Optimal. Leaf size=72 \[ -\frac {a^5 c^4}{4 x^4}+\frac {a^4 b c^4}{x^3}-\frac {a^3 b^2 c^4}{x^2}-\frac {2 a^2 b^3 c^4}{x}-3 a b^4 c^4 \log (x)+b^5 c^4 x \]
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Rubi [A] time = 0.03, antiderivative size = 72, 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} \[ -\frac {a^3 b^2 c^4}{x^2}-\frac {2 a^2 b^3 c^4}{x}+\frac {a^4 b c^4}{x^3}-\frac {a^5 c^4}{4 x^4}-3 a b^4 c^4 \log (x)+b^5 c^4 x \]
Antiderivative was successfully verified.
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Rule 75
Rubi steps
\begin {align*} \int \frac {(a+b x) (a c-b c x)^4}{x^5} \, dx &=\int \left (b^5 c^4+\frac {a^5 c^4}{x^5}-\frac {3 a^4 b c^4}{x^4}+\frac {2 a^3 b^2 c^4}{x^3}+\frac {2 a^2 b^3 c^4}{x^2}-\frac {3 a b^4 c^4}{x}\right ) \, dx\\ &=-\frac {a^5 c^4}{4 x^4}+\frac {a^4 b c^4}{x^3}-\frac {a^3 b^2 c^4}{x^2}-\frac {2 a^2 b^3 c^4}{x}+b^5 c^4 x-3 a b^4 c^4 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 72, normalized size = 1.00 \[ -\frac {a^5 c^4}{4 x^4}+\frac {a^4 b c^4}{x^3}-\frac {a^3 b^2 c^4}{x^2}-\frac {2 a^2 b^3 c^4}{x}-3 a b^4 c^4 \log (x)+b^5 c^4 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 77, normalized size = 1.07 \[ \frac {4 \, b^{5} c^{4} x^{5} - 12 \, a b^{4} c^{4} x^{4} \log \relax (x) - 8 \, a^{2} b^{3} c^{4} x^{3} - 4 \, a^{3} b^{2} c^{4} x^{2} + 4 \, a^{4} b c^{4} x - a^{5} c^{4}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 72, normalized size = 1.00 \[ b^{5} c^{4} x - 3 \, a b^{4} c^{4} \log \left ({\left | x \right |}\right ) - \frac {8 \, a^{2} b^{3} c^{4} x^{3} + 4 \, a^{3} b^{2} c^{4} x^{2} - 4 \, a^{4} b c^{4} x + a^{5} c^{4}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 71, normalized size = 0.99 \[ -3 a \,b^{4} c^{4} \ln \relax (x )+b^{5} c^{4} x -\frac {2 a^{2} b^{3} c^{4}}{x}-\frac {a^{3} b^{2} c^{4}}{x^{2}}+\frac {a^{4} b \,c^{4}}{x^{3}}-\frac {a^{5} c^{4}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 71, normalized size = 0.99 \[ b^{5} c^{4} x - 3 \, a b^{4} c^{4} \log \relax (x) - \frac {8 \, a^{2} b^{3} c^{4} x^{3} + 4 \, a^{3} b^{2} c^{4} x^{2} - 4 \, a^{4} b c^{4} x + a^{5} c^{4}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 60, normalized size = 0.83 \[ -\frac {c^4\,\left (a^5-4\,b^5\,x^5+4\,a^3\,b^2\,x^2+8\,a^2\,b^3\,x^3-4\,a^4\,b\,x+12\,a\,b^4\,x^4\,\ln \relax (x)\right )}{4\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 75, normalized size = 1.04 \[ - 3 a b^{4} c^{4} \log {\relax (x )} + b^{5} c^{4} x + \frac {- a^{5} c^{4} + 4 a^{4} b c^{4} x - 4 a^{3} b^{2} c^{4} x^{2} - 8 a^{2} b^{3} c^{4} x^{3}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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