Optimal. Leaf size=44 \[ \frac {a^2 x^4}{4}+\frac {2 a b x^{n+4}}{n+4}+\frac {b^2 x^{2 (n+2)}}{2 (n+2)} \]
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Rubi [A] time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \frac {a^2 x^4}{4}+\frac {2 a b x^{n+4}}{n+4}+\frac {b^2 x^{2 (n+2)}}{2 (n+2)} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int x^3 \left (a+b x^n\right )^2 \, dx &=\int \left (a^2 x^3+2 a b x^{3+n}+b^2 x^{3+2 n}\right ) \, dx\\ &=\frac {a^2 x^4}{4}+\frac {b^2 x^{2 (2+n)}}{2 (2+n)}+\frac {2 a b x^{4+n}}{4+n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 38, normalized size = 0.86 \[ \frac {1}{4} x^4 \left (a^2+\frac {8 a b x^n}{n+4}+\frac {2 b^2 x^{2 n}}{n+2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 74, normalized size = 1.68 \[ \frac {2 \, {\left (b^{2} n + 4 \, b^{2}\right )} x^{4} x^{2 \, n} + 8 \, {\left (a b n + 2 \, a b\right )} x^{4} x^{n} + {\left (a^{2} n^{2} + 6 \, a^{2} n + 8 \, a^{2}\right )} x^{4}}{4 \, {\left (n^{2} + 6 \, n + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 88, normalized size = 2.00 \[ \frac {2 \, b^{2} n x^{4} x^{2 \, n} + 8 \, a b n x^{4} x^{n} + a^{2} n^{2} x^{4} + 8 \, b^{2} x^{4} x^{2 \, n} + 16 \, a b x^{4} x^{n} + 6 \, a^{2} n x^{4} + 8 \, a^{2} x^{4}}{4 \, {\left (n^{2} + 6 \, n + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 47, normalized size = 1.07 \[ \frac {2 a b \,x^{4} {\mathrm e}^{n \ln \relax (x )}}{n +4}+\frac {b^{2} x^{4} {\mathrm e}^{2 n \ln \relax (x )}}{4+2 n}+\frac {a^{2} x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 40, normalized size = 0.91 \[ \frac {1}{4} \, a^{2} x^{4} + \frac {b^{2} x^{2 \, n + 4}}{2 \, {\left (n + 2\right )}} + \frac {2 \, a b x^{n + 4}}{n + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 43, normalized size = 0.98 \[ \frac {a^2\,x^4}{4}+\frac {b^2\,x^{2\,n}\,x^4}{2\,n+4}+\frac {2\,a\,b\,x^n\,x^4}{n+4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.97, size = 202, normalized size = 4.59 \[ \begin {cases} \frac {a^{2} x^{4}}{4} + 2 a b \log {\relax (x )} - \frac {b^{2}}{4 x^{4}} & \text {for}\: n = -4 \\\frac {a^{2} x^{4}}{4} + a b x^{2} + b^{2} \log {\relax (x )} & \text {for}\: n = -2 \\\frac {a^{2} n^{2} x^{4}}{4 n^{2} + 24 n + 32} + \frac {6 a^{2} n x^{4}}{4 n^{2} + 24 n + 32} + \frac {8 a^{2} x^{4}}{4 n^{2} + 24 n + 32} + \frac {8 a b n x^{4} x^{n}}{4 n^{2} + 24 n + 32} + \frac {16 a b x^{4} x^{n}}{4 n^{2} + 24 n + 32} + \frac {2 b^{2} n x^{4} x^{2 n}}{4 n^{2} + 24 n + 32} + \frac {8 b^{2} x^{4} x^{2 n}}{4 n^{2} + 24 n + 32} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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