Optimal. Leaf size=45 \[ -\frac {a^4 c^3}{3 x^3}+\frac {a^3 b c^3}{x^2}+2 a b^3 c^3 \log (x)-b^4 c^3 x \]
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Rubi [A] time = 0.02, antiderivative size = 45, 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} \[ \frac {a^3 b c^3}{x^2}-\frac {a^4 c^3}{3 x^3}+2 a b^3 c^3 \log (x)-b^4 c^3 x \]
Antiderivative was successfully verified.
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Rule 75
Rubi steps
\begin {align*} \int \frac {(a+b x) (a c-b c x)^3}{x^4} \, dx &=\int \left (-b^4 c^3+\frac {a^4 c^3}{x^4}-\frac {2 a^3 b c^3}{x^3}+\frac {2 a b^3 c^3}{x}\right ) \, dx\\ &=-\frac {a^4 c^3}{3 x^3}+\frac {a^3 b c^3}{x^2}-b^4 c^3 x+2 a b^3 c^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 0.82 \[ c^3 \left (-\frac {a^4}{3 x^3}+\frac {a^3 b}{x^2}+2 a b^3 \log (x)-b^4 x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 48, normalized size = 1.07 \[ -\frac {3 \, b^{4} c^{3} x^{4} - 6 \, a b^{3} c^{3} x^{3} \log \relax (x) - 3 \, a^{3} b c^{3} x + a^{4} c^{3}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 46, normalized size = 1.02 \[ -b^{4} c^{3} x + 2 \, a b^{3} c^{3} \log \left ({\left | x \right |}\right ) + \frac {3 \, a^{3} b c^{3} x - a^{4} c^{3}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 44, normalized size = 0.98 \[ 2 a \,b^{3} c^{3} \ln \relax (x )-b^{4} c^{3} x +\frac {a^{3} b \,c^{3}}{x^{2}}-\frac {a^{4} c^{3}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 45, normalized size = 1.00 \[ -b^{4} c^{3} x + 2 \, a b^{3} c^{3} \log \relax (x) + \frac {3 \, a^{3} b c^{3} x - a^{4} c^{3}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 38, normalized size = 0.84 \[ -\frac {c^3\,\left (a^4+3\,b^4\,x^4-3\,a^3\,b\,x-6\,a\,b^3\,x^3\,\ln \relax (x)\right )}{3\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 44, normalized size = 0.98 \[ 2 a b^{3} c^{3} \log {\relax (x )} - b^{4} c^{3} x - \frac {a^{4} c^{3} - 3 a^{3} b c^{3} x}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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