Optimal. Leaf size=98 \[ \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{4 b^{5/2} n}-\frac {3 a x^{n/2} \sqrt {a+b x^n}}{4 b^2 n}+\frac {x^{3 n/2} \sqrt {a+b x^n}}{2 b n} \]
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Rubi [A] time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {355, 288, 206} \[ \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{4 b^{5/2} n}-\frac {3 a x^{n/2} \sqrt {a+b x^n}}{4 b^2 n}+\frac {x^{3 n/2} \sqrt {a+b x^n}}{2 b n} \]
Antiderivative was successfully verified.
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Rule 206
Rule 288
Rule 355
Rubi steps
\begin {align*} \int \frac {x^{-1+\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx &=\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1-b x^2\right )^3} \, dx,x,\frac {x^{n/2}}{\sqrt {a+b x^n}}\right )}{n}\\ &=\frac {x^{3 n/2} \sqrt {a+b x^n}}{2 b n}-\frac {\left (3 a^2\right ) \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 )}{2 b n}\\ &=-\frac {3 a x^{n/2} \sqrt {a+b x^n}}{4 b^2 n}+\frac {x^{3 n/2} \sqrt {a+b x^n}}{2 b n}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{n/2}}{\sqrt {a+b x^n}}\right )}{4 b^2 n}\\ &=-\frac {3 a x^{n/2} \sqrt {a+b x^n}}{4 b^2 n}+\frac {x^{3 n/2} \sqrt {a+b x^n}}{2 b n}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{4 b^{5/2} n}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 100, normalized size = 1.02 \[ \frac {3 a^{5/2} \sqrt {\frac {b x^n}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a}}\right )+\sqrt {b} x^{n/2} \left (-3 a^2-a b x^n+2 b^2 x^{2 n}\right )}{4 b^{5/2} n \sqrt {a+b x^n}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 150, normalized size = 1.53 \[ \left [\frac {3 \, a^{2} \sqrt {b} \log \left (-2 \, \sqrt {b x^{n} + a} \sqrt {b} x^{\frac {1}{2} \, n} - 2 \, b x^{n} - a\right ) + 2 \, {\left (2 \, b^{2} x^{\frac {3}{2} \, n} - 3 \, a b x^{\frac {1}{2} \, n}\right )} \sqrt {b x^{n} + a}}{8 \, b^{3} n}, -\frac {3 \, a^{2} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x^{\frac {1}{2} \, n}}{\sqrt {b x^{n} + a}}\right ) - {\left (2 \, b^{2} x^{\frac {3}{2} \, n} - 3 \, a b x^{\frac {1}{2} \, n}\right )} \sqrt {b x^{n} + a}}{4 \, b^{3} n}\right ] \]
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 \[ \int \frac {x^{\frac {5}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 82, normalized size = 0.84 \[ \frac {3 a^{2} \ln \left (\sqrt {b}\, {\mathrm e}^{\frac {n \ln \relax (x )}{2}}+\sqrt {b \,{\mathrm e}^{n \ln \relax (x )}+a}\right )}{4 b^{\frac {5}{2}} n}-\frac {\left (-2 b \,{\mathrm e}^{n \ln \relax (x )}+3 a \right ) \sqrt {b \,{\mathrm e}^{n \ln \relax (x )}+a}\, {\mathrm e}^{\frac {n \ln \relax (x )}{2}}}{4 b^{2} n} \]
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 \[ \int \frac {x^{\frac {5}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^{\frac {5\,n}{2}-1}}{\sqrt {a+b\,x^n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.74, size = 116, normalized size = 1.18 \[ - \frac {3 a^{\frac {3}{2}} x^{\frac {n}{2}}}{4 b^{2} n \sqrt {1 + \frac {b x^{n}}{a}}} - \frac {\sqrt {a} x^{\frac {3 n}{2}}}{4 b n \sqrt {1 + \frac {b x^{n}}{a}}} + \frac {3 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{4 b^{\frac {5}{2}} n} + \frac {x^{\frac {5 n}{2}}}{2 \sqrt {a} n \sqrt {1 + \frac {b x^{n}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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