Optimal. Leaf size=51 \[ \frac{2 \sqrt{a+b x^n}}{c n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{c n} \]
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Rubi [A] time = 0.0764388, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{2 \sqrt{a+b x^n}}{c n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{c n} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x^n]/(c*x),x]
[Out]
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Rubi in Sympy [A] time = 9.40493, size = 41, normalized size = 0.8 \[ - \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{n}}}{\sqrt{a}} \right )}}{c n} + \frac{2 \sqrt{a + b x^{n}}}{c n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)**(1/2)/c/x,x)
[Out]
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Mathematica [A] time = 0.0347111, size = 45, normalized size = 0.88 \[ \frac{2 \left (\sqrt{a+b x^n}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )\right )}{c n} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x^n]/(c*x),x]
[Out]
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Maple [A] time = 0.001, size = 39, normalized size = 0.8 \[{\frac{1}{cn} \left ( 2\,\sqrt{a+b{x}^{n}}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{a+b{x}^{n}}}{\sqrt{a}}} \right ) \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)^(1/2)/c/x,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(sqrt(b*x^n + a)/(c*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241772, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{a} \log \left (\frac{b x^{n} - 2 \, \sqrt{b x^{n} + a} \sqrt{a} + 2 \, a}{x^{n}}\right ) + 2 \, \sqrt{b x^{n} + a}}{c n}, -\frac{2 \,{\left (\sqrt{-a} \arctan \left (\frac{\sqrt{b x^{n} + a}}{\sqrt{-a}}\right ) - \sqrt{b x^{n} + a}\right )}}{c n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)/(c*x),x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*x**n)**(1/2)/c/x,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{n} + a}}{c x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)/(c*x),x, algorithm="giac")
[Out]