3.510 \(\int \frac{1}{x \sqrt{a+b (c x)^m}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^m}}{\sqrt{a}}\right )}{\sqrt{a} m} \]

[Out]

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

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Rubi [A]  time = 0.0777392, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^m}}{\sqrt{a}}\right )}{\sqrt{a} m} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[a + b*(c*x)^m]),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.68997, size = 27, normalized size = 0.9 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b \left (c x\right )^{m}}}{\sqrt{a}} \right )}}{\sqrt{a} m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0335413, size = 30, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^m}}{\sqrt{a}}\right )}{\sqrt{a} m} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[a + b*(c*x)^m]),x]

[Out]

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

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Maple [A]  time = 0.009, size = 25, normalized size = 0.8 \[ -2\,{\frac{1}{m\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( cx \right ) ^{m}}}{\sqrt{a}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*(c*x)^m)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282901, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{\left (c x\right )^{m} \sqrt{a} b - 2 \, \sqrt{\left (c x\right )^{m} b + a} a + 2 \, a^{\frac{3}{2}}}{\left (c x\right )^{m}}\right )}{\sqrt{a} m}, \frac{2 \, \arctan \left (\frac{a}{\sqrt{\left (c x\right )^{m} b + a} \sqrt{-a}}\right )}{\sqrt{-a} m}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*sqrt(a + b*(c*x)**m)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt((c*x)^m*b + a)*x), x)