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

Optimal. Leaf size=86 \[ \frac{d (2 d-e x) \sqrt{d^2-e^2 x^2}}{2 e^3}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac{d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^3} \]

[Out]

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

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Rubi [A]  time = 0.253148, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{d (2 d-e x) \sqrt{d^2-e^2 x^2}}{2 e^3}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac{d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.5293, size = 87, normalized size = 1.01 \[ \frac{d^{3} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e^{3}} + \frac{d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{e^{3}} - \frac{d x \sqrt{d^{2} - e^{2} x^{2}}}{2 e^{2}} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0665955, size = 69, normalized size = 0.8 \[ \frac{\sqrt{d^2-e^2 x^2} \left (4 d^2-3 d e x+2 e^2 x^2\right )+3 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{6 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(4*d^2 - 3*d*e*x + 2*e^2*x^2) + 3*d^3*ArcTan[(e*x)/Sqrt[d^2
- e^2*x^2]])/(6*e^3)

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Maple [B]  time = 0.016, size = 160, normalized size = 1.9 \[ -{\frac{1}{3\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{dx}{2\,{e}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{{d}^{3}}{2\,{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{{d}^{2}}{{e}^{3}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{\frac{{d}^{3}}{{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283985, size = 301, normalized size = 3.5 \[ \frac{2 \, e^{6} x^{6} - 3 \, d e^{5} x^{5} - 6 \, d^{2} e^{4} x^{4} + 15 \, d^{3} e^{3} x^{3} - 12 \, d^{5} e x - 6 \,{\left (3 \, d^{4} e^{2} x^{2} - 4 \, d^{6} -{\left (d^{3} e^{2} x^{2} - 4 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 3 \,{\left (2 \, d e^{4} x^{4} - 3 \, d^{2} e^{3} x^{3} + 4 \, d^{4} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (3 \, d e^{5} x^{2} - 4 \, d^{3} e^{3} -{\left (e^{5} x^{2} - 4 \, d^{2} e^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*e^6*x^6 - 3*d*e^5*x^5 - 6*d^2*e^4*x^4 + 15*d^3*e^3*x^3 - 12*d^5*e*x - 6*(
3*d^4*e^2*x^2 - 4*d^6 - (d^3*e^2*x^2 - 4*d^5)*sqrt(-e^2*x^2 + d^2))*arctan(-(d -
sqrt(-e^2*x^2 + d^2))/(e*x)) + 3*(2*d*e^4*x^4 - 3*d^2*e^3*x^3 + 4*d^4*e*x)*sqrt
(-e^2*x^2 + d^2))/(3*d*e^5*x^2 - 4*d^3*e^3 - (e^5*x^2 - 4*d^2*e^3)*sqrt(-e^2*x^2
+ d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2*sqrt(-(-d + e*x)*(d + e*x))/(d + e*x), x)

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GIAC/XCAS [A]  time = 0.28904, size = 73, normalized size = 0.85 \[ \frac{1}{2} \, d^{3} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-3\right )}{\rm sign}\left (d\right ) + \frac{1}{6} \, \sqrt{-x^{2} e^{2} + d^{2}}{\left (4 \, d^{2} e^{\left (-3\right )} +{\left (2 \, x e^{\left (-1\right )} - 3 \, d e^{\left (-2\right )}\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d^3*arcsin(x*e/d)*e^(-3)*sign(d) + 1/6*sqrt(-x^2*e^2 + d^2)*(4*d^2*e^(-3) +
(2*x*e^(-1) - 3*d*e^(-2))*x)