Optimal. Leaf size=62 \[ \frac {x^{n/2} \sqrt {a+b x^n}}{b n}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{b^{3/2} n} \]
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Rubi [A] time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {355, 288, 206} \[ \frac {x^{n/2} \sqrt {a+b x^n}}{b n}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{b^{3/2} n} \]
Antiderivative was successfully verified.
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Rule 206
Rule 288
Rule 355
Rubi steps
\begin {align*} \int \frac {x^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx &=\frac {(2 a) \operatorname {Subst}\left (\int \frac {x^2}{\left (1-b x^2\right )^2} \, dx,x,\frac {x^{n/2}}{\sqrt {a+b x^n}}\right )}{n}\\ &=\frac {x^{n/2} \sqrt {a+b x^n}}{b n}-\frac {a \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{n/2}}{\sqrt {a+b x^n}}\right )}{b n}\\ &=\frac {x^{n/2} \sqrt {a+b x^n}}{b n}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{b^{3/2} n}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 81, normalized size = 1.31 \[ \frac {\sqrt {b} x^{n/2} \left (a+b x^n\right )-a^{3/2} \sqrt {\frac {b x^n}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a}}\right )}{b^{3/2} n \sqrt {a+b x^n}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 114, normalized size = 1.84 \[ \left [\frac {2 \, \sqrt {b x^{n} + a} b x^{\frac {1}{2} \, n} + a \sqrt {b} \log \left (2 \, \sqrt {b x^{n} + a} \sqrt {b} x^{\frac {1}{2} \, n} - 2 \, b x^{n} - a\right )}{2 \, b^{2} n}, \frac {\sqrt {b x^{n} + a} b x^{\frac {1}{2} \, n} + a \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x^{\frac {1}{2} \, n}}{\sqrt {b x^{n} + a}}\right )}{b^{2} n}\right ] \]
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 \[ \int \frac {x^{\frac {3}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 64, normalized size = 1.03 \[ -\frac {a \ln \left (\sqrt {b}\, {\mathrm e}^{\frac {n \ln \relax (x )}{2}}+\sqrt {b \,{\mathrm e}^{n \ln \relax (x )}+a}\right )}{b^{\frac {3}{2}} n}+\frac {\sqrt {b \,{\mathrm e}^{n \ln \relax (x )}+a}\, {\mathrm e}^{\frac {n \ln \relax (x )}{2}}}{b n} \]
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 \[ \int \frac {x^{\frac {3}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]
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 \[ \int \frac {x^{\frac {3\,n}{2}-1}}{\sqrt {a+b\,x^n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.86, size = 49, normalized size = 0.79 \[ \frac {\sqrt {a} x^{\frac {n}{2}} \sqrt {1 + \frac {b x^{n}}{a}}}{b n} - \frac {a \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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