Optimal. Leaf size=129 \[ \frac{e^4 x (5 b d-4 a e)}{b^5}-\frac{10 e^2 (b d-a e)^3}{b^6 (a+b x)}+\frac{10 e^3 (b d-a e)^2 \log (a+b x)}{b^6}-\frac{5 e (b d-a e)^4}{2 b^6 (a+b x)^2}-\frac{(b d-a e)^5}{3 b^6 (a+b x)^3}+\frac{e^5 x^2}{2 b^4} \]
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Rubi [A] time = 0.124658, antiderivative size = 129, 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} \[ \frac{e^4 x (5 b d-4 a e)}{b^5}-\frac{10 e^2 (b d-a e)^3}{b^6 (a+b x)}+\frac{10 e^3 (b d-a e)^2 \log (a+b x)}{b^6}-\frac{5 e (b d-a e)^4}{2 b^6 (a+b x)^2}-\frac{(b d-a e)^5}{3 b^6 (a+b x)^3}+\frac{e^5 x^2}{2 b^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^5}{(a+b x)^4} \, dx\\ &=\int \left (\frac{e^4 (5 b d-4 a e)}{b^5}+\frac{e^5 x}{b^4}+\frac{(b d-a e)^5}{b^5 (a+b x)^4}+\frac{5 e (b d-a e)^4}{b^5 (a+b x)^3}+\frac{10 e^2 (b d-a e)^3}{b^5 (a+b x)^2}+\frac{10 e^3 (b d-a e)^2}{b^5 (a+b x)}\right ) \, dx\\ &=\frac{e^4 (5 b d-4 a e) x}{b^5}+\frac{e^5 x^2}{2 b^4}-\frac{(b d-a e)^5}{3 b^6 (a+b x)^3}-\frac{5 e (b d-a e)^4}{2 b^6 (a+b x)^2}-\frac{10 e^2 (b d-a e)^3}{b^6 (a+b x)}+\frac{10 e^3 (b d-a e)^2 \log (a+b x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0818163, size = 228, normalized size = 1.77 \[ \frac{-a^2 b^3 e^2 \left (-270 d^2 e x+20 d^3+90 d e^2 x^2+63 e^3 x^3\right )+a^3 b^2 e^3 \left (110 d^2-270 d e x-9 e^2 x^2\right )+a^4 b e^4 (81 e x-130 d)+47 a^5 e^5-5 a b^4 e \left (-36 d^2 e^2 x^2+12 d^3 e x+d^4-18 d e^3 x^3+3 e^4 x^4\right )+60 e^3 (a+b x)^3 (b d-a e)^2 \log (a+b x)+b^5 \left (-60 d^3 e^2 x^2-15 d^4 e x-2 d^5+30 d e^4 x^4+3 e^5 x^5\right )}{6 b^6 (a+b x)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 361, normalized size = 2.8 \begin{align*}{\frac{{e}^{5}{x}^{2}}{2\,{b}^{4}}}-4\,{\frac{a{e}^{5}x}{{b}^{5}}}+5\,{\frac{{e}^{4}xd}{{b}^{4}}}-{\frac{5\,{a}^{4}{e}^{5}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}+10\,{\frac{{e}^{4}{a}^{3}d}{{b}^{5} \left ( bx+a \right ) ^{2}}}-15\,{\frac{{e}^{3}{d}^{2}{a}^{2}}{{b}^{4} \left ( bx+a \right ) ^{2}}}+10\,{\frac{a{e}^{2}{d}^{3}}{{b}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{5\,e{d}^{4}}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}+{\frac{{a}^{5}{e}^{5}}{3\,{b}^{6} \left ( bx+a \right ) ^{3}}}-{\frac{5\,{a}^{4}d{e}^{4}}{3\,{b}^{5} \left ( bx+a \right ) ^{3}}}+{\frac{10\,{a}^{3}{d}^{2}{e}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{10\,{a}^{2}{d}^{3}{e}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{5\,a{d}^{4}e}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{{d}^{5}}{3\,b \left ( bx+a \right ) ^{3}}}+10\,{\frac{{e}^{5}\ln \left ( bx+a \right ){a}^{2}}{{b}^{6}}}-20\,{\frac{{e}^{4}\ln \left ( bx+a \right ) ad}{{b}^{5}}}+10\,{\frac{{e}^{3}\ln \left ( bx+a \right ){d}^{2}}{{b}^{4}}}+10\,{\frac{{a}^{3}{e}^{5}}{{b}^{6} \left ( bx+a \right ) }}-30\,{\frac{{a}^{2}d{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}+30\,{\frac{a{e}^{3}{d}^{2}}{{b}^{4} \left ( bx+a \right ) }}-10\,{\frac{{e}^{2}{d}^{3}}{{b}^{3} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14849, size = 379, normalized size = 2.94 \begin{align*} -\frac{2 \, b^{5} d^{5} + 5 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} - 110 \, a^{3} b^{2} d^{2} e^{3} + 130 \, a^{4} b d e^{4} - 47 \, a^{5} e^{5} + 60 \,{\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 15 \,{\left (b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} - 18 \, a^{2} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{2} d e^{4} - 7 \, a^{4} b e^{5}\right )} x}{6 \,{\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} + \frac{b e^{5} x^{2} + 2 \,{\left (5 \, b d e^{4} - 4 \, a e^{5}\right )} x}{2 \, b^{5}} + \frac{10 \,{\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} \log \left (b x + a\right )}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.66791, size = 867, normalized size = 6.72 \begin{align*} \frac{3 \, b^{5} e^{5} x^{5} - 2 \, b^{5} d^{5} - 5 \, a b^{4} d^{4} e - 20 \, a^{2} b^{3} d^{3} e^{2} + 110 \, a^{3} b^{2} d^{2} e^{3} - 130 \, a^{4} b d e^{4} + 47 \, a^{5} e^{5} + 15 \,{\left (2 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 9 \,{\left (10 \, a b^{4} d e^{4} - 7 \, a^{2} b^{3} e^{5}\right )} x^{3} - 3 \,{\left (20 \, b^{5} d^{3} e^{2} - 60 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} + 3 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \,{\left (5 \, b^{5} d^{4} e + 20 \, a b^{4} d^{3} e^{2} - 90 \, a^{2} b^{3} d^{2} e^{3} + 90 \, a^{3} b^{2} d e^{4} - 27 \, a^{4} b e^{5}\right )} x + 60 \,{\left (a^{3} b^{2} d^{2} e^{3} - 2 \, a^{4} b d e^{4} + a^{5} e^{5} +{\left (b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 3 \,{\left (a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + 3 \,{\left (a^{2} b^{3} d^{2} e^{3} - 2 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.74392, size = 282, normalized size = 2.19 \begin{align*} \frac{47 a^{5} e^{5} - 130 a^{4} b d e^{4} + 110 a^{3} b^{2} d^{2} e^{3} - 20 a^{2} b^{3} d^{3} e^{2} - 5 a b^{4} d^{4} e - 2 b^{5} d^{5} + x^{2} \left (60 a^{3} b^{2} e^{5} - 180 a^{2} b^{3} d e^{4} + 180 a b^{4} d^{2} e^{3} - 60 b^{5} d^{3} e^{2}\right ) + x \left (105 a^{4} b e^{5} - 300 a^{3} b^{2} d e^{4} + 270 a^{2} b^{3} d^{2} e^{3} - 60 a b^{4} d^{3} e^{2} - 15 b^{5} d^{4} e\right )}{6 a^{3} b^{6} + 18 a^{2} b^{7} x + 18 a b^{8} x^{2} + 6 b^{9} x^{3}} + \frac{e^{5} x^{2}}{2 b^{4}} - \frac{x \left (4 a e^{5} - 5 b d e^{4}\right )}{b^{5}} + \frac{10 e^{3} \left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12406, size = 332, normalized size = 2.57 \begin{align*} \frac{10 \,{\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{b^{4} x^{2} e^{5} + 10 \, b^{4} d x e^{4} - 8 \, a b^{3} x e^{5}}{2 \, b^{8}} - \frac{2 \, b^{5} d^{5} + 5 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} - 110 \, a^{3} b^{2} d^{2} e^{3} + 130 \, a^{4} b d e^{4} - 47 \, a^{5} e^{5} + 60 \,{\left (b^{5} d^{3} e^{2} - 3 \, a b^{4} d^{2} e^{3} + 3 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 15 \,{\left (b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} - 18 \, a^{2} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{2} d e^{4} - 7 \, a^{4} b e^{5}\right )} x}{6 \,{\left (b x + a\right )}^{3} b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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