Optimal. Leaf size=62 \[ -\frac{(b+c x)^5 (2 b B-A c)}{5 c^3}+\frac{b (b+c x)^4 (b B-A c)}{4 c^3}+\frac{B (b+c x)^6}{6 c^3} \]
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Rubi [A] time = 0.0419726, antiderivative size = 62, 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} \[ -\frac{(b+c x)^5 (2 b B-A c)}{5 c^3}+\frac{b (b+c x)^4 (b B-A c)}{4 c^3}+\frac{B (b+c x)^6}{6 c^3} \]
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^2} \, dx &=\int \left (\frac{b (b B-A c) (b+c x)^3}{c^2}+\frac{(-2 b B+A c) (b+c x)^4}{c^2}+\frac{B (b+c x)^5}{c^2}\right ) \, dx\\ &=\frac{b (b B-A c) (b+c x)^4}{4 c^3}-\frac{(2 b B-A c) (b+c x)^5}{5 c^3}+\frac{B (b+c x)^6}{6 c^3}\\ \end{align*}
Mathematica [A] time = 0.0144274, size = 67, normalized size = 1.08 \[ \frac{1}{60} x^2 \left (20 b^2 x (3 A c+b B)+30 A b^3+12 c^2 x^3 (A c+3 b B)+45 b c x^2 (A c+b B)+10 B c^3 x^4\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 76, normalized size = 1.2 \begin{align*}{\frac{B{c}^{3}{x}^{6}}{6}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,Ab{c}^{2}+3\,B{b}^{2}c \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,A{b}^{2}c+{b}^{3}B \right ){x}^{3}}{3}}+{\frac{A{b}^{3}{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12939, size = 99, normalized size = 1.6 \begin{align*} \frac{1}{6} \, B c^{3} x^{6} + \frac{1}{2} \, A b^{3} x^{2} + \frac{1}{5} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} + \frac{3}{4} \,{\left (B b^{2} c + A b c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64776, size = 163, normalized size = 2.63 \begin{align*} \frac{1}{6} \, B c^{3} x^{6} + \frac{1}{2} \, A b^{3} x^{2} + \frac{1}{5} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} + \frac{3}{4} \,{\left (B b^{2} c + A b c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.124298, size = 80, normalized size = 1.29 \begin{align*} \frac{A b^{3} x^{2}}{2} + \frac{B c^{3} x^{6}}{6} + x^{5} \left (\frac{A c^{3}}{5} + \frac{3 B b c^{2}}{5}\right ) + x^{4} \left (\frac{3 A b c^{2}}{4} + \frac{3 B b^{2} c}{4}\right ) + x^{3} \left (A b^{2} c + \frac{B b^{3}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14505, size = 103, normalized size = 1.66 \begin{align*} \frac{1}{6} \, B c^{3} x^{6} + \frac{3}{5} \, B b c^{2} x^{5} + \frac{1}{5} \, A c^{3} x^{5} + \frac{3}{4} \, B b^{2} c x^{4} + \frac{3}{4} \, A b c^{2} x^{4} + \frac{1}{3} \, B b^{3} x^{3} + A b^{2} c x^{3} + \frac{1}{2} \, A b^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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