Optimal. Leaf size=49 \[ \frac{e x (b d-a e)}{b^2}+\frac{(b d-a e)^2 \log (a+b x)}{b^3}+\frac{(d+e x)^2}{2 b} \]
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Rubi [A] time = 0.0201425, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ \frac{e x (b d-a e)}{b^2}+\frac{(b d-a e)^2 \log (a+b x)}{b^3}+\frac{(d+e x)^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x) (d+e x)^2}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^2}{a+b x} \, dx\\ &=\int \left (\frac{e (b d-a e)}{b^2}+\frac{(b d-a e)^2}{b^2 (a+b x)}+\frac{e (d+e x)}{b}\right ) \, dx\\ &=\frac{e (b d-a e) x}{b^2}+\frac{(d+e x)^2}{2 b}+\frac{(b d-a e)^2 \log (a+b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0160117, size = 43, normalized size = 0.88 \[ \frac{b e x (-2 a e+4 b d+b e x)+2 (b d-a e)^2 \log (a+b x)}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 74, normalized size = 1.5 \begin{align*}{\frac{{e}^{2}{x}^{2}}{2\,b}}-{\frac{a{e}^{2}x}{{b}^{2}}}+2\,{\frac{edx}{b}}+{\frac{\ln \left ( bx+a \right ){a}^{2}{e}^{2}}{{b}^{3}}}-2\,{\frac{\ln \left ( bx+a \right ) ade}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){d}^{2}}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.954582, size = 82, normalized size = 1.67 \begin{align*} \frac{b e^{2} x^{2} + 2 \,{\left (2 \, b d e - a e^{2}\right )} x}{2 \, b^{2}} + \frac{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (b x + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5328, size = 135, normalized size = 2.76 \begin{align*} \frac{b^{2} e^{2} x^{2} + 2 \,{\left (2 \, b^{2} d e - a b e^{2}\right )} x + 2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (b x + a\right )}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.386651, size = 44, normalized size = 0.9 \begin{align*} \frac{e^{2} x^{2}}{2 b} - \frac{x \left (a e^{2} - 2 b d e\right )}{b^{2}} + \frac{\left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10599, size = 80, normalized size = 1.63 \begin{align*} \frac{b x^{2} e^{2} + 4 \, b d x e - 2 \, a x e^{2}}{2 \, b^{2}} + \frac{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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