Optimal. Leaf size=70 \[ -\frac {x^{-4-3 n} \left (a x^2+b x^3\right )^{1+n}}{a (2+n)}+\frac {b x^{-3 (1+n)} \left (a x^2+b x^3\right )^{1+n}}{a^2 (1+n) (2+n)} \]
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Rubi [A]
time = 0.04, 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 = {2041, 2039}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rule 2041
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 = 0.05, size = 44, normalized size = 0.63 \begin {gather*} -\frac {x^{-4-3 n} (a+a n-b x) \left (x^2 (a+b x)\right )^{1+n}}{a^2 (1+n) (2+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 50, normalized size = 0.71
method | result | size |
gosper | \(-\frac {\left (b \,x^{3}+a \,x^{2}\right )^{n} x^{-2-3 n} \left (a n -b x +a \right ) \left (b x +a \right )}{\left (2+n \right ) \left (1+n \right ) a^{2}}\) | \(50\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.06, size = 70, normalized size = 1.00 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \int x^{- 3 n - 3} \left (x^{2} \left (a + b x\right )\right )^{n}\, dx \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*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -{\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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