Optimal. Leaf size=31 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{\sqrt {a} c n} \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {12, 266, 63, 208} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{\sqrt {a} c n} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{c x \sqrt {a+b x^n}} \, dx &=\frac {\int \frac {1}{x \sqrt {a+b x^n}} \, dx}{c}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{c n}\\ &=\frac {2 \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 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{\sqrt {a} c n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{\sqrt {a} c n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 76, normalized size = 2.45 \[ \left [\frac {\log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right )}{\sqrt {a} c n}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right )}{a c n}\right ] \]
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 \[ \int \frac {1}{\sqrt {b x^{n} + a} c x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 26, normalized size = 0.84 \[ -\frac {2 \arctanh \left (\frac {\sqrt {b \,x^{n}+a}}{\sqrt {a}}\right )}{\sqrt {a}\, c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 42, normalized size = 1.35 \[ \frac {\log \left (\frac {\sqrt {b x^{n} + a} - \sqrt {a}}{\sqrt {b x^{n} + a} + \sqrt {a}}\right )}{\sqrt {a} c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{c\,x\,\sqrt {a+b\,x^n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.98, size = 27, normalized size = 0.87 \[ - \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {a} x^{- \frac {n}{2}}}{\sqrt {b}} \right )}}{\sqrt {a} c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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