Optimal. Leaf size=78 \[ \frac{3 e^2 (b d-a e) \log (a+b x)}{b^4}-\frac{3 e (b d-a e)^2}{b^4 (a+b x)}-\frac{(b d-a e)^3}{2 b^4 (a+b x)^2}+\frac{e^3 x}{b^3} \]
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Rubi [A] time = 0.0639198, antiderivative size = 78, 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} \[ \frac{3 e^2 (b d-a e) \log (a+b x)}{b^4}-\frac{3 e (b d-a e)^2}{b^4 (a+b x)}-\frac{(b d-a e)^3}{2 b^4 (a+b x)^2}+\frac{e^3 x}{b^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x) (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)^3} \, dx\\ &=\int \left (\frac{e^3}{b^3}+\frac{(b d-a e)^3}{b^3 (a+b x)^3}+\frac{3 e (b d-a e)^2}{b^3 (a+b x)^2}+\frac{3 e^2 (b d-a e)}{b^3 (a+b x)}\right ) \, dx\\ &=\frac{e^3 x}{b^3}-\frac{(b d-a e)^3}{2 b^4 (a+b x)^2}-\frac{3 e (b d-a e)^2}{b^4 (a+b x)}+\frac{3 e^2 (b d-a e) \log (a+b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.044099, size = 114, normalized size = 1.46 \[ \frac{a^2 b e^2 (9 d-4 e x)-5 a^3 e^3+a b^2 e \left (-3 d^2+12 d e x+4 e^2 x^2\right )-6 e^2 (a+b x)^2 (a e-b d) \log (a+b x)+b^3 \left (-\left (6 d^2 e x+d^3-2 e^3 x^3\right )\right )}{2 b^4 (a+b x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 160, normalized size = 2.1 \begin{align*}{\frac{{e}^{3}x}{{b}^{3}}}-3\,{\frac{{a}^{2}{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}+6\,{\frac{a{e}^{2}d}{{b}^{3} \left ( bx+a \right ) }}-3\,{\frac{e{d}^{2}}{{b}^{2} \left ( bx+a \right ) }}+{\frac{{a}^{3}{e}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{a}^{2}d{e}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{3\,a{d}^{2}e}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{{d}^{3}}{2\,b \left ( bx+a \right ) ^{2}}}-3\,{\frac{{e}^{3}\ln \left ( bx+a \right ) a}{{b}^{4}}}+3\,{\frac{{e}^{2}\ln \left ( bx+a \right ) d}{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975972, size = 169, normalized size = 2.17 \begin{align*} \frac{e^{3} x}{b^{3}} - \frac{b^{3} d^{3} + 3 \, a b^{2} d^{2} e - 9 \, a^{2} b d e^{2} + 5 \, a^{3} e^{3} + 6 \,{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac{3 \,{\left (b d e^{2} - a 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 = 1.47337, size = 375, normalized size = 4.81 \begin{align*} \frac{2 \, b^{3} e^{3} x^{3} + 4 \, a b^{2} e^{3} x^{2} - b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3} - 2 \,{\left (3 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 2 \, a^{2} b e^{3}\right )} x + 6 \,{\left (a^{2} b d e^{2} - a^{3} e^{3} +{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.0577, size = 128, normalized size = 1.64 \begin{align*} - \frac{5 a^{3} e^{3} - 9 a^{2} b d e^{2} + 3 a b^{2} d^{2} e + b^{3} d^{3} + x \left (6 a^{2} b e^{3} - 12 a b^{2} d e^{2} + 6 b^{3} d^{2} e\right )}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} + \frac{e^{3} x}{b^{3}} - \frac{3 e^{2} \left (a e - b d\right ) \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 = 1.12623, size = 144, normalized size = 1.85 \begin{align*} \frac{x e^{3}}{b^{3}} + \frac{3 \,{\left (b d e^{2} - a e^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac{b^{3} d^{3} + 3 \, a b^{2} d^{2} e - 9 \, a^{2} b d e^{2} + 5 \, a^{3} e^{3} + 6 \,{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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