Optimal. Leaf size=49 \[ \frac{2 \sqrt{a+b (c x)^n}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{n} \]
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Rubi [A] time = 0.101153, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{2 \sqrt{a+b (c x)^n}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{n} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*(c*x)^n]/x,x]
[Out]
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Rubi in Sympy [A] time = 4.67254, size = 41, normalized size = 0.84 \[ - \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{n} + \frac{2 \sqrt{a + b \left (c x\right )^{n}}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(c*x)**n)**(1/2)/x,x)
[Out]
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Mathematica [A] time = 0.0294167, size = 46, normalized size = 0.94 \[ \frac{2 \left (\sqrt{a+b (c x)^n}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )\right )}{n} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*(c*x)^n]/x,x]
[Out]
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Maple [A] time = 0.002, size = 40, normalized size = 0.8 \[{\frac{1}{n} \left ( 2\,\sqrt{a+b \left ( cx \right ) ^{n}}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(c*x)^n)^(1/2)/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((c*x)^n*b + a)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.284205, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{a} \log \left (\frac{\left (c x\right )^{n} b - 2 \, \sqrt{\left (c x\right )^{n} b + a} \sqrt{a} + 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \, \sqrt{\left (c x\right )^{n} b + a}}{n}, -\frac{2 \,{\left (\sqrt{-a} \arctan \left (\frac{\sqrt{\left (c x\right )^{n} b + a}}{\sqrt{-a}}\right ) - \sqrt{\left (c x\right )^{n} b + a}\right )}}{n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((c*x)^n*b + a)/x,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \left (c x\right )^{n}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(c*x)**n)**(1/2)/x,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (c x\right )^{n} b + a}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((c*x)^n*b + a)/x,x, algorithm="giac")
[Out]