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 = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {12, 266, 50, 63, 208} \[ \frac {2 \sqrt {a+b x^n}}{c n}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{c n} \]
Antiderivative was successfully verified.
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Rule 12
Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^n}}{c x} \, dx &=\frac {\int \frac {\sqrt {a+b x^n}}{x} \, dx}{c}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^n\right )}{c n}\\ &=\frac {2 \sqrt {a+b x^n}}{c n}+\frac {a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{c n}\\ &=\frac {2 \sqrt {a+b x^n}}{c n}+\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^n}\right )}{b c n}\\ &=\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}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 0.90 \[ \frac {2 \sqrt {a+b x^n}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{c n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 97, normalized size = 1.90 \[ \left [\frac {\sqrt {a} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, \sqrt {b x^{n} + a}}{c n}, \frac {2 \, {\left (\sqrt {-a} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right ) + \sqrt {b x^{n} + a}\right )}}{c n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b x^{n} + a}}{c x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 39, normalized size = 0.76 \[ \frac {-2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {b \,x^{n}+a}}{\sqrt {a}}\right )+2 \sqrt {b \,x^{n}+a}}{c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 58, normalized size = 1.14 \[ \frac {\frac {\sqrt {a} \log \left (\frac {\sqrt {b x^{n} + a} - \sqrt {a}}{\sqrt {b x^{n} + a} + \sqrt {a}}\right )}{n} + \frac {2 \, \sqrt {b x^{n} + a}}{n}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a+b\,x^n}}{c\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.78, size = 78, normalized size = 1.53 \[ \frac {- \frac {2 \sqrt {a} \operatorname {asinh}{\left (\frac {\sqrt {a} x^{- \frac {n}{2}}}{\sqrt {b}} \right )}}{n} + \frac {2 a x^{- \frac {n}{2}}}{\sqrt {b} n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {2 \sqrt {b} x^{\frac {n}{2}}}{n \sqrt {\frac {a x^{- n}}{b} + 1}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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