Optimal. Leaf size=111 \[ -\frac{x \left (A e (c d-b e)-B \left (c d^2-e (b d-a e)\right )\right )}{e^3}-\frac{(B d-A e) \log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac{x^2 (-A c e-b B e+B c d)}{2 e^2}+\frac{B c x^3}{3 e} \]
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Rubi [A] time = 0.132888, antiderivative size = 109, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {771} \[ \frac{x \left (-B e (b d-a e)-A e (c d-b e)+B c d^2\right )}{e^3}-\frac{(B d-A e) \log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac{x^2 (-A c e-b B e+B c d)}{2 e^2}+\frac{B c x^3}{3 e} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )}{d+e x} \, dx &=\int \left (\frac{B c d^2-B e (b d-a e)-A e (c d-b e)}{e^3}+\frac{(-B c d+b B e+A c e) x}{e^2}+\frac{B c x^2}{e}+\frac{(-B d+A e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{\left (B c d^2-B e (b d-a e)-A e (c d-b e)\right ) x}{e^3}-\frac{(B c d-b B e-A c e) x^2}{2 e^2}+\frac{B c x^3}{3 e}-\frac{(B d-A e) \left (c d^2-b d e+a e^2\right ) \log (d+e x)}{e^4}\\ \end{align*}
Mathematica [A] time = 0.058619, size = 100, normalized size = 0.9 \[ \frac{e x \left (3 B e (2 a e-2 b d+b e x)+3 A e (2 b e-2 c d+c e x)+B c \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 (B d-A e) \log (d+e x) \left (e (a e-b d)+c d^2\right )}{6 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 171, normalized size = 1.5 \begin{align*}{\frac{Bc{x}^{3}}{3\,e}}+{\frac{Ac{x}^{2}}{2\,e}}+{\frac{B{x}^{2}b}{2\,e}}-{\frac{Bc{x}^{2}d}{2\,{e}^{2}}}+{\frac{Abx}{e}}-{\frac{Acdx}{{e}^{2}}}+{\frac{aBx}{e}}-{\frac{Bbdx}{{e}^{2}}}+{\frac{Bc{d}^{2}x}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) aA}{e}}-{\frac{\ln \left ( ex+d \right ) Abd}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ) Ac{d}^{2}}{{e}^{3}}}-{\frac{\ln \left ( ex+d \right ) aBd}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ) Bb{d}^{2}}{{e}^{3}}}-{\frac{\ln \left ( ex+d \right ) Bc{d}^{3}}{{e}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02376, size = 165, normalized size = 1.49 \begin{align*} \frac{2 \, B c e^{2} x^{3} - 3 \,{\left (B c d e -{\left (B b + A c\right )} e^{2}\right )} x^{2} + 6 \,{\left (B c d^{2} -{\left (B b + A c\right )} d e +{\left (B a + A b\right )} e^{2}\right )} x}{6 \, e^{3}} - \frac{{\left (B c d^{3} - A a e^{3} -{\left (B b + A c\right )} d^{2} e +{\left (B a + A b\right )} d e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20967, size = 265, normalized size = 2.39 \begin{align*} \frac{2 \, B c e^{3} x^{3} - 3 \,{\left (B c d e^{2} -{\left (B b + A c\right )} e^{3}\right )} x^{2} + 6 \,{\left (B c d^{2} e -{\left (B b + A c\right )} d e^{2} +{\left (B a + A b\right )} e^{3}\right )} x - 6 \,{\left (B c d^{3} - A a e^{3} -{\left (B b + A c\right )} d^{2} e +{\left (B a + A b\right )} d e^{2}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.701952, size = 104, normalized size = 0.94 \begin{align*} \frac{B c x^{3}}{3 e} + \frac{x^{2} \left (A c e + B b e - B c d\right )}{2 e^{2}} + \frac{x \left (A b e^{2} - A c d e + B a e^{2} - B b d e + B c d^{2}\right )}{e^{3}} - \frac{\left (- A e + B d\right ) \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13566, size = 184, normalized size = 1.66 \begin{align*} -{\left (B c d^{3} - B b d^{2} e - A c d^{2} e + B a d e^{2} + A b d e^{2} - A a e^{3}\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, B c x^{3} e^{2} - 3 \, B c d x^{2} e + 6 \, B c d^{2} x + 3 \, B b x^{2} e^{2} + 3 \, A c x^{2} e^{2} - 6 \, B b d x e - 6 \, A c d x e + 6 \, B a x e^{2} + 6 \, A b x e^{2}\right )} e^{\left (-3\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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