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

Optimal. Leaf size=77 \[ -\frac{d \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{e^3}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3} \]

[Out]

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

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Rubi [A]  time = 0.198237, antiderivative size = 77, 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 \[ -\frac{d \sqrt{d^2-e^2 x^2}}{e^3 (d+e x)}-\frac{\sqrt{d^2-e^2 x^2}}{e^3}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.1858, size = 65, normalized size = 0.84 \[ - \frac{d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{3}} - \frac{d \sqrt{d^{2} - e^{2} x^{2}}}{e^{3} \left (d + e x\right )} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.116152, size = 59, normalized size = 0.77 \[ -\frac{\frac{\sqrt{d^2-e^2 x^2} (2 d+e x)}{d+e x}+d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 97, normalized size = 1.3 \[ -{\frac{1}{{e}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{d}{{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{d}{{e}^{4}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.293096, size = 243, normalized size = 3.16 \[ \frac{e^{3} x^{3} + d e^{2} x^{2} + 2 \, d^{2} e x + 2 \,{\left (d e^{2} x^{2} - d^{2} e x - 2 \, d^{3} + \sqrt{-e^{2} x^{2} + d^{2}}{\left (d e x + 2 \, d^{2}\right )}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (e^{2} x^{2} + 2 \, d e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{e^{5} x^{2} - d e^{4} x - 2 \, d^{2} e^{3} +{\left (e^{4} x + 2 \, d e^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError