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 = 0.31, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 = {6873, 1830}
\begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Rule 1830
Rule 6873
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*}
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Mathematica [A]
time = 0.67, size = 45, normalized size = 1.00 \begin {gather*} \frac {2 b f x^{n/2}-2 a \left (g+2 h x^{n/4}\right )}{a n \sqrt {a+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {-a h \,x^{-1+\frac {n}{4}}+b f \,x^{-1+\frac {n}{2}}+b g \,x^{-1+n}+b h \,x^{-1+\frac {5 n}{4}}}{\left (a +b \,x^{n}\right )^{\frac {3}{2}}}\, dx\]
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 = 0.38, size = 66, normalized size = 1.47 \begin {gather*} \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 {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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {b\,f\,x^{\frac {n}{2}-1}-a\,h\,x^{\frac {n}{4}-1}+b\,h\,x^{\frac {5\,n}{4}-1}+b\,g\,x^{n-1}}{{\left (a+b\,x^n\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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