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

Optimal. Leaf size=103 \[ -\frac{d x \sqrt{d^2-e^2 x^2}}{2 e^2}-\frac{d^2 \sqrt{d^2-e^2 x^2}}{e^3}+\frac{\left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac{d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^3} \]

[Out]

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

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Rubi [A]  time = 0.159127, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{d x \sqrt{d^2-e^2 x^2}}{2 e^2}-\frac{d^2 \sqrt{d^2-e^2 x^2}}{e^3}+\frac{\left (d^2-e^2 x^2\right )^{3/2}}{3 e^3}+\frac{d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.5057, size = 87, normalized size = 0.84 \[ \frac{d^{3} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e^{3}} - \frac{d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{e^{3}} - \frac{d x \sqrt{d^{2} - e^{2} x^{2}}}{2 e^{2}} + \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(e*x+d)/(-e**2*x**2+d**2)**(1/2),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 = 0.0692683, size = 70, normalized size = 0.68 \[ \frac{3 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\sqrt{d^2-e^2 x^2} \left (4 d^2+3 d e x+2 e^2 x^2\right )}{6 e^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(d + e*x))/Sqrt[d^2 - e^2*x^2],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 [A]  time = 0.02, size = 102, normalized size = 1. \[ -{\frac{dx}{2\,{e}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{{d}^{3}}{2\,{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{x}^{2}}{3\,e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{2\,{d}^{2}}{3\,{e}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}} \]

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]

-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))-1/3*x^2/e*(-e^2*x^2+d^2)^(1/2)-2/3*d^2*(-e^2*x^2+d^2)^(1/2)
/e^3

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Maxima [A]  time = 0.804686, size = 127, normalized size = 1.23 \[ -\frac{\sqrt{-e^{2} x^{2} + d^{2}} x^{2}}{3 \, e} + \frac{d^{3} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}} e^{2}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} d x}{2 \, e^{2}} - \frac{2 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2}}{3 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.276598, size = 301, normalized size = 2.92 \[ -\frac{2 \, e^{6} x^{6} + 3 \, d e^{5} x^{5} - 6 \, d^{2} e^{4} x^{4} - 15 \, d^{3} e^{3} x^{3} + 12 \, d^{5} e x + 6 \,{\left (3 \, d^{4} e^{2} x^{2} - 4 \, d^{6} -{\left (d^{3} e^{2} x^{2} - 4 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 3 \,{\left (2 \, d e^{4} x^{4} + 3 \, d^{2} e^{3} x^{3} - 4 \, d^{4} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (3 \, d e^{5} x^{2} - 4 \, d^{3} e^{3} -{\left (e^{5} x^{2} - 4 \, d^{2} e^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*x^2/sqrt(-e^2*x^2 + d^2),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)*sqr
t(-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 [A]  time = 11.3763, size = 177, normalized size = 1.72 \[ d \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e^{3}} - \frac{i d x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}}{2 e^{2}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e^{3}} - \frac{d x}{2 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{x^{3}}{2 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e \left (\begin{cases} - \frac{2 d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 e^{4}} - \frac{x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 e^{2}} & \text{for}\: e \neq 0 \\\frac{x^{4}}{4 \sqrt{d^{2}}} & \text{otherwise} \end{cases}\right ) \]

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]

d*Piecewise((-I*d**2*acosh(e*x/d)/(2*e**3) - I*d*x*sqrt(-1 + e**2*x**2/d**2)/(2*
e**2), Abs(e**2*x**2/d**2) > 1), (d**2*asin(e*x/d)/(2*e**3) - d*x/(2*e**2*sqrt(1
- e**2*x**2/d**2)) + x**3/(2*d*sqrt(1 - e**2*x**2/d**2)), True)) + e*Piecewise(
(-2*d**2*sqrt(d**2 - e**2*x**2)/(3*e**4) - x**2*sqrt(d**2 - e**2*x**2)/(3*e**2),
Ne(e, 0)), (x**4/(4*sqrt(d**2)), True))

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GIAC/XCAS [A]  time = 0.291206, size = 73, normalized size = 0.71 \[ \frac{1}{2} \, d^{3} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-3\right )}{\rm sign}\left (d\right ) - \frac{1}{6} \, \sqrt{-x^{2} e^{2} + d^{2}}{\left (4 \, d^{2} e^{\left (-3\right )} +{\left (2 \, x e^{\left (-1\right )} + 3 \, d e^{\left (-2\right )}\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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