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

Optimal. Leaf size=51 \[ \frac{4 \sqrt{a x+1}}{\sqrt{1-a x}}-\sin ^{-1}(a x)-\tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.15832, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304 \[ \frac{4 \sqrt{a x+1}}{\sqrt{1-a x}}-\sin ^{-1}(a x)-\tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.14738, size = 41, normalized size = 0.8 \[ - \operatorname{asin}{\left (a x \right )} - \operatorname{atanh}{\left (\sqrt{- a x + 1} \sqrt{a x + 1} \right )} + \frac{4 \sqrt{a x + 1}}{\sqrt{- a x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.112888, size = 74, normalized size = 1.45 \[ \frac{4 \sqrt{1-a^2 x^2}}{1-a x}-\log \left (\sqrt{1-a^2 x^2}+1\right )-i \log \left (2 \left (\sqrt{1-a^2 x^2}-i a x\right )\right )+\log (x) \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [C]  time = 0.045, size = 130, normalized size = 2.6 \[{\frac{{\it csgn} \left ( a \right ) }{ax-1} \left ( -{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ){\it csgn} \left ( a \right ) xa-\arctan \left ({{\it csgn} \left ( a \right ) xa{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) xa+{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ){\it csgn} \left ( a \right ) -4\,\sqrt{-{a}^{2}{x}^{2}+1}{\it csgn} \left ( a \right ) +\arctan \left ({{\it csgn} \left ( a \right ) xa{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) \right ) \sqrt{-ax+1}\sqrt{ax+1}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.760395, size = 105, normalized size = 2.06 \[ \frac{4 \, a x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{a \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} + \frac{4}{\sqrt{-a^{2} x^{2} + 1}} - \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 1)^(3/2)/((-a*x + 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 = 0.290707, size = 166, normalized size = 3.25 \[ \frac{8 \, a x + 2 \,{\left (a x + \sqrt{a x + 1} \sqrt{-a x + 1} - 1\right )} \arctan \left (\frac{\sqrt{a x + 1} \sqrt{-a x + 1} - 1}{a x}\right ) +{\left (a x + \sqrt{a x + 1} \sqrt{-a x + 1} - 1\right )} \log \left (\frac{\sqrt{a x + 1} \sqrt{-a x + 1} - 1}{x}\right )}{a x + \sqrt{a x + 1} \sqrt{-a x + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError