3.511 \(\int \frac{1}{x \sqrt{a+b \left (c (d x)^m\right )^n}} \, dx\)

Optimal. Leaf size=37 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c (d x)^m\right )^n}}{\sqrt{a}}\right )}{\sqrt{a} m n} \]

[Out]

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

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Rubi [A]  time = 0.301416, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c (d x)^m\right )^n}}{\sqrt{a}}\right )}{\sqrt{a} m n} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.112403, size = 37, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c (d x)^m\right )^n}}{\sqrt{a}}\right )}{\sqrt{a} m n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 32, normalized size = 0.9 \[ -2\,{\frac{1}{mn\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( c \left ( dx \right ) ^{m} \right ) ^{n}}}{\sqrt{a}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*(c*(d*x)^m)^n)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282608, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left ({\left (\sqrt{a} b e^{\left (m n \log \left (d x\right ) + n \log \left (c\right )\right )} - 2 \, \sqrt{b e^{\left (m n \log \left (d x\right ) + n \log \left (c\right )\right )} + a} a + 2 \, a^{\frac{3}{2}}\right )} e^{\left (-m n \log \left (d x\right ) - n \log \left (c\right )\right )}\right )}{\sqrt{a} m n}, \frac{2 \, \arctan \left (\frac{a}{\sqrt{b e^{\left (m n \log \left (d x\right ) + n \log \left (c\right )\right )} + a} \sqrt{-a}}\right )}{\sqrt{-a} m n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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