Optimal. Leaf size=86 \[ \frac{3 b^2 (b d-a e)}{e^4 (d+e x)}-\frac{3 b (b d-a e)^2}{2 e^4 (d+e x)^2}+\frac{(b d-a e)^3}{3 e^4 (d+e x)^3}+\frac{b^3 \log (d+e x)}{e^4} \]
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Rubi [A] time = 0.0572354, antiderivative size = 86, 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} \[ \frac{3 b^2 (b d-a e)}{e^4 (d+e x)}-\frac{3 b (b d-a e)^2}{2 e^4 (d+e x)^2}+\frac{(b d-a e)^3}{3 e^4 (d+e x)^3}+\frac{b^3 \log (d+e x)}{e^4} \]
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)^4} \, dx &=\int \frac{(a+b x)^3}{(d+e x)^4} \, dx\\ &=\int \left (\frac{(-b d+a e)^3}{e^3 (d+e x)^4}+\frac{3 b (b d-a e)^2}{e^3 (d+e x)^3}-\frac{3 b^2 (b d-a e)}{e^3 (d+e x)^2}+\frac{b^3}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{(b d-a e)^3}{3 e^4 (d+e x)^3}-\frac{3 b (b d-a e)^2}{2 e^4 (d+e x)^2}+\frac{3 b^2 (b d-a e)}{e^4 (d+e x)}+\frac{b^3 \log (d+e x)}{e^4}\\ \end{align*}
Mathematica [A] time = 0.0436969, size = 79, normalized size = 0.92 \[ \frac{\frac{(b d-a e) \left (2 a^2 e^2+a b e (5 d+9 e x)+b^2 \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )}{(d+e x)^3}+6 b^3 \log (d+e x)}{6 e^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 166, normalized size = 1.9 \begin{align*} -{\frac{3\,{a}^{2}b}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{b}^{2}da}{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{b}^{3}{d}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{3}\ln \left ( ex+d \right ) }{{e}^{4}}}-3\,{\frac{{b}^{2}a}{{e}^{3} \left ( ex+d \right ) }}+3\,{\frac{{b}^{3}d}{{e}^{4} \left ( ex+d \right ) }}-{\frac{{a}^{3}}{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 ) ^{3}}}+{\frac{{b}^{3}{d}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987621, size = 193, normalized size = 2.24 \begin{align*} \frac{11 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} + 18 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (3 \, b^{3} d^{2} e - 2 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} + \frac{b^{3} \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 = 1.45371, size = 359, normalized size = 4.17 \begin{align*} \frac{11 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} + 18 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (3 \, b^{3} d^{2} e - 2 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, b^{3} d e^{2} x^{2} + 3 \, b^{3} d^{2} e x + b^{3} d^{3}\right )} \log \left (e x + d\right )}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.33141, size = 148, normalized size = 1.72 \begin{align*} \frac{b^{3} \log{\left (d + e x \right )}}{e^{4}} - \frac{2 a^{3} e^{3} + 3 a^{2} b d e^{2} + 6 a b^{2} d^{2} e - 11 b^{3} d^{3} + x^{2} \left (18 a b^{2} e^{3} - 18 b^{3} d e^{2}\right ) + x \left (9 a^{2} b e^{3} + 18 a b^{2} d e^{2} - 27 b^{3} d^{2} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18953, size = 158, normalized size = 1.84 \begin{align*} b^{3} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (18 \,{\left (b^{3} d e - a b^{2} e^{2}\right )} x^{2} + 9 \,{\left (3 \, b^{3} d^{2} - 2 \, a b^{2} d e - a^{2} b e^{2}\right )} x +{\left (11 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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