Optimal. Leaf size=84 \[ \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{4 a^{3/2} n}-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 n}-\frac {b x^{-n} \sqrt {a+b x^n}}{4 a n} \]
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Rubi [A] time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {266, 47, 51, 63, 208} \[ \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{4 a^{3/2} n}-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 n}-\frac {b x^{-n} \sqrt {a+b x^n}}{4 a n} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int x^{-1-2 n} \sqrt {a+b x^n} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^3} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 n}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^n\right )}{4 n}\\ &=-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 n}-\frac {b x^{-n} \sqrt {a+b x^n}}{4 a n}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{8 a n}\\ &=-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 n}-\frac {b x^{-n} \sqrt {a+b x^n}}{4 a n}-\frac {b \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^n}\right )}{4 a n}\\ &=-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 n}-\frac {b x^{-n} \sqrt {a+b x^n}}{4 a n}+\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{4 a^{3/2} n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 42, normalized size = 0.50 \[ -\frac {2 b^2 \left (a+b x^n\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {b x^n}{a}+1\right )}{3 a^3 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 153, normalized size = 1.82 \[ \left [\frac {\sqrt {a} b^{2} x^{2 \, n} \log \left (\frac {b x^{n} + 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) - 2 \, {\left (a b x^{n} + 2 \, a^{2}\right )} \sqrt {b x^{n} + a}}{8 \, a^{2} n x^{2 \, n}}, -\frac {\sqrt {-a} b^{2} x^{2 \, n} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right ) + {\left (a b x^{n} + 2 \, a^{2}\right )} \sqrt {b x^{n} + a}}{4 \, a^{2} n x^{2 \, n}}\right ] \]
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 \[ \int \sqrt {b x^{n} + a} x^{-2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \sqrt {b \,x^{n}+a}\, x^{-2 n -1}\, dx \]
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 \[ \int \sqrt {b x^{n} + a} x^{-2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+b\,x^n}}{x^{2\,n+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 54.20, size = 112, normalized size = 1.33 \[ - \frac {a x^{- \frac {5 n}{2}}}{2 \sqrt {b} n \sqrt {\frac {a x^{- n}}{b} + 1}} - \frac {3 \sqrt {b} x^{- \frac {3 n}{2}}}{4 n \sqrt {\frac {a x^{- n}}{b} + 1}} - \frac {b^{\frac {3}{2}} x^{- \frac {n}{2}}}{4 a n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} x^{- \frac {n}{2}}}{\sqrt {b}} \right )}}{4 a^{\frac {3}{2}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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