Optimal. Leaf size=131 \[ \frac{5 e^4 (a+b x)^3 (b d-a e)}{3 b^6}+\frac{5 e^3 (a+b x)^2 (b d-a e)^2}{b^6}+\frac{10 e^2 x (b d-a e)^3}{b^5}-\frac{(b d-a e)^5}{b^6 (a+b x)}+\frac{5 e (b d-a e)^4 \log (a+b x)}{b^6}+\frac{e^5 (a+b x)^4}{4 b^6} \]
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Rubi [A] time = 0.146367, antiderivative size = 131, 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} \[ \frac{5 e^4 (a+b x)^3 (b d-a e)}{3 b^6}+\frac{5 e^3 (a+b x)^2 (b d-a e)^2}{b^6}+\frac{10 e^2 x (b d-a e)^3}{b^5}-\frac{(b d-a e)^5}{b^6 (a+b x)}+\frac{5 e (b d-a e)^4 \log (a+b x)}{b^6}+\frac{e^5 (a+b x)^4}{4 b^6} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^5}{(a+b x)^2} \, dx\\ &=\int \left (\frac{10 e^2 (b d-a e)^3}{b^5}+\frac{(b d-a e)^5}{b^5 (a+b x)^2}+\frac{5 e (b d-a e)^4}{b^5 (a+b x)}+\frac{10 e^3 (b d-a e)^2 (a+b x)}{b^5}+\frac{5 e^4 (b d-a e) (a+b x)^2}{b^5}+\frac{e^5 (a+b x)^3}{b^5}\right ) \, dx\\ &=\frac{10 e^2 (b d-a e)^3 x}{b^5}-\frac{(b d-a e)^5}{b^6 (a+b x)}+\frac{5 e^3 (b d-a e)^2 (a+b x)^2}{b^6}+\frac{5 e^4 (b d-a e) (a+b x)^3}{3 b^6}+\frac{e^5 (a+b x)^4}{4 b^6}+\frac{5 e (b d-a e)^4 \log (a+b x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0783108, size = 230, normalized size = 1.76 \[ \frac{10 a^2 b^3 e^2 \left (-24 d^2 e x-12 d^3+12 d e^2 x^2+e^3 x^3\right )+30 a^3 b^2 e^3 \left (4 d^2+6 d e x-e^2 x^2\right )-12 a^4 b e^4 (5 d+4 e x)+12 a^5 e^5-5 a b^4 e \left (36 d^2 e^2 x^2-24 d^3 e x-12 d^4+8 d e^3 x^3+e^4 x^4\right )+60 e (a+b x) (b d-a e)^4 \log (a+b x)+b^5 \left (120 d^3 e^2 x^2+60 d^2 e^3 x^3-12 d^5+20 d e^4 x^4+3 e^5 x^5\right )}{12 b^6 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 326, normalized size = 2.5 \begin{align*}{\frac{{e}^{5}{x}^{4}}{4\,{b}^{2}}}-{\frac{2\,{e}^{5}{x}^{3}a}{3\,{b}^{3}}}+{\frac{5\,{e}^{4}{x}^{3}d}{3\,{b}^{2}}}+{\frac{3\,{e}^{5}{x}^{2}{a}^{2}}{2\,{b}^{4}}}-5\,{\frac{{e}^{4}{x}^{2}ad}{{b}^{3}}}+5\,{\frac{{e}^{3}{x}^{2}{d}^{2}}{{b}^{2}}}-4\,{\frac{{a}^{3}{e}^{5}x}{{b}^{5}}}+15\,{\frac{{a}^{2}d{e}^{4}x}{{b}^{4}}}-20\,{\frac{a{d}^{2}{e}^{3}x}{{b}^{3}}}+10\,{\frac{{d}^{3}{e}^{2}x}{{b}^{2}}}+5\,{\frac{{e}^{5}\ln \left ( bx+a \right ){a}^{4}}{{b}^{6}}}-20\,{\frac{{e}^{4}\ln \left ( bx+a \right ){a}^{3}d}{{b}^{5}}}+30\,{\frac{{e}^{3}\ln \left ( bx+a \right ){d}^{2}{a}^{2}}{{b}^{4}}}-20\,{\frac{{e}^{2}\ln \left ( bx+a \right ) a{d}^{3}}{{b}^{3}}}+5\,{\frac{e\ln \left ( bx+a \right ){d}^{4}}{{b}^{2}}}+{\frac{{a}^{5}{e}^{5}}{{b}^{6} \left ( bx+a \right ) }}-5\,{\frac{{a}^{4}d{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}+10\,{\frac{{a}^{3}{d}^{2}{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}-10\,{\frac{{a}^{2}{d}^{3}{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}+5\,{\frac{a{d}^{4}e}{{b}^{2} \left ( bx+a \right ) }}-{\frac{{d}^{5}}{b \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.1607, size = 358, normalized size = 2.73 \begin{align*} -\frac{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}}{b^{7} x + a b^{6}} + \frac{3 \, b^{3} e^{5} x^{4} + 4 \,{\left (5 \, b^{3} d e^{4} - 2 \, a b^{2} e^{5}\right )} x^{3} + 6 \,{\left (10 \, b^{3} d^{2} e^{3} - 10 \, a b^{2} d e^{4} + 3 \, a^{2} b e^{5}\right )} x^{2} + 12 \,{\left (10 \, b^{3} d^{3} e^{2} - 20 \, a b^{2} d^{2} e^{3} + 15 \, a^{2} b d e^{4} - 4 \, a^{3} e^{5}\right )} x}{12 \, b^{5}} + \frac{5 \,{\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} 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 = 1.81274, size = 767, normalized size = 5.85 \begin{align*} \frac{3 \, b^{5} e^{5} x^{5} - 12 \, b^{5} d^{5} + 60 \, a b^{4} d^{4} e - 120 \, a^{2} b^{3} d^{3} e^{2} + 120 \, a^{3} b^{2} d^{2} e^{3} - 60 \, a^{4} b d e^{4} + 12 \, a^{5} e^{5} + 5 \,{\left (4 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (6 \, b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 30 \,{\left (4 \, b^{5} d^{3} e^{2} - 6 \, a b^{4} d^{2} e^{3} + 4 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 12 \,{\left (10 \, a b^{4} d^{3} e^{2} - 20 \, a^{2} b^{3} d^{2} e^{3} + 15 \, a^{3} b^{2} d e^{4} - 4 \, a^{4} b e^{5}\right )} x + 60 \,{\left (a b^{4} d^{4} e - 4 \, a^{2} b^{3} d^{3} e^{2} + 6 \, a^{3} b^{2} d^{2} e^{3} - 4 \, a^{4} b d e^{4} + a^{5} e^{5} +{\left (b^{5} d^{4} e - 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (b x + a\right )}{12 \,{\left (b^{7} x + a b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.20108, size = 224, normalized size = 1.71 \begin{align*} \frac{a^{5} e^{5} - 5 a^{4} b d e^{4} + 10 a^{3} b^{2} d^{2} e^{3} - 10 a^{2} b^{3} d^{3} e^{2} + 5 a b^{4} d^{4} e - b^{5} d^{5}}{a b^{6} + b^{7} x} + \frac{e^{5} x^{4}}{4 b^{2}} - \frac{x^{3} \left (2 a e^{5} - 5 b d e^{4}\right )}{3 b^{3}} + \frac{x^{2} \left (3 a^{2} e^{5} - 10 a b d e^{4} + 10 b^{2} d^{2} e^{3}\right )}{2 b^{4}} - \frac{x \left (4 a^{3} e^{5} - 15 a^{2} b d e^{4} + 20 a b^{2} d^{2} e^{3} - 10 b^{3} d^{3} e^{2}\right )}{b^{5}} + \frac{5 e \left (a e - b d\right )^{4} \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 [B] time = 1.15016, size = 347, normalized size = 2.65 \begin{align*} \frac{5 \,{\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}}{{\left (b x + a\right )} b^{6}} + \frac{3 \, b^{6} x^{4} e^{5} + 20 \, b^{6} d x^{3} e^{4} + 60 \, b^{6} d^{2} x^{2} e^{3} + 120 \, b^{6} d^{3} x e^{2} - 8 \, a b^{5} x^{3} e^{5} - 60 \, a b^{5} d x^{2} e^{4} - 240 \, a b^{5} d^{2} x e^{3} + 18 \, a^{2} b^{4} x^{2} e^{5} + 180 \, a^{2} b^{4} d x e^{4} - 48 \, a^{3} b^{3} x e^{5}}{12 \, b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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