Optimal. Leaf size=157 \[ a^2 x (a B+3 A b)+a^3 A \log (x)+\frac{1}{4} x^4 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{3}{5} c x^5 \left (a B c+A b c+b^2 B\right )+\frac{1}{3} x^3 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{3}{2} a x^2 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{6} c^2 x^6 (A c+3 b B)+\frac{1}{7} B c^3 x^7 \]
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Rubi [A] time = 0.110567, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {765} \[ a^2 x (a B+3 A b)+a^3 A \log (x)+\frac{1}{4} x^4 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{3}{5} c x^5 \left (a B c+A b c+b^2 B\right )+\frac{1}{3} x^3 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{3}{2} a x^2 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{6} c^2 x^6 (A c+3 b B)+\frac{1}{7} B c^3 x^7 \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )^3}{x} \, dx &=\int \left (a^2 (3 A b+a B)+\frac{a^3 A}{x}+3 a \left (a b B+A \left (b^2+a c\right )\right ) x+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^2+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^3+3 c \left (b^2 B+A b c+a B c\right ) x^4+c^2 (3 b B+A c) x^5+B c^3 x^6\right ) \, dx\\ &=a^2 (3 A b+a B) x+\frac{3}{2} a \left (a b B+A \left (b^2+a c\right )\right ) x^2+\frac{1}{3} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^3+\frac{1}{4} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^4+\frac{3}{5} c \left (b^2 B+A b c+a B c\right ) x^5+\frac{1}{6} c^2 (3 b B+A c) x^6+\frac{1}{7} B c^3 x^7+a^3 A \log (x)\\ \end{align*}
Mathematica [A] time = 0.0650266, size = 157, normalized size = 1. \[ a^2 x (a B+3 A b)+a^3 A \log (x)+\frac{1}{4} x^4 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{3}{5} c x^5 \left (a B c+A b c+b^2 B\right )+\frac{1}{3} x^3 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{3}{2} a x^2 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{6} c^2 x^6 (A c+3 b B)+\frac{1}{7} B c^3 x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 185, normalized size = 1.2 \begin{align*}{\frac{B{c}^{3}{x}^{7}}{7}}+{\frac{A{c}^{3}{x}^{6}}{6}}+{\frac{B{x}^{6}b{c}^{2}}{2}}+{\frac{3\,A{x}^{5}b{c}^{2}}{5}}+{\frac{3\,aB{c}^{2}{x}^{5}}{5}}+{\frac{3\,B{x}^{5}{b}^{2}c}{5}}+{\frac{3\,aA{c}^{2}{x}^{4}}{4}}+{\frac{3\,A{x}^{4}{b}^{2}c}{4}}+{\frac{3\,B{x}^{4}abc}{2}}+{\frac{B{x}^{4}{b}^{3}}{4}}+2\,A{x}^{3}abc+{\frac{A{b}^{3}{x}^{3}}{3}}+{a}^{2}Bc{x}^{3}+B{x}^{3}a{b}^{2}+{\frac{3\,{a}^{2}Ac{x}^{2}}{2}}+{\frac{3\,A{x}^{2}a{b}^{2}}{2}}+{\frac{3\,B{x}^{2}{a}^{2}b}{2}}+3\,A{a}^{2}bx+{a}^{3}Bx+{a}^{3}A\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04642, size = 217, normalized size = 1.38 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{6} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + \frac{3}{5} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + A a^{3} \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + \frac{3}{2} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26851, size = 365, normalized size = 2.32 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{6} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + \frac{3}{5} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + A a^{3} \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + \frac{3}{2} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.491705, size = 192, normalized size = 1.22 \begin{align*} A a^{3} \log{\left (x \right )} + \frac{B c^{3} x^{7}}{7} + x^{6} \left (\frac{A c^{3}}{6} + \frac{B b c^{2}}{2}\right ) + x^{5} \left (\frac{3 A b c^{2}}{5} + \frac{3 B a c^{2}}{5} + \frac{3 B b^{2} c}{5}\right ) + x^{4} \left (\frac{3 A a c^{2}}{4} + \frac{3 A b^{2} c}{4} + \frac{3 B a b c}{2} + \frac{B b^{3}}{4}\right ) + x^{3} \left (2 A a b c + \frac{A b^{3}}{3} + B a^{2} c + B a b^{2}\right ) + x^{2} \left (\frac{3 A a^{2} c}{2} + \frac{3 A a b^{2}}{2} + \frac{3 B a^{2} b}{2}\right ) + x \left (3 A a^{2} b + B a^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17793, size = 250, normalized size = 1.59 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{2} \, B b c^{2} x^{6} + \frac{1}{6} \, A c^{3} x^{6} + \frac{3}{5} \, B b^{2} c x^{5} + \frac{3}{5} \, B a c^{2} x^{5} + \frac{3}{5} \, A b c^{2} x^{5} + \frac{1}{4} \, B b^{3} x^{4} + \frac{3}{2} \, B a b c x^{4} + \frac{3}{4} \, A b^{2} c x^{4} + \frac{3}{4} \, A a c^{2} x^{4} + B a b^{2} x^{3} + \frac{1}{3} \, A b^{3} x^{3} + B a^{2} c x^{3} + 2 \, A a b c x^{3} + \frac{3}{2} \, B a^{2} b x^{2} + \frac{3}{2} \, A a b^{2} x^{2} + \frac{3}{2} \, A a^{2} c x^{2} + B a^{3} x + 3 \, A a^{2} b x + A a^{3} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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