Optimal. Leaf size=103 \[ -\frac{6 b^2 (b d-a e)^2}{e^5 (d+e x)}-\frac{4 b^3 (b d-a e) \log (d+e x)}{e^5}+\frac{2 b (b d-a e)^3}{e^5 (d+e x)^2}-\frac{(b d-a e)^4}{3 e^5 (d+e x)^3}+\frac{b^4 x}{e^4} \]
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Rubi [A] time = 0.0798661, antiderivative size = 103, 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} \[ -\frac{6 b^2 (b d-a e)^2}{e^5 (d+e x)}-\frac{4 b^3 (b d-a e) \log (d+e x)}{e^5}+\frac{2 b (b d-a e)^3}{e^5 (d+e x)^2}-\frac{(b d-a e)^4}{3 e^5 (d+e x)^3}+\frac{b^4 x}{e^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx &=\int \frac{(a+b x)^4}{(d+e x)^4} \, dx\\ &=\int \left (\frac{b^4}{e^4}+\frac{(-b d+a e)^4}{e^4 (d+e x)^4}-\frac{4 b (b d-a e)^3}{e^4 (d+e x)^3}+\frac{6 b^2 (b d-a e)^2}{e^4 (d+e x)^2}-\frac{4 b^3 (b d-a e)}{e^4 (d+e x)}\right ) \, dx\\ &=\frac{b^4 x}{e^4}-\frac{(b d-a e)^4}{3 e^5 (d+e x)^3}+\frac{2 b (b d-a e)^3}{e^5 (d+e x)^2}-\frac{6 b^2 (b d-a e)^2}{e^5 (d+e x)}-\frac{4 b^3 (b d-a e) \log (d+e x)}{e^5}\\ \end{align*}
Mathematica [A] time = 0.0746552, size = 163, normalized size = 1.58 \[ -\frac{6 a^2 b^2 e^2 \left (d^2+3 d e x+3 e^2 x^2\right )+2 a^3 b e^3 (d+3 e x)+a^4 e^4-2 a b^3 d e \left (11 d^2+27 d e x+18 e^2 x^2\right )+12 b^3 (d+e x)^3 (b d-a e) \log (d+e x)+b^4 \left (9 d^2 e^2 x^2+27 d^3 e x+13 d^4-9 d e^3 x^3-3 e^4 x^4\right )}{3 e^5 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 255, normalized size = 2.5 \begin{align*}{\frac{{b}^{4}x}{{e}^{4}}}-{\frac{{a}^{4}}{3\,e \left ( ex+d \right ) ^{3}}}+{\frac{4\,d{a}^{3}b}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-2\,{\frac{{b}^{2}{d}^{2}{a}^{2}}{{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{4\,{d}^{3}a{b}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{4}{d}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-2\,{\frac{{a}^{3}b}{{e}^{2} \left ( ex+d \right ) ^{2}}}+6\,{\frac{{b}^{2}{a}^{2}d}{{e}^{3} \left ( ex+d \right ) ^{2}}}-6\,{\frac{{d}^{2}a{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{b}^{4}{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+4\,{\frac{{b}^{3}\ln \left ( ex+d \right ) a}{{e}^{4}}}-4\,{\frac{{b}^{4}\ln \left ( ex+d \right ) d}{{e}^{5}}}-6\,{\frac{{b}^{2}{a}^{2}}{{e}^{3} \left ( ex+d \right ) }}+12\,{\frac{ad{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}-6\,{\frac{{b}^{4}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04734, size = 271, normalized size = 2.63 \begin{align*} \frac{b^{4} x}{e^{4}} - \frac{13 \, b^{4} d^{4} - 22 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} b d e^{3} + a^{4} e^{4} + 18 \,{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \,{\left (5 \, b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} - \frac{4 \,{\left (b^{4} d - a b^{3} e\right )} \log \left (e x + d\right )}{e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.03765, size = 581, normalized size = 5.64 \begin{align*} \frac{3 \, b^{4} e^{4} x^{4} + 9 \, b^{4} d e^{3} x^{3} - 13 \, b^{4} d^{4} + 22 \, a b^{3} d^{3} e - 6 \, a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} - a^{4} e^{4} - 9 \,{\left (b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + 2 \, a^{2} b^{2} e^{4}\right )} x^{2} - 3 \,{\left (9 \, b^{4} d^{3} e - 18 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 2 \, a^{3} b e^{4}\right )} x - 12 \,{\left (b^{4} d^{4} - a b^{3} d^{3} e +{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (b^{4} d^{2} e^{2} - a b^{3} d e^{3}\right )} x^{2} + 3 \,{\left (b^{4} d^{3} e - a b^{3} d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.67267, size = 209, normalized size = 2.03 \begin{align*} \frac{b^{4} x}{e^{4}} + \frac{4 b^{3} \left (a e - b d\right ) \log{\left (d + e x \right )}}{e^{5}} - \frac{a^{4} e^{4} + 2 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} - 22 a b^{3} d^{3} e + 13 b^{4} d^{4} + x^{2} \left (18 a^{2} b^{2} e^{4} - 36 a b^{3} d e^{3} + 18 b^{4} d^{2} e^{2}\right ) + x \left (6 a^{3} b e^{4} + 18 a^{2} b^{2} d e^{3} - 54 a b^{3} d^{2} e^{2} + 30 b^{4} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17378, size = 230, normalized size = 2.23 \begin{align*} b^{4} x e^{\left (-4\right )} - 4 \,{\left (b^{4} d - a b^{3} e\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (13 \, b^{4} d^{4} - 22 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} b d e^{3} + a^{4} e^{4} + 18 \,{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \,{\left (5 \, b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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