Optimal. Leaf size=48 \[ \frac {x^{-1-3 n} \left (a x^2+b x^3\right )^{1+n} \, _2F_1\left (1,2;2-n;-\frac {b x}{a}\right )}{a (1-n)} \]
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Rubi [A]
time = 0.03, 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 = {2057, 68, 66}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 68
Rule 2057
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 = 0.02, size = 59, normalized size = 1.23 \begin {gather*} \frac {x^{1-3 n} \left (x^2 (a+b x)\right )^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (1-n,-n;2-n;-\frac {b x}{a}\right )}{1-n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \left (b \,x^{3}+a \,x^{2}\right )^{n} x^{-3 n}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{- 3 n} \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (b\,x^3+a\,x^2\right )}^n}{x^{3\,n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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