Optimal. Leaf size=65 \[ \frac {a^3 x^4}{4}+\frac {3 a^2 b x^{n+4}}{n+4}+\frac {3 a b^2 x^{2 (n+2)}}{2 (n+2)}+\frac {b^3 x^{3 n+4}}{3 n+4} \]
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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \frac {3 a^2 b x^{n+4}}{n+4}+\frac {a^3 x^4}{4}+\frac {3 a b^2 x^{2 (n+2)}}{2 (n+2)}+\frac {b^3 x^{3 n+4}}{3 n+4} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int x^3 \left (a+b x^n\right )^3 \, dx &=\int \left (a^3 x^3+b^3 x^{3 (1+n)}+3 a^2 b x^{3+n}+3 a b^2 x^{3+2 n}\right ) \, dx\\ &=\frac {a^3 x^4}{4}+\frac {3 a b^2 x^{2 (2+n)}}{2 (2+n)}+\frac {3 a^2 b x^{4+n}}{4+n}+\frac {b^3 x^{4+3 n}}{4+3 n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 58, normalized size = 0.89 \[ \frac {1}{4} x^4 \left (a^3+\frac {12 a^2 b x^n}{n+4}+\frac {6 a b^2 x^{2 n}}{n+2}+\frac {4 b^3 x^{3 n}}{3 n+4}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 145, normalized size = 2.23 \[ \frac {4 \, {\left (b^{3} n^{2} + 6 \, b^{3} n + 8 \, b^{3}\right )} x^{4} x^{3 \, n} + 6 \, {\left (3 \, a b^{2} n^{2} + 16 \, a b^{2} n + 16 \, a b^{2}\right )} x^{4} x^{2 \, n} + 12 \, {\left (3 \, a^{2} b n^{2} + 10 \, a^{2} b n + 8 \, a^{2} b\right )} x^{4} x^{n} + {\left (3 \, a^{3} n^{3} + 22 \, a^{3} n^{2} + 48 \, a^{3} n + 32 \, a^{3}\right )} x^{4}}{4 \, {\left (3 \, n^{3} + 22 \, n^{2} + 48 \, n + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 188, normalized size = 2.89 \[ \frac {4 \, b^{3} n^{2} x^{4} x^{3 \, n} + 18 \, a b^{2} n^{2} x^{4} x^{2 \, n} + 36 \, a^{2} b n^{2} x^{4} x^{n} + 3 \, a^{3} n^{3} x^{4} + 24 \, b^{3} n x^{4} x^{3 \, n} + 96 \, a b^{2} n x^{4} x^{2 \, n} + 120 \, a^{2} b n x^{4} x^{n} + 22 \, a^{3} n^{2} x^{4} + 32 \, b^{3} x^{4} x^{3 \, n} + 96 \, a b^{2} x^{4} x^{2 \, n} + 96 \, a^{2} b x^{4} x^{n} + 48 \, a^{3} n x^{4} + 32 \, a^{3} x^{4}}{4 \, {\left (3 \, n^{3} + 22 \, n^{2} + 48 \, n + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 65, normalized size = 1.00 \[ \frac {3 a^{2} b \,x^{4} x^{n}}{n +4}+\frac {3 a \,b^{2} x^{4} x^{2 n}}{2 \left (n +2\right )}+\frac {b^{3} x^{4} x^{3 n}}{3 n +4}+\frac {a^{3} x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 61, normalized size = 0.94 \[ \frac {1}{4} \, a^{3} x^{4} + \frac {b^{3} x^{3 \, n + 4}}{3 \, n + 4} + \frac {3 \, a b^{2} x^{2 \, n + 4}}{2 \, {\left (n + 2\right )}} + \frac {3 \, a^{2} b x^{n + 4}}{n + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 66, normalized size = 1.02 \[ \frac {a^3\,x^4}{4}+\frac {b^3\,x^{3\,n}\,x^4}{3\,n+4}+\frac {3\,a\,b^2\,x^{2\,n}\,x^4}{2\,n+4}+\frac {3\,a^2\,b\,x^n\,x^4}{n+4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.03, size = 507, normalized size = 7.80 \[ \begin {cases} \frac {a^{3} x^{4}}{4} + 3 a^{2} b \log {\relax (x )} - \frac {3 a b^{2}}{4 x^{4}} - \frac {b^{3}}{8 x^{8}} & \text {for}\: n = -4 \\\frac {a^{3} x^{4}}{4} + \frac {3 a^{2} b x^{2}}{2} + 3 a b^{2} \log {\relax (x )} - \frac {b^{3}}{2 x^{2}} & \text {for}\: n = -2 \\\frac {a^{3} x^{4}}{4} + \frac {9 a^{2} b x^{\frac {8}{3}}}{8} + \frac {9 a b^{2} x^{\frac {4}{3}}}{4} + b^{3} \log {\relax (x )} & \text {for}\: n = - \frac {4}{3} \\\frac {3 a^{3} n^{3} x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {22 a^{3} n^{2} x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {48 a^{3} n x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {32 a^{3} x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {36 a^{2} b n^{2} x^{4} x^{n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {120 a^{2} b n x^{4} x^{n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {96 a^{2} b x^{4} x^{n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {18 a b^{2} n^{2} x^{4} x^{2 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {96 a b^{2} n x^{4} x^{2 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {96 a b^{2} x^{4} x^{2 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {4 b^{3} n^{2} x^{4} x^{3 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {24 b^{3} n x^{4} x^{3 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac {32 b^{3} x^{4} x^{3 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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