Optimal. Leaf size=75 \[ -\frac{2 \left (h x^{n/2} \left (b^2-4 a c\right )+c (b f-2 a g)+c x^n (2 c f-b g)\right )}{n \left (b^2-4 a c\right ) \sqrt{a+b x^n+c x^{2 n}}} \]
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Rubi [A] time = 0.530568, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 63, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032, Rules used = {6741, 1753} \[ -\frac{2 \left (h x^{n/2} \left (b^2-4 a c\right )+c (b f-2 a g)+c x^n (2 c f-b g)\right )}{n \left (b^2-4 a c\right ) \sqrt{a+b x^n+c x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 6741
Rule 1753
Rubi steps
\begin{align*} \int \frac{-a h x^{-1+\frac{n}{2}}+c f x^{-1+n}+c g x^{-1+2 n}+c h x^{-1+\frac{5 n}{2}}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx &=\int \frac{x^{-1+\frac{n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx\\ &=-\frac{2 \left (c (b f-2 a g)+\left (b^2-4 a c\right ) h x^{n/2}+c (2 c f-b g) x^n\right )}{\left (b^2-4 a c\right ) n \sqrt{a+b x^n+c x^{2 n}}}\\ \end{align*}
Mathematica [F] time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -ah{x}^{-1+{\frac{n}{2}}}+cf{x}^{-1+n}+cg{x}^{-1+2\,n}+ch{x}^{-1+{\frac{5\,n}{2}}} \right ) \left ( a+b{x}^{n}+c{x}^{2\,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{c h x^{\frac{5}{2} \, n - 1} + c g x^{2 \, n - 1} + c f x^{n - 1} - a h x^{\frac{1}{2} \, n - 1}}{{\left (c x^{2 \, n} + 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 = 1.96787, size = 311, normalized size = 4.15 \begin{align*} -\frac{2 \, \sqrt{c x^{4} x^{2 \, n - 4} + b x^{2} x^{n - 2} + a}{\left ({\left (2 \, c^{2} f - b c g\right )} x^{2} x^{n - 2} +{\left (b^{2} - 4 \, a c\right )} h x x^{\frac{1}{2} \, n - 1} + b c f - 2 \, a c g\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} n x^{4} x^{2 \, n - 4} +{\left (b^{3} - 4 \, a b c\right )} n x^{2} x^{n - 2} +{\left (a b^{2} - 4 \, a^{2} c\right )} 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{c h x^{\frac{5}{2} \, n - 1} + c g x^{2 \, n - 1} + c f x^{n - 1} - a h x^{\frac{1}{2} \, n - 1}}{{\left (c x^{2 \, n} + 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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