Optimal. Leaf size=51 \[ -\frac{2 x^{\frac{n-1}{2}} (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a x^{n-1}+b x^n+c x^{n+1}}} \]
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Rubi [A] time = 0.0499673, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {1915} \[ -\frac{2 x^{\frac{n-1}{2}} (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a x^{n-1}+b x^n+c x^{n+1}}} \]
Antiderivative was successfully verified.
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Rule 1915
Rubi steps
\begin{align*} \int \frac{x^{\frac{3}{2} (-1+n)}}{\left (a x^{-1+n}+b x^n+c x^{1+n}\right )^{3/2}} \, dx &=-\frac{2 x^{\frac{1}{2} (-1+n)} (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{a x^{-1+n}+b x^n+c x^{1+n}}}\\ \end{align*}
Mathematica [A] time = 0.0883845, size = 46, normalized size = 0.9 \[ -\frac{2 x^{\frac{n-1}{2}} (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt{x^{n-1} (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.074, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-{\frac{3}{2}}+{\frac{3\,n}{2}}} \left ( a{x}^{-1+n}+b{x}^{n}+c{x}^{1+n} \right ) ^{-{\frac{3}{2}}}}\, 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{x^{\frac{3}{2} \, n - \frac{3}{2}}}{{\left (c x^{n + 1} + a x^{n - 1} + b x^{n}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65691, size = 186, normalized size = 3.65 \begin{align*} -\frac{2 \,{\left (2 \, c x^{2} + b x\right )} \sqrt{\frac{{\left (c x^{2} + b x + a\right )} x^{n + 1}}{x^{2}}}}{{\left (a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x\right )} x^{\frac{1}{2} \, n + \frac{1}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{3}{2} \, n - \frac{3}{2}}}{{\left (c x^{n + 1} + a x^{n - 1} + b x^{n}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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