Optimal. Leaf size=45 \[ -\frac{2 \left (a g+2 a h x^{n/4}-c f x^{n/2}\right )}{a n \sqrt{a+c x^n}} \]
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Rubi [A] time = 0.07873, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 52, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.019, Rules used = {1816} \[ -\frac{2 \left (a g+2 a h x^{n/4}-c f x^{n/2}\right )}{a n \sqrt{a+c x^n}} \]
Antiderivative was successfully verified.
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Rule 1816
Rubi steps
\begin{align*} \int \frac{x^{-1+\frac{n}{4}} \left (-a h+c f x^{n/4}+c g x^{3 n/4}+c h x^n\right )}{\left (a+c x^n\right )^{3/2}} \, dx &=-\frac{2 \left (a g+2 a h x^{n/4}-c f x^{n/2}\right )}{a n \sqrt{a+c x^n}}\\ \end{align*}
Mathematica [A] time = 0.222079, size = 45, normalized size = 1. \[ \frac{2 c f x^{n/2}-2 a \left (g+2 h x^{n/4}\right )}{a n \sqrt{a+c x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1+{\frac{n}{4}}} \left ( -ah+cf{x}^{{\frac{n}{4}}}+cg{x}^{{\frac{3\,n}{4}}}+ch{x}^{n} \right ) \left ( a+c{x}^{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{{\left (c g x^{\frac{3}{4} \, n} + c f x^{\frac{1}{4} \, n} + c h x^{n} - a h\right )} x^{\frac{1}{4} \, n - 1}}{{\left (c x^{n} + a\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 = 1.3892, size = 107, normalized size = 2.38 \begin{align*} \frac{2 \,{\left (c f x^{\frac{1}{2} \, n} - 2 \, a h x^{\frac{1}{4} \, n} - a g\right )} \sqrt{c x^{n} + a}}{a c n x^{n} + a^{2} n} \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{{\left (c g x^{\frac{3}{4} \, n} + c f x^{\frac{1}{4} \, n} + c h x^{n} - a h\right )} x^{\frac{1}{4} \, n - 1}}{{\left (c x^{n} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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