Optimal. Leaf size=82 \[ -\frac {a^6 c^5}{6 x^6}+\frac {4 a^5 b c^5}{5 x^5}-\frac {5 a^4 b^2 c^5}{4 x^4}+\frac {5 a^2 b^4 c^5}{2 x^2}-\frac {4 a b^5 c^5}{x}-b^6 c^5 \log (x) \]
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Rubi [A] time = 0.04, antiderivative size = 82, 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} \[ -\frac {5 a^4 b^2 c^5}{4 x^4}+\frac {5 a^2 b^4 c^5}{2 x^2}+\frac {4 a^5 b c^5}{5 x^5}-\frac {a^6 c^5}{6 x^6}-\frac {4 a b^5 c^5}{x}-b^6 c^5 \log (x) \]
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^7} \, dx &=\int \left (\frac {a^6 c^5}{x^7}-\frac {4 a^5 b c^5}{x^6}+\frac {5 a^4 b^2 c^5}{x^5}-\frac {5 a^2 b^4 c^5}{x^3}+\frac {4 a b^5 c^5}{x^2}-\frac {b^6 c^5}{x}\right ) \, dx\\ &=-\frac {a^6 c^5}{6 x^6}+\frac {4 a^5 b c^5}{5 x^5}-\frac {5 a^4 b^2 c^5}{4 x^4}+\frac {5 a^2 b^4 c^5}{2 x^2}-\frac {4 a b^5 c^5}{x}-b^6 c^5 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 68, normalized size = 0.83 \[ c^5 \left (-\frac {a^6}{6 x^6}+\frac {4 a^5 b}{5 x^5}-\frac {5 a^4 b^2}{4 x^4}+\frac {5 a^2 b^4}{2 x^2}-\frac {4 a b^5}{x}-b^6 \log (x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 77, normalized size = 0.94 \[ -\frac {60 \, b^{6} c^{5} x^{6} \log \relax (x) + 240 \, a b^{5} c^{5} x^{5} - 150 \, a^{2} b^{4} c^{5} x^{4} + 75 \, a^{4} b^{2} c^{5} x^{2} - 48 \, a^{5} b c^{5} x + 10 \, a^{6} c^{5}}{60 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 76, normalized size = 0.93 \[ -b^{6} c^{5} \log \left ({\left | x \right |}\right ) - \frac {240 \, a b^{5} c^{5} x^{5} - 150 \, a^{2} b^{4} c^{5} x^{4} + 75 \, a^{4} b^{2} c^{5} x^{2} - 48 \, a^{5} b c^{5} x + 10 \, a^{6} c^{5}}{60 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 75, normalized size = 0.91 \[ -b^{6} c^{5} \ln \relax (x )-\frac {4 a \,b^{5} c^{5}}{x}+\frac {5 a^{2} b^{4} c^{5}}{2 x^{2}}-\frac {5 a^{4} b^{2} c^{5}}{4 x^{4}}+\frac {4 a^{5} b \,c^{5}}{5 x^{5}}-\frac {a^{6} c^{5}}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 75, normalized size = 0.91 \[ -b^{6} c^{5} \log \relax (x) - \frac {240 \, a b^{5} c^{5} x^{5} - 150 \, a^{2} b^{4} c^{5} x^{4} + 75 \, a^{4} b^{2} c^{5} x^{2} - 48 \, a^{5} b c^{5} x + 10 \, a^{6} c^{5}}{60 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 62, normalized size = 0.76 \[ -\frac {c^5\,\left (10\,a^6+240\,a\,b^5\,x^5+75\,a^4\,b^2\,x^2-150\,a^2\,b^4\,x^4+60\,b^6\,x^6\,\ln \relax (x)-48\,a^5\,b\,x\right )}{60\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 80, normalized size = 0.98 \[ - b^{6} c^{5} \log {\relax (x )} - \frac {10 a^{6} c^{5} - 48 a^{5} b c^{5} x + 75 a^{4} b^{2} c^{5} x^{2} - 150 a^{2} b^{4} c^{5} x^{4} + 240 a b^{5} c^{5} x^{5}}{60 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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