Optimal. Leaf size=79 \[ a^6 c^5 \log (x)-4 a^5 b c^5 x+\frac {5}{2} a^4 b^2 c^5 x^2-\frac {5}{4} a^2 b^4 c^5 x^4+\frac {4}{5} a b^5 c^5 x^5-\frac {1}{6} b^6 c^5 x^6 \]
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Rubi [A] time = 0.03, antiderivative size = 79, 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} \[ \frac {5}{2} a^4 b^2 c^5 x^2-\frac {5}{4} a^2 b^4 c^5 x^4-4 a^5 b c^5 x+a^6 c^5 \log (x)+\frac {4}{5} a b^5 c^5 x^5-\frac {1}{6} b^6 c^5 x^6 \]
Antiderivative was successfully verified.
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Rule 75
Rubi steps
\begin {align*} \int \frac {(a+b x) (a c-b c x)^5}{x} \, dx &=\int \left (-4 a^5 b c^5+\frac {a^6 c^5}{x}+5 a^4 b^2 c^5 x-5 a^2 b^4 c^5 x^3+4 a b^5 c^5 x^4-b^6 c^5 x^5\right ) \, dx\\ &=-4 a^5 b c^5 x+\frac {5}{2} a^4 b^2 c^5 x^2-\frac {5}{4} a^2 b^4 c^5 x^4+\frac {4}{5} a b^5 c^5 x^5-\frac {1}{6} b^6 c^5 x^6+a^6 c^5 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 75, normalized size = 0.95 \[ c^5 \left (a^6 \log (-b x)+\frac {127 a^6}{60}-4 a^5 b x+\frac {5}{2} a^4 b^2 x^2-\frac {5}{4} a^2 b^4 x^4+\frac {4}{5} a b^5 x^5-\frac {b^6 x^6}{6}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 71, normalized size = 0.90 \[ -\frac {1}{6} \, b^{6} c^{5} x^{6} + \frac {4}{5} \, a b^{5} c^{5} x^{5} - \frac {5}{4} \, a^{2} b^{4} c^{5} x^{4} + \frac {5}{2} \, a^{4} b^{2} c^{5} x^{2} - 4 \, a^{5} b c^{5} x + a^{6} c^{5} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 72, normalized size = 0.91 \[ -\frac {1}{6} \, b^{6} c^{5} x^{6} + \frac {4}{5} \, a b^{5} c^{5} x^{5} - \frac {5}{4} \, a^{2} b^{4} c^{5} x^{4} + \frac {5}{2} \, a^{4} b^{2} c^{5} x^{2} - 4 \, a^{5} b c^{5} x + a^{6} c^{5} \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 72, normalized size = 0.91 \[ -\frac {b^{6} c^{5} x^{6}}{6}+\frac {4 a \,b^{5} c^{5} x^{5}}{5}-\frac {5 a^{2} b^{4} c^{5} x^{4}}{4}+\frac {5 a^{4} b^{2} c^{5} x^{2}}{2}+a^{6} c^{5} \ln \relax (x )-4 a^{5} b \,c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 71, normalized size = 0.90 \[ -\frac {1}{6} \, b^{6} c^{5} x^{6} + \frac {4}{5} \, a b^{5} c^{5} x^{5} - \frac {5}{4} \, a^{2} b^{4} c^{5} x^{4} + \frac {5}{2} \, a^{4} b^{2} c^{5} x^{2} - 4 \, a^{5} b c^{5} x + a^{6} c^{5} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 71, normalized size = 0.90 \[ a^6\,c^5\,\ln \relax (x)-\frac {b^6\,c^5\,x^6}{6}+\frac {4\,a\,b^5\,c^5\,x^5}{5}+\frac {5\,a^4\,b^2\,c^5\,x^2}{2}-\frac {5\,a^2\,b^4\,c^5\,x^4}{4}-4\,a^5\,b\,c^5\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 82, normalized size = 1.04 \[ a^{6} c^{5} \log {\relax (x )} - 4 a^{5} b c^{5} x + \frac {5 a^{4} b^{2} c^{5} x^{2}}{2} - \frac {5 a^{2} b^{4} c^{5} x^{4}}{4} + \frac {4 a b^{5} c^{5} x^{5}}{5} - \frac {b^{6} c^{5} x^{6}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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