Optimal. Leaf size=75 \[ -\frac{b^2 x (2 b d-3 a e)}{e^3}+\frac{(b d-a e)^3}{e^4 (d+e x)}+\frac{3 b (b d-a e)^2 \log (d+e x)}{e^4}+\frac{b^3 x^2}{2 e^2} \]
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Rubi [A] time = 0.0617914, antiderivative size = 75, 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, 43} \[ -\frac{b^2 x (2 b d-3 a e)}{e^3}+\frac{(b d-a e)^3}{e^4 (d+e x)}+\frac{3 b (b d-a e)^2 \log (d+e x)}{e^4}+\frac{b^3 x^2}{2 e^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^2} \, dx &=\int \frac{(a+b x)^3}{(d+e x)^2} \, dx\\ &=\int \left (-\frac{b^2 (2 b d-3 a e)}{e^3}+\frac{b^3 x}{e^2}+\frac{(-b d+a e)^3}{e^3 (d+e x)^2}+\frac{3 b (b d-a e)^2}{e^3 (d+e x)}\right ) \, dx\\ &=-\frac{b^2 (2 b d-3 a e) x}{e^3}+\frac{b^3 x^2}{2 e^2}+\frac{(b d-a e)^3}{e^4 (d+e x)}+\frac{3 b (b d-a e)^2 \log (d+e x)}{e^4}\\ \end{align*}
Mathematica [A] time = 0.0390258, size = 114, normalized size = 1.52 \[ \frac{3 a^2 b d e^2-a^3 e^3-3 a b^2 d^2 e+b^3 d^3}{e^4 (d+e x)}+\frac{3 \left (a^2 b e^2-2 a b^2 d e+b^3 d^2\right ) \log (d+e x)}{e^4}-\frac{b^2 x (2 b d-3 a e)}{e^3}+\frac{b^3 x^2}{2 e^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 149, normalized size = 2. \begin{align*}{\frac{{b}^{3}{x}^{2}}{2\,{e}^{2}}}+3\,{\frac{{b}^{2}ax}{{e}^{2}}}-2\,{\frac{{b}^{3}dx}{{e}^{3}}}+3\,{\frac{b\ln \left ( ex+d \right ){a}^{2}}{{e}^{2}}}-6\,{\frac{{b}^{2}\ln \left ( ex+d \right ) ad}{{e}^{3}}}+3\,{\frac{{b}^{3}\ln \left ( ex+d \right ){d}^{2}}{{e}^{4}}}-{\frac{{a}^{3}}{e \left ( ex+d \right ) }}+3\,{\frac{{a}^{2}db}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{{b}^{2}{d}^{2}a}{{e}^{3} \left ( ex+d \right ) }}+{\frac{{b}^{3}{d}^{3}}{{e}^{4} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959432, size = 158, normalized size = 2.11 \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}}{e^{5} x + d e^{4}} + \frac{b^{3} e x^{2} - 2 \,{\left (2 \, b^{3} d - 3 \, a b^{2} e\right )} x}{2 \, e^{3}} + \frac{3 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b 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 [B] time = 1.52022, 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 (b^{3} d e^{2} - 2 \, a b^{2} e^{3}\right )} x^{2} - 2 \,{\left (2 \, b^{3} d^{2} e - 3 \, a b^{2} d e^{2}\right )} x + 6 \,{\left (b^{3} d^{3} - 2 \, a b^{2} d^{2} e + a^{2} b d e^{2} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x + d e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.712109, size = 100, normalized size = 1.33 \begin{align*} \frac{b^{3} x^{2}}{2 e^{2}} + \frac{3 b \left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{4}} - \frac{a^{3} e^{3} - 3 a^{2} b d e^{2} + 3 a b^{2} d^{2} e - b^{3} d^{3}}{d e^{4} + e^{5} x} + \frac{x \left (3 a b^{2} e - 2 b^{3} d\right )}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17438, size = 221, normalized size = 2.95 \begin{align*} \frac{1}{2} \,{\left (b^{3} - \frac{6 \,{\left (b^{3} d e - a b^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-4\right )} - 3 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} e^{\left (-4\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{b^{3} d^{3} e^{2}}{x e + d} - \frac{3 \, a b^{2} d^{2} e^{3}}{x e + d} + \frac{3 \, a^{2} b d e^{4}}{x e + d} - \frac{a^{3} e^{5}}{x e + d}\right )} e^{\left (-6\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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