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