Optimal. Leaf size=129 \[ \frac{5 a^2 x^{n/2} \sqrt{a+b x^n}}{8 b^3 n}-\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{8 b^{7/2} n}-\frac{5 a x^{3 n/2} \sqrt{a+b x^n}}{12 b^2 n}+\frac{x^{5 n/2} \sqrt{a+b x^n}}{3 b n} \]
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Rubi [A] time = 0.0590889, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {355, 288, 206} \[ \frac{5 a^2 x^{n/2} \sqrt{a+b x^n}}{8 b^3 n}-\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{8 b^{7/2} n}-\frac{5 a x^{3 n/2} \sqrt{a+b x^n}}{12 b^2 n}+\frac{x^{5 n/2} \sqrt{a+b x^n}}{3 b n} \]
Antiderivative was successfully verified.
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Rule 355
Rule 288
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{-1+\frac{7 n}{2}}}{\sqrt{a+b x^n}} \, dx &=\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{x^6}{\left (1-b x^2\right )^4} \, dx,x,\frac{x^{n/2}}{\sqrt{a+b x^n}}\right )}{n}\\ &=\frac{x^{5 n/2} \sqrt{a+b x^n}}{3 b n}-\frac{\left (5 a^3\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 )}{3 b n}\\ &=-\frac{5 a x^{3 n/2} \sqrt{a+b x^n}}{12 b^2 n}+\frac{x^{5 n/2} \sqrt{a+b x^n}}{3 b n}+\frac{\left (5 a^3\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 )}{4 b^2 n}\\ &=\frac{5 a^2 x^{n/2} \sqrt{a+b x^n}}{8 b^3 n}-\frac{5 a x^{3 n/2} \sqrt{a+b x^n}}{12 b^2 n}+\frac{x^{5 n/2} \sqrt{a+b x^n}}{3 b n}-\frac{\left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^{n/2}}{\sqrt{a+b x^n}}\right )}{8 b^3 n}\\ &=\frac{5 a^2 x^{n/2} \sqrt{a+b x^n}}{8 b^3 n}-\frac{5 a x^{3 n/2} \sqrt{a+b x^n}}{12 b^2 n}+\frac{x^{5 n/2} \sqrt{a+b x^n}}{3 b n}-\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{8 b^{7/2} n}\\ \end{align*}
Mathematica [A] time = 0.206233, size = 100, normalized size = 0.78 \[ \frac{\sqrt{a+b x^n} \left (\sqrt{b} x^{n/2} \left (15 a^2-10 a b x^n+8 b^2 x^{2 n}\right )-\frac{15 a^{5/2} \sinh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a}}\right )}{\sqrt{\frac{b x^n}{a}+1}}\right )}{24 b^{7/2} n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.094, size = 98, normalized size = 0.8 \begin{align*}{\frac{1}{24\,{b}^{3}n}{{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \left ( 8\, \left ({{\rm e}^{1/2\,n\ln \left ( x \right ) }} \right ) ^{4}{b}^{2}-10\,a \left ({{\rm e}^{1/2\,n\ln \left ( x \right ) }} \right ) ^{2}b+15\,{a}^{2} \right ) \sqrt{a+b \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \right ) ^{2}}}-{\frac{5\,{a}^{3}}{8\,n}\ln \left ( \sqrt{b}{{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}}+\sqrt{a+b \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \right ) ^{2}} \right ){b}^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{7}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26573, size = 436, normalized size = 3.38 \begin{align*} \left [\frac{15 \, a^{3} \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 (8 \, b^{3} x^{\frac{5}{2} \, n} - 10 \, a b^{2} x^{\frac{3}{2} \, n} + 15 \, a^{2} b x^{\frac{1}{2} \, n}\right )} \sqrt{b x^{n} + a}}{48 \, b^{4} n}, \frac{15 \, a^{3} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x^{\frac{1}{2} \, n}}{\sqrt{b x^{n} + a}}\right ) +{\left (8 \, b^{3} x^{\frac{5}{2} \, n} - 10 \, a b^{2} x^{\frac{3}{2} \, n} + 15 \, a^{2} b x^{\frac{1}{2} \, n}\right )} \sqrt{b x^{n} + a}}{24 \, b^{4} n}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 25.4921, size = 148, normalized size = 1.15 \begin{align*} \frac{5 a^{\frac{5}{2}} x^{\frac{n}{2}}}{8 b^{3} n \sqrt{1 + \frac{b x^{n}}{a}}} + \frac{5 a^{\frac{3}{2}} x^{\frac{3 n}{2}}}{24 b^{2} n \sqrt{1 + \frac{b x^{n}}{a}}} - \frac{\sqrt{a} x^{\frac{5 n}{2}}}{12 b n \sqrt{1 + \frac{b x^{n}}{a}}} - \frac{5 a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a}} \right )}}{8 b^{\frac{7}{2}} n} + \frac{x^{\frac{7 n}{2}}}{3 \sqrt{a} n \sqrt{1 + \frac{b x^{n}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{7}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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