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

Optimal. Leaf size=811 \[ -\frac{(a e-c d x) (d+e x)^{3/2}}{2 a c \left (c x^2+a\right )}-\frac{d e \sqrt{d+e x}}{2 a c}+\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]

[Out]

-(d*e*Sqrt[d + e*x])/(2*a*c) - ((a*e - c*d*x)*(d + e*x)^(3/2))/(2*a*c*(a + c*x^2)) + (e*(c^(3/2)*d^3 + a*Sqrt[
c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*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^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt
[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c^(3/2)*d^3 + a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*a*e^2))*Arc
Tanh[(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^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c^(3/2)*d^3 + a*S
qrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*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^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt
[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(c^(3/2)*d^3 + a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*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^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

________________________________________________________________________________________

Rubi [A]  time = 2.83939, antiderivative size = 811, 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 = {739, 825, 827, 1169, 634, 618, 206, 628} \[ -\frac{(a e-c d x) (d+e x)^{3/2}}{2 a c \left (c x^2+a\right )}-\frac{d e \sqrt{d+e x}}{2 a c}+\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)/(a + c*x^2)^2,x]

[Out]

-(d*e*Sqrt[d + e*x])/(2*a*c) - ((a*e - c*d*x)*(d + e*x)^(3/2))/(2*a*c*(a + c*x^2)) + (e*(c^(3/2)*d^3 + a*Sqrt[
c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*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^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt
[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c^(3/2)*d^3 + a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*a*e^2))*Arc
Tanh[(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^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c^(3/2)*d^3 + a*S
qrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*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^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt
[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(c^(3/2)*d^3 + a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*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^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

Rule 739

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*c*(p + 1)), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 825

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

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{(d+e x)^{5/2}}{\left (a+c x^2\right )^2} \, dx &=-\frac{(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac{\int \frac{\sqrt{d+e x} \left (\frac{1}{2} \left (2 c d^2+3 a e^2\right )-\frac{1}{2} c d e x\right )}{a+c x^2} \, dx}{2 a c}\\ &=-\frac{d e \sqrt{d+e x}}{2 a c}-\frac{(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac{\int \frac{c d \left (c d^2+2 a e^2\right )+\frac{1}{2} c e \left (c d^2+3 a e^2\right ) x}{\sqrt{d+e x} \left (a+c x^2\right )} \, dx}{2 a c^2}\\ &=-\frac{d e \sqrt{d+e x}}{2 a c}-\frac{(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{c d e \left (c d^2+2 a e^2\right )-\frac{1}{2} c d e \left (c d^2+3 a e^2\right )+\frac{1}{2} c e \left (c d^2+3 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 c^2}\\ &=-\frac{d e \sqrt{d+e x}}{2 a c}-\frac{(a e-c d x) (d+e x)^{3/2}}{2 a c \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 (c d e \left (c d^2+2 a e^2\right )-\frac{1}{2} c d e \left (c d^2+3 a e^2\right )\right )}{\sqrt [4]{c}}-\left (c d e \left (c d^2+2 a e^2\right )-\frac{1}{2} c d e \left (c d^2+3 a e^2\right )-\frac{1}{2} \sqrt{c} e \sqrt{c d^2+a e^2} \left (c d^2+3 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 c^{9/4} \sqrt{c d^2+a e^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 (c d e \left (c d^2+2 a e^2\right )-\frac{1}{2} c d e \left (c d^2+3 a e^2\right )\right )}{\sqrt [4]{c}}+\left (c d e \left (c d^2+2 a e^2\right )-\frac{1}{2} c d e \left (c d^2+3 a e^2\right )-\frac{1}{2} \sqrt{c} e \sqrt{c d^2+a e^2} \left (c d^2+3 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 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ &=-\frac{d e \sqrt{d+e x}}{2 a c}-\frac{(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}-\frac{\left (e \left (c^{3/2} d^3+a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\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 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\left (e \left (c^{3/2} d^3+a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\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 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\left (e \left (c^{3/2} d^3+a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\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 c^2 \sqrt{c d^2+a e^2}}+\frac{\left (e \left (c^{3/2} d^3+a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\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 c^2 \sqrt{c d^2+a e^2}}\\ &=-\frac{d e \sqrt{d+e x}}{2 a c}-\frac{(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}-\frac{e \left (c^{3/2} d^3+a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\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 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (c^{3/2} d^3+a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\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 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}-\frac{\left (e \left (c^{3/2} d^3+a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\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 c^2 \sqrt{c d^2+a e^2}}-\frac{\left (e \left (c^{3/2} d^3+a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\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 c^2 \sqrt{c d^2+a e^2}}\\ &=-\frac{d e \sqrt{d+e x}}{2 a c}-\frac{(a e-c d x) (d+e x)^{3/2}}{2 a c \left (a+c x^2\right )}+\frac{e \left (c^{3/2} d^3+a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\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 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^{3/2} d^3+a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\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 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^{3/2} d^3+a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\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 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (c^{3/2} d^3+a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\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 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ \end{align*}

Mathematica [A]  time = 0.611967, size = 248, normalized size = 0.31 \[ \frac{\frac{2 c^{3/4} \sqrt{d+e x} \left (c d^2 x-a e (2 d+e x)\right )}{a \left (a+c x^2\right )}+\frac{a \sqrt{\sqrt{c} d-\sqrt{-a} e} \left (\sqrt{-a} \sqrt{c} d e+3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{(-a)^{5/2}}+\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \left (-\sqrt{-a} \sqrt{c} d e+3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{(-a)^{3/2}}}{4 c^{7/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*c^(3/4)*Sqrt[d + e*x]*(c*d^2*x - a*e*(2*d + e*x)))/(a*(a + c*x^2)) + (a*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(2*c*
d^2 + Sqrt[-a]*Sqrt[c]*d*e + 3*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]])/(-a)^(5/2
) + (Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(2*c*d^2 - Sqrt[-a]*Sqrt[c]*d*e + 3*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/S
qrt[Sqrt[c]*d + Sqrt[-a]*e]])/(-a)^(3/2))/(4*c^(7/4))

________________________________________________________________________________________

Maple [B]  time = 0.238, size = 4065, normalized size = 5. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*e/a/c^(5/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d
)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*
d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1
/2)-3/8*e/a/c^(3/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e
*x+d)^(1/2)-(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-
2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d+3/8*e/a/c^(5/2
)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)-(2*(c*
(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(
c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)-3/8*e/a/c^
(3/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2
*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*
(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d+1/16/e/a^2/c^2*ln((e*x+d)*c^
(1/2)+(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e
^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d-1/16/e/a^2/c^2*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*(c
*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^
(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d-1/8/e/a^2/c^(1/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c
*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)-(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1
/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)
+2*c*d)^(1/2)*d^3-1/8/e/a^2/c^(1/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arct
an((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^
2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*
d^3-1/2*e^3/(c*e^2*x^2+a*e^2)/c*(e*x+d)^(3/2)+1/2*e/a/c/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/
2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2
)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(a*e^2+c*d^2)^(1/2)*d+1/16/e/a^2/c*ln((e*x+d)*c^(1/2)-(e*x+d
)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^
(1/2)+2*c*d)^(1/2)*d^2+1/2*e/a/c/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan(
(2*c^(1/2)*(e*x+d)^(1/2)-(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c
*d^2))^(1/2)-2*c*d)^(1/2))*(a*e^2+c*d^2)^(1/2)*d-1/16/e/a^2/c*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*(c*(a*e^2+c*
d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^2-
1/16/e/a^2/c^(3/2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2
))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^2+1/16/e/a^2/c^(3/2)*ln((e*x+d)*c^(1/2)-(
e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1
/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^2-1/8/e/a^2/c^2/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^
(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-
2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^
2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d-1/8/e/a^2/c^2/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e
^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)-(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e
^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(a*e^2
+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d+1/8/e/a^2/c/(4*(a*e^2+c*d^2)^(
1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)-(2*(c*(a*e^2+c*d^2))^(1/2)
+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/
2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^2+1/8/e/a^2/c^(3/2)/(4*(a*e^2+c*
d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)-(2*(c*(a*e^2+c*d^2))
^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2
))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^2+1/8/e/a^2/c^(3/2)/(4
*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e
^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a
*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^2+1/8/e/a^2/
c/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*
(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(
c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^2-1/2*e^3/(c
*e^2*x^2+a*e^2)*d/c*(e*x+d)^(1/2)+1/2*e/(c*e^2*x^2+a*e^2)/a*(e*x+d)^(3/2)*d^2-1/2*e/(c*e^2*x^2+a*e^2)*d^3/a*(e
*x+d)^(1/2)-1/16/e/a^2/c^(1/2)*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2
+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^3+3/16*e/a/c^(5/2)*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(
2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c
^2*d^2)^(1/2)+3/16*e/a/c^(3/2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2
+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d-3/16*e/a/c^(5/2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*
(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2
*d^2)^(1/2)-3/16*e/a/c^(3/2)*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c
*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d+1/16/e/a^2/c^(1/2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*
(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.83791, size = 2917, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((a*c^2*x^2 + a^2*c)*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90
*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))*log((20*c^3*d^6*e^3 + 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7
+ 81*a^3*e^9)*sqrt(e*x + d) + (5*a^2*c^3*d^3*e^4 + 9*a^3*c^2*d*e^6 - (2*a^3*c^6*d^2 + 3*a^4*c^5*e^2)*sqrt(-(2
5*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a
^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))) - (a*c^2*x^2 + a^2*c)*sqr
t(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/
(a^3*c^7)))/(a^3*c^3))*log((20*c^3*d^6*e^3 + 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7 + 81*a^3*e^9)*sqrt(e*x + d)
- (5*a^2*c^3*d^3*e^4 + 9*a^3*c^2*d*e^6 - (2*a^3*c^6*d^2 + 3*a^4*c^5*e^2)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e
^8 + 81*a^2*e^10)/(a^3*c^7)))*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*c^3*sqrt(-(25*c^2*d^4*e^6
+ 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))) + (a*c^2*x^2 + a^2*c)*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e
^2 + 15*a^2*d*e^4 - a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))*log((
20*c^3*d^6*e^3 + 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7 + 81*a^3*e^9)*sqrt(e*x + d) + (5*a^2*c^3*d^3*e^4 + 9*a^
3*c^2*d*e^6 + (2*a^3*c^6*d^2 + 3*a^4*c^5*e^2)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7))
)*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e
^10)/(a^3*c^7)))/(a^3*c^3))) - (a*c^2*x^2 + a^2*c)*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - a^3*c^3*
sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))*log((20*c^3*d^6*e^3 + 101*a*c^2*d
^4*e^5 + 162*a^2*c*d^2*e^7 + 81*a^3*e^9)*sqrt(e*x + d) - (5*a^2*c^3*d^3*e^4 + 9*a^3*c^2*d*e^6 + (2*a^3*c^6*d^2
+ 3*a^4*c^5*e^2)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))*sqrt(-(4*c^2*d^5 + 15*a*c*
d^3*e^2 + 15*a^2*d*e^4 - a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3)))
- 4*(2*a*d*e - (c*d^2 - a*e^2)*x)*sqrt(e*x + d))/(a*c^2*x^2 + a^2*c)

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

[Out]

Timed out