### 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]

(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]])

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Rubi [A]  time = 6.48863, antiderivative size = 930, normalized size of antiderivative = 1., 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 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 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 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 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]

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 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 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 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]

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 = 0.538045, 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))

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Maple [B]  time = 0.268, size = 10403, normalized size = 11.2 \begin{align*} \text{output too large to display} \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + a\right )}^{2}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

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)

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Fricas [B]  time = 18.9744, size = 18290, normalized size = 19.67 \begin{align*} \text{result too large to display} \end{align*}

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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out