Optimal. Leaf size=32 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{\sqrt{a} n} \]
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Rubi [A] time = 0.0811627, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{\sqrt{a} n} \]
Antiderivative was successfully verified.
[In] Int[1/(x*Sqrt[-a + b*(c*x)^n]),x]
[Out]
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Rubi in Sympy [A] time = 4.07476, size = 26, normalized size = 0.81 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{\sqrt{a} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(-a+b*(c*x)**n)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0256412, size = 32, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{\sqrt{a} n} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*Sqrt[-a + b*(c*x)^n]),x]
[Out]
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Maple [A] time = 0.007, size = 27, normalized size = 0.8 \[ 2\,{\frac{1}{n\sqrt{a}}\arctan \left ({\frac{\sqrt{-a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(-a+b*(c*x)^n)^(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)^n*b - a)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.289236, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{\left (c x\right )^{n} \sqrt{-a} b + 2 \, \sqrt{\left (c x\right )^{n} b - a} a - 2 \, \sqrt{-a} a}{\left (c x\right )^{n}}\right )}{\sqrt{-a} n}, -\frac{2 \, \arctan \left (\frac{\sqrt{a}}{\sqrt{\left (c x\right )^{n} b - a}}\right )}{\sqrt{a} n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt((c*x)^n*b - a)*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{- a + b \left (c x\right )^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(-a+b*(c*x)**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (c x\right )^{n} b - a} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt((c*x)^n*b - a)*x),x, algorithm="giac")
[Out]