Optimal. Leaf size=38 \[ \frac{B (b+c x)^4}{4 c^2}-\frac{(b+c x)^3 (b B-A c)}{3 c^2} \]
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Rubi [A] time = 0.027039, antiderivative size = 38, 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} \[ \frac{B (b+c x)^4}{4 c^2}-\frac{(b+c x)^3 (b B-A c)}{3 c^2} \]
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 )^2}{x^2} \, dx &=\int \left (\frac{(-b B+A c) (b+c x)^2}{c}+\frac{B (b+c x)^3}{c}\right ) \, dx\\ &=-\frac{(b B-A c) (b+c x)^3}{3 c^2}+\frac{B (b+c x)^4}{4 c^2}\\ \end{align*}
Mathematica [A] time = 0.0099015, size = 47, normalized size = 1.24 \[ \frac{1}{12} x \left (12 A b^2+4 c x^2 (A c+2 b B)+6 b x (2 A c+b B)+3 B c^2 x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 49, normalized size = 1.3 \begin{align*}{\frac{B{c}^{2}{x}^{4}}{4}}+{\frac{ \left ( A{c}^{2}+2\,Bbc \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,Abc+{b}^{2}B \right ){x}^{2}}{2}}+A{b}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13911, size = 65, normalized size = 1.71 \begin{align*} \frac{1}{4} \, B c^{2} x^{4} + A b^{2} x + \frac{1}{3} \,{\left (2 \, B b c + A c^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B b^{2} + 2 \, A b c\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80916, size = 109, normalized size = 2.87 \begin{align*} \frac{1}{4} \, B c^{2} x^{4} + A b^{2} x + \frac{1}{3} \,{\left (2 \, B b c + A c^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B b^{2} + 2 \, A b c\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.07274, size = 49, normalized size = 1.29 \begin{align*} A b^{2} x + \frac{B c^{2} x^{4}}{4} + x^{3} \left (\frac{A c^{2}}{3} + \frac{2 B b c}{3}\right ) + x^{2} \left (A b c + \frac{B b^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15935, size = 66, normalized size = 1.74 \begin{align*} \frac{1}{4} \, B c^{2} x^{4} + \frac{2}{3} \, B b c x^{3} + \frac{1}{3} \, A c^{2} x^{3} + \frac{1}{2} \, B b^{2} x^{2} + A b c x^{2} + A b^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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