Optimal. Leaf size=75 \[ -\frac {2 \left (c (b f-2 a g)+\left (b^2-4 a c\right ) h x^{n/2}+c (2 c f-b g) x^n\right )}{\left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}} \]
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Rubi [A]
time = 0.35, antiderivative size = 75, normalized size of antiderivative = 1.00, 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 = {6873, 1767}
\begin {gather*} -\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}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1767
Rule 6873
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*}
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Mathematica [A]
time = 1.62, size = 84, normalized size = 1.12 \begin {gather*} -\frac {2 \left (b c f-2 a c g+b^2 h x^{n/2}-4 a c h x^{n/2}+2 c^2 f x^n-b c g x^n\right )}{\left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {-a h \,x^{-1+\frac {n}{2}}+c f \,x^{-1+n}+c g \,x^{-1+2 n}+c h \,x^{-1+\frac {5 n}{2}}}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{\frac {3}{2}}}\, dx\]
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 = 137, normalized size = 1.83 \begin {gather*} -\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 {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 = -0.01 \begin {gather*} \int \frac {c\,g\,x^{2\,n-1}-a\,h\,x^{\frac {n}{2}-1}+c\,h\,x^{\frac {5\,n}{2}-1}+c\,f\,x^{n-1}}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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