Optimal. Leaf size=70 \[ \frac {b x^{-3 (n+1)} \left (a x^2+b x^3\right )^{n+1}}{a^2 (n+1) (n+2)}-\frac {x^{-3 n-4} \left (a x^2+b x^3\right )^{n+1}}{a (n+2)} \]
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Rubi [A] time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2016, 2014} \[ \frac {b x^{-3 (n+1)} \left (a x^2+b x^3\right )^{n+1}}{a^2 (n+1) (n+2)}-\frac {x^{-3 n-4} \left (a x^2+b x^3\right )^{n+1}}{a (n+2)} \]
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int x^{-3-3 n} \left (a x^2+b x^3\right )^n \, dx &=-\frac {x^{-4-3 n} \left (a x^2+b x^3\right )^{1+n}}{a (2+n)}-\frac {b \int x^{-2-3 n} \left (a x^2+b x^3\right )^n \, dx}{a (2+n)}\\ &=-\frac {x^{-4-3 n} \left (a x^2+b x^3\right )^{1+n}}{a (2+n)}+\frac {b x^{-3 (1+n)} \left (a x^2+b x^3\right )^{1+n}}{a^2 (1+n) (2+n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.63 \[ -\frac {x^{-3 n-4} (a n+a-b x) \left (x^2 (a+b x)\right )^{n+1}}{a^2 (n+1) (n+2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 70, normalized size = 1.00 \[ -\frac {{\left (a b n x^{2} - b^{2} x^{3} + {\left (a^{2} n + a^{2}\right )} x\right )} {\left (b x^{3} + a x^{2}\right )}^{n} x^{-3 \, n - 3}}{a^{2} n^{2} + 3 \, a^{2} n + 2 \, a^{2}} \]
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 \[ \int {\left (b x^{3} + a x^{2}\right )}^{n} x^{-3 \, n - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 50, normalized size = 0.71 \[ -\frac {\left (a n -b x +a \right ) \left (b x +a \right ) x^{-3 n -2} \left (b \,x^{3}+a \,x^{2}\right )^{n}}{\left (n +2\right ) \left (n +1\right ) a^{2}} \]
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 \[ \int {\left (b x^{3} + a x^{2}\right )}^{n} x^{-3 \, n - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.28, size = 98, normalized size = 1.40 \[ -{\left (b\,x^3+a\,x^2\right )}^n\,\left (\frac {x\,\left (n+1\right )}{x^{3\,n+3}\,\left (n^2+3\,n+2\right )}-\frac {b^2\,x^3}{a^2\,x^{3\,n+3}\,\left (n^2+3\,n+2\right )}+\frac {b\,n\,x^2}{a\,x^{3\,n+3}\,\left (n^2+3\,n+2\right )}\right ) \]
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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