Optimal. Leaf size=47 \[ -\frac {a^4 c^3}{x}-2 a^3 b c^3 \log (x)+a b^3 c^3 x^2-\frac {1}{3} b^4 c^3 x^3 \]
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Rubi [A] time = 0.02, antiderivative size = 47, 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*} -2 a^3 b c^3 \log (x)-\frac {a^4 c^3}{x}+a b^3 c^3 x^2-\frac {1}{3} b^4 c^3 x^3 \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)^3}{x^2} \, dx &=\int \left (\frac {a^4 c^3}{x^2}-\frac {2 a^3 b c^3}{x}+2 a b^3 c^3 x-b^4 c^3 x^2\right ) \, dx\\ &=-\frac {a^4 c^3}{x}+a b^3 c^3 x^2-\frac {1}{3} b^4 c^3 x^3-2 a^3 b c^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.83 \begin {gather*} c^3 \left (-\frac {a^4}{x}-2 a^3 b \log (x)+a b^3 x^2-\frac {b^4 x^3}{3}\right ) \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)^3}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.33, size = 48, normalized size = 1.02 \begin {gather*} -\frac {b^{4} c^{3} x^{4} - 3 \, a b^{3} c^{3} x^{3} + 6 \, a^{3} b c^{3} x \log \relax (x) + 3 \, a^{4} c^{3}}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 46, normalized size = 0.98 \begin {gather*} -\frac {1}{3} \, b^{4} c^{3} x^{3} + a b^{3} c^{3} x^{2} - 2 \, a^{3} b c^{3} \log \left ({\left | x \right |}\right ) - \frac {a^{4} c^{3}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 46, normalized size = 0.98 \begin {gather*} -\frac {b^{4} c^{3} x^{3}}{3}+a \,b^{3} c^{3} x^{2}-2 a^{3} b \,c^{3} \ln \relax (x )-\frac {a^{4} c^{3}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 45, normalized size = 0.96 \begin {gather*} -\frac {1}{3} \, b^{4} c^{3} x^{3} + a b^{3} c^{3} x^{2} - 2 \, a^{3} b c^{3} \log \relax (x) - \frac {a^{4} c^{3}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 45, normalized size = 0.96 \begin {gather*} a\,b^3\,c^3\,x^2-\frac {b^4\,c^3\,x^3}{3}-\frac {a^4\,c^3}{x}-2\,a^3\,b\,c^3\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 44, normalized size = 0.94 \begin {gather*} - \frac {a^{4} c^{3}}{x} - 2 a^{3} b c^{3} \log {\relax (x )} + a b^{3} c^{3} x^{2} - \frac {b^{4} c^{3} x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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