Optimal. Leaf size=79 \[ -\frac {a^5 c^4}{5 x^5}+\frac {3 a^4 b c^4}{4 x^4}-\frac {2 a^3 b^2 c^4}{3 x^3}-\frac {a^2 b^3 c^4}{x^2}+\frac {3 a b^4 c^4}{x}+b^5 c^4 \log (x) \]
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Rubi [A] time = 0.04, 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} \begin {gather*} -\frac {2 a^3 b^2 c^4}{3 x^3}-\frac {a^2 b^3 c^4}{x^2}+\frac {3 a^4 b c^4}{4 x^4}-\frac {a^5 c^4}{5 x^5}+\frac {3 a b^4 c^4}{x}+b^5 c^4 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 75
Rubi steps
\begin {align*} \int \frac {(a+b x) (a c-b c x)^4}{x^6} \, dx &=\int \left (\frac {a^5 c^4}{x^6}-\frac {3 a^4 b c^4}{x^5}+\frac {2 a^3 b^2 c^4}{x^4}+\frac {2 a^2 b^3 c^4}{x^3}-\frac {3 a b^4 c^4}{x^2}+\frac {b^5 c^4}{x}\right ) \, dx\\ &=-\frac {a^5 c^4}{5 x^5}+\frac {3 a^4 b c^4}{4 x^4}-\frac {2 a^3 b^2 c^4}{3 x^3}-\frac {a^2 b^3 c^4}{x^2}+\frac {3 a b^4 c^4}{x}+b^5 c^4 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 79, normalized size = 1.00 \begin {gather*} -\frac {a^5 c^4}{5 x^5}+\frac {3 a^4 b c^4}{4 x^4}-\frac {2 a^3 b^2 c^4}{3 x^3}-\frac {a^2 b^3 c^4}{x^2}+\frac {3 a b^4 c^4}{x}+b^5 c^4 \log (x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (a c-b c x)^4}{x^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.80, size = 77, normalized size = 0.97 \begin {gather*} \frac {60 \, b^{5} c^{4} x^{5} \log \relax (x) + 180 \, a b^{4} c^{4} x^{4} - 60 \, a^{2} b^{3} c^{4} x^{3} - 40 \, a^{3} b^{2} c^{4} x^{2} + 45 \, a^{4} b c^{4} x - 12 \, a^{5} c^{4}}{60 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 75, normalized size = 0.95 \begin {gather*} b^{5} c^{4} \log \left ({\left | x \right |}\right ) + \frac {180 \, a b^{4} c^{4} x^{4} - 60 \, a^{2} b^{3} c^{4} x^{3} - 40 \, a^{3} b^{2} c^{4} x^{2} + 45 \, a^{4} b c^{4} x - 12 \, a^{5} c^{4}}{60 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 0.94 \begin {gather*} b^{5} c^{4} \ln \relax (x )+\frac {3 a \,b^{4} c^{4}}{x}-\frac {a^{2} b^{3} c^{4}}{x^{2}}-\frac {2 a^{3} b^{2} c^{4}}{3 x^{3}}+\frac {3 a^{4} b \,c^{4}}{4 x^{4}}-\frac {a^{5} c^{4}}{5 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 74, normalized size = 0.94 \begin {gather*} b^{5} c^{4} \log \relax (x) + \frac {180 \, a b^{4} c^{4} x^{4} - 60 \, a^{2} b^{3} c^{4} x^{3} - 40 \, a^{3} b^{2} c^{4} x^{2} + 45 \, a^{4} b c^{4} x - 12 \, a^{5} c^{4}}{60 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 61, normalized size = 0.77 \begin {gather*} -\frac {c^4\,\left (\frac {a^5}{5}-3\,a\,b^4\,x^4+\frac {2\,a^3\,b^2\,x^2}{3}+a^2\,b^3\,x^3-b^5\,x^5\,\ln \relax (x)-\frac {3\,a^4\,b\,x}{4}\right )}{x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 78, normalized size = 0.99 \begin {gather*} b^{5} c^{4} \log {\relax (x )} + \frac {- 12 a^{5} c^{4} + 45 a^{4} b c^{4} x - 40 a^{3} b^{2} c^{4} x^{2} - 60 a^{2} b^{3} c^{4} x^{3} + 180 a b^{4} c^{4} x^{4}}{60 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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