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

Optimal. Leaf size=52 \[ \frac{2}{a n \sqrt{a+b (c x)^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{a^{3/2} n} \]

[Out]

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

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Rubi [A]  time = 0.112795, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{2}{a n \sqrt{a+b (c x)^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{a^{3/2} n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.9284, size = 42, normalized size = 0.81 \[ \frac{2}{a n \sqrt{a + b \left (c x\right )^{n}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0604115, size = 52, normalized size = 1. \[ \frac{2}{a n \sqrt{a+b (c x)^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{a^{3/2} n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 43, normalized size = 0.8 \[{\frac{1}{n} \left ( -2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) }+2\,{\frac{1}{\sqrt{a+b \left ( cx \right ) ^{n}}a}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*(c*x)^n)^(3/2),x)

[Out]

1/n*(-2/a^(3/2)*arctanh((a+b*(c*x)^n)^(1/2)/a^(1/2))+2/a/(a+b*(c*x)^n)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.285816, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{\left (c x\right )^{n} b + a} \log \left (\frac{\left (c x\right )^{n} \sqrt{a} b - 2 \, \sqrt{\left (c x\right )^{n} b + a} a + 2 \, a^{\frac{3}{2}}}{\left (c x\right )^{n}}\right ) + 2 \, \sqrt{a}}{\sqrt{\left (c x\right )^{n} b + a} a^{\frac{3}{2}} n}, \frac{2 \,{\left (\sqrt{\left (c x\right )^{n} b + a} \arctan \left (\frac{a}{\sqrt{\left (c x\right )^{n} b + a} \sqrt{-a}}\right ) + \sqrt{-a}\right )}}{\sqrt{\left (c x\right )^{n} b + a} \sqrt{-a} a n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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