3.150 \(\int \frac{x^3}{(1+a x) \sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=66 \[ \frac{3 \sin ^{-1}(a x)}{2 a^4}+\frac{x^2 (1-a x)}{a^2 \sqrt{1-a^2 x^2}}+\frac{(4-3 a x) \sqrt{1-a^2 x^2}}{2 a^4} \]

[Out]

(x^2*(1 - a*x))/(a^2*Sqrt[1 - a^2*x^2]) + ((4 - 3*a*x)*Sqrt[1 - a^2*x^2])/(2*a^4
) + (3*ArcSin[a*x])/(2*a^4)

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Rubi [A]  time = 0.222484, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{3 \sin ^{-1}(a x)}{2 a^4}+\frac{x^2 (1-a x)}{a^2 \sqrt{1-a^2 x^2}}+\frac{(4-3 a x) \sqrt{1-a^2 x^2}}{2 a^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3/((1 + a*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

(x^2*(1 - a*x))/(a^2*Sqrt[1 - a^2*x^2]) + ((4 - 3*a*x)*Sqrt[1 - a^2*x^2])/(2*a^4
) + (3*ArcSin[a*x])/(2*a^4)

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Rubi in Sympy [A]  time = 17.9313, size = 65, normalized size = 0.98 \[ - \frac{x \sqrt{- a^{2} x^{2} + 1}}{2 a^{3}} + \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{4}} + \frac{3 \operatorname{asin}{\left (a x \right )}}{2 a^{4}} + \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{4} \left (a x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*sqrt(-a**2*x**2 + 1)/(2*a**3) + sqrt(-a**2*x**2 + 1)/a**4 + 3*asin(a*x)/(2*a*
*4) + sqrt(-a**2*x**2 + 1)/(a**4*(a*x + 1))

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Mathematica [A]  time = 0.0909491, size = 44, normalized size = 0.67 \[ \frac{\sqrt{1-a^2 x^2} \left (-a x+\frac{2}{a x+1}+2\right )+3 \sin ^{-1}(a x)}{2 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/((1 + a*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

(Sqrt[1 - a^2*x^2]*(2 - a*x + 2/(1 + a*x)) + 3*ArcSin[a*x])/(2*a^4)

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Maple [A]  time = 0.018, size = 100, normalized size = 1.5 \[{\frac{3}{2\,{a}^{3}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{x}{2\,{a}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{1}{{a}^{4}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{1}{{a}^{5} \left ( x+{a}^{-1} \right ) }\sqrt{- \left ( x+{a}^{-1} \right ) ^{2}{a}^{2}+2\, \left ( x+{a}^{-1} \right ) a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2/a^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-1/2/a^3*x*(-a^2*x^2
+1)^(1/2)+1/a^4*(-a^2*x^2+1)^(1/2)+1/a^5/(x+1/a)*(-(x+1/a)^2*a^2+2*(x+1/a)*a)^(1
/2)

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Maxima [A]  time = 0.790013, size = 92, normalized size = 1.39 \[ \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{5} x + a^{4}} - \frac{\sqrt{-a^{2} x^{2} + 1} x}{2 \, a^{3}} + \frac{3 \, \arcsin \left (a x\right )}{2 \, a^{4}} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(sqrt(-a^2*x^2 + 1)*(a*x + 1)),x, algorithm="maxima")

[Out]

sqrt(-a^2*x^2 + 1)/(a^5*x + a^4) - 1/2*sqrt(-a^2*x^2 + 1)*x/a^3 + 3/2*arcsin(a*x
)/a^4 + sqrt(-a^2*x^2 + 1)/a^4

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Fricas [A]  time = 0.292551, size = 286, normalized size = 4.33 \[ -\frac{a^{5} x^{5} - 4 \, a^{4} x^{4} - 7 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 12 \, a x + 6 \,{\left (a^{3} x^{3} + 3 \, a^{2} x^{2} - 2 \, a x -{\left (a^{2} x^{2} - 2 \, a x - 4\right )} \sqrt{-a^{2} x^{2} + 1} - 4\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (a^{4} x^{4} + a^{3} x^{3} - 6 \, a^{2} x^{2} - 12 \, a x\right )} \sqrt{-a^{2} x^{2} + 1}}{2 \,{\left (a^{7} x^{3} + 3 \, a^{6} x^{2} - 2 \, a^{5} x - 4 \, a^{4} -{\left (a^{6} x^{2} - 2 \, a^{5} x - 4 \, a^{4}\right )} \sqrt{-a^{2} x^{2} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(sqrt(-a^2*x^2 + 1)*(a*x + 1)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**3/(sqrt(-(a*x - 1)*(a*x + 1))*(a*x + 1)), x)

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GIAC/XCAS [A]  time = 0.284264, size = 105, normalized size = 1.59 \[ -\frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{x}{a^{3}} - \frac{2}{a^{4}}\right )} + \frac{3 \, \arcsin \left (a x\right ){\rm sign}\left (a\right )}{2 \, a^{3}{\left | a \right |}} - \frac{2}{a^{3}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(sqrt(-a^2*x^2 + 1)*(a*x + 1)),x, algorithm="giac")

[Out]

-1/2*sqrt(-a^2*x^2 + 1)*(x/a^3 - 2/a^4) + 3/2*arcsin(a*x)*sign(a)/(a^3*abs(a)) -
2/(a^3*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a))