Optimal. Leaf size=66 \[ b^2 x (3 A c+b B)+A b^3 \log (x)+\frac{1}{3} c^2 x^3 (A c+3 b B)+\frac{3}{2} b c x^2 (A c+b B)+\frac{1}{4} B c^3 x^4 \]
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Rubi [A] time = 0.0414899, antiderivative size = 66, 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 = {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*}
Mathematica [A] time = 0.0245278, size = 63, normalized size = 0.95 \[ \frac{1}{12} x \left (18 b^2 c (2 A+B x)+6 b c^2 x (3 A+2 B x)+c^3 x^2 (4 A+3 B x)+12 b^3 B\right )+A b^3 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 70, normalized size = 1.1 \begin{align*}{\frac{B{c}^{3}{x}^{4}}{4}}+{\frac{A{x}^{3}{c}^{3}}{3}}+B{x}^{3}b{c}^{2}+{\frac{3\,A{x}^{2}b{c}^{2}}{2}}+{\frac{3\,B{x}^{2}{b}^{2}c}{2}}+3\,A{b}^{2}cx+{b}^{3}Bx+A{b}^{3}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0865, size = 92, normalized size = 1.39 \begin{align*} \frac{1}{4} \, B c^{3} x^{4} + A b^{3} \log \left (x\right ) + \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77382, size = 154, normalized size = 2.33 \begin{align*} \frac{1}{4} \, B c^{3} x^{4} + A b^{3} \log \left (x\right ) + \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.414911, size = 73, normalized size = 1.11 \begin{align*} A b^{3} \log{\left (x \right )} + \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12582, size = 95, normalized size = 1.44 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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