3.181 \(\int \frac{x^3}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=120 \[ \frac{d^2 (d-e x)^3}{5 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{13 d (d-e x)^2}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{32 (d-e x)}{15 e^4 \sqrt{d^2-e^2 x^2}}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^4} \]

[Out]

(d^2*(d - e*x)^3)/(5*e^4*(d^2 - e^2*x^2)^(5/2)) - (13*d*(d - e*x)^2)/(15*e^4*(d^
2 - e^2*x^2)^(3/2)) + (32*(d - e*x))/(15*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 = 0.482841, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{d^2 (d-e x)^3}{5 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{13 d (d-e x)^2}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{32 (d-e x)}{15 e^4 \sqrt{d^2-e^2 x^2}}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^4} \]

Antiderivative was successfully verified.

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

[Out]

(d^2*(d - e*x)^3)/(5*e^4*(d^2 - e^2*x^2)^(5/2)) - (13*d*(d - e*x)^2)/(15*e^4*(d^
2 - e^2*x^2)^(3/2)) + (32*(d - e*x))/(15*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 = 40.5074, size = 104, normalized size = 0.87 \[ \frac{d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{5 e^{4} \left (d + e x\right )^{3}} - \frac{13 d \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{4} \left (d + e x\right )^{2}} + \frac{\operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{4}} + \frac{32 \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{4} \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*sqrt(d**2 - e**2*x**2)/(5*e**4*(d + e*x)**3) - 13*d*sqrt(d**2 - e**2*x**2)/
(15*e**4*(d + e*x)**2) + atan(e*x/sqrt(d**2 - e**2*x**2))/e**4 + 32*sqrt(d**2 -
e**2*x**2)/(15*e**4*(d + e*x))

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Mathematica [A]  time = 0.110382, size = 73, normalized size = 0.61 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (22 d^2+51 d e x+32 e^2 x^2\right )}{(d+e x)^3}+15 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{15 e^4} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(22*d^2 + 51*d*e*x + 32*e^2*x^2))/(d + e*x)^3 + 15*ArcTan[
(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e^4)

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Maple [A]  time = 0.016, size = 163, normalized size = 1.4 \[{\frac{1}{{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{32}{15\,{e}^{5}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-1}}-{\frac{13\,d}{15\,{e}^{6}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-2}}+{\frac{{d}^{2}}{5\,{e}^{7}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/e^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+32/15/e^5/(x+d/e)*(
-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-13/15/e^6*d/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*
(x+d/e))^(1/2)+1/5*d^2/e^7/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.291176, size = 456, normalized size = 3.8 \[ \frac{54 \, e^{5} x^{5} + 65 \, d e^{4} x^{4} - 85 \, d^{2} e^{3} x^{3} - 150 \, d^{3} e^{2} x^{2} - 60 \, d^{4} e x - 30 \,{\left (e^{5} x^{5} + 5 \, d e^{4} x^{4} + 5 \, d^{2} e^{3} x^{3} - 5 \, d^{3} e^{2} x^{2} - 10 \, d^{4} e x - 4 \, d^{5} -{\left (e^{4} x^{4} - 7 \, d^{2} e^{2} x^{2} - 10 \, d^{3} e x - 4 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 5 \,{\left (2 \, e^{4} x^{4} + 23 \, d e^{3} x^{3} + 30 \, d^{2} e^{2} x^{2} + 12 \, d^{3} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 5 \, d^{2} e^{7} x^{3} - 5 \, d^{3} e^{6} x^{2} - 10 \, d^{4} e^{5} x - 4 \, d^{5} e^{4} -{\left (e^{8} x^{4} - 7 \, d^{2} e^{6} x^{2} - 10 \, d^{3} e^{5} x - 4 \, d^{4} e^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(54*e^5*x^5 + 65*d*e^4*x^4 - 85*d^2*e^3*x^3 - 150*d^3*e^2*x^2 - 60*d^4*e*x
- 30*(e^5*x^5 + 5*d*e^4*x^4 + 5*d^2*e^3*x^3 - 5*d^3*e^2*x^2 - 10*d^4*e*x - 4*d^5
- (e^4*x^4 - 7*d^2*e^2*x^2 - 10*d^3*e*x - 4*d^4)*sqrt(-e^2*x^2 + d^2))*arctan(-
(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 5*(2*e^4*x^4 + 23*d*e^3*x^3 + 30*d^2*e^2*x^2
+ 12*d^3*e*x)*sqrt(-e^2*x^2 + d^2))/(e^9*x^5 + 5*d*e^8*x^4 + 5*d^2*e^7*x^3 - 5*
d^3*e^6*x^2 - 10*d^4*e^5*x - 4*d^5*e^4 - (e^8*x^4 - 7*d^2*e^6*x^2 - 10*d^3*e^5*x
- 4*d^4*e^4)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError