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 = 0.0118678, antiderivative size = 38, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.002617, size = 64, normalized size = 1.68 \[ c^5 \left (-a^2 b^4 x^5+\frac{5}{3} a^4 b^2 x^3-2 a^5 b x^2+a^6 x+\frac{2}{3} a b^5 x^6-\frac{1}{7} b^6 x^7\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.001, size = 73, normalized size = 1.9 \begin{align*} -{\frac{{b}^{6}{c}^{5}{x}^{7}}{7}}+{\frac{2\,a{b}^{5}{c}^{5}{x}^{6}}{3}}-{a}^{2}{b}^{4}{c}^{5}{x}^{5}+{\frac{5\,{a}^{4}{b}^{2}{c}^{5}{x}^{3}}{3}}-2\,{a}^{5}b{c}^{5}{x}^{2}+{a}^{6}{c}^{5}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03502, size = 97, normalized size = 2.55 \begin{align*} -\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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72362, size = 146, normalized size = 3.84 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.080583, size = 78, normalized size = 2.05 \begin{align*} 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21807, size = 97, normalized size = 2.55 \begin{align*} -\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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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