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

Optimal. Leaf size=118 \[ -\frac{d x^2 \sqrt{d^2-e^2 x^2}}{3 e^2}+\frac{x^3 \sqrt{d^2-e^2 x^2}}{4 e}-\frac{d^2 (16 d-9 e x) \sqrt{d^2-e^2 x^2}}{24 e^4}-\frac{3 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^4} \]

[Out]

-(d*x^2*Sqrt[d^2 - e^2*x^2])/(3*e^2) + (x^3*Sqrt[d^2 - e^2*x^2])/(4*e) - (d^2*(1
6*d - 9*e*x)*Sqrt[d^2 - e^2*x^2])/(24*e^4) - (3*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*
x^2]])/(8*e^4)

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Rubi [A]  time = 0.351921, antiderivative size = 118, 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 \[ -\frac{d x^2 \sqrt{d^2-e^2 x^2}}{3 e^2}+\frac{x^3 \sqrt{d^2-e^2 x^2}}{4 e}-\frac{d^2 (16 d-9 e x) \sqrt{d^2-e^2 x^2}}{24 e^4}-\frac{3 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^4} \]

Antiderivative was successfully verified.

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

[Out]

-(d*x^2*Sqrt[d^2 - e^2*x^2])/(3*e^2) + (x^3*Sqrt[d^2 - e^2*x^2])/(4*e) - (d^2*(1
6*d - 9*e*x)*Sqrt[d^2 - e^2*x^2])/(24*e^4) - (3*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*
x^2]])/(8*e^4)

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Rubi in Sympy [A]  time = 40.4693, size = 102, normalized size = 0.86 \[ - \frac{3 d^{4} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{8 e^{4}} - \frac{d^{2} \left (16 d - 9 e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{24 e^{4}} - \frac{d x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 e^{2}} + \frac{x^{3} \sqrt{d^{2} - e^{2} x^{2}}}{4 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*d**4*atan(e*x/sqrt(d**2 - e**2*x**2))/(8*e**4) - d**2*(16*d - 9*e*x)*sqrt(d**
2 - e**2*x**2)/(24*e**4) - d*x**2*sqrt(d**2 - e**2*x**2)/(3*e**2) + x**3*sqrt(d*
*2 - e**2*x**2)/(4*e)

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Mathematica [A]  time = 0.0789366, size = 80, normalized size = 0.68 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-16 d^3+9 d^2 e x-8 d e^2 x^2+6 e^3 x^3\right )-9 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{24 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-16*d^3 + 9*d^2*e*x - 8*d*e^2*x^2 + 6*e^3*x^3) - 9*d^4*Arc
Tan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(24*e^4)

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Maple [A]  time = 0.015, size = 185, normalized size = 1.6 \[ -{\frac{x}{4\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{d}^{2}x}{8\,{e}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{5\,{d}^{4}}{8\,{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{d}{3\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{{d}^{3}}{{e}^{4}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{{d}^{4}}{{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/e^3*x*(-e^2*x^2+d^2)^(3/2)+5/8*d^2/e^3*x*(-e^2*x^2+d^2)^(1/2)+5/8/e^3*d^4/(
e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+1/3*d/e^4*(-e^2*x^2+d^2)^(
3/2)-d^3/e^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-d^4/e^3/(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^3/(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28909, size = 387, normalized size = 3.28 \[ -\frac{24 \, d e^{7} x^{7} - 32 \, d^{2} e^{6} x^{6} - 36 \, d^{3} e^{5} x^{5} + 48 \, d^{4} e^{4} x^{4} - 60 \, d^{5} e^{3} x^{3} + 72 \, d^{7} e x - 18 \,{\left (d^{4} e^{4} x^{4} - 8 \, d^{6} e^{2} x^{2} + 8 \, d^{8} + 4 \,{\left (d^{5} e^{2} x^{2} - 2 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (6 \, e^{7} x^{7} - 8 \, d e^{6} x^{6} - 39 \, d^{2} e^{5} x^{5} + 48 \, d^{3} e^{4} x^{4} - 24 \, d^{4} e^{3} x^{3} + 72 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \,{\left (e^{8} x^{4} - 8 \, d^{2} e^{6} x^{2} + 8 \, d^{4} e^{4} + 4 \,{\left (d e^{6} x^{2} - 2 \, d^{3} e^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(24*d*e^7*x^7 - 32*d^2*e^6*x^6 - 36*d^3*e^5*x^5 + 48*d^4*e^4*x^4 - 60*d^5*
e^3*x^3 + 72*d^7*e*x - 18*(d^4*e^4*x^4 - 8*d^6*e^2*x^2 + 8*d^8 + 4*(d^5*e^2*x^2
- 2*d^7)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (6*e^
7*x^7 - 8*d*e^6*x^6 - 39*d^2*e^5*x^5 + 48*d^3*e^4*x^4 - 24*d^4*e^3*x^3 + 72*d^6*
e*x)*sqrt(-e^2*x^2 + d^2))/(e^8*x^4 - 8*d^2*e^6*x^2 + 8*d^4*e^4 + 4*(d*e^6*x^2 -
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. \[ \int \frac{x^{3} \sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.294434, size = 89, normalized size = 0.75 \[ -\frac{3}{8} \, d^{4} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-4\right )}{\rm sign}\left (d\right ) - \frac{1}{24} \,{\left (16 \, d^{3} e^{\left (-4\right )} -{\left (9 \, d^{2} e^{\left (-3\right )} + 2 \,{\left (3 \, x e^{\left (-1\right )} - 4 \, d e^{\left (-2\right )}\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/8*d^4*arcsin(x*e/d)*e^(-4)*sign(d) - 1/24*(16*d^3*e^(-4) - (9*d^2*e^(-3) + 2*
(3*x*e^(-1) - 4*d*e^(-2))*x)*x)*sqrt(-x^2*e^2 + d^2)