3.140 \(\int \frac{x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=91 \[ \frac{x^2 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 d-3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{x}{5 d^2 e^3 \sqrt{d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.250177, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{x^2 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 d-3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{x}{5 d^2 e^3 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.4255, size = 80, normalized size = 0.88 \[ - \frac{x^{3} \left (d - e x\right )}{5 d e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{2}{15 d e^{4} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{x^{2} \left (d + 3 e x\right )}{15 d^{2} e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0579947, size = 82, normalized size = 0.9 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-2 d^4-2 d^3 e x+3 d^2 e^2 x^2+3 d e^3 x^3+3 e^4 x^4\right )}{15 d^2 e^4 (d-e x)^2 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/((d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x)

[Out]

-1/15*(-e*x+d)*(-3*e^4*x^4-3*d*e^3*x^3-3*d^2*e^2*x^2+2*d^3*e*x+2*d^4)/d^2/e^4/(-
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^3/((-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 = 0.285406, size = 367, normalized size = 4.03 \[ \frac{3 \, e^{4} x^{8} - 5 \, d e^{3} x^{7} - 29 \, d^{2} e^{2} x^{6} + 6 \, d^{3} e x^{5} + 30 \, d^{4} x^{4} + 2 \,{\left (e^{3} x^{7} + 7 \, d e^{2} x^{6} - 3 \, d^{2} e x^{5} - 15 \, d^{3} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (4 \, d^{3} e^{7} x^{7} + 4 \, d^{4} e^{6} x^{6} - 16 \, d^{5} e^{5} x^{5} - 16 \, d^{6} e^{4} x^{4} + 20 \, d^{7} e^{3} x^{3} + 20 \, d^{8} e^{2} x^{2} - 8 \, d^{9} e x - 8 \, d^{10} -{\left (d^{2} e^{7} x^{7} + d^{3} e^{6} x^{6} - 9 \, d^{4} e^{5} x^{5} - 9 \, d^{5} e^{4} x^{4} + 16 \, d^{6} e^{3} x^{3} + 16 \, d^{7} e^{2} x^{2} - 8 \, d^{8} e x - 8 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)),x, algorithm="fricas")

[Out]

1/15*(3*e^4*x^8 - 5*d*e^3*x^7 - 29*d^2*e^2*x^6 + 6*d^3*e*x^5 + 30*d^4*x^4 + 2*(e
^3*x^7 + 7*d*e^2*x^6 - 3*d^2*e*x^5 - 15*d^3*x^4)*sqrt(-e^2*x^2 + d^2))/(4*d^3*e^
7*x^7 + 4*d^4*e^6*x^6 - 16*d^5*e^5*x^5 - 16*d^6*e^4*x^4 + 20*d^7*e^3*x^3 + 20*d^
8*e^2*x^2 - 8*d^9*e*x - 8*d^10 - (d^2*e^7*x^7 + d^3*e^6*x^6 - 9*d^4*e^5*x^5 - 9*
d^5*e^4*x^4 + 16*d^6*e^3*x^3 + 16*d^7*e^2*x^2 - 8*d^8*e*x - 8*d^9)*sqrt(-e^2*x^2
+ d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(x**3/((-(-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^3/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)),x, algorithm="giac")

[Out]

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