Optimal. Leaf size=95 \[ -\frac {2 x^{1-\frac {n}{2}} (d x)^{\frac {n-2}{2}} \left (h x^{n/2} \left (b^2-4 a c\right )+c (b f-2 a g)+c x^n (2 c f-b g)\right )}{n \left (b^2-4 a c\right ) \sqrt {a+b x^n+c x^{2 n}}} \]
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Rubi [A] time = 0.22, antiderivative size = 95, 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 = {1754, 1753} \[ -\frac {2 x^{1-\frac {n}{2}} (d x)^{\frac {n-2}{2}} \left (h x^{n/2} \left (b^2-4 a c\right )+c (b f-2 a g)+c x^n (2 c f-b g)\right )}{n \left (b^2-4 a c\right ) \sqrt {a+b x^n+c x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 1753
Rule 1754
Rubi steps
\begin {align*} \int \frac {(d 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 &=\left (x^{1-\frac {n}{2}} (d x)^{-1+\frac {n}{2}}\right ) \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 x^{1-\frac {n}{2}} (d x)^{\frac {1}{2} (-2+n)} \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 [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [A] time = 0.64, size = 132, normalized size = 1.39 \[ -\frac {2 \, {\left ({\left (b^{2} - 4 \, a c\right )} d^{\frac {1}{2} \, n - 1} h x^{\frac {1}{2} \, n} + {\left (2 \, c^{2} f - b c g\right )} d^{\frac {1}{2} \, n - 1} x^{n} + {\left (b c f - 2 \, a c g\right )} d^{\frac {1}{2} \, n - 1}\right )} \sqrt {c x^{2 \, n} + b x^{n} + a}}{{\left (b^{2} c - 4 \, a c^{2}\right )} n x^{2 \, n} + {\left (b^{3} - 4 \, a b c\right )} n x^{n} + {\left (a b^{2} - 4 \, a^{2} c\right )} n} \]
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 \[ \int \frac {{\left (c h x^{2 \, n} + c g x^{\frac {3}{2} \, n} + c f x^{\frac {1}{2} \, n} - a h\right )} \left (d x\right )^{\frac {1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {\left (c f \,x^{\frac {n}{2}}+c g \,x^{\frac {3 n}{2}}+c h \,x^{2 n}-a h \right ) \left (d x \right )^{\frac {n}{2}-1}}{\left (b \,x^{n}+c \,x^{2 n}+a \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (c h x^{2 \, n} + c g x^{\frac {3}{2} \, n} + c f x^{\frac {1}{2} \, n} - a h\right )} \left (d x\right )^{\frac {1}{2} \, n - 1}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (d\,x\right )}^{\frac {n}{2}-1}\,\left (c\,f\,x^{n/2}-a\,h+c\,g\,x^{\frac {3\,n}{2}}+c\,h\,x^{2\,n}\right )}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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