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

Optimal. Leaf size=91 \[ \frac{x^2 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}+\frac{(4 d-3 e x) \sqrt{d^2-e^2 x^2}}{2 e^4}+\frac{3 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^4} \]

[Out]

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

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Rubi [A]  time = 0.247766, antiderivative size = 91, 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 \[ \frac{x^2 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}+\frac{(4 d-3 e x) \sqrt{d^2-e^2 x^2}}{2 e^4}+\frac{3 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.4651, size = 92, normalized size = 1.01 \[ \frac{3 d^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e^{4}} + \frac{d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{e^{4} \left (d + e x\right )} + \frac{d \sqrt{d^{2} - e^{2} x^{2}}}{e^{4}} - \frac{x \sqrt{d^{2} - e^{2} x^{2}}}{2 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.115552, size = 70, normalized size = 0.77 \[ \frac{\sqrt{d^2-e^2 x^2} \left (\frac{2 d^2}{d+e x}+2 d-e x\right )+3 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 120, normalized size = 1.3 \[{\frac{3\,{d}^{2}}{2\,{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{x}{2\,{e}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{d}{{e}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{{d}^{2}}{{e}^{5}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*d^2/e^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/2/e^3*x*(-e
^2*x^2+d^2)^(1/2)+d/e^4*(-e^2*x^2+d^2)^(1/2)+d^2/e^5/(x+d/e)*(-(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)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283686, size = 370, normalized size = 4.07 \[ -\frac{e^{5} x^{5} - 4 \, d e^{4} x^{4} - 7 \, d^{2} e^{3} x^{3} + 6 \, d^{3} e^{2} x^{2} + 12 \, d^{4} e x + 6 \,{\left (d^{2} e^{3} x^{3} + 3 \, d^{3} e^{2} x^{2} - 2 \, d^{4} e x - 4 \, d^{5} -{\left (d^{2} e^{2} x^{2} - 2 \, d^{3} e x - 4 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (e^{4} x^{4} + d e^{3} x^{3} - 6 \, d^{2} e^{2} x^{2} - 12 \, d^{3} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{2 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} - 2 \, d^{2} e^{5} x - 4 \, d^{3} e^{4} -{\left (e^{6} x^{2} - 2 \, d e^{5} x - 4 \, d^{2} e^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(e^5*x^5 - 4*d*e^4*x^4 - 7*d^2*e^3*x^3 + 6*d^3*e^2*x^2 + 12*d^4*e*x + 6*(d^
2*e^3*x^3 + 3*d^3*e^2*x^2 - 2*d^4*e*x - 4*d^5 - (d^2*e^2*x^2 - 2*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)) + (e^4*x^4 + d
*e^3*x^3 - 6*d^2*e^2*x^2 - 12*d^3*e*x)*sqrt(-e^2*x^2 + d^2))/(e^7*x^3 + 3*d*e^6*
x^2 - 2*d^2*e^5*x - 4*d^3*e^4 - (e^6*x^2 - 2*d*e^5*x - 4*d^2*e^4)*sqrt(-e^2*x^2
+ d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError