Optimal. Leaf size=116 \[ -\frac {2 b^2 x^{-3 (n+1)} \left (a x^2+b x^3\right )^{n+1}}{a^3 (n+1) (n+2) (n+3)}+\frac {2 b x^{-3 n-4} \left (a x^2+b x^3\right )^{n+1}}{a^2 (n+2) (n+3)}-\frac {x^{-3 n-5} \left (a x^2+b x^3\right )^{n+1}}{a (n+3)} \]
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Rubi [A] time = 0.09, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2016, 2014} \begin {gather*} -\frac {2 b^2 x^{-3 (n+1)} \left (a x^2+b x^3\right )^{n+1}}{a^3 (n+1) (n+2) (n+3)}+\frac {2 b x^{-3 n-4} \left (a x^2+b x^3\right )^{n+1}}{a^2 (n+2) (n+3)}-\frac {x^{-3 n-5} \left (a x^2+b x^3\right )^{n+1}}{a (n+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int x^{-4-3 n} \left (a x^2+b x^3\right )^n \, dx &=-\frac {x^{-5-3 n} \left (a x^2+b x^3\right )^{1+n}}{a (3+n)}-\frac {(2 b) \int x^{-3-3 n} \left (a x^2+b x^3\right )^n \, dx}{a (3+n)}\\ &=-\frac {x^{-5-3 n} \left (a x^2+b x^3\right )^{1+n}}{a (3+n)}+\frac {2 b x^{-4-3 n} \left (a x^2+b x^3\right )^{1+n}}{a^2 (2+n) (3+n)}+\frac {\left (2 b^2\right ) \int x^{-2-3 n} \left (a x^2+b x^3\right )^n \, dx}{a^2 (2+n) (3+n)}\\ &=-\frac {x^{-5-3 n} \left (a x^2+b x^3\right )^{1+n}}{a (3+n)}+\frac {2 b x^{-4-3 n} \left (a x^2+b x^3\right )^{1+n}}{a^2 (2+n) (3+n)}-\frac {2 b^2 x^{-3 (1+n)} \left (a x^2+b x^3\right )^{1+n}}{a^3 (1+n) (2+n) (3+n)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 72, normalized size = 0.62 \begin {gather*} -\frac {x^{-3 (n+1)} (a+b x) \left (x^2 (a+b x)\right )^n \left (a^2 \left (n^2+3 n+2\right )-2 a b (n+1) x+2 b^2 x^2\right )}{a^3 (n+1) (n+2) (n+3)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int x^{-4-3 n} \left (a x^2+b x^3\right )^n \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 111, normalized size = 0.96 \begin {gather*} \frac {{\left (2 \, a b^{2} n x^{3} - 2 \, b^{3} x^{4} - {\left (a^{2} b n^{2} + a^{2} b n\right )} x^{2} - {\left (a^{3} n^{2} + 3 \, a^{3} n + 2 \, a^{3}\right )} x\right )} {\left (b x^{3} + a x^{2}\right )}^{n} x^{-3 \, n - 4}}{a^{3} n^{3} + 6 \, a^{3} n^{2} + 11 \, a^{3} n + 6 \, a^{3}} \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 {\left (b x^{3} + a x^{2}\right )}^{n} x^{-3 \, n - 4}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 84, normalized size = 0.72 \begin {gather*} -\frac {\left (b x +a \right ) \left (a^{2} n^{2}-2 a b n x +2 b^{2} x^{2}+3 a^{2} n -2 a b x +2 a^{2}\right ) x^{-3 n -3} \left (b \,x^{3}+a \,x^{2}\right )^{n}}{\left (n +3\right ) \left (n +2\right ) \left (n +1\right ) a^{3}} \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 {\left (b x^{3} + a x^{2}\right )}^{n} x^{-3 \, n - 4}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.36, size = 157, normalized size = 1.35 \begin {gather*} -{\left (b\,x^3+a\,x^2\right )}^n\,\left (\frac {x\,\left (n^2+3\,n+2\right )}{x^{3\,n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {2\,b^3\,x^4}{a^3\,x^{3\,n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}-\frac {2\,b^2\,n\,x^3}{a^2\,x^{3\,n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {b\,n\,x^2\,\left (n+1\right )}{a\,x^{3\,n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}\right ) \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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