Optimal. Leaf size=82 \[ -\frac {a^6 c^5}{2 x^2}+\frac {4 a^5 b c^5}{x}+5 a^4 b^2 c^5 \log (x)-\frac {5}{2} a^2 b^4 c^5 x^2+\frac {4}{3} a b^5 c^5 x^3-\frac {1}{4} b^6 c^5 x^4 \]
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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} \[ -\frac {5}{2} a^2 b^4 c^5 x^2+5 a^4 b^2 c^5 \log (x)+\frac {4 a^5 b c^5}{x}-\frac {a^6 c^5}{2 x^2}+\frac {4}{3} a b^5 c^5 x^3-\frac {1}{4} b^6 c^5 x^4 \]
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^3} \, dx &=\int \left (\frac {a^6 c^5}{x^3}-\frac {4 a^5 b c^5}{x^2}+\frac {5 a^4 b^2 c^5}{x}-5 a^2 b^4 c^5 x+4 a b^5 c^5 x^2-b^6 c^5 x^3\right ) \, dx\\ &=-\frac {a^6 c^5}{2 x^2}+\frac {4 a^5 b c^5}{x}-\frac {5}{2} a^2 b^4 c^5 x^2+\frac {4}{3} a b^5 c^5 x^3-\frac {1}{4} b^6 c^5 x^4+5 a^4 b^2 c^5 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 68, normalized size = 0.83 \[ c^5 \left (-\frac {a^6}{2 x^2}+\frac {4 a^5 b}{x}+5 a^4 b^2 \log (x)-\frac {5}{2} a^2 b^4 x^2+\frac {4}{3} a b^5 x^3-\frac {b^6 x^4}{4}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 77, normalized size = 0.94 \[ -\frac {3 \, b^{6} c^{5} x^{6} - 16 \, a b^{5} c^{5} x^{5} + 30 \, a^{2} b^{4} c^{5} x^{4} - 60 \, a^{4} b^{2} c^{5} x^{2} \log \relax (x) - 48 \, a^{5} b c^{5} x + 6 \, a^{6} c^{5}}{12 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 76, normalized size = 0.93 \[ -\frac {1}{4} \, b^{6} c^{5} x^{4} + \frac {4}{3} \, a b^{5} c^{5} x^{3} - \frac {5}{2} \, a^{2} b^{4} c^{5} x^{2} + 5 \, a^{4} b^{2} c^{5} \log \left ({\left | x \right |}\right ) + \frac {8 \, a^{5} b c^{5} x - a^{6} c^{5}}{2 \, x^{2}} \]
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 \[ -\frac {b^{6} c^{5} x^{4}}{4}+\frac {4 a \,b^{5} c^{5} x^{3}}{3}-\frac {5 a^{2} b^{4} c^{5} x^{2}}{2}+5 a^{4} b^{2} c^{5} \ln \relax (x )+\frac {4 a^{5} b \,c^{5}}{x}-\frac {a^{6} c^{5}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 75, normalized size = 0.91 \[ -\frac {1}{4} \, b^{6} c^{5} x^{4} + \frac {4}{3} \, a b^{5} c^{5} x^{3} - \frac {5}{2} \, a^{2} b^{4} c^{5} x^{2} + 5 \, a^{4} b^{2} c^{5} \log \relax (x) + \frac {8 \, a^{5} b c^{5} x - a^{6} c^{5}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 75, normalized size = 0.91 \[ \frac {4\,a\,b^5\,c^5\,x^3}{3}-\frac {b^6\,c^5\,x^4}{4}-\frac {\frac {a^6\,c^5}{2}-4\,a^5\,b\,c^5\,x}{x^2}-\frac {5\,a^2\,b^4\,c^5\,x^2}{2}+5\,a^4\,b^2\,c^5\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 82, normalized size = 1.00 \[ 5 a^{4} b^{2} c^{5} \log {\relax (x )} - \frac {5 a^{2} b^{4} c^{5} x^{2}}{2} + \frac {4 a b^{5} c^{5} x^{3}}{3} - \frac {b^{6} c^{5} x^{4}}{4} - \frac {a^{6} c^{5} - 8 a^{5} b c^{5} x}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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