Optimal. Leaf size=86 \[ -\frac{3 e^2 (b d-a e)}{b^4 (a+b x)}-\frac{3 e (b d-a e)^2}{2 b^4 (a+b x)^2}-\frac{(b d-a e)^3}{3 b^4 (a+b x)^3}+\frac{e^3 \log (a+b x)}{b^4} \]
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Rubi [A] time = 0.0595502, antiderivative size = 86, 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} \[ -\frac{3 e^2 (b d-a e)}{b^4 (a+b x)}-\frac{3 e (b d-a e)^2}{2 b^4 (a+b x)^2}-\frac{(b d-a e)^3}{3 b^4 (a+b x)^3}+\frac{e^3 \log (a+b x)}{b^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^3}{(a+b x)^4} \, dx\\ &=\int \left (\frac{(b d-a e)^3}{b^3 (a+b x)^4}+\frac{3 e (b d-a e)^2}{b^3 (a+b x)^3}+\frac{3 e^2 (b d-a e)}{b^3 (a+b x)^2}+\frac{e^3}{b^3 (a+b x)}\right ) \, dx\\ &=-\frac{(b d-a e)^3}{3 b^4 (a+b x)^3}-\frac{3 e (b d-a e)^2}{2 b^4 (a+b x)^2}-\frac{3 e^2 (b d-a e)}{b^4 (a+b x)}+\frac{e^3 \log (a+b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0414583, size = 80, normalized size = 0.93 \[ \frac{6 e^3 \log (a+b x)-\frac{(b d-a e) \left (11 a^2 e^2+a b e (5 d+27 e x)+b^2 \left (2 d^2+9 d e x+18 e^2 x^2\right )\right )}{(a+b x)^3}}{6 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 166, normalized size = 1.9 \begin{align*} -{\frac{3\,{a}^{2}{e}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}+3\,{\frac{ad{e}^{2}}{{b}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{d}^{2}e}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}+{\frac{{a}^{3}{e}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{d{e}^{2}{a}^{2}}{{b}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{{d}^{2}ea}{{b}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{{d}^{3}}{3\,b \left ( bx+a \right ) ^{3}}}+{\frac{{e}^{3}\ln \left ( bx+a \right ) }{{b}^{4}}}+3\,{\frac{a{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}-3\,{\frac{d{e}^{2}}{{b}^{3} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20806, size = 192, normalized size = 2.23 \begin{align*} -\frac{2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 18 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x}{6 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac{e^{3} \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 = 1.74703, size = 360, normalized size = 4.19 \begin{align*} -\frac{2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 18 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x - 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (b x + a\right )}{6 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.52887, size = 148, normalized size = 1.72 \begin{align*} \frac{11 a^{3} e^{3} - 6 a^{2} b d e^{2} - 3 a b^{2} d^{2} e - 2 b^{3} d^{3} + x^{2} \left (18 a b^{2} e^{3} - 18 b^{3} d e^{2}\right ) + x \left (27 a^{2} b e^{3} - 18 a b^{2} d e^{2} - 9 b^{3} d^{2} e\right )}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac{e^{3} \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 = 1.16156, size = 153, normalized size = 1.78 \begin{align*} \frac{e^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac{18 \,{\left (b^{2} d e^{2} - a b e^{3}\right )} x^{2} + 9 \,{\left (b^{2} d^{2} e + 2 \, a b d e^{2} - 3 \, a^{2} e^{3}\right )} x + \frac{2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3}}{b}}{6 \,{\left (b x + a\right )}^{3} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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