3.711 \(\int \frac{(1+a x)^3}{x \left (1-a^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ \frac{4 (a x+1)}{\sqrt{1-a^2 x^2}}-\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\sin ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.196041, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{4 (a x+1)}{\sqrt{1-a^2 x^2}}-\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\sin ^{-1}(a x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.32408, size = 36, normalized size = 0.8 \[ - \operatorname{asin}{\left (a x \right )} - \operatorname{atanh}{\left (\sqrt{- a^{2} x^{2} + 1} \right )} + \frac{4 \sqrt{- a^{2} x^{2} + 1}}{- a x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0780291, size = 51, normalized size = 1.13 \[ -\frac{4 \sqrt{1-a^2 x^2}}{a x-1}-\log \left (\sqrt{1-a^2 x^2}+1\right )-\sin ^{-1}(a x)+\log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 75, normalized size = 1.7 \[ 4\,{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}-{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +4\,{\frac{ax}{\sqrt{-{a}^{2}{x}^{2}+1}}}-{a\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.755662, size = 105, normalized size = 2.33 \[ \frac{4 \, a x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{a \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} + \frac{4}{\sqrt{-a^{2} x^{2} + 1}} - \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.284308, size = 139, normalized size = 3.09 \[ \frac{8 \, a x + 2 \,{\left (a x + \sqrt{-a^{2} x^{2} + 1} - 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (a x + \sqrt{-a^{2} x^{2} + 1} - 1\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right )}{a x + \sqrt{-a^{2} x^{2} + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(8*a*x + 2*(a*x + sqrt(-a^2*x^2 + 1) - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x))
+ (a*x + sqrt(-a^2*x^2 + 1) - 1)*log((sqrt(-a^2*x^2 + 1) - 1)/x))/(a*x + sqrt(-
a^2*x^2 + 1) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a x + 1\right )^{3}}{x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a*x + 1)**3/(x*(-(a*x - 1)*(a*x + 1))**(3/2)), x)

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GIAC/XCAS [A]  time = 0.278451, size = 117, normalized size = 2.6 \[ -\frac{a \arcsin \left (a x\right ){\rm sign}\left (a\right )}{{\left | a \right |}} - \frac{a{\rm ln}\left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{{\left | a \right |}} + \frac{8 \, a}{{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*arcsin(a*x)*sign(a)/abs(a) - a*ln(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)
/(a^2*abs(x)))/abs(a) + 8*a/(((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)*abs(a
))