Optimal. Leaf size=75 \[ \frac{e^2 x (3 b d-2 a e)}{b^3}-\frac{(b d-a e)^3}{b^4 (a+b x)}+\frac{3 e (b d-a e)^2 \log (a+b x)}{b^4}+\frac{e^3 x^2}{2 b^2} \]
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Rubi [A] time = 0.0641327, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ \frac{e^2 x (3 b d-2 a e)}{b^3}-\frac{(b d-a e)^3}{b^4 (a+b x)}+\frac{3 e (b d-a e)^2 \log (a+b x)}{b^4}+\frac{e^3 x^2}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^3}{(a+b x)^2} \, dx\\ &=\int \left (\frac{e^2 (3 b d-2 a e)}{b^3}+\frac{e^3 x}{b^2}+\frac{(b d-a e)^3}{b^3 (a+b x)^2}+\frac{3 e (b d-a e)^2}{b^3 (a+b x)}\right ) \, dx\\ &=\frac{e^2 (3 b d-2 a e) x}{b^3}+\frac{e^3 x^2}{2 b^2}-\frac{(b d-a e)^3}{b^4 (a+b x)}+\frac{3 e (b d-a e)^2 \log (a+b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0519787, size = 72, normalized size = 0.96 \[ \frac{2 b e^2 x (3 b d-2 a e)-\frac{2 (b d-a e)^3}{a+b x}+6 e (b d-a e)^2 \log (a+b x)+b^2 e^3 x^2}{2 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 149, normalized size = 2. \begin{align*}{\frac{{e}^{3}{x}^{2}}{2\,{b}^{2}}}-2\,{\frac{a{e}^{3}x}{{b}^{3}}}+3\,{\frac{{e}^{2}xd}{{b}^{2}}}+3\,{\frac{{e}^{3}\ln \left ( bx+a \right ){a}^{2}}{{b}^{4}}}-6\,{\frac{{e}^{2}\ln \left ( bx+a \right ) ad}{{b}^{3}}}+3\,{\frac{e\ln \left ( bx+a \right ){d}^{2}}{{b}^{2}}}+{\frac{{a}^{3}{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}-3\,{\frac{{a}^{2}d{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}+3\,{\frac{a{d}^{2}e}{{b}^{2} \left ( bx+a \right ) }}-{\frac{{d}^{3}}{b \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11418, size = 159, normalized size = 2.12 \begin{align*} -\frac{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}}{b^{5} x + a b^{4}} + \frac{b e^{3} x^{2} + 2 \,{\left (3 \, b d e^{2} - 2 \, a e^{3}\right )} x}{2 \, b^{3}} + \frac{3 \,{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} \log \left (b x + a\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.72801, size = 354, normalized size = 4.72 \begin{align*} \frac{b^{3} e^{3} x^{3} - 2 \, b^{3} d^{3} + 6 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3} + 3 \,{\left (2 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (3 \, a b^{2} d e^{2} - 2 \, a^{2} b e^{3}\right )} x + 6 \,{\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x + a b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.773353, size = 100, normalized size = 1.33 \begin{align*} \frac{a^{3} e^{3} - 3 a^{2} b d e^{2} + 3 a b^{2} d^{2} e - b^{3} d^{3}}{a b^{4} + b^{5} x} + \frac{e^{3} x^{2}}{2 b^{2}} - \frac{x \left (2 a e^{3} - 3 b d e^{2}\right )}{b^{3}} + \frac{3 e \left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11367, size = 154, normalized size = 2.05 \begin{align*} \frac{3 \,{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} + \frac{b^{2} x^{2} e^{3} + 6 \, b^{2} d x e^{2} - 4 \, a b x e^{3}}{2 \, b^{4}} - \frac{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}}{{\left (b x + a\right )} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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