Optimal. Leaf size=48 \[ \frac {x^{-3 n-1} \left (a x^2+b x^3\right )^{n+1} \, _2F_1\left (1,2;2-n;-\frac {b x}{a}\right )}{a (1-n)} \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.27, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2032, 66, 64} \[ \frac {x^{1-3 n} \left (\frac {b x}{a}+1\right )^{-n} \left (a x^2+b x^3\right )^n \, _2F_1\left (1-n,-n;2-n;-\frac {b x}{a}\right )}{1-n} \]
Antiderivative was successfully verified.
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Rule 64
Rule 66
Rule 2032
Rubi steps
\begin {align*} \int x^{-3 n} \left (a x^2+b x^3\right )^n \, dx &=\left (x^{-2 n} (a+b x)^{-n} \left (a x^2+b x^3\right )^n\right ) \int x^{-n} (a+b x)^n \, dx\\ &=\left (x^{-2 n} \left (1+\frac {b x}{a}\right )^{-n} \left (a x^2+b x^3\right )^n\right ) \int x^{-n} \left (1+\frac {b x}{a}\right )^n \, dx\\ &=\frac {x^{1-3 n} \left (1+\frac {b x}{a}\right )^{-n} \left (a x^2+b x^3\right )^n \, _2F_1\left (1-n,-n;2-n;-\frac {b x}{a}\right )}{1-n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 59, normalized size = 1.23 \[ \frac {x^{1-3 n} \left (x^2 (a+b x)\right )^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (1-n,-n;2-n;-\frac {b x}{a}\right )}{1-n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{3} + a x^{2}\right )}^{n}}{x^{3 \, n}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{3} + a x^{2}\right )}^{n}}{x^{3 \, n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int x^{-3 n} \left (b \,x^{3}+a \,x^{2}\right )^{n}\, dx \]
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 \[ \int \frac {{\left (b x^{3} + a x^{2}\right )}^{n}}{x^{3 \, n}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\,x^2\right )}^n}{x^{3\,n}} \,d x \]
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 \[ \int x^{- 3 n} \left (x^{2} \left (a + b x\right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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