Optimal. Leaf size=65 \[ \frac{4 (c x)^{-3 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}-\frac{2 (c x)^{-3 n/2} \sqrt{a+b x^n}}{a c n} \]
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Rubi [A] time = 0.0174776, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {273, 264} \[ \frac{4 (c x)^{-3 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}-\frac{2 (c x)^{-3 n/2} \sqrt{a+b x^n}}{a c n} \]
Antiderivative was successfully verified.
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Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{(c x)^{-1-\frac{3 n}{2}}}{\sqrt{a+b x^n}} \, dx &=-\frac{2 (c x)^{-3 n/2} \sqrt{a+b x^n}}{a c n}-\frac{2 \int (c x)^{-1-\frac{3 n}{2}} \sqrt{a+b x^n} \, dx}{a}\\ &=-\frac{2 (c x)^{-3 n/2} \sqrt{a+b x^n}}{a c n}+\frac{4 (c x)^{-3 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}\\ \end{align*}
Mathematica [A] time = 0.0149682, size = 41, normalized size = 0.63 \[ -\frac{2 (c x)^{-3 n/2} \left (a-2 b x^n\right ) \sqrt{a+b x^n}}{3 a^2 c n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.055, 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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.0699, size = 68, normalized size = 1.05 \begin{align*} - \frac{2 \sqrt{b} c^{- \frac{3 n}{2}} x^{- n} \sqrt{\frac{a x^{- n}}{b} + 1}}{3 a c n} + \frac{4 b^{\frac{3}{2}} c^{- \frac{3 n}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{3 a^{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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