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

Optimal. Leaf size=311 \[ -\frac{c^{3/4} \left (2 \sqrt{c} d-7 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} \left (\sqrt{c} d-\sqrt{a} e\right )^{7/2}}+\frac{c^{3/4} \left (7 \sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} \left (\sqrt{a} e+\sqrt{c} d\right )^{7/2}}-\frac{a e-c d x}{2 a \left (a-c x^2\right ) (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{e \left (7 a e^2+3 c d^2\right )}{6 a (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}-\frac{c d e \left (19 a e^2+c d^2\right )}{2 a \sqrt{d+e x} \left (c d^2-a e^2\right )^3} \]

[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)*(2*Sqrt[c]*d - 7*Sqrt[a]*e)*Arc
Tanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*(Sqrt[c]*d
- Sqrt[a]*e)^(7/2)) + (c^(3/4)*(2*Sqrt[c]*d + 7*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqr
t[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*(Sqrt[c]*d + Sqrt[a]*e)^(7/
2))

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Rubi [A]  time = 1.80152, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{c^{3/4} \left (2 \sqrt{c} d-7 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} \left (\sqrt{c} d-\sqrt{a} e\right )^{7/2}}+\frac{c^{3/4} \left (7 \sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} \left (\sqrt{a} e+\sqrt{c} d\right )^{7/2}}-\frac{a e-c d x}{2 a \left (a-c x^2\right ) (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{e \left (7 a e^2+3 c d^2\right )}{6 a (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}-\frac{c d e \left (19 a e^2+c d^2\right )}{2 a \sqrt{d+e x} \left (c d^2-a e^2\right )^3} \]

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)*(2*Sqrt[c]*d - 7*Sqrt[a]*e)*Arc
Tanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*(Sqrt[c]*d
- Sqrt[a]*e)^(7/2)) + (c^(3/4)*(2*Sqrt[c]*d + 7*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqr
t[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*(Sqrt[c]*d + Sqrt[a]*e)^(7/
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)**(5/2)/(-c*x**2+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.919442, size = 334, normalized size = 1.07 \[ -\frac{c \left (2 \sqrt{c} d-7 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{4 a^{3/2} \left (\sqrt{c} d-\sqrt{a} e\right )^3 \sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{c \left (7 \sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{4 a^{3/2} \left (\sqrt{a} e+\sqrt{c} d\right )^3 \sqrt{\sqrt{a} \sqrt{c} e+c d}}+\frac{-4 a^3 e^5+a^2 c e^3 \left (55 d^2+54 d e x+7 e^2 x^2\right )+a c^2 d e \left (9 d^3+9 d^2 e x-61 d e^2 x^2-57 e^3 x^3\right )-3 c^3 d^3 x (d+e x)^2}{6 a \left (c x^2-a\right ) (d+e x)^{3/2} \left (c d^2-a e^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(-4*a^3*e^5 - 3*c^3*d^3*x*(d + e*x)^2 + a^2*c*e^3*(55*d^2 + 54*d*e*x + 7*e^2*x^2
) + a*c^2*d*e*(9*d^3 + 9*d^2*e*x - 61*d*e^2*x^2 - 57*e^3*x^3))/(6*a*(c*d^2 - a*e
^2)^3*(d + e*x)^(3/2)*(-a + c*x^2)) - (c*(2*Sqrt[c]*d - 7*Sqrt[a]*e)*ArcTanh[(Sq
rt[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/(4*a^(3/2)*(Sqrt[c]*d - Sqr
t[a]*e)^3*Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]) + (c*(2*Sqrt[c]*d + 7*Sqrt[a]*e)*ArcTan
h[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/(4*a^(3/2)*(Sqrt[c]*d
+ Sqrt[a]*e)^3*Sqrt[c*d + Sqrt[a]*Sqrt[c]*e])

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Maple [B]  time = 0.066, size = 1203, normalized size = 3.9 \[ \text{result too large to display} \]

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]

-2/3*e^3/(a*e^2-c*d^2)^2/(e*x+d)^(3/2)+8*e^3/(a*e^2-c*d^2)^3*c*d/(e*x+d)^(1/2)+3
/2*e^3/(a*e^2-c*d^2)^3*c^2/(c*e^2*x^2-a*e^2)*d*(e*x+d)^(3/2)+1/2*e/(a*e^2-c*d^2)
^3*c^3/(c*e^2*x^2-a*e^2)*d^3/a*(e*x+d)^(3/2)-1/2*e^5/(a*e^2-c*d^2)^3*c/(c*e^2*x^
2-a*e^2)*a*(e*x+d)^(1/2)-3*e^3/(a*e^2-c*d^2)^3*c^2/(c*e^2*x^2-a*e^2)*(e*x+d)^(1/
2)*d^2-1/2*e/(a*e^2-c*d^2)^3*c^3/(c*e^2*x^2-a*e^2)/a*(e*x+d)^(1/2)*d^4+7/4*e^8/(
a*e^2-c*d^2)^3*c^2*a^3/(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))+15/4*e
^6/(a*e^2-c*d^2)^3*c^3*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^
2-1/2*e^4/(a*e^2-c*d^2)^3*c^4*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^4-19/4*e^4/(a*e^2-c*d^2)^3*c^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))*d-1/4*e^2
/(a*e^2-c*d^2)^3*c^3/((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+7/4*e^8/(a*e^2-c*d^2)^
3*c^2*a^3/(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))+15/4*e^6/(a*e^2-c*
d^2)^3*c^3*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)*arct
an(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^2-1/2*e^4/(
a*e^2-c*d^2)^3*c^4*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^4+19/
4*e^4/(a*e^2-c*d^2)^3*c^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))*d+1/4*e^2/(a*e^2-c
*d^2)^3*c^3/((-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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 2.38883, size = 10791, normalized size = 34.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(36*a*c^2*d^4*e + 220*a^2*c*d^2*e^3 - 16*a^3*e^5 - 12*(c^3*d^3*e^2 + 19*a*
c^2*d*e^4)*x^3 - 4*(6*c^3*d^4*e + 61*a*c^2*d^2*e^3 - 7*a^2*c*e^5)*x^2 - 3*(a^2*c
^3*d^7 - 3*a^3*c^2*d^5*e^2 + 3*a^4*c*d^3*e^4 - a^5*d*e^6 - (a*c^4*d^6*e - 3*a^2*
c^3*d^4*e^3 + 3*a^3*c^2*d^2*e^5 - a^4*c*e^7)*x^3 - (a*c^4*d^7 - 3*a^2*c^3*d^5*e^
2 + 3*a^3*c^2*d^3*e^4 - a^4*c*d*e^6)*x^2 + (a^2*c^3*d^6*e - 3*a^3*c^2*d^4*e^3 +
3*a^4*c*d^2*e^5 - a^5*e^7)*x)*sqrt(e*x + d)*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))*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^2
4*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 - 2
002*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^1
5*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^7 - 3*a^3*c^2*d^5*e^2 + 3*a^4*c*d^3*e^4 - a^
5*d*e^6 - (a*c^4*d^6*e - 3*a^2*c^3*d^4*e^3 + 3*a^3*c^2*d^2*e^5 - a^4*c*e^7)*x^3
- (a*c^4*d^7 - 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 - a^4*c*d*e^6)*x^2 + (a^2*c
^3*d^6*e - 3*a^3*c^2*d^4*e^3 + 3*a^4*c*d^2*e^5 - a^5*e^7)*x)*sqrt(e*x + d)*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 - 20
02*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^2
0*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 + 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^7 - 3*a^3*c^2
*d^5*e^2 + 3*a^4*c*d^3*e^4 - a^5*d*e^6 - (a*c^4*d^6*e - 3*a^2*c^3*d^4*e^3 + 3*a^
3*c^2*d^2*e^5 - a^4*c*e^7)*x^3 - (a*c^4*d^7 - 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*
e^4 - a^4*c*d*e^6)*x^2 + (a^2*c^3*d^6*e - 3*a^3*c^2*d^4*e^3 + 3*a^4*c*d^2*e^5 -
a^5*e^7)*x)*sqrt(e*x + d)*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))*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((1
1025*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^1
6*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 - 1
4*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 - 17199
0*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 - 1
4*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^7 - 3*a^3*c^2*d^5*e^2 + 3*a^4*c*d^3*e^4 - a^5*d*e^6 - (a*c^4*d
^6*e - 3*a^2*c^3*d^4*e^3 + 3*a^3*c^2*d^2*e^5 - a^4*c*e^7)*x^3 - (a*c^4*d^7 - 3*a
^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 - a^4*c*d*e^6)*x^2 + (a^2*c^3*d^6*e - 3*a^3*c
^2*d^4*e^3 + 3*a^4*c*d^2*e^5 - a^5*e^7)*x)*sqrt(e*x + d)*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 + 3
5*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((11
025*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^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^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 + 49499
1*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)*s
qrt((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))) - 12*(c^3*d^5 - 3*a*c^2*d^3*e^2 - 18*a^2*c*d*e
^4)*x)/((a^2*c^3*d^7 - 3*a^3*c^2*d^5*e^2 + 3*a^4*c*d^3*e^4 - a^5*d*e^6 - (a*c^4*
d^6*e - 3*a^2*c^3*d^4*e^3 + 3*a^3*c^2*d^2*e^5 - a^4*c*e^7)*x^3 - (a*c^4*d^7 - 3*
a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 - a^4*c*d*e^6)*x^2 + (a^2*c^3*d^6*e - 3*a^3*
c^2*d^4*e^3 + 3*a^4*c*d^2*e^5 - a^5*e^7)*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)**2,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)^2*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

Timed out