2-(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^3*d^5)/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*(14*a^4*c*e^8-54*
a^3*c^2*d^2*e^6-70*a^2*c^3*d^4*e^4-2*a*c^4*d^6*e^2+1/2*(-c^(9/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^7-3
5*c^(7/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*d^5*e^2+c^(7/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^
2)^(1/2)+2*c*d)^(1/2)*d^6-27*c^(5/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a^2*d^3*e^4+35*c^(5/2)*(a*c*e^2+c
^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*d^4*e^2+7*c^(3/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1
/2)*a^3*d*e^6+27*c^(3/2)*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a^2*d^2*e^4-7*c^(1/2)
*(a*c*e^2+c^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a^3*e^6-19*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^
2*d^2)^(1/2)+2*c*d)^(1/2)*a^2*c^2*d^2*e^4-18*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*a*c^3
*d^4*e^2+(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^4*d^6+19*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c
^2*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)...
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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)^(5/2)/(c*x^2+a)^2,x, algorithm="maxima")
[Out]
integrate(1/((c*x^2 + a)^2*(x*e + d)^(5/2)), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 7818 vs.
\(2 (759) = 1518\).
time = 5.43, size = 7818, normalized size = 8.41 \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)^(5/2)/(c*x^2+a)^2,x, algorithm="fricas")
[Out]
1/24*(3*(a*c^4*d^8*x^2 + a^2*c^3*d^8 + (a^4*c*x^4 + a^5*x^2)*e^8 + 2*(a^4*c*d*x^3 + a^5*d*x)*e^7 + (3*a^3*c^2*
d^2*x^4 + 4*a^4*c*d^2*x^2 + a^5*d^2)*e^6 + 6*(a^3*c^2*d^3*x^3 + a^4*c*d^3*x)*e^5 + 3*(a^2*c^3*d^4*x^4 + 2*a^3*
c^2*d^4*x^2 + a^4*c*d^4)*e^4 + 6*(a^2*c^3*d^5*x^3 + a^3*c^2*d^5*x)*e^3 + (a*c^4*d^6*x^4 + 4*a^2*c^3*d^6*x^2 +
3*a^3*c^2*d^6)*e^2 + 2*(a*c^4*d^7*x^3 + a^2*c^3*d^7*x)*e)*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))*l
og((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(x*e + 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^1
4*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*c^4*d^8*x^2 + a^2*c^3*d^8 + (a^4*c*x^4 + a^
5*x^2)*e^8 + 2*(a^4*c*d*x^3 + a^5*d*x)*e^7 + (3*a^3*c^2*d^2*x^4 + 4*a^4*c*d^2*x^2 + a^5*d^2)*e^6 + 6*(a^3*c^2*
d^3*x^3 + a^4*c*d^3*x)*e^5 + 3*(a^2*c^3*d^4*x^4 + 2*a^3*c^2*d^4*x^2 + a^4*c*d^4)*e^4 + 6*(a^2*c^3*d^5*x^3 + a^
3*c^2*d^5*x)*e^3 + (a*c^4*d^6*x^4 + 4*a^2*c^3*d^6*x^2 + 3*a^3*c^2*d^6)*e^2 + 2*(a*c^4*d^7*x^3 + a^2*c^3*d^7*x)
*e)*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 + 200
2*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(x*e + 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^...
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1928 vs.
\(2 (759) = 1518\).
time = 2.19, size = 1928, normalized size = 2.07 \begin {gather*} -\frac {{\left ({\left (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}\right )}^{2} {\left (c^{2} d^{3} e - 19 \, a c d e^{3}\right )} {\left | c \right |} - {\left (\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}\right )} {\left | -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} \right |} {\left | c \right |} + {\left (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}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {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 {{\left (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}\right )}^{2} - {\left (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}\right )} {\left (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}\right )}}}{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}}}}\right )}{4 \, {\left (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}\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | -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} \right |}} - \frac {{\left ({\left (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}\right )}^{2} {\left (c^{2} d^{3} e - 19 \, a c d e^{3}\right )} {\left | c \right |} + {\left (\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}\right )} {\left | -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} \right |} {\left | c \right |} + {\left (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}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {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 {{\left (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}\right )}^{2} - {\left (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}\right )} {\left (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}\right )}}}{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}}}}\right )}{4 \, {\left (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}\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | -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} \right |}} + \frac {{\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e - \sqrt {x e + d} c^{3} d^{4} e - 3 \, {\left (x e + d\right )}^{\frac {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}}{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 ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + a e^{2}\right )}} - \frac {2 \, {\left (12 \, {\left (x e + d\right )} c d e^{3} + c d^{2} e^{3} + a e^{5}\right )}}{3 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
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))
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Mupad [B]
time = 4.74, size = 2500, normalized size = 2.69 \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)^(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*...
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