Optimal. Leaf size=82 \[ -\frac{5}{2} a^2 b^4 c^5 x^2+5 a^4 b^2 c^5 \log (x)+\frac{4 a^5 b c^5}{x}-\frac{a^6 c^5}{2 x^2}+\frac{4}{3} a b^5 c^5 x^3-\frac{1}{4} b^6 c^5 x^4 \]
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Rubi [A] time = 0.036217, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, 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*}
Mathematica [A] time = 0.006879, size = 68, normalized size = 0.83 \[ c^5 \left (-\frac{5}{2} a^2 b^4 x^2+5 a^4 b^2 \log (x)+\frac{4 a^5 b}{x}-\frac{a^6}{2 x^2}+\frac{4}{3} a b^5 x^3-\frac{b^6 x^4}{4}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 75, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}{c}^{5}}{2\,{x}^{2}}}+4\,{\frac{{a}^{5}b{c}^{5}}{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}}+5\,{a}^{4}{b}^{2}{c}^{5}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06547, size = 101, normalized size = 1.23 \begin{align*} -\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 (x\right ) + \frac{8 \, a^{5} b c^{5} x - a^{6} c^{5}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8758, size = 167, normalized size = 2.04 \begin{align*} -\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 \left (x\right ) - 48 \, a^{5} b c^{5} x + 6 \, a^{6} c^{5}}{12 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.422046, size = 82, normalized size = 1. \begin{align*} 5 a^{4} b^{2} c^{5} \log{\left (x \right )} - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24559, size = 103, normalized size = 1.26 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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