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

Optimal. Leaf size=85 \[ \frac{x}{15 d^2 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 x}{15 d^4 e \sqrt{d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0973462, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{x}{15 d^2 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 x}{15 d^4 e \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.70396, size = 70, normalized size = 0.82 \[ \frac{1}{5 e^{2} \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{x}{15 d^{2} e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{2 x}{15 d^{4} e \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0543011, size = 82, normalized size = 0.96 \[ \frac{\sqrt{d^2-e^2 x^2} \left (3 d^4+3 d^3 e x+3 d^2 e^2 x^2-2 d e^3 x^3-2 e^4 x^4\right )}{15 d^4 e^2 (d-e x)^2 (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 70, normalized size = 0.8 \[{\frac{ \left ( -ex+d \right ) \left ( -2\,{e}^{4}{x}^{4}-2\,{e}^{3}{x}^{3}d+3\,{x}^{2}{d}^{2}{e}^{2}+3\,x{d}^{3}e+3\,{d}^{4} \right ) }{15\,{d}^{4}{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(-e*x+d)*(-2*e^4*x^4-2*d*e^3*x^3+3*d^2*e^2*x^2+3*d^3*e*x+3*d^4)/d^4/e^2/(-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(x/((-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 = 0.296427, size = 427, normalized size = 5.02 \[ -\frac{2 \, e^{6} x^{8} - 10 \, d e^{5} x^{7} - 31 \, d^{2} e^{4} x^{6} + 29 \, d^{3} e^{3} x^{5} + 85 \, d^{4} e^{2} x^{4} - 20 \, d^{5} e x^{3} - 60 \, d^{6} x^{2} +{\left (3 \, e^{5} x^{7} + 11 \, d e^{4} x^{6} - 19 \, d^{2} e^{3} x^{5} - 55 \, d^{3} e^{2} x^{4} + 20 \, d^{4} e x^{3} + 60 \, d^{5} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (4 \, d^{5} e^{7} x^{7} + 4 \, d^{6} e^{6} x^{6} - 16 \, d^{7} e^{5} x^{5} - 16 \, d^{8} e^{4} x^{4} + 20 \, d^{9} e^{3} x^{3} + 20 \, d^{10} e^{2} x^{2} - 8 \, d^{11} e x - 8 \, d^{12} -{\left (d^{4} e^{7} x^{7} + d^{5} e^{6} x^{6} - 9 \, d^{6} e^{5} x^{5} - 9 \, d^{7} e^{4} x^{4} + 16 \, d^{8} e^{3} x^{3} + 16 \, d^{9} e^{2} x^{2} - 8 \, d^{10} e x - 8 \, d^{11}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(2*e^6*x^8 - 10*d*e^5*x^7 - 31*d^2*e^4*x^6 + 29*d^3*e^3*x^5 + 85*d^4*e^2*x
^4 - 20*d^5*e*x^3 - 60*d^6*x^2 + (3*e^5*x^7 + 11*d*e^4*x^6 - 19*d^2*e^3*x^5 - 55
*d^3*e^2*x^4 + 20*d^4*e*x^3 + 60*d^5*x^2)*sqrt(-e^2*x^2 + d^2))/(4*d^5*e^7*x^7 +
4*d^6*e^6*x^6 - 16*d^7*e^5*x^5 - 16*d^8*e^4*x^4 + 20*d^9*e^3*x^3 + 20*d^10*e^2*
x^2 - 8*d^11*e*x - 8*d^12 - (d^4*e^7*x^7 + d^5*e^6*x^6 - 9*d^6*e^5*x^5 - 9*d^7*e
^4*x^4 + 16*d^8*e^3*x^3 + 16*d^9*e^2*x^2 - 8*d^10*e*x - 8*d^11)*sqrt(-e^2*x^2 +
d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/((-(-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(x/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)),x, algorithm="giac")

[Out]

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