Optimal. Leaf size=38 \[ \frac {c^5 (a-b x)^7}{7 b}-\frac {a c^5 (a-b x)^6}{3 b} \]
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Rubi [A] time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {43} \[ \frac {c^5 (a-b x)^7}{7 b}-\frac {a c^5 (a-b x)^6}{3 b} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (a+b x) (a c-b c x)^5 \, dx &=\int \left (2 a (a c-b c x)^5-\frac {(a c-b c x)^6}{c}\right ) \, dx\\ &=-\frac {a c^5 (a-b x)^6}{3 b}+\frac {c^5 (a-b x)^7}{7 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 64, normalized size = 1.68 \[ c^5 \left (a^6 x-2 a^5 b x^2+\frac {5}{3} a^4 b^2 x^3-a^2 b^4 x^5+\frac {2}{3} a b^5 x^6-\frac {1}{7} b^6 x^7\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 72, normalized size = 1.89 \[ -\frac {1}{7} x^{7} c^{5} b^{6} + \frac {2}{3} x^{6} c^{5} b^{5} a - x^{5} c^{5} b^{4} a^{2} + \frac {5}{3} x^{3} c^{5} b^{2} a^{4} - 2 x^{2} c^{5} b a^{5} + x c^{5} a^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 72, normalized size = 1.89 \[ -\frac {1}{7} \, b^{6} c^{5} x^{7} + \frac {2}{3} \, a b^{5} c^{5} x^{6} - a^{2} b^{4} c^{5} x^{5} + \frac {5}{3} \, a^{4} b^{2} c^{5} x^{3} - 2 \, a^{5} b c^{5} x^{2} + a^{6} c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 73, normalized size = 1.92 \[ -\frac {1}{7} b^{6} c^{5} x^{7}+\frac {2}{3} a \,b^{5} c^{5} x^{6}-a^{2} b^{4} c^{5} x^{5}+\frac {5}{3} a^{4} b^{2} c^{5} x^{3}-2 a^{5} b \,c^{5} x^{2}+a^{6} c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 72, normalized size = 1.89 \[ -\frac {1}{7} \, b^{6} c^{5} x^{7} + \frac {2}{3} \, a b^{5} c^{5} x^{6} - a^{2} b^{4} c^{5} x^{5} + \frac {5}{3} \, a^{4} b^{2} c^{5} x^{3} - 2 \, a^{5} b c^{5} x^{2} + a^{6} c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 72, normalized size = 1.89 \[ a^6\,c^5\,x-2\,a^5\,b\,c^5\,x^2+\frac {5\,a^4\,b^2\,c^5\,x^3}{3}-a^2\,b^4\,c^5\,x^5+\frac {2\,a\,b^5\,c^5\,x^6}{3}-\frac {b^6\,c^5\,x^7}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.09, size = 78, normalized size = 2.05 \[ a^{6} c^{5} x - 2 a^{5} b c^{5} x^{2} + \frac {5 a^{4} b^{2} c^{5} x^{3}}{3} - a^{2} b^{4} c^{5} x^{5} + \frac {2 a b^{5} c^{5} x^{6}}{3} - \frac {b^{6} c^{5} x^{7}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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