3.618 \(\int \frac{1}{(d+e x)^{3/2} \left (a-c x^2\right )^2} \, dx\)

Optimal. Leaf size=265 \[ -\frac{\sqrt [4]{c} \left (2 \sqrt{c} d-5 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{\sqrt [4]{c} \left (5 \sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}}-\frac{a e-c d x}{2 a \left (a-c x^2\right ) \sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{e \left (5 a e^2+c d^2\right )}{2 a \sqrt{d+e x} \left (c d^2-a e^2\right )^2} \]

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

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Rubi [A]  time = 1.27302, antiderivative size = 265, 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 \[ -\frac{\sqrt [4]{c} \left (2 \sqrt{c} d-5 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{\sqrt [4]{c} \left (5 \sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}}-\frac{a e-c d x}{2 a \left (a-c x^2\right ) \sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{e \left (5 a e^2+c d^2\right )}{2 a \sqrt{d+e x} \left (c d^2-a e^2\right )^2} \]

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

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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)**(3/2)/(-c*x**2+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.543638, size = 298, normalized size = 1.12 \[ -\frac{\sqrt{c} \left (2 \sqrt{c} d-5 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{4 a^{3/2} \left (\sqrt{c} d-\sqrt{a} e\right )^2 \sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{\sqrt{c} \left (5 \sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{4 a^{3/2} \left (\sqrt{a} e+\sqrt{c} d\right )^2 \sqrt{\sqrt{a} \sqrt{c} e+c d}}+\frac{4 a^2 e^3+a c e \left (2 d^2+d e x-5 e^2 x^2\right )-c^2 d^2 x (d+e x)}{2 a \left (c x^2-a\right ) \sqrt{d+e x} \left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.062, size = 925, normalized size = 3.5 \[ \text{result too large to display} \]

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]

-2*e^3/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)-1/2*e^3*c/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)
*(e*x+d)^(3/2)-1/2*e*c^2/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)/a*(e*x+d)^(3/2)*d^2+3
/2*e^3*c/(a*e^2-c*d^2)^2/(c*e^2*x^2-a*e^2)*d*(e*x+d)^(1/2)+1/2*e*c^2/(a*e^2-c*d^
2)^2/(c*e^2*x^2-a*e^2)*d^3/a*(e*x+d)^(1/2)-2*e^6*c^2/(a*e^2-c*d^2)^2*a^2/(a^3*c*
e^6)^(1/2)/((a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)
/((a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2))*d+1/2*e^4*c^3/(a*e^2-c*d^2)^2*a/(a^3
*c*e^6)^(1/2)/((a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2)*arctanh(a*c*e*(e*x+d)^(1
/2)/((a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2))*d^3+5/4*e^4*c/(a*e^2-c*d^2)^2*a/(
(a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((a*d*e^2*c
+(a^3*c*e^6)^(1/2))*a*c)^(1/2))+1/4*e^2*c^2/(a*e^2-c*d^2)^2/((a*d*e^2*c+(a^3*c*e
^6)^(1/2))*a*c)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((a*d*e^2*c+(a^3*c*e^6)^(1/2))
*a*c)^(1/2))*d^2-2*e^6*c^2/(a*e^2-c*d^2)^2*a^2/(a^3*c*e^6)^(1/2)/((-a*d*e^2*c+(a
^3*c*e^6)^(1/2))*a*c)^(1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*d*e^2*c+(a^3*c*e^6)^
(1/2))*a*c)^(1/2))*d+1/2*e^4*c^3/(a*e^2-c*d^2)^2*a/(a^3*c*e^6)^(1/2)/((-a*d*e^2*
c+(a^3*c*e^6)^(1/2))*a*c)^(1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*d*e^2*c+(a^3*c*e
^6)^(1/2))*a*c)^(1/2))*d^3-5/4*e^4*c/(a*e^2-c*d^2)^2*a/((-a*d*e^2*c+(a^3*c*e^6)^
(1/2))*a*c)^(1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c
)^(1/2))-1/4*e^2*c^2/(a*e^2-c*d^2)^2/((-a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2)*
arctan(a*c*e*(e*x+d)^(1/2)/((-a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2))*d^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} - a\right )}^{2}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 - a)^2*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 - a)^2*(e*x + d)^(3/2)), x)

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Fricas [A]  time = 0.80801, size = 7247, normalized size = 27.35 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 - a)^2*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

-1/8*(8*a*c*d^2*e + 16*a^2*e^3 - 4*(c^2*d^2*e + 5*a*c*e^3)*x^2 - (a^2*c^2*d^4 -
2*a^3*c*d^2*e^2 + a^4*e^4 - (a*c^3*d^4 - 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^2)*sqr
t(e*x + d)*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 + 2196
6*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^1
2*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 + 4
5*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)*sq
rt((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^4 - 2*a^3*c*d^
2*e^2 + a^4*e^4 - (a*c^3*d^4 - 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^2)*sqrt(e*x + d)
*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)*s
qrt(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 - 25
2*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^1
2 + 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^4 - 2*a^3*c*d^2*e^2 + a^
4*e^4 - (a*c^3*d^4 - 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^2)*sqrt(e*x + d)*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)))*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^1
2 - 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))) + (a^2*c^2*d^4 - 2*a^3*c*d^2*e^2 + a^4*e^4 - (a
*c^3*d^4 - 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^2)*sqrt(e*x + d)*sqrt((4*c^4*d^7 - 3
5*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^1
0)*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)*s
qrt((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 - 1
0*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*(c^2*d^3 - a*c*d*e^2)*x)/((a^2*c^2*d^4 - 2*a^3*c*d^2*
e^2 + a^4*e^4 - (a*c^3*d^4 - 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^2)*sqrt(e*x + d))

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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)**(3/2)/(-c*x**2+a)**2,x)

[Out]

Timed out

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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)^2*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

Timed out