Optimal. Leaf size=32 \[ -\frac {x^{-3 (n+1)} \left (a x^2+b x^3\right )^{n+1}}{a (n+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2014} \[ -\frac {x^{-3 (n+1)} \left (a x^2+b x^3\right )^{n+1}}{a (n+1)} \]
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin {align*} \int x^{-2-3 n} \left (a x^2+b x^3\right )^n \, dx &=-\frac {x^{-3 (1+n)} \left (a x^2+b x^3\right )^{1+n}}{a (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.94 \[ -\frac {x^{-3 (n+1)} \left (x^2 (a+b x)\right )^{n+1}}{a (n+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 38, normalized size = 1.19 \[ -\frac {{\left (b x^{2} + a x\right )} {\left (b x^{3} + a x^{2}\right )}^{n} x^{-3 \, n - 2}}{a n + a} \]
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 \[ \int {\left (b x^{3} + a x^{2}\right )}^{n} x^{-3 \, n - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 36, normalized size = 1.12 \[ -\frac {\left (b x +a \right ) x^{-3 n -1} \left (b \,x^{3}+a \,x^{2}\right )^{n}}{\left (n +1\right ) a} \]
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 \[ \int {\left (b x^{3} + a x^{2}\right )}^{n} x^{-3 \, n - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.28, size = 54, normalized size = 1.69 \[ -{\left (b\,x^3+a\,x^2\right )}^n\,\left (\frac {x}{x^{3\,n+2}\,\left (n+1\right )}+\frac {b\,x^2}{a\,x^{3\,n+2}\,\left (n+1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{- 3 n - 2} \left (x^{2} \left (a + b x\right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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