Optimal. Leaf size=66 \[ A b^3 \log (x)+b^2 x (3 A c+b B)+\frac {1}{3} c^2 x^3 (A c+3 b B)+\frac {3}{2} b c x^2 (A c+b B)+\frac {1}{4} B c^3 x^4 \]
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Rubi [A] time = 0.04, antiderivative size = 66, 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 = {765} \[ b^2 x (3 A c+b B)+A b^3 \log (x)+\frac {1}{3} c^2 x^3 (A c+3 b B)+\frac {3}{2} b c x^2 (A c+b B)+\frac {1}{4} B c^3 x^4 \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^4} \, dx &=\int \left (b^2 (b B+3 A c)+\frac {A b^3}{x}+3 b c (b B+A c) x+c^2 (3 b B+A c) x^2+B c^3 x^3\right ) \, dx\\ &=b^2 (b B+3 A c) x+\frac {3}{2} b c (b B+A c) x^2+\frac {1}{3} c^2 (3 b B+A c) x^3+\frac {1}{4} B c^3 x^4+A b^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 63, normalized size = 0.95 \[ A b^3 \log (x)+\frac {1}{12} x \left (18 b^2 c (2 A+B x)+6 b c^2 x (3 A+2 B x)+c^3 x^2 (4 A+3 B x)+12 b^3 B\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 68, normalized size = 1.03 \[ \frac {1}{4} \, B c^{3} x^{4} + A b^{3} \log \relax (x) + \frac {1}{3} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + \frac {3}{2} \, {\left (B b^{2} c + A b c^{2}\right )} x^{2} + {\left (B b^{3} + 3 \, A b^{2} c\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 70, normalized size = 1.06 \[ \frac {1}{4} \, B c^{3} x^{4} + B b c^{2} x^{3} + \frac {1}{3} \, A c^{3} x^{3} + \frac {3}{2} \, B b^{2} c x^{2} + \frac {3}{2} \, A b c^{2} x^{2} + B b^{3} x + 3 \, A b^{2} c x + A b^{3} \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 70, normalized size = 1.06 \[ \frac {B \,c^{3} x^{4}}{4}+\frac {A \,c^{3} x^{3}}{3}+B b \,c^{2} x^{3}+\frac {3 A b \,c^{2} x^{2}}{2}+\frac {3 B \,b^{2} c \,x^{2}}{2}+A \,b^{3} \ln \relax (x )+3 A \,b^{2} c x +B \,b^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 68, normalized size = 1.03 \[ \frac {1}{4} \, B c^{3} x^{4} + A b^{3} \log \relax (x) + \frac {1}{3} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + \frac {3}{2} \, {\left (B b^{2} c + A b c^{2}\right )} x^{2} + {\left (B b^{3} + 3 \, A b^{2} c\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 63, normalized size = 0.95 \[ x\,\left (B\,b^3+3\,A\,c\,b^2\right )+x^3\,\left (\frac {A\,c^3}{3}+B\,b\,c^2\right )+\frac {B\,c^3\,x^4}{4}+A\,b^3\,\ln \relax (x)+\frac {3\,b\,c\,x^2\,\left (A\,c+B\,b\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 73, normalized size = 1.11 \[ A b^{3} \log {\relax (x )} + \frac {B c^{3} x^{4}}{4} + x^{3} \left (\frac {A c^{3}}{3} + B b c^{2}\right ) + x^{2} \left (\frac {3 A b c^{2}}{2} + \frac {3 B b^{2} c}{2}\right ) + x \left (3 A b^{2} c + B b^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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