Optimal. Leaf size=65 \[ \frac {a^3 x^2}{2}+\frac {3 a^2 b x^{n+2}}{n+2}+\frac {3 a b^2 x^{2 (n+1)}}{2 (n+1)}+\frac {b^3 x^{3 n+2}}{3 n+2} \]
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Rubi [A] time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {270} \[ \frac {3 a^2 b x^{n+2}}{n+2}+\frac {a^3 x^2}{2}+\frac {3 a b^2 x^{2 (n+1)}}{2 (n+1)}+\frac {b^3 x^{3 n+2}}{3 n+2} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int x \left (a+b x^n\right )^3 \, dx &=\int \left (a^3 x+3 a^2 b x^{1+n}+3 a b^2 x^{1+2 n}+b^3 x^{1+3 n}\right ) \, dx\\ &=\frac {a^3 x^2}{2}+\frac {3 a b^2 x^{2 (1+n)}}{2 (1+n)}+\frac {3 a^2 b x^{2+n}}{2+n}+\frac {b^3 x^{2+3 n}}{2+3 n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 58, normalized size = 0.89 \[ \frac {1}{2} x^2 \left (a^3+\frac {6 a^2 b x^n}{n+2}+\frac {3 a b^2 x^{2 n}}{n+1}+\frac {2 b^3 x^{3 n}}{3 n+2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 145, normalized size = 2.23 \[ \frac {2 \, {\left (b^{3} n^{2} + 3 \, b^{3} n + 2 \, b^{3}\right )} x^{2} x^{3 \, n} + 3 \, {\left (3 \, a b^{2} n^{2} + 8 \, a b^{2} n + 4 \, a b^{2}\right )} x^{2} x^{2 \, n} + 6 \, {\left (3 \, a^{2} b n^{2} + 5 \, a^{2} b n + 2 \, a^{2} b\right )} x^{2} x^{n} + {\left (3 \, a^{3} n^{3} + 11 \, a^{3} n^{2} + 12 \, a^{3} n + 4 \, a^{3}\right )} x^{2}}{2 \, {\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 188, normalized size = 2.89 \[ \frac {2 \, b^{3} n^{2} x^{2} x^{3 \, n} + 9 \, a b^{2} n^{2} x^{2} x^{2 \, n} + 18 \, a^{2} b n^{2} x^{2} x^{n} + 3 \, a^{3} n^{3} x^{2} + 6 \, b^{3} n x^{2} x^{3 \, n} + 24 \, a b^{2} n x^{2} x^{2 \, n} + 30 \, a^{2} b n x^{2} x^{n} + 11 \, a^{3} n^{2} x^{2} + 4 \, b^{3} x^{2} x^{3 \, n} + 12 \, a b^{2} x^{2} x^{2 \, n} + 12 \, a^{2} b x^{2} x^{n} + 12 \, a^{3} n x^{2} + 4 \, a^{3} x^{2}}{2 \, {\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 71, normalized size = 1.09 \[ \frac {3 a^{2} b \,x^{2} {\mathrm e}^{n \ln \relax (x )}}{n +2}+\frac {3 a \,b^{2} x^{2} {\mathrm e}^{2 n \ln \relax (x )}}{2 \left (n +1\right )}+\frac {b^{3} x^{2} {\mathrm e}^{3 n \ln \relax (x )}}{3 n +2}+\frac {a^{3} x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 61, normalized size = 0.94 \[ \frac {1}{2} \, a^{3} x^{2} + \frac {b^{3} x^{3 \, n + 2}}{3 \, n + 2} + \frac {3 \, a b^{2} x^{2 \, n + 2}}{2 \, {\left (n + 1\right )}} + \frac {3 \, a^{2} b x^{n + 2}}{n + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 66, normalized size = 1.02 \[ \frac {a^3\,x^2}{2}+\frac {b^3\,x^{3\,n}\,x^2}{3\,n+2}+\frac {3\,a\,b^2\,x^{2\,n}\,x^2}{2\,n+2}+\frac {3\,a^2\,b\,x^n\,x^2}{n+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.78, size = 500, normalized size = 7.69 \[ \begin {cases} \frac {a^{3} x^{2}}{2} + 3 a^{2} b \log {\relax (x )} - \frac {3 a b^{2}}{2 x^{2}} - \frac {b^{3}}{4 x^{4}} & \text {for}\: n = -2 \\\frac {a^{3} x^{2}}{2} + 3 a^{2} b x + 3 a b^{2} \log {\relax (x )} - \frac {b^{3}}{x} & \text {for}\: n = -1 \\\frac {a^{3} x^{2}}{2} + \frac {9 a^{2} b x^{\frac {4}{3}}}{4} + \frac {9 a b^{2} x^{\frac {2}{3}}}{2} + b^{3} \log {\relax (x )} & \text {for}\: n = - \frac {2}{3} \\\frac {3 a^{3} n^{3} x^{2}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac {11 a^{3} n^{2} x^{2}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac {12 a^{3} n x^{2}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac {4 a^{3} x^{2}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac {18 a^{2} b n^{2} x^{2} x^{n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac {30 a^{2} b n x^{2} x^{n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac {12 a^{2} b x^{2} x^{n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac {9 a b^{2} n^{2} x^{2} x^{2 n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac {24 a b^{2} n x^{2} x^{2 n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac {12 a b^{2} x^{2} x^{2 n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac {2 b^{3} n^{2} x^{2} x^{3 n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac {6 b^{3} n x^{2} x^{3 n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac {4 b^{3} x^{2} x^{3 n}}{6 n^{3} + 22 n^{2} + 24 n + 8} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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