Optimal. Leaf size=51 \[ -\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{\sqrt {a} n}-\frac {x^{-n} \sqrt {a+b x^n}}{n} \]
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Rubi [A] time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {266, 47, 63, 208} \[ -\frac {x^{-n} \sqrt {a+b x^n}}{n}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{\sqrt {a} n} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int x^{-1-n} \sqrt {a+b x^n} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n} \sqrt {a+b x^n}}{n}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{2 n}\\ &=-\frac {x^{-n} \sqrt {a+b x^n}}{n}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^n}\right )}{n}\\ &=-\frac {x^{-n} \sqrt {a+b x^n}}{n}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{\sqrt {a} n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 62, normalized size = 1.22 \[ -\frac {x^{-n} \left (b x^n \sqrt {\frac {b x^n}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b x^n}{a}+1}\right )+a+b x^n\right )}{n \sqrt {a+b x^n}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 119, normalized size = 2.33 \[ \left [\frac {\sqrt {a} b x^{n} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) - 2 \, \sqrt {b x^{n} + a} a}{2 \, a n x^{n}}, \frac {\sqrt {-a} b x^{n} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right ) - \sqrt {b x^{n} + a} a}{a n x^{n}}\right ] \]
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 \[ \int \sqrt {b x^{n} + a} x^{-n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \sqrt {b \,x^{n}+a}\, x^{-n -1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{n} + a} x^{-n - 1}\,{d x} \]
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 \[ \int \frac {\sqrt {a+b\,x^n}}{x^{n+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 25.08, size = 49, normalized size = 0.96 \[ - \frac {\sqrt {b} x^{- \frac {n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{n} - \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a} x^{- \frac {n}{2}}}{\sqrt {b}} \right )}}{\sqrt {a} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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