Optimal. Leaf size=138 \[ -\frac{5 e^4 (b d-a e)}{b^6 (a+b x)}-\frac{5 e^3 (b d-a e)^2}{b^6 (a+b x)^2}-\frac{10 e^2 (b d-a e)^3}{3 b^6 (a+b x)^3}-\frac{5 e (b d-a e)^4}{4 b^6 (a+b x)^4}-\frac{(b d-a e)^5}{5 b^6 (a+b x)^5}+\frac{e^5 \log (a+b x)}{b^6} \]
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Rubi [A] time = 0.116599, antiderivative size = 138, 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} \[ -\frac{5 e^4 (b d-a e)}{b^6 (a+b x)}-\frac{5 e^3 (b d-a e)^2}{b^6 (a+b x)^2}-\frac{10 e^2 (b d-a e)^3}{3 b^6 (a+b x)^3}-\frac{5 e (b d-a e)^4}{4 b^6 (a+b x)^4}-\frac{(b d-a e)^5}{5 b^6 (a+b x)^5}+\frac{e^5 \log (a+b x)}{b^6} \]
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 )^3} \, dx &=\int \frac{(d+e x)^5}{(a+b x)^6} \, dx\\ &=\int \left (\frac{(b d-a e)^5}{b^5 (a+b x)^6}+\frac{5 e (b d-a e)^4}{b^5 (a+b x)^5}+\frac{10 e^2 (b d-a e)^3}{b^5 (a+b x)^4}+\frac{10 e^3 (b d-a e)^2}{b^5 (a+b x)^3}+\frac{5 e^4 (b d-a e)}{b^5 (a+b x)^2}+\frac{e^5}{b^5 (a+b x)}\right ) \, dx\\ &=-\frac{(b d-a e)^5}{5 b^6 (a+b x)^5}-\frac{5 e (b d-a e)^4}{4 b^6 (a+b x)^4}-\frac{10 e^2 (b d-a e)^3}{3 b^6 (a+b x)^3}-\frac{5 e^3 (b d-a e)^2}{b^6 (a+b x)^2}-\frac{5 e^4 (b d-a e)}{b^6 (a+b x)}+\frac{e^5 \log (a+b x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0886145, size = 171, normalized size = 1.24 \[ \frac{e^5 \log (a+b x)}{b^6}-\frac{(b d-a e) \left (a^2 b^2 e^2 \left (47 d^2+325 d e x+1100 e^2 x^2\right )+a^3 b e^3 (77 d+625 e x)+137 a^4 e^4+a b^3 e \left (175 d^2 e x+27 d^3+500 d e^2 x^2+900 e^3 x^3\right )+b^4 \left (200 d^2 e^2 x^2+75 d^3 e x+12 d^4+300 d e^3 x^3+300 e^4 x^4\right )\right )}{60 b^6 (a+b x)^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 377, normalized size = 2.7 \begin{align*} -5\,{\frac{{a}^{2}{e}^{5}}{{b}^{6} \left ( bx+a \right ) ^{2}}}+10\,{\frac{d{e}^{4}a}{{b}^{5} \left ( bx+a \right ) ^{2}}}-5\,{\frac{{d}^{2}{e}^{3}}{{b}^{4} \left ( bx+a \right ) ^{2}}}+{\frac{{a}^{5}{e}^{5}}{5\,{b}^{6} \left ( bx+a \right ) ^{5}}}-{\frac{d{e}^{4}{a}^{4}}{{b}^{5} \left ( bx+a \right ) ^{5}}}+2\,{\frac{{d}^{2}{e}^{3}{a}^{3}}{{b}^{4} \left ( bx+a \right ) ^{5}}}-2\,{\frac{{d}^{3}{e}^{2}{a}^{2}}{{b}^{3} \left ( bx+a \right ) ^{5}}}+{\frac{{d}^{4}ea}{{b}^{2} \left ( bx+a \right ) ^{5}}}-{\frac{{d}^{5}}{5\,b \left ( bx+a \right ) ^{5}}}+{\frac{10\,{a}^{3}{e}^{5}}{3\,{b}^{6} \left ( bx+a \right ) ^{3}}}-10\,{\frac{{a}^{2}d{e}^{4}}{{b}^{5} \left ( bx+a \right ) ^{3}}}+10\,{\frac{{d}^{2}{e}^{3}a}{{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{10\,{d}^{3}{e}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{{e}^{5}\ln \left ( bx+a \right ) }{{b}^{6}}}-{\frac{5\,{e}^{5}{a}^{4}}{4\,{b}^{6} \left ( bx+a \right ) ^{4}}}+5\,{\frac{d{e}^{4}{a}^{3}}{{b}^{5} \left ( bx+a \right ) ^{4}}}-{\frac{15\,{d}^{2}{e}^{3}{a}^{2}}{2\,{b}^{4} \left ( bx+a \right ) ^{4}}}+5\,{\frac{{d}^{3}{e}^{2}a}{{b}^{3} \left ( bx+a \right ) ^{4}}}-{\frac{5\,{d}^{4}e}{4\,{b}^{2} \left ( bx+a \right ) ^{4}}}+5\,{\frac{{e}^{5}a}{{b}^{6} \left ( bx+a \right ) }}-5\,{\frac{d{e}^{4}}{{b}^{5} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.18595, size = 419, normalized size = 3.04 \begin{align*} -\frac{12 \, b^{5} d^{5} + 15 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 300 \,{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 300 \,{\left (b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 100 \,{\left (2 \, b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} + 25 \,{\left (3 \, b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} + 12 \, a^{3} b^{2} d e^{4} - 25 \, a^{4} b e^{5}\right )} x}{60 \,{\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} + \frac{e^{5} \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 = 2.09062, size = 775, normalized size = 5.62 \begin{align*} -\frac{12 \, b^{5} d^{5} + 15 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 300 \,{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 300 \,{\left (b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 100 \,{\left (2 \, b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} + 25 \,{\left (3 \, b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} + 12 \, a^{3} b^{2} d e^{4} - 25 \, a^{4} b e^{5}\right )} x - 60 \,{\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \log \left (b x + a\right )}{60 \,{\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 14.3819, size = 326, normalized size = 2.36 \begin{align*} \frac{137 a^{5} e^{5} - 60 a^{4} b d e^{4} - 30 a^{3} b^{2} d^{2} e^{3} - 20 a^{2} b^{3} d^{3} e^{2} - 15 a b^{4} d^{4} e - 12 b^{5} d^{5} + x^{4} \left (300 a b^{4} e^{5} - 300 b^{5} d e^{4}\right ) + x^{3} \left (900 a^{2} b^{3} e^{5} - 600 a b^{4} d e^{4} - 300 b^{5} d^{2} e^{3}\right ) + x^{2} \left (1100 a^{3} b^{2} e^{5} - 600 a^{2} b^{3} d e^{4} - 300 a b^{4} d^{2} e^{3} - 200 b^{5} d^{3} e^{2}\right ) + x \left (625 a^{4} b e^{5} - 300 a^{3} b^{2} d e^{4} - 150 a^{2} b^{3} d^{2} e^{3} - 100 a b^{4} d^{3} e^{2} - 75 b^{5} d^{4} e\right )}{60 a^{5} b^{6} + 300 a^{4} b^{7} x + 600 a^{3} b^{8} x^{2} + 600 a^{2} b^{9} x^{3} + 300 a b^{10} x^{4} + 60 b^{11} x^{5}} + \frac{e^{5} \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 = 1.1774, size = 335, normalized size = 2.43 \begin{align*} \frac{e^{5} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac{300 \,{\left (b^{4} d e^{4} - a b^{3} e^{5}\right )} x^{4} + 300 \,{\left (b^{4} d^{2} e^{3} + 2 \, a b^{3} d e^{4} - 3 \, a^{2} b^{2} e^{5}\right )} x^{3} + 100 \,{\left (2 \, b^{4} d^{3} e^{2} + 3 \, a b^{3} d^{2} e^{3} + 6 \, a^{2} b^{2} d e^{4} - 11 \, a^{3} b e^{5}\right )} x^{2} + 25 \,{\left (3 \, b^{4} d^{4} e + 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} + 12 \, a^{3} b d e^{4} - 25 \, a^{4} e^{5}\right )} x + \frac{12 \, b^{5} d^{5} + 15 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5}}{b}}{60 \,{\left (b x + a\right )}^{5} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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