3.402 \(\int \frac{1}{c x \left (a+b x^n\right )^{3/2}} \, dx\)

Optimal. Leaf size=54 \[ \frac{2}{a c n \sqrt{a+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} c n} \]

[Out]

2/(a*c*n*Sqrt[a + b*x^n]) - (2*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(a^(3/2)*c*n)

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Rubi [A]  time = 0.0874904, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{2}{a c n \sqrt{a+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} c n} \]

Antiderivative was successfully verified.

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

[Out]

2/(a*c*n*Sqrt[a + b*x^n]) - (2*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(a^(3/2)*c*n)

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Rubi in Sympy [A]  time = 10.2156, size = 42, normalized size = 0.78 \[ \frac{2}{a c n \sqrt{a + b x^{n}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{n}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/c/x/(a+b*x**n)**(3/2),x)

[Out]

2/(a*c*n*sqrt(a + b*x**n)) - 2*atanh(sqrt(a + b*x**n)/sqrt(a))/(a**(3/2)*c*n)

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Mathematica [A]  time = 0.073822, size = 52, normalized size = 0.96 \[ \frac{\frac{2}{a n \sqrt{a+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} n}}{c} \]

Antiderivative was successfully verified.

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

[Out]

(2/(a*n*Sqrt[a + b*x^n]) - (2*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(a^(3/2)*n))/c

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Maple [A]  time = 0., size = 42, normalized size = 0.8 \[{\frac{1}{cn} \left ( -2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{a+b{x}^{n}}}{\sqrt{a}}} \right ) }+2\,{\frac{1}{a\sqrt{a+b{x}^{n}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/c/x/(a+b*x^n)^(3/2),x)

[Out]

1/c/n*(-2/a^(3/2)*arctanh((a+b*x^n)^(1/2)/a^(1/2))+2/a/(a+b*x^n)^(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(1/((b*x^n + a)^(3/2)*c*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255504, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b x^{n} + a} \log \left (\frac{\sqrt{a} b x^{n} - 2 \, \sqrt{b x^{n} + a} a + 2 \, a^{\frac{3}{2}}}{x^{n}}\right ) + 2 \, \sqrt{a}}{\sqrt{b x^{n} + a} a^{\frac{3}{2}} c n}, \frac{2 \,{\left (\sqrt{b x^{n} + a} \arctan \left (\frac{a}{\sqrt{b x^{n} + a} \sqrt{-a}}\right ) + \sqrt{-a}\right )}}{\sqrt{b x^{n} + a} \sqrt{-a} a c n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(sqrt(b*x^n + a)*log((sqrt(a)*b*x^n - 2*sqrt(b*x^n + a)*a + 2*a^(3/2))/x^n) + 2
*sqrt(a))/(sqrt(b*x^n + a)*a^(3/2)*c*n), 2*(sqrt(b*x^n + a)*arctan(a/(sqrt(b*x^n
+ a)*sqrt(-a))) + sqrt(-a))/(sqrt(b*x^n + a)*sqrt(-a)*a*c*n)]

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/c/x/(a+b*x**n)**(3/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{3}{2}} c x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^n + a)^(3/2)*c*x), x)