### 3.636 $$\int \frac{1}{(d+e x)^{3/2} (a+c x^2)^2} \, dx$$

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

[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]])

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Rubi [A]  time = 3.34351, antiderivative size = 845, normalized size of antiderivative = 1., 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 = {741, 829, 827, 1169, 634, 618, 206, 628} $\frac{e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt{d+e x}}+\frac{\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt{c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt{c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt{c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt{c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt{c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt{c d^2+a e^2}\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt{c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt{c d^2+a e^2}\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt{d+e x} \left (c x^2+a\right )}$

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 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)^{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{\operatorname{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{\operatorname{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{\operatorname{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 ) \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 )^{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 ) \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 )^{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 ) \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 )^{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 ) \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 )^{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 ) \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 )^{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 ) \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 )^{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*}

Mathematica [C]  time = 0.558981, size = 346, normalized size = 0.41 $\frac{\frac{3 c^{3/4} d \left (\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{2 \sqrt{-a}}-\frac{\left (5 a e^2-c d^2\right ) \left (\left (a e-\sqrt{-a} \sqrt{c} d\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )+\left (\sqrt{-a} \sqrt{c} d+a e\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )\right )}{2 a \sqrt{d+e x} \left (a e^2+c d^2\right )}+\frac{a e+c d x}{\left (a+c x^2\right ) \sqrt{d+e x}}}{2 a \left (a e^2+c d^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((a*e + c*d*x)/(Sqrt[d + e*x]*(a + c*x^2)) + (3*c^(3/4)*d*(ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sq
rt[-a]*e]]/Sqrt[Sqrt[c]*d - Sqrt[-a]*e] - ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]]/Sqrt[S
qrt[c]*d + Sqrt[-a]*e]))/(2*Sqrt[-a]) - ((-(c*d^2) + 5*a*e^2)*((-(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)*Hypergeometric2F1[-1
/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]))/(2*a*(c*d^2 + a*e^2)*Sqrt[d + e*x]))/(2*a*(c*d^2 +
a*e^2))

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Maple [B]  time = 0.281, size = 8744, normalized size = 10.4 \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)^(3/2)/(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{3}{2}}}\,{d x} \end{align*}

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*(e*x + d)^(3/2)), x)

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Fricas [B]  time = 7.46447, size = 12104, normalized size = 14.32 \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)^(3/2)/(c*x^2+a)^2,x, algorithm="fricas")

[Out]

-1/8*((a^2*c^2*d^5 + 2*a^3*c*d^3*e^2 + a^4*d*e^4 + (a*c^3*d^4*e + 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 + (a*c^3*
d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e + 2*a^3*c*d^2*e^3 + a^4*e^5)*x)*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 + 1
0*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(e*x + 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 + 1
0780*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 + 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^2*c^2*d^5 + 2*a^3*c*d^3*e
^2 + a^4*d*e^4 + (a*c^3*d^4*e + 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 + (a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*
e^4)*x^2 + (a^2*c^2*d^4*e + 2*a^3*c*d^2*e^3 + a^4*e^5)*x)*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 + 21
0*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(e*x + 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 + 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^2*c^2*d^5 + 2*a^3*c*d^3*e^2 + a^4*d*e^4 + (a*c^3*d^4*e + 2*
a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 + (a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e + 2*a^3*
c*d^2*e^3 + a^4*e^5)*x)*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^2
0)))/(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(e*x + 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 + 1
0*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^2*c^2*d^5 + 2*a^3*c*d^3*e^2 + a^4*d*e^4 + (a*c^3*d^4*e + 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 +
(a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e + 2*a^3*c*d^2*e^3 + a^4*e^5)*x)*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(e*x + 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)))*s
qrt(-(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))) - 4*(2*a*c*d^2*e - 4*a
^2*e^3 + (c^2*d^2*e - 5*a*c*e^3)*x^2 + (c^2*d^3 + a*c*d*e^2)*x)*sqrt(e*x + d))/(a^2*c^2*d^5 + 2*a^3*c*d^3*e^2
+ a^4*d*e^4 + (a*c^3*d^4*e + 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 + (a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4
)*x^2 + (a^2*c^2*d^4*e + 2*a^3*c*d^2*e^3 + a^4*e^5)*x)

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

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 [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)^(3/2)/(c*x^2+a)^2,x, algorithm="giac")

[Out]

Timed out