Optimal. Leaf size=178 \[ \frac{5 a^2 x^{-3 n} (c x)^{7 n/2} \sqrt{a+b x^n}}{8 b^3 c n}-\frac{5 a^3 x^{-7 n/2} (c x)^{7 n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{8 b^{7/2} c n}-\frac{5 a x^{-2 n} (c x)^{7 n/2} \sqrt{a+b x^n}}{12 b^2 c n}+\frac{x^{-n} (c x)^{7 n/2} \sqrt{a+b x^n}}{3 b c n} \]
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Rubi [A] time = 0.0815932, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {357, 355, 288, 206} \[ \frac{5 a^2 x^{-3 n} (c x)^{7 n/2} \sqrt{a+b x^n}}{8 b^3 c n}-\frac{5 a^3 x^{-7 n/2} (c x)^{7 n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{8 b^{7/2} c n}-\frac{5 a x^{-2 n} (c x)^{7 n/2} \sqrt{a+b x^n}}{12 b^2 c n}+\frac{x^{-n} (c x)^{7 n/2} \sqrt{a+b x^n}}{3 b c n} \]
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{7 n}{2}}}{\sqrt{a+b x^n}} \, dx &=\frac{\left (x^{-7 n/2} (c x)^{7 n/2}\right ) \int \frac{x^{-1+\frac{7 n}{2}}}{\sqrt{a+b x^n}} \, dx}{c}\\ &=\frac{\left (2 a^3 x^{-7 n/2} (c x)^{7 n/2}\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 )}{c n}\\ &=\frac{x^{-n} (c x)^{7 n/2} \sqrt{a+b x^n}}{3 b c n}-\frac{\left (5 a^3 x^{-7 n/2} (c x)^{7 n/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 )}{3 b c n}\\ &=-\frac{5 a x^{-2 n} (c x)^{7 n/2} \sqrt{a+b x^n}}{12 b^2 c n}+\frac{x^{-n} (c x)^{7 n/2} \sqrt{a+b x^n}}{3 b c n}+\frac{\left (5 a^3 x^{-7 n/2} (c x)^{7 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 )}{4 b^2 c n}\\ &=\frac{5 a^2 x^{-3 n} (c x)^{7 n/2} \sqrt{a+b x^n}}{8 b^3 c n}-\frac{5 a x^{-2 n} (c x)^{7 n/2} \sqrt{a+b x^n}}{12 b^2 c n}+\frac{x^{-n} (c x)^{7 n/2} \sqrt{a+b x^n}}{3 b c n}-\frac{\left (5 a^3 x^{-7 n/2} (c x)^{7 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 )}{8 b^3 c n}\\ &=\frac{5 a^2 x^{-3 n} (c x)^{7 n/2} \sqrt{a+b x^n}}{8 b^3 c n}-\frac{5 a x^{-2 n} (c x)^{7 n/2} \sqrt{a+b x^n}}{12 b^2 c n}+\frac{x^{-n} (c x)^{7 n/2} \sqrt{a+b x^n}}{3 b c n}-\frac{5 a^3 x^{-7 n/2} (c x)^{7 n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{8 b^{7/2} c n}\\ \end{align*}
Mathematica [A] time = 0.176455, size = 133, normalized size = 0.75 \[ \frac{x^{-7 n/2} (c x)^{7 n/2} \sqrt{a+b x^n} \left (\sqrt{b} x^{n/2} \sqrt{\frac{b x^n}{a}+1} \left (15 a^2-10 a b x^n+8 b^2 x^{2 n}\right )-15 a^{5/2} \sinh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a}}\right )\right )}{24 b^{7/2} c n \sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.061, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-1+{\frac{7\,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{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 = 1.50267, size = 587, normalized size = 3.3 \begin{align*} \left [\frac{15 \, a^{3} \sqrt{b} c^{\frac{7}{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 \,{\left (8 \, b^{3} c^{\frac{7}{2} \, n - 1} x^{\frac{5}{2} \, n} - 10 \, a b^{2} c^{\frac{7}{2} \, n - 1} x^{\frac{3}{2} \, n} + 15 \, a^{2} b c^{\frac{7}{2} \, n - 1} x^{\frac{1}{2} \, n}\right )} \sqrt{b x^{n} + a}}{48 \, b^{4} n}, \frac{15 \, a^{3} \sqrt{-b} c^{\frac{7}{2} \, n - 1} \arctan \left (\frac{\sqrt{-b} x^{\frac{1}{2} \, n}}{\sqrt{b x^{n} + a}}\right ) +{\left (8 \, b^{3} c^{\frac{7}{2} \, n - 1} x^{\frac{5}{2} \, n} - 10 \, a b^{2} c^{\frac{7}{2} \, n - 1} x^{\frac{3}{2} \, n} + 15 \, a^{2} b c^{\frac{7}{2} \, n - 1} 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 = 19.5634, size = 190, normalized size = 1.07 \begin{align*} \frac{5 a^{\frac{5}{2}} c^{\frac{7 n}{2}} x^{\frac{n}{2}}}{8 b^{3} c n \sqrt{1 + \frac{b x^{n}}{a}}} + \frac{5 a^{\frac{3}{2}} c^{\frac{7 n}{2}} x^{\frac{3 n}{2}}}{24 b^{2} c n \sqrt{1 + \frac{b x^{n}}{a}}} - \frac{\sqrt{a} c^{\frac{7 n}{2}} x^{\frac{5 n}{2}}}{12 b c n \sqrt{1 + \frac{b x^{n}}{a}}} - \frac{5 a^{3} c^{\frac{7 n}{2}} \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a}} \right )}}{8 b^{\frac{7}{2}} c n} + \frac{c^{\frac{7 n}{2}} x^{\frac{7 n}{2}}}{3 \sqrt{a} c 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{\left (c x\right )^{\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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