Optimal. Leaf size=155 \[ -\frac{b^5 x (5 b d-6 a e)}{e^6}+\frac{20 b^3 (b d-a e)^3}{e^7 (d+e x)}-\frac{15 b^2 (b d-a e)^4}{2 e^7 (d+e x)^2}+\frac{15 b^4 (b d-a e)^2 \log (d+e x)}{e^7}+\frac{2 b (b d-a e)^5}{e^7 (d+e x)^3}-\frac{(b d-a e)^6}{4 e^7 (d+e x)^4}+\frac{b^6 x^2}{2 e^5} \]
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Rubi [A] time = 0.146443, antiderivative size = 155, 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} \[ -\frac{b^5 x (5 b d-6 a e)}{e^6}+\frac{20 b^3 (b d-a e)^3}{e^7 (d+e x)}-\frac{15 b^2 (b d-a e)^4}{2 e^7 (d+e x)^2}+\frac{15 b^4 (b d-a e)^2 \log (d+e x)}{e^7}+\frac{2 b (b d-a e)^5}{e^7 (d+e x)^3}-\frac{(b d-a e)^6}{4 e^7 (d+e x)^4}+\frac{b^6 x^2}{2 e^5} \]
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 )^3}{(d+e x)^5} \, dx &=\int \frac{(a+b x)^6}{(d+e x)^5} \, dx\\ &=\int \left (-\frac{b^5 (5 b d-6 a e)}{e^6}+\frac{b^6 x}{e^5}+\frac{(-b d+a e)^6}{e^6 (d+e x)^5}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^4}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^3}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)^2}+\frac{15 b^4 (b d-a e)^2}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac{b^5 (5 b d-6 a e) x}{e^6}+\frac{b^6 x^2}{2 e^5}-\frac{(b d-a e)^6}{4 e^7 (d+e x)^4}+\frac{2 b (b d-a e)^5}{e^7 (d+e x)^3}-\frac{15 b^2 (b d-a e)^4}{2 e^7 (d+e x)^2}+\frac{20 b^3 (b d-a e)^3}{e^7 (d+e x)}+\frac{15 b^4 (b d-a e)^2 \log (d+e x)}{e^7}\\ \end{align*}
Mathematica [A] time = 0.121313, size = 301, normalized size = 1.94 \[ -\frac{-5 a^2 b^4 d e^2 \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )+20 a^3 b^3 e^3 \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )+5 a^4 b^2 e^4 \left (d^2+4 d e x+6 e^2 x^2\right )+2 a^5 b e^5 (d+4 e x)+a^6 e^6+2 a b^5 e \left (252 d^3 e^2 x^2+48 d^2 e^3 x^3+248 d^4 e x+77 d^5-48 d e^4 x^4-12 e^5 x^5\right )-60 b^4 (d+e x)^4 (b d-a e)^2 \log (d+e x)+b^6 \left (-\left (132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4+168 d^5 e x+57 d^6-12 d e^5 x^5+2 e^6 x^6\right )\right )}{4 e^7 (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 498, normalized size = 3.2 \begin{align*} 60\,{\frac{{a}^{2}{b}^{4}d}{{e}^{5} \left ( ex+d \right ) }}-60\,{\frac{a{b}^{5}{d}^{2}}{{e}^{6} \left ( ex+d \right ) }}+{\frac{{b}^{6}{x}^{2}}{2\,{e}^{5}}}-45\,{\frac{{b}^{4}{d}^{2}{a}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+30\,{\frac{a{b}^{5}{d}^{3}}{{e}^{6} \left ( ex+d \right ) ^{2}}}-30\,{\frac{{b}^{5}\ln \left ( ex+d \right ) ad}{{e}^{6}}}+30\,{\frac{{a}^{3}{b}^{3}d}{{e}^{4} \left ( ex+d \right ) ^{2}}}-20\,{\frac{{a}^{3}{b}^{3}{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{3}}}+20\,{\frac{{a}^{2}{b}^{4}{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{3}}}-10\,{\frac{a{b}^{5}{d}^{4}}{{e}^{6} \left ( ex+d \right ) ^{3}}}+{\frac{3\,d{a}^{5}b}{2\,{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{d}^{2}{a}^{4}{b}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}+5\,{\frac{{a}^{3}{d}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{d}^{4}{a}^{2}{b}^{4}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{3\,a{b}^{5}{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{{d}^{6}{b}^{6}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{a}^{4}{b}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{b}^{6}{d}^{4}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+6\,{\frac{a{b}^{5}x}{{e}^{5}}}-5\,{\frac{{b}^{6}xd}{{e}^{6}}}-2\,{\frac{{a}^{5}b}{{e}^{2} \left ( ex+d \right ) ^{3}}}+2\,{\frac{{b}^{6}{d}^{5}}{{e}^{7} \left ( ex+d \right ) ^{3}}}+15\,{\frac{{b}^{4}\ln \left ( ex+d \right ){a}^{2}}{{e}^{5}}}+15\,{\frac{{b}^{6}\ln \left ( ex+d \right ){d}^{2}}{{e}^{7}}}-20\,{\frac{{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}+20\,{\frac{{b}^{6}{d}^{3}}{{e}^{7} \left ( ex+d \right ) }}-{\frac{{a}^{6}}{4\,e \left ( ex+d \right ) ^{4}}}+10\,{\frac{{a}^{4}{b}^{2}d}{{e}^{3} \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08471, size = 522, normalized size = 3.37 \begin{align*} \frac{57 \, b^{6} d^{6} - 154 \, a b^{5} d^{5} e + 125 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} - a^{6} e^{6} + 80 \,{\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 30 \,{\left (7 \, b^{6} d^{4} e^{2} - 20 \, a b^{5} d^{3} e^{3} + 18 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - a^{4} b^{2} e^{6}\right )} x^{2} + 4 \,{\left (47 \, b^{6} d^{5} e - 130 \, a b^{5} d^{4} e^{2} + 110 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 5 \, a^{4} b^{2} d e^{5} - 2 \, a^{5} b e^{6}\right )} x}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} + \frac{b^{6} e x^{2} - 2 \,{\left (5 \, b^{6} d - 6 \, a b^{5} e\right )} x}{2 \, e^{6}} + \frac{15 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} \log \left (e x + d\right )}{e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.83083, size = 1158, normalized size = 7.47 \begin{align*} \frac{2 \, b^{6} e^{6} x^{6} + 57 \, b^{6} d^{6} - 154 \, a b^{5} d^{5} e + 125 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} - a^{6} e^{6} - 12 \,{\left (b^{6} d e^{5} - 2 \, a b^{5} e^{6}\right )} x^{5} - 4 \,{\left (17 \, b^{6} d^{2} e^{4} - 24 \, a b^{5} d e^{5}\right )} x^{4} - 16 \,{\left (2 \, b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} - 15 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 6 \,{\left (22 \, b^{6} d^{4} e^{2} - 84 \, a b^{5} d^{3} e^{3} + 90 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 4 \,{\left (42 \, b^{6} d^{5} e - 124 \, a b^{5} d^{4} e^{2} + 110 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 5 \, a^{4} b^{2} d e^{5} - 2 \, a^{5} b e^{6}\right )} x + 60 \,{\left (b^{6} d^{6} - 2 \, a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} +{\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 4 \,{\left (b^{6} d^{3} e^{3} - 2 \, a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5}\right )} x^{3} + 6 \,{\left (b^{6} d^{4} e^{2} - 2 \, a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (b^{6} d^{5} e - 2 \, a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 13.3998, size = 393, normalized size = 2.54 \begin{align*} \frac{b^{6} x^{2}}{2 e^{5}} + \frac{15 b^{4} \left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{6} e^{6} + 2 a^{5} b d e^{5} + 5 a^{4} b^{2} d^{2} e^{4} + 20 a^{3} b^{3} d^{3} e^{3} - 125 a^{2} b^{4} d^{4} e^{2} + 154 a b^{5} d^{5} e - 57 b^{6} d^{6} + x^{3} \left (80 a^{3} b^{3} e^{6} - 240 a^{2} b^{4} d e^{5} + 240 a b^{5} d^{2} e^{4} - 80 b^{6} d^{3} e^{3}\right ) + x^{2} \left (30 a^{4} b^{2} e^{6} + 120 a^{3} b^{3} d e^{5} - 540 a^{2} b^{4} d^{2} e^{4} + 600 a b^{5} d^{3} e^{3} - 210 b^{6} d^{4} e^{2}\right ) + x \left (8 a^{5} b e^{6} + 20 a^{4} b^{2} d e^{5} + 80 a^{3} b^{3} d^{2} e^{4} - 440 a^{2} b^{4} d^{3} e^{3} + 520 a b^{5} d^{4} e^{2} - 188 b^{6} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} + \frac{x \left (6 a b^{5} e - 5 b^{6} d\right )}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14891, size = 694, normalized size = 4.48 \begin{align*} \frac{1}{2} \,{\left (b^{6} - \frac{12 \,{\left (b^{6} d e - a b^{5} e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-7\right )} - 15 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} e^{\left (-7\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{4} \,{\left (\frac{80 \, b^{6} d^{3} e^{29}}{x e + d} - \frac{30 \, b^{6} d^{4} e^{29}}{{\left (x e + d\right )}^{2}} + \frac{8 \, b^{6} d^{5} e^{29}}{{\left (x e + d\right )}^{3}} - \frac{b^{6} d^{6} e^{29}}{{\left (x e + d\right )}^{4}} - \frac{240 \, a b^{5} d^{2} e^{30}}{x e + d} + \frac{120 \, a b^{5} d^{3} e^{30}}{{\left (x e + d\right )}^{2}} - \frac{40 \, a b^{5} d^{4} e^{30}}{{\left (x e + d\right )}^{3}} + \frac{6 \, a b^{5} d^{5} e^{30}}{{\left (x e + d\right )}^{4}} + \frac{240 \, a^{2} b^{4} d e^{31}}{x e + d} - \frac{180 \, a^{2} b^{4} d^{2} e^{31}}{{\left (x e + d\right )}^{2}} + \frac{80 \, a^{2} b^{4} d^{3} e^{31}}{{\left (x e + d\right )}^{3}} - \frac{15 \, a^{2} b^{4} d^{4} e^{31}}{{\left (x e + d\right )}^{4}} - \frac{80 \, a^{3} b^{3} e^{32}}{x e + d} + \frac{120 \, a^{3} b^{3} d e^{32}}{{\left (x e + d\right )}^{2}} - \frac{80 \, a^{3} b^{3} d^{2} e^{32}}{{\left (x e + d\right )}^{3}} + \frac{20 \, a^{3} b^{3} d^{3} e^{32}}{{\left (x e + d\right )}^{4}} - \frac{30 \, a^{4} b^{2} e^{33}}{{\left (x e + d\right )}^{2}} + \frac{40 \, a^{4} b^{2} d e^{33}}{{\left (x e + d\right )}^{3}} - \frac{15 \, a^{4} b^{2} d^{2} e^{33}}{{\left (x e + d\right )}^{4}} - \frac{8 \, a^{5} b e^{34}}{{\left (x e + d\right )}^{3}} + \frac{6 \, a^{5} b d e^{34}}{{\left (x e + d\right )}^{4}} - \frac{a^{6} e^{35}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-36\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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