Optimal. Leaf size=104 \[ \frac{2 e^3 (a+b x)^2 (b d-a e)}{b^5}+\frac{6 e^2 x (b d-a e)^2}{b^4}-\frac{(b d-a e)^4}{b^5 (a+b x)}+\frac{4 e (b d-a e)^3 \log (a+b x)}{b^5}+\frac{e^4 (a+b x)^3}{3 b^5} \]
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Rubi [A] time = 0.102178, antiderivative size = 104, 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} \[ \frac{2 e^3 (a+b x)^2 (b d-a e)}{b^5}+\frac{6 e^2 x (b d-a e)^2}{b^4}-\frac{(b d-a e)^4}{b^5 (a+b x)}+\frac{4 e (b d-a e)^3 \log (a+b x)}{b^5}+\frac{e^4 (a+b x)^3}{3 b^5} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^4}{(a+b x)^2} \, dx\\ &=\int \left (\frac{6 e^2 (b d-a e)^2}{b^4}+\frac{(b d-a e)^4}{b^4 (a+b x)^2}+\frac{4 e (b d-a e)^3}{b^4 (a+b x)}+\frac{4 e^3 (b d-a e) (a+b x)}{b^4}+\frac{e^4 (a+b x)^2}{b^4}\right ) \, dx\\ &=\frac{6 e^2 (b d-a e)^2 x}{b^4}-\frac{(b d-a e)^4}{b^5 (a+b x)}+\frac{2 e^3 (b d-a e) (a+b x)^2}{b^5}+\frac{e^4 (a+b x)^3}{3 b^5}+\frac{4 e (b d-a e)^3 \log (a+b x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0637961, size = 166, normalized size = 1.6 \[ \frac{6 a^2 b^2 e^2 \left (-3 d^2-4 d e x+e^2 x^2\right )+3 a^3 b e^3 (4 d+3 e x)-3 a^4 e^4-2 a b^3 e \left (-9 d^2 e x-6 d^3+9 d e^2 x^2+e^3 x^3\right )-12 e (a+b x) (a e-b d)^3 \log (a+b x)+b^4 \left (18 d^2 e^2 x^2-3 d^4+6 d e^3 x^3+e^4 x^4\right )}{3 b^5 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 230, normalized size = 2.2 \begin{align*}{\frac{{e}^{4}{x}^{3}}{3\,{b}^{2}}}-{\frac{{e}^{4}{x}^{2}a}{{b}^{3}}}+2\,{\frac{{e}^{3}{x}^{2}d}{{b}^{2}}}+3\,{\frac{{a}^{2}{e}^{4}x}{{b}^{4}}}-8\,{\frac{ad{e}^{3}x}{{b}^{3}}}+6\,{\frac{{d}^{2}{e}^{2}x}{{b}^{2}}}-4\,{\frac{{e}^{4}\ln \left ( bx+a \right ){a}^{3}}{{b}^{5}}}+12\,{\frac{{e}^{3}\ln \left ( bx+a \right ){a}^{2}d}{{b}^{4}}}-12\,{\frac{{e}^{2}\ln \left ( bx+a \right ) a{d}^{2}}{{b}^{3}}}+4\,{\frac{e\ln \left ( bx+a \right ){d}^{3}}{{b}^{2}}}-{\frac{{a}^{4}{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}+4\,{\frac{{a}^{3}d{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}-6\,{\frac{{d}^{2}{e}^{2}{a}^{2}}{{b}^{3} \left ( bx+a \right ) }}+4\,{\frac{a{d}^{3}e}{{b}^{2} \left ( bx+a \right ) }}-{\frac{{d}^{4}}{b \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1323, size = 248, normalized size = 2.38 \begin{align*} -\frac{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}}{b^{6} x + a b^{5}} + \frac{b^{2} e^{4} x^{3} + 3 \,{\left (2 \, b^{2} d e^{3} - a b e^{4}\right )} x^{2} + 3 \,{\left (6 \, b^{2} d^{2} e^{2} - 8 \, a b d e^{3} + 3 \, a^{2} e^{4}\right )} x}{3 \, b^{4}} + \frac{4 \,{\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} \log \left (b x + a\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75765, size = 540, normalized size = 5.19 \begin{align*} \frac{b^{4} e^{4} x^{4} - 3 \, b^{4} d^{4} + 12 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 2 \,{\left (3 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (3 \, b^{4} d^{2} e^{2} - 3 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 3 \,{\left (6 \, a b^{3} d^{2} e^{2} - 8 \, a^{2} b^{2} d e^{3} + 3 \, a^{3} b e^{4}\right )} x + 12 \,{\left (a b^{3} d^{3} e - 3 \, a^{2} b^{2} d^{2} e^{2} + 3 \, a^{3} b d e^{3} - a^{4} e^{4} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right )}{3 \,{\left (b^{6} x + a b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.978266, size = 151, normalized size = 1.45 \begin{align*} - \frac{a^{4} e^{4} - 4 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e + b^{4} d^{4}}{a b^{5} + b^{6} x} + \frac{e^{4} x^{3}}{3 b^{2}} - \frac{x^{2} \left (a e^{4} - 2 b d e^{3}\right )}{b^{3}} + \frac{x \left (3 a^{2} e^{4} - 8 a b d e^{3} + 6 b^{2} d^{2} e^{2}\right )}{b^{4}} - \frac{4 e \left (a e - b d\right )^{3} \log{\left (a + b x \right )}}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19414, size = 240, normalized size = 2.31 \begin{align*} \frac{4 \,{\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac{b^{4} x^{3} e^{4} + 6 \, b^{4} d x^{2} e^{3} + 18 \, b^{4} d^{2} x e^{2} - 3 \, a b^{3} x^{2} e^{4} - 24 \, a b^{3} d x e^{3} + 9 \, a^{2} b^{2} x e^{4}}{3 \, b^{6}} - \frac{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}}{{\left (b x + a\right )} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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