3.606 \(\int \frac{1}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx\)

Optimal. Leaf size=190 \[ -\frac{c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}}+\frac{4 c d e}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}+\frac{2 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 1.04574, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}}+\frac{4 c d e}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}+\frac{2 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(5/2)*(a - c*x^2)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(5/2)/(-c*x**2+a),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.668091, size = 211, normalized size = 1.11 \[ \frac{2 e \left (c d (7 d+6 e x)-a e^2\right )}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-\sqrt{a} e\right )^2 \sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right )^2 \sqrt{\sqrt{a} \sqrt{c} e+c d}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*(a - c*x^2)),x]

[Out]

(2*e*(-(a*e^2) + c*d*(7*d + 6*e*x)))/(3*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) - (c*
ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*(Sqrt[c
]*d - Sqrt[a]*e)^2*Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]) + (c*ArcTanh[(Sqrt[c]*Sqrt[d +
e*x])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*(Sqrt[c]*d + Sqrt[a]*e)^2*Sqrt[c
*d + Sqrt[a]*Sqrt[c]*e])

_______________________________________________________________________________________

Maple [B]  time = 0.033, size = 472, normalized size = 2.5 \[ -{\frac{2\,e}{3\,a{e}^{2}-3\,c{d}^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{ced}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}\sqrt{ex+d}}}+{\frac{a{c}^{2}{e}^{3}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{\frac{e{c}^{3}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}-2\,{\frac{e{c}^{2}d}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}{\it Artanh} \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}} \right ) }+{\frac{a{c}^{2}{e}^{3}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{\frac{e{c}^{3}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}+2\,{\frac{e{c}^{2}d}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(5/2)/(-c*x^2+a),x)

[Out]

-2/3*e/(a*e^2-c*d^2)/(e*x+d)^(3/2)+4*e*c*d/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)+c^2/(a*
e^2-c*d^2)^2/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(
1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*a*e^3+e*c^3/(a*e^2-c*d^2)^2/(a*c*e^2)^(1/2
)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))
*c)^(1/2))*d^2-2*e*c^2/(a*e^2-c*d^2)^2/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c
*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*d+c^2/(a*e^2-c*d^2)^2/(a*c*e^2)^
(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(
1/2))*c)^(1/2))*a*e^3+e*c^3/(a*e^2-c*d^2)^2/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/
2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*d^2+2*e*c^
2/(a*e^2-c*d^2)^2/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((-c*d
+(a*c*e^2)^(1/2))*c)^(1/2))*d

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((c*x^2 - a)*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

-integrate(1/((c*x^2 - a)*(e*x + d)^(5/2)), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.318972, size = 6732, normalized size = 35.43 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((c*x^2 - a)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

1/6*(24*c*d*e^2*x + 28*c*d^2*e - 4*a*e^3 + 3*(c^2*d^5 - 2*a*c*d^3*e^2 + a^2*d*e^
4 + (c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(e*x + d)*sqrt((c^4*d^5 + 10*a*
c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6
*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 1
00*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c
^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 21
0*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*
d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 -
5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 -
a^6*e^10))*log((5*c^4*d^4*e + 10*a*c^3*d^2*e^3 + a^2*c^2*e^5)*sqrt(e*x + d) + (
15*a*c^4*d^6*e^2 + 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 + a^4*c*e^8 - (a*c^6*
d^13 - 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 + 20*a^4*c^3*d^7*e^6 - 25*a^5*c^2*
d^5*e^8 + 14*a^6*c*d^3*e^10 - 3*a^7*d*e^12)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6
*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 1
0*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^
12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 4
5*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt((c^4*d^5 + 10*a*c^3*
d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4
- 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a
*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*
d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^
5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*
e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a
^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6
*e^10))) - 3*(c^2*d^5 - 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e - 2*a*c*d^2*e^3 +
a^2*e^5)*x)*sqrt(e*x + d)*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 +
(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^
5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d
^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 +
45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5
*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 -
10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*
d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))*log((5*c^4*d^4*e + 1
0*a*c^3*d^2*e^3 + a^2*c^2*e^5)*sqrt(e*x + d) - (15*a*c^4*d^6*e^2 + 35*a^2*c^3*d^
4*e^4 + 13*a^3*c^2*d^2*e^6 + a^4*c*e^8 - (a*c^6*d^13 - 2*a^2*c^5*d^11*e^2 - 5*a^
3*c^4*d^9*e^4 + 20*a^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 + 14*a^6*c*d^3*e^10 - 3*
a^7*d*e^12)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*
a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*
d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 +
210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d
^2*e^18 + a^11*e^20)))*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c
^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*
d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e
^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*
a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^1
0*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*
a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*
e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))) + 3*(c^2*d^5 - 2*a*c*d^
3*e^2 + a^2*d*e^4 + (c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(e*x + d)*sqrt(
(c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2
+ 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((25
*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^
4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c
^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^1
2 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)
))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5
*a^5*c*d^2*e^8 - a^6*e^10))*log((5*c^4*d^4*e + 10*a*c^3*d^2*e^3 + a^2*c^2*e^5)*s
qrt(e*x + d) + (15*a*c^4*d^6*e^2 + 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 + a^4
*c*e^8 + (a*c^6*d^13 - 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 + 20*a^4*c^3*d^7*e
^6 - 25*a^5*c^2*d^5*e^8 + 14*a^6*c*d^3*e^10 - 3*a^7*d*e^12)*sqrt((25*c^7*d^8*e^2
+ 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/
(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6
+ 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*
c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt((c^4
*d^5 + 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10
*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7
*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^
3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d
^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 -
120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(
a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5
*c*d^2*e^8 - a^6*e^10))) - 3*(c^2*d^5 - 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e -
2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(e*x + d)*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*
a^2*c^2*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^
2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4
+ 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^
2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e
^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^
9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e
^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))*log(
(5*c^4*d^4*e + 10*a*c^3*d^2*e^3 + a^2*c^2*e^5)*sqrt(e*x + d) - (15*a*c^4*d^6*e^2
+ 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 + a^4*c*e^8 + (a*c^6*d^13 - 2*a^2*c^5
*d^11*e^2 - 5*a^3*c^4*d^9*e^4 + 20*a^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 + 14*a^6
*c*d^3*e^10 - 3*a^7*d*e^12)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c
^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e
^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6
*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^
16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*a^2*
c^2*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^
4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 1
10*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^
9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 -
252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^
2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 +
10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))))/((c^2*
d^5 - 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(
e*x + d))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(5/2)/(-c*x**2+a),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((c*x^2 - a)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

Timed out