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

Optimal. Leaf size=89 \[ \frac{x^2 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 d-3 e x}{3 e^4 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^4} \]

[Out]

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

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Rubi [A]  time = 0.238978, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{x^2 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 d-3 e x}{3 e^4 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.5979, size = 88, normalized size = 0.99 \[ \frac{d^{2}}{3 e^{4} \left (d + e x\right ) \sqrt{d^{2} - e^{2} x^{2}}} - \frac{d}{e^{4} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{4 x}{3 e^{3} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{\operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.133322, size = 80, normalized size = 0.9 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (-2 d^2+d e x+4 e^2 x^2\right )}{(d-e x) (d+e x)^2}-3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{3 e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 153, normalized size = 1.7 \[ 2\,{\frac{x}{{e}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}-{\frac{1}{{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{d}{{e}^{4}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}+{\frac{{d}^{2}}{3\,{e}^{5}} \left ( x+{\frac{d}{e}} \right ) ^{-1}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}}-{\frac{2\,x}{3\,{e}^{3}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/e^3*x/(-e^2*x^2+d^2)^(1/2)-1/e^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^
2)^(1/2))-d/e^4/(-e^2*x^2+d^2)^(1/2)+1/3*d^2/e^5/(x+d/e)/(-(x+d/e)^2*e^2+2*d*e*(
x+d/e))^(1/2)-2/3/e^3/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290756, size = 369, normalized size = 4.15 \[ -\frac{4 \, e^{4} x^{4} + 5 \, d e^{3} x^{3} - 6 \, d^{2} e^{2} x^{2} - 6 \, d^{3} e x - 6 \,{\left (2 \, d e^{3} x^{3} + 2 \, d^{2} e^{2} x^{2} - 2 \, d^{3} e x - 2 \, d^{4} -{\left (e^{3} x^{3} + d e^{2} x^{2} - 2 \, d^{2} e x - 2 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 2 \,{\left (e^{3} x^{3} - 3 \, d e^{2} x^{2} - 3 \, d^{2} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (2 \, d e^{7} x^{3} + 2 \, d^{2} e^{6} x^{2} - 2 \, d^{3} e^{5} x - 2 \, d^{4} e^{4} -{\left (e^{7} x^{3} + d e^{6} x^{2} - 2 \, d^{2} e^{5} x - 2 \, d^{3} e^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**3/((-(-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. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[undef, undef, undef, 1]