Optimal. Leaf size=45 \[ -\frac{2 \left (a g+2 a h x^{n/4}-b f x^{n/2}\right )}{a n \sqrt{a+b x^n}} \]
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Rubi [A] time = 0.394729, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 58, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {6741, 1816} \[ -\frac{2 \left (a g+2 a h x^{n/4}-b f x^{n/2}\right )}{a n \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Rule 6741
Rule 1816
Rubi steps
\begin{align*} \int \frac{-a h x^{-1+\frac{n}{4}}+b f x^{-1+\frac{n}{2}}+b g x^{-1+n}+b h x^{-1+\frac{5 n}{4}}}{\left (a+b x^n\right )^{3/2}} \, dx &=\int \frac{x^{-1+\frac{n}{4}} \left (-a h+b f x^{n/4}+b g x^{3 n/4}+b h x^n\right )}{\left (a+b x^n\right )^{3/2}} \, dx\\ &=-\frac{2 \left (a g+2 a h x^{n/4}-b f x^{n/2}\right )}{a n \sqrt{a+b x^n}}\\ \end{align*}
Mathematica [A] time = 0.21546, size = 45, normalized size = 1. \[ \frac{2 b f x^{n/2}-2 a \left (g+2 h x^{n/4}\right )}{a n \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.487, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -ah{x}^{-1+{\frac{n}{4}}}+bf{x}^{-1+{\frac{n}{2}}}+bg{x}^{-1+n}+bh{x}^{-1+{\frac{5\,n}{4}}} \right ) \left ( a+b{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{b h x^{\frac{5}{4} \, n - 1} + b g x^{n - 1} + b f x^{\frac{1}{2} \, n - 1} - a h x^{\frac{1}{4} \, n - 1}}{{\left (b 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 = 1.895, size = 153, normalized size = 3.4 \begin{align*} \frac{2 \, \sqrt{b x^{4} x^{n - 4} + a}{\left (b f x^{2} x^{\frac{1}{2} \, n - 2} - 2 \, a h x x^{\frac{1}{4} \, n - 1} - a g\right )}}{a b n x^{4} x^{n - 4} + 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{b h x^{\frac{5}{4} \, n - 1} + b g x^{n - 1} + b f x^{\frac{1}{2} \, n - 1} - a h x^{\frac{1}{4} \, n - 1}}{{\left (b 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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