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

Optimal. Leaf size=58 \[ \frac{2 x}{3 d^3 \sqrt{d^2-e^2 x^2}}-\frac{1}{3 d e (d+e x) \sqrt{d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0473002, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 x}{3 d^3 \sqrt{d^2-e^2 x^2}}-\frac{1}{3 d e (d+e x) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.19475, size = 46, normalized size = 0.79 \[ - \frac{1}{3 d e \left (d + e x\right ) \sqrt{d^{2} - e^{2} x^{2}}} + \frac{2 x}{3 d^{3} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(-e**2*x**2+d**2)**(3/2),x)

[Out]

-1/(3*d*e*(d + e*x)*sqrt(d**2 - e**2*x**2)) + 2*x/(3*d**3*sqrt(d**2 - e**2*x**2)
)

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Mathematica [A]  time = 0.0396245, size = 58, normalized size = 1. \[ -\frac{\left (d^2-2 d e x-2 e^2 x^2\right ) \sqrt{d^2-e^2 x^2}}{3 d^3 e (d-e x) (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-((d^2 - 2*d*e*x - 2*e^2*x^2)*Sqrt[d^2 - e^2*x^2])/(3*d^3*e*(d - e*x)*(d + e*x)^
2)

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Maple [A]  time = 0.009, size = 46, normalized size = 0.8 \[ -{\frac{ \left ( -ex+d \right ) \left ( -2\,{e}^{2}{x}^{2}-2\,dex+{d}^{2} \right ) }{3\,{d}^{3}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(-e*x+d)*(-2*e^2*x^2-2*d*e*x+d^2)/d^3/e/(-e^2*x^2+d^2)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283863, size = 213, normalized size = 3.67 \[ -\frac{2 \, e^{3} x^{4} + 4 \, d e^{2} x^{3} - 3 \, d^{2} e x^{2} - 6 \, d^{3} x -{\left (e^{2} x^{3} - 3 \, d e x^{2} - 6 \, d^{2} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (2 \, d^{4} e^{3} x^{3} + 2 \, d^{5} e^{2} x^{2} - 2 \, d^{6} e x - 2 \, d^{7} -{\left (d^{3} e^{3} x^{3} + d^{4} e^{2} x^{2} - 2 \, d^{5} e x - 2 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(2*e^3*x^4 + 4*d*e^2*x^3 - 3*d^2*e*x^2 - 6*d^3*x - (e^2*x^3 - 3*d*e*x^2 - 6
*d^2*x)*sqrt(-e^2*x^2 + d^2))/(2*d^4*e^3*x^3 + 2*d^5*e^2*x^2 - 2*d^6*e*x - 2*d^7
- (d^3*e^3*x^3 + d^4*e^2*x^2 - 2*d^5*e*x - 2*d^6)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(-e**2*x**2+d**2)**(3/2),x)

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(3/2)*(d + e*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

undef