Optimal. Leaf size=98 \[ -\frac{b x (b d-a e)^3}{e^4}+\frac{(a+b x)^2 (b d-a e)^2}{2 e^3}-\frac{(a+b x)^3 (b d-a e)}{3 e^2}+\frac{(b d-a e)^4 \log (d+e x)}{e^5}+\frac{(a+b x)^4}{4 e} \]
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Rubi [A] time = 0.0423296, antiderivative size = 98, 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} \[ -\frac{b x (b d-a e)^3}{e^4}+\frac{(a+b x)^2 (b d-a e)^2}{2 e^3}-\frac{(a+b x)^3 (b d-a e)}{3 e^2}+\frac{(b d-a e)^4 \log (d+e x)}{e^5}+\frac{(a+b x)^4}{4 e} \]
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} \, dx &=\int \frac{(a+b x)^4}{d+e x} \, dx\\ &=\int \left (-\frac{b (b d-a e)^3}{e^4}+\frac{b (b d-a e)^2 (a+b x)}{e^3}-\frac{b (b d-a e) (a+b x)^2}{e^2}+\frac{b (a+b x)^3}{e}+\frac{(-b d+a e)^4}{e^4 (d+e x)}\right ) \, dx\\ &=-\frac{b (b d-a e)^3 x}{e^4}+\frac{(b d-a e)^2 (a+b x)^2}{2 e^3}-\frac{(b d-a e) (a+b x)^3}{3 e^2}+\frac{(a+b x)^4}{4 e}+\frac{(b d-a e)^4 \log (d+e x)}{e^5}\\ \end{align*}
Mathematica [A] time = 0.0452689, size = 115, normalized size = 1.17 \[ \frac{b e x \left (36 a^2 b e^2 (e x-2 d)+48 a^3 e^3+8 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^3 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )+12 (b d-a e)^4 \log (d+e x)}{12 e^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 209, normalized size = 2.1 \begin{align*}{\frac{{b}^{4}{x}^{4}}{4\,e}}+{\frac{4\,a{b}^{3}{x}^{3}}{3\,e}}-{\frac{{b}^{4}{x}^{3}d}{3\,{e}^{2}}}+3\,{\frac{{b}^{2}{x}^{2}{a}^{2}}{e}}-2\,{\frac{{b}^{3}{x}^{2}ad}{{e}^{2}}}+{\frac{{b}^{4}{x}^{2}{d}^{2}}{2\,{e}^{3}}}+4\,{\frac{x{a}^{3}b}{e}}-6\,{\frac{{b}^{2}{a}^{2}dx}{{e}^{2}}}+4\,{\frac{{d}^{2}a{b}^{3}x}{{e}^{3}}}-{\frac{{b}^{4}{d}^{3}x}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ){a}^{4}}{e}}-4\,{\frac{\ln \left ( ex+d \right ){a}^{3}bd}{{e}^{2}}}+6\,{\frac{\ln \left ( ex+d \right ){d}^{2}{b}^{2}{a}^{2}}{{e}^{3}}}-4\,{\frac{\ln \left ( ex+d \right ){d}^{3}a{b}^{3}}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ){b}^{4}{d}^{4}}{{e}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15193, size = 239, normalized size = 2.44 \begin{align*} \frac{3 \, b^{4} e^{3} x^{4} - 4 \,{\left (b^{4} d e^{2} - 4 \, a b^{3} e^{3}\right )} x^{3} + 6 \,{\left (b^{4} d^{2} e - 4 \, a b^{3} d e^{2} + 6 \, a^{2} b^{2} e^{3}\right )} x^{2} - 12 \,{\left (b^{4} d^{3} - 4 \, a b^{3} d^{2} e + 6 \, a^{2} b^{2} d e^{2} - 4 \, a^{3} b e^{3}\right )} x}{12 \, e^{4}} + \frac{{\left (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}\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 [A] time = 1.78922, size = 369, normalized size = 3.77 \begin{align*} \frac{3 \, b^{4} e^{4} x^{4} - 4 \,{\left (b^{4} d e^{3} - 4 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} - 12 \,{\left (b^{4} d^{3} e - 4 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 4 \, a^{3} b e^{4}\right )} x + 12 \,{\left (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}\right )} \log \left (e x + d\right )}{12 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.706945, size = 134, normalized size = 1.37 \begin{align*} \frac{b^{4} x^{4}}{4 e} + \frac{x^{3} \left (4 a b^{3} e - b^{4} d\right )}{3 e^{2}} + \frac{x^{2} \left (6 a^{2} b^{2} e^{2} - 4 a b^{3} d e + b^{4} d^{2}\right )}{2 e^{3}} + \frac{x \left (4 a^{3} b e^{3} - 6 a^{2} b^{2} d e^{2} + 4 a b^{3} d^{2} e - b^{4} d^{3}\right )}{e^{4}} + \frac{\left (a e - b d\right )^{4} \log{\left (d + e x \right )}}{e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15053, size = 238, normalized size = 2.43 \begin{align*}{\left (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}\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (3 \, b^{4} x^{4} e^{3} - 4 \, b^{4} d x^{3} e^{2} + 6 \, b^{4} d^{2} x^{2} e - 12 \, b^{4} d^{3} x + 16 \, a b^{3} x^{3} e^{3} - 24 \, a b^{3} d x^{2} e^{2} + 48 \, a b^{3} d^{2} x e + 36 \, a^{2} b^{2} x^{2} e^{3} - 72 \, a^{2} b^{2} d x e^{2} + 48 \, a^{3} b x e^{3}\right )} e^{\left (-4\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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