Optimal. Leaf size=92 \[ -\frac{(a+b x)^2 (B d-A e)}{2 e^2}+\frac{b x (b d-a e) (B d-A e)}{e^3}-\frac{(b d-a e)^2 (B d-A e) \log (d+e x)}{e^4}+\frac{B (a+b x)^3}{3 b e} \]
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Rubi [A] time = 0.0628983, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 77} \[ -\frac{(a+b x)^2 (B d-A e)}{2 e^2}+\frac{b x (b d-a e) (B d-A e)}{e^3}-\frac{(b d-a e)^2 (B d-A e) \log (d+e x)}{e^4}+\frac{B (a+b x)^3}{3 b e} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{d+e x} \, dx &=\int \frac{(a+b x)^2 (A+B x)}{d+e x} \, dx\\ &=\int \left (-\frac{b (b d-a e) (-B d+A e)}{e^3}+\frac{b (-B d+A e) (a+b x)}{e^2}+\frac{B (a+b x)^2}{e}+\frac{(-b d+a e)^2 (-B d+A e)}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{b (b d-a e) (B d-A e) x}{e^3}-\frac{(B d-A e) (a+b x)^2}{2 e^2}+\frac{B (a+b x)^3}{3 b e}-\frac{(b d-a e)^2 (B d-A e) \log (d+e x)}{e^4}\\ \end{align*}
Mathematica [A] time = 0.0570592, size = 102, normalized size = 1.11 \[ \frac{e x \left (6 a^2 B e^2+6 a b e (2 A e-2 B d+B e x)+b^2 \left (3 A e (e x-2 d)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )\right )-6 (b d-a e)^2 (B d-A e) \log (d+e x)}{6 e^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 197, normalized size = 2.1 \begin{align*}{\frac{{b}^{2}B{x}^{3}}{3\,e}}+{\frac{A{b}^{2}{x}^{2}}{2\,e}}+{\frac{B{x}^{2}ab}{e}}-{\frac{{b}^{2}B{x}^{2}d}{2\,{e}^{2}}}+2\,{\frac{aAbx}{e}}-{\frac{Ad{b}^{2}x}{{e}^{2}}}+{\frac{{a}^{2}Bx}{e}}-2\,{\frac{abBdx}{{e}^{2}}}+{\frac{B{b}^{2}{d}^{2}x}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) A{a}^{2}}{e}}-2\,{\frac{\ln \left ( ex+d \right ) Aabd}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ) A{b}^{2}{d}^{2}}{{e}^{3}}}-{\frac{\ln \left ( ex+d \right ) B{a}^{2}d}{{e}^{2}}}+2\,{\frac{\ln \left ( ex+d \right ) Bab{d}^{2}}{{e}^{3}}}-{\frac{\ln \left ( ex+d \right ) B{b}^{2}{d}^{3}}{{e}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22968, size = 205, normalized size = 2.23 \begin{align*} \frac{2 \, B b^{2} e^{2} x^{3} - 3 \,{\left (B b^{2} d e -{\left (2 \, B a b + A b^{2}\right )} e^{2}\right )} x^{2} + 6 \,{\left (B b^{2} d^{2} -{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} x}{6 \, e^{3}} - \frac{{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, 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 = 1.43392, size = 319, normalized size = 3.47 \begin{align*} \frac{2 \, B b^{2} e^{3} x^{3} - 3 \,{\left (B b^{2} d e^{2} -{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 6 \,{\left (B b^{2} d^{2} e -{\left (2 \, B a b + A b^{2}\right )} d e^{2} +{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x - 6 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, 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 = 0.730056, size = 117, normalized size = 1.27 \begin{align*} \frac{B b^{2} x^{3}}{3 e} + \frac{x^{2} \left (A b^{2} e + 2 B a b e - B b^{2} d\right )}{2 e^{2}} + \frac{x \left (2 A a b e^{2} - A b^{2} d e + B a^{2} e^{2} - 2 B a b d e + B b^{2} d^{2}\right )}{e^{3}} - \frac{\left (- A e + B d\right ) \left (a e - b d\right )^{2} \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 = 1.14308, size = 219, normalized size = 2.38 \begin{align*} -{\left (B b^{2} d^{3} - 2 \, B a b d^{2} e - A b^{2} d^{2} e + B a^{2} d e^{2} + 2 \, A a b d e^{2} - A a^{2} e^{3}\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, B b^{2} x^{3} e^{2} - 3 \, B b^{2} d x^{2} e + 6 \, B b^{2} d^{2} x + 6 \, B a b x^{2} e^{2} + 3 \, A b^{2} x^{2} e^{2} - 12 \, B a b d x e - 6 \, A b^{2} d x e + 6 \, B a^{2} x e^{2} + 12 \, A 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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