Optimal. Leaf size=75 \[ \frac{1}{4} b^2 x^4 (3 A c+b B)+\frac{1}{3} A b^3 x^3+\frac{1}{6} c^2 x^6 (A c+3 b B)+\frac{3}{5} b c x^5 (A c+b B)+\frac{1}{7} B c^3 x^7 \]
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Rubi [A] time = 0.0439684, antiderivative size = 75, 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} \[ \frac{1}{4} b^2 x^4 (3 A c+b B)+\frac{1}{3} A b^3 x^3+\frac{1}{6} c^2 x^6 (A c+3 b B)+\frac{3}{5} b c x^5 (A c+b B)+\frac{1}{7} B c^3 x^7 \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^3}{x} \, dx &=\int \left (A b^3 x^2+b^2 (b B+3 A c) x^3+3 b c (b B+A c) x^4+c^2 (3 b B+A c) x^5+B c^3 x^6\right ) \, dx\\ &=\frac{1}{3} A b^3 x^3+\frac{1}{4} b^2 (b B+3 A c) x^4+\frac{3}{5} b c (b B+A c) x^5+\frac{1}{6} c^2 (3 b B+A c) x^6+\frac{1}{7} B c^3 x^7\\ \end{align*}
Mathematica [A] time = 0.0105688, size = 75, normalized size = 1. \[ \frac{1}{4} b^2 x^4 (3 A c+b B)+\frac{1}{3} A b^3 x^3+\frac{1}{6} c^2 x^6 (A c+3 b B)+\frac{3}{5} b c x^5 (A c+b B)+\frac{1}{7} B c^3 x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 76, normalized size = 1. \begin{align*}{\frac{B{c}^{3}{x}^{7}}{7}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,Ab{c}^{2}+3\,B{b}^{2}c \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,A{b}^{2}c+{b}^{3}B \right ){x}^{4}}{4}}+{\frac{A{b}^{3}{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02121, size = 99, normalized size = 1.32 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{3} \, A b^{3} x^{3} + \frac{1}{6} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + \frac{3}{5} \,{\left (B b^{2} c + A b c^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81004, size = 163, normalized size = 2.17 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{3} \, A b^{3} x^{3} + \frac{1}{6} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + \frac{3}{5} \,{\left (B b^{2} c + A b c^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.076226, size = 82, normalized size = 1.09 \begin{align*} \frac{A b^{3} x^{3}}{3} + \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 b^{2} c}{5}\right ) + x^{4} \left (\frac{3 A b^{2} c}{4} + \frac{B b^{3}}{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16755, size = 104, normalized size = 1.39 \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} \, A b c^{2} x^{5} + \frac{1}{4} \, B b^{3} x^{4} + \frac{3}{4} \, A b^{2} c x^{4} + \frac{1}{3} \, A b^{3} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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