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

Optimal. Leaf size=82 \[ -\frac{1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(4*x)/(15*d^3*(d^2 - e^2*x^2)^(3/2)) - 1/(5*d*e*(d + e*x)*(d^2 - e^2*x^2)^(3/2))
+ (8*x)/(15*d^5*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.0683842, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 x}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*x)/(15*d^3*(d^2 - e^2*x^2)^(3/2)) - 1/(5*d*e*(d + e*x)*(d^2 - e^2*x^2)^(3/2))
+ (8*x)/(15*d^5*Sqrt[d^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 7.90402, size = 68, normalized size = 0.83 \[ - \frac{1}{5 d e \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{4 x}{15 d^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{8 x}{15 d^{5} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(5*d*e*(d + e*x)*(d**2 - e**2*x**2)**(3/2)) + 4*x/(15*d**3*(d**2 - e**2*x**2)
**(3/2)) + 8*x/(15*d**5*sqrt(d**2 - e**2*x**2))

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Mathematica [A]  time = 0.064921, size = 82, normalized size = 1. \[ -\frac{\sqrt{d^2-e^2 x^2} \left (3 d^4-12 d^3 e x-12 d^2 e^2 x^2+8 d e^3 x^3+8 e^4 x^4\right )}{15 d^5 e (d-e x)^2 (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d^2 - e^2*x^2]*(3*d^4 - 12*d^3*e*x - 12*d^2*e^2*x^2 + 8*d*e^3*x^3 + 8*e^4
*x^4))/(15*d^5*e*(d - e*x)^2*(d + e*x)^3)

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Maple [A]  time = 0.011, size = 70, normalized size = 0.9 \[ -{\frac{ \left ( -ex+d \right ) \left ( 8\,{e}^{4}{x}^{4}+8\,{e}^{3}{x}^{3}d-12\,{e}^{2}{x}^{2}{d}^{2}-12\,x{d}^{3}e+3\,{d}^{4} \right ) }{15\,{d}^{5}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(-e*x+d)*(8*e^4*x^4+8*d*e^3*x^3-12*d^2*e^2*x^2-12*d^3*e*x+3*d^4)/d^5/e/(-e
^2*x^2+d^2)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240352, size = 452, normalized size = 5.51 \[ -\frac{8 \, e^{7} x^{8} + 20 \, d e^{6} x^{7} - 64 \, d^{2} e^{5} x^{6} - 124 \, d^{3} e^{4} x^{5} + 115 \, d^{4} e^{3} x^{4} + 220 \, d^{5} e^{2} x^{3} - 60 \, d^{6} e x^{2} - 120 \, d^{7} x -{\left (3 \, e^{6} x^{7} - 29 \, d e^{5} x^{6} - 59 \, d^{2} e^{4} x^{5} + 85 \, d^{3} e^{3} x^{4} + 160 \, d^{4} e^{2} x^{3} - 60 \, d^{5} e x^{2} - 120 \, d^{6} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (4 \, d^{6} e^{7} x^{7} + 4 \, d^{7} e^{6} x^{6} - 16 \, d^{8} e^{5} x^{5} - 16 \, d^{9} e^{4} x^{4} + 20 \, d^{10} e^{3} x^{3} + 20 \, d^{11} e^{2} x^{2} - 8 \, d^{12} e x - 8 \, d^{13} -{\left (d^{5} e^{7} x^{7} + d^{6} e^{6} x^{6} - 9 \, d^{7} e^{5} x^{5} - 9 \, d^{8} e^{4} x^{4} + 16 \, d^{9} e^{3} x^{3} + 16 \, d^{10} e^{2} x^{2} - 8 \, d^{11} e x - 8 \, d^{12}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(8*e^7*x^8 + 20*d*e^6*x^7 - 64*d^2*e^5*x^6 - 124*d^3*e^4*x^5 + 115*d^4*e^3
*x^4 + 220*d^5*e^2*x^3 - 60*d^6*e*x^2 - 120*d^7*x - (3*e^6*x^7 - 29*d*e^5*x^6 -
59*d^2*e^4*x^5 + 85*d^3*e^3*x^4 + 160*d^4*e^2*x^3 - 60*d^5*e*x^2 - 120*d^6*x)*sq
rt(-e^2*x^2 + d^2))/(4*d^6*e^7*x^7 + 4*d^7*e^6*x^6 - 16*d^8*e^5*x^5 - 16*d^9*e^4
*x^4 + 20*d^10*e^3*x^3 + 20*d^11*e^2*x^2 - 8*d^12*e*x - 8*d^13 - (d^5*e^7*x^7 +
d^6*e^6*x^6 - 9*d^7*e^5*x^5 - 9*d^8*e^4*x^4 + 16*d^9*e^3*x^3 + 16*d^10*e^2*x^2 -
8*d^11*e*x - 8*d^12)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[undef, undef, undef, undef, undef, 1]