Optimal. Leaf size=91 \[ \frac{x^{-n} (c x)^{3 n/2} \sqrt{a+b x^n}}{b c n}-\frac{a x^{-3 n/2} (c x)^{3 n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{b^{3/2} c n} \]
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Rubi [A] time = 0.0399805, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {357, 355, 288, 206} \[ \frac{x^{-n} (c x)^{3 n/2} \sqrt{a+b x^n}}{b c n}-\frac{a x^{-3 n/2} (c x)^{3 n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{b^{3/2} c n} \]
Antiderivative was successfully verified.
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Rule 357
Rule 355
Rule 288
Rule 206
Rubi steps
\begin{align*} \int \frac{(c x)^{-1+\frac{3 n}{2}}}{\sqrt{a+b x^n}} \, dx &=\frac{\left (x^{-3 n/2} (c x)^{3 n/2}\right ) \int \frac{x^{-1+\frac{3 n}{2}}}{\sqrt{a+b x^n}} \, dx}{c}\\ &=\frac{\left (2 a x^{-3 n/2} (c x)^{3 n/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 )}{c n}\\ &=\frac{x^{-n} (c x)^{3 n/2} \sqrt{a+b x^n}}{b c n}-\frac{\left (a x^{-3 n/2} (c x)^{3 n/2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^{n/2}}{\sqrt{a+b x^n}}\right )}{b c n}\\ &=\frac{x^{-n} (c x)^{3 n/2} \sqrt{a+b x^n}}{b c n}-\frac{a x^{-3 n/2} (c x)^{3 n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{b^{3/2} c n}\\ \end{align*}
Mathematica [A] time = 0.0561301, size = 110, normalized size = 1.21 \[ \frac{a x^{1-\frac{3 n}{2}} (c x)^{\frac{3 n}{2}-1} \sqrt{\frac{b x^n}{a}+1} \left (\sqrt{b} x^{n/2} \sqrt{\frac{a+b x^n}{a}}-\sqrt{a} \sinh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a}}\right )\right )}{b^{3/2} n \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-1+{\frac{3\,n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \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{\left (c x\right )^{\frac{3}{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 = 1.45142, size = 360, normalized size = 3.96 \begin{align*} \left [\frac{2 \, \sqrt{b x^{n} + a} b c^{\frac{3}{2} \, n - 1} x^{\frac{1}{2} \, n} + a \sqrt{b} c^{\frac{3}{2} \, n - 1} \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 c^{\frac{3}{2} \, n - 1} x^{\frac{1}{2} \, n} + a \sqrt{-b} c^{\frac{3}{2} \, n - 1} \arctan \left (\frac{\sqrt{-b} x^{\frac{1}{2} \, n}}{\sqrt{b x^{n} + a}}\right )}{b^{2} n}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.7052, size = 66, normalized size = 0.73 \begin{align*} \frac{\sqrt{a} c^{\frac{3 n}{2}} x^{\frac{n}{2}} \sqrt{1 + \frac{b x^{n}}{a}}}{b c n} - \frac{a c^{\frac{3 n}{2}} \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a}} \right )}}{b^{\frac{3}{2}} c n} \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{\left (c x\right )^{\frac{3}{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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