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 = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 = {1830}
\begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Rule 1830
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*}
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Mathematica [A]
time = 0.69, size = 45, normalized size = 1.00 \begin {gather*} \frac {2 c f x^{n/2}-2 a \left (g+2 h x^{n/4}\right )}{a n \sqrt {a+c x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{-1+\frac {n}{4}} \left (-a h +c f \,x^{\frac {n}{4}}+c g \,x^{\frac {3 n}{4}}+c h \,x^{n}\right )}{\left (a +c \,x^{n}\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 48, normalized size = 1.07 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 15.44, size = 39, normalized size = 0.87 \begin {gather*} \frac {2 \, {\left ({\left (\frac {c f {\left (x^{n}\right )}^{\frac {1}{4}}}{a} - 2 \, h\right )} {\left (x^{n}\right )}^{\frac {1}{4}} - g\right )}}{\sqrt {c x^{n} + a} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.56, size = 39, normalized size = 0.87 \begin {gather*} -\frac {2\,\left (a\,g-c\,f\,x^{n/2}+2\,a\,h\,x^{n/4}\right )}{a\,n\,\sqrt {a+c\,x^n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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