Optimal. Leaf size=51 \[ -\frac {2 x^{\frac {n-1}{2}} (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a x^{n-1}+b x^n+c x^{n+1}}} \]
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Rubi [A] time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {1915} \begin {gather*} -\frac {2 x^{\frac {n-1}{2}} (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a x^{n-1}+b x^n+c x^{n+1}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1915
Rubi steps
\begin {align*} \int \frac {x^{\frac {3}{2} (-1+n)}}{\left (a x^{-1+n}+b x^n+c x^{1+n}\right )^{3/2}} \, dx &=-\frac {2 x^{\frac {1}{2} (-1+n)} (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {a x^{-1+n}+b x^n+c x^{1+n}}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 46, normalized size = 0.90 \begin {gather*} -\frac {2 x^{\frac {n-1}{2}} (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt {x^{n-1} (a+x (b+c x))}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 73, normalized size = 1.43 \begin {gather*} -\frac {2 x^{\frac {3 (n-1)}{2}} (b+2 c x) (a+x (b+c x))^{3/2}}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (x^{n-1} (a+x (b+c x))\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 83, normalized size = 1.63 \begin {gather*} -\frac {2 \, {\left (2 \, c x^{2} + b x\right )} \sqrt {\frac {{\left (c x^{2} + b x + a\right )} x^{n + 1}}{x^{2}}}}{{\left (a b^{2} - 4 \, a^{2} c + {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} + {\left (b^{3} - 4 \, a b c\right )} x\right )} x^{\frac {1}{2} \, n + \frac {1}{2}}} \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*} \int \frac {x^{\frac {3}{2} \, n - \frac {3}{2}}}{{\left (c x^{n + 1} + a x^{n - 1} + b x^{n}\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {3 n}{2}-\frac {3}{2}}}{\left (a \,x^{n -1}+b \,x^{n}+c \,x^{n +1}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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*} \int \frac {x^{\frac {3}{2} \, n - \frac {3}{2}}}{{\left (c x^{n + 1} + a x^{n - 1} + b x^{n}\right )}^{\frac {3}{2}}}\,{d x} \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 {x^{\frac {3\,n}{2}-\frac {3}{2}}}{{\left (b\,x^n+a\,x^{n-1}+c\,x^{n+1}\right )}^{3/2}} \,d x \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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