Optimal. Leaf size=87 \[ \frac {1}{4} a^6 c^5 x^4-\frac {4}{5} a^5 b c^5 x^5+\frac {5}{6} a^4 b^2 c^5 x^6-\frac {5}{8} a^2 b^4 c^5 x^8+\frac {4}{9} a b^5 c^5 x^9-\frac {1}{10} b^6 c^5 x^{10} \]
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Rubi [A] time = 0.04, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {75} \[ -\frac {5}{8} a^2 b^4 c^5 x^8+\frac {5}{6} a^4 b^2 c^5 x^6-\frac {4}{5} a^5 b c^5 x^5+\frac {1}{4} a^6 c^5 x^4+\frac {4}{9} a b^5 c^5 x^9-\frac {1}{10} b^6 c^5 x^{10} \]
Antiderivative was successfully verified.
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Rule 75
Rubi steps
\begin {align*} \int x^3 (a+b x) (a c-b c x)^5 \, dx &=\int \left (a^6 c^5 x^3-4 a^5 b c^5 x^4+5 a^4 b^2 c^5 x^5-5 a^2 b^4 c^5 x^7+4 a b^5 c^5 x^8-b^6 c^5 x^9\right ) \, dx\\ &=\frac {1}{4} a^6 c^5 x^4-\frac {4}{5} a^5 b c^5 x^5+\frac {5}{6} a^4 b^2 c^5 x^6-\frac {5}{8} a^2 b^4 c^5 x^8+\frac {4}{9} a b^5 c^5 x^9-\frac {1}{10} b^6 c^5 x^{10}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 73, normalized size = 0.84 \[ c^5 \left (\frac {a^6 x^4}{4}-\frac {4}{5} a^5 b x^5+\frac {5}{6} a^4 b^2 x^6-\frac {5}{8} a^2 b^4 x^8+\frac {4}{9} a b^5 x^9-\frac {1}{10} b^6 x^{10}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 75, normalized size = 0.86 \[ -\frac {1}{10} x^{10} c^{5} b^{6} + \frac {4}{9} x^{9} c^{5} b^{5} a - \frac {5}{8} x^{8} c^{5} b^{4} a^{2} + \frac {5}{6} x^{6} c^{5} b^{2} a^{4} - \frac {4}{5} x^{5} c^{5} b a^{5} + \frac {1}{4} x^{4} c^{5} a^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 75, normalized size = 0.86 \[ -\frac {1}{10} \, b^{6} c^{5} x^{10} + \frac {4}{9} \, a b^{5} c^{5} x^{9} - \frac {5}{8} \, a^{2} b^{4} c^{5} x^{8} + \frac {5}{6} \, a^{4} b^{2} c^{5} x^{6} - \frac {4}{5} \, a^{5} b c^{5} x^{5} + \frac {1}{4} \, a^{6} c^{5} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 76, normalized size = 0.87 \[ -\frac {1}{10} b^{6} c^{5} x^{10}+\frac {4}{9} a \,b^{5} c^{5} x^{9}-\frac {5}{8} a^{2} b^{4} c^{5} x^{8}+\frac {5}{6} a^{4} b^{2} c^{5} x^{6}-\frac {4}{5} a^{5} b \,c^{5} x^{5}+\frac {1}{4} a^{6} c^{5} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 75, normalized size = 0.86 \[ -\frac {1}{10} \, b^{6} c^{5} x^{10} + \frac {4}{9} \, a b^{5} c^{5} x^{9} - \frac {5}{8} \, a^{2} b^{4} c^{5} x^{8} + \frac {5}{6} \, a^{4} b^{2} c^{5} x^{6} - \frac {4}{5} \, a^{5} b c^{5} x^{5} + \frac {1}{4} \, a^{6} c^{5} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 75, normalized size = 0.86 \[ \frac {a^6\,c^5\,x^4}{4}-\frac {4\,a^5\,b\,c^5\,x^5}{5}+\frac {5\,a^4\,b^2\,c^5\,x^6}{6}-\frac {5\,a^2\,b^4\,c^5\,x^8}{8}+\frac {4\,a\,b^5\,c^5\,x^9}{9}-\frac {b^6\,c^5\,x^{10}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 87, normalized size = 1.00 \[ \frac {a^{6} c^{5} x^{4}}{4} - \frac {4 a^{5} b c^{5} x^{5}}{5} + \frac {5 a^{4} b^{2} c^{5} x^{6}}{6} - \frac {5 a^{2} b^{4} c^{5} x^{8}}{8} + \frac {4 a b^{5} c^{5} x^{9}}{9} - \frac {b^{6} c^{5} x^{10}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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