Optimal. Leaf size=158 \[ -\frac{2 b^5 (d+e x)^3 (b d-a e)}{e^7}+\frac{15 b^4 (d+e x)^2 (b d-a e)^2}{2 e^7}-\frac{20 b^3 x (b d-a e)^3}{e^6}+\frac{15 b^2 (b d-a e)^4 \log (d+e x)}{e^7}+\frac{6 b (b d-a e)^5}{e^7 (d+e x)}-\frac{(b d-a e)^6}{2 e^7 (d+e x)^2}+\frac{b^6 (d+e x)^4}{4 e^7} \]
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Rubi [A] time = 0.179208, antiderivative size = 158, 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} \[ -\frac{2 b^5 (d+e x)^3 (b d-a e)}{e^7}+\frac{15 b^4 (d+e x)^2 (b d-a e)^2}{2 e^7}-\frac{20 b^3 x (b d-a e)^3}{e^6}+\frac{15 b^2 (b d-a e)^4 \log (d+e x)}{e^7}+\frac{6 b (b d-a e)^5}{e^7 (d+e x)}-\frac{(b d-a e)^6}{2 e^7 (d+e x)^2}+\frac{b^6 (d+e x)^4}{4 e^7} \]
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)^3} \, dx &=\int \frac{(a+b x)^6}{(d+e x)^3} \, dx\\ &=\int \left (-\frac{20 b^3 (b d-a e)^3}{e^6}+\frac{(-b d+a e)^6}{e^6 (d+e x)^3}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^2}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)}+\frac{15 b^4 (b d-a e)^2 (d+e x)}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^2}{e^6}+\frac{b^6 (d+e x)^3}{e^6}\right ) \, dx\\ &=-\frac{20 b^3 (b d-a e)^3 x}{e^6}-\frac{(b d-a e)^6}{2 e^7 (d+e x)^2}+\frac{6 b (b d-a e)^5}{e^7 (d+e x)}+\frac{15 b^4 (b d-a e)^2 (d+e x)^2}{2 e^7}-\frac{2 b^5 (b d-a e) (d+e x)^3}{e^7}+\frac{b^6 (d+e x)^4}{4 e^7}+\frac{15 b^2 (b d-a e)^4 \log (d+e x)}{e^7}\\ \end{align*}
Mathematica [A] time = 0.108777, size = 303, normalized size = 1.92 \[ \frac{30 a^2 b^4 e^2 \left (-11 d^2 e^2 x^2+2 d^3 e x+7 d^4-4 d e^3 x^3+e^4 x^4\right )+40 a^3 b^3 e^3 \left (-4 d^2 e x-5 d^3+4 d e^2 x^2+2 e^3 x^3\right )+30 a^4 b^2 d e^4 (3 d+4 e x)-12 a^5 b e^5 (d+2 e x)-2 a^6 e^6+4 a b^5 e \left (63 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x-27 d^5-5 d e^4 x^4+2 e^5 x^5\right )+60 b^2 (d+e x)^2 (b d-a e)^4 \log (d+e x)+b^6 \left (-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-16 d^5 e x+22 d^6-2 d e^5 x^5+e^6 x^6\right )}{4 e^7 (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 464, normalized size = 2.9 \begin{align*}{\frac{{b}^{6}{x}^{4}}{4\,{e}^{3}}}-{\frac{{a}^{6}}{2\,e \left ( ex+d \right ) ^{2}}}-9\,{\frac{{b}^{5}{x}^{2}ad}{{e}^{4}}}-45\,{\frac{{a}^{2}{b}^{4}dx}{{e}^{4}}}-{\frac{15\,{d}^{2}{a}^{4}{b}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+10\,{\frac{{a}^{3}{b}^{3}{d}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{a}^{2}{b}^{4}{d}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+3\,{\frac{a{b}^{5}{d}^{5}}{{e}^{6} \left ( ex+d \right ) ^{2}}}-60\,{\frac{{b}^{3}\ln \left ( ex+d \right ){a}^{3}d}{{e}^{4}}}+90\,{\frac{{b}^{4}\ln \left ( ex+d \right ){d}^{2}{a}^{2}}{{e}^{5}}}-60\,{\frac{{b}^{5}\ln \left ( ex+d \right ) a{d}^{3}}{{e}^{6}}}+30\,{\frac{{a}^{4}{b}^{2}d}{{e}^{3} \left ( ex+d \right ) }}-60\,{\frac{{a}^{3}{b}^{3}{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}+60\,{\frac{{a}^{2}{b}^{4}{d}^{3}}{{e}^{5} \left ( ex+d \right ) }}-30\,{\frac{a{b}^{5}{d}^{4}}{{e}^{6} \left ( ex+d \right ) }}+36\,{\frac{a{b}^{5}{d}^{2}x}{{e}^{5}}}+3\,{\frac{d{a}^{5}b}{{e}^{2} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{b}^{5}{x}^{3}a}{{e}^{3}}}-{\frac{{b}^{6}{x}^{3}d}{{e}^{4}}}+{\frac{15\,{b}^{4}{x}^{2}{a}^{2}}{2\,{e}^{3}}}+3\,{\frac{{b}^{6}{x}^{2}{d}^{2}}{{e}^{5}}}+20\,{\frac{x{a}^{3}{b}^{3}}{{e}^{3}}}-10\,{\frac{{b}^{6}{d}^{3}x}{{e}^{6}}}-{\frac{{d}^{6}{b}^{6}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+15\,{\frac{{b}^{2}\ln \left ( ex+d \right ){a}^{4}}{{e}^{3}}}+15\,{\frac{{b}^{6}\ln \left ( ex+d \right ){d}^{4}}{{e}^{7}}}-6\,{\frac{{a}^{5}b}{{e}^{2} \left ( ex+d \right ) }}+6\,{\frac{{b}^{6}{d}^{5}}{{e}^{7} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16169, size = 491, normalized size = 3.11 \begin{align*} \frac{11 \, b^{6} d^{6} - 54 \, a b^{5} d^{5} e + 105 \, a^{2} b^{4} d^{4} e^{2} - 100 \, a^{3} b^{3} d^{3} e^{3} + 45 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} - a^{6} e^{6} + 12 \,{\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x}{2 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac{b^{6} e^{3} x^{4} - 4 \,{\left (b^{6} d e^{2} - 2 \, a b^{5} e^{3}\right )} x^{3} + 6 \,{\left (2 \, b^{6} d^{2} e - 6 \, a b^{5} d e^{2} + 5 \, a^{2} b^{4} e^{3}\right )} x^{2} - 4 \,{\left (10 \, b^{6} d^{3} - 36 \, a b^{5} d^{2} e + 45 \, a^{2} b^{4} d e^{2} - 20 \, a^{3} b^{3} e^{3}\right )} x}{4 \, e^{6}} + \frac{15 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\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 = 1.75028, size = 1110, normalized size = 7.03 \begin{align*} \frac{b^{6} e^{6} x^{6} + 22 \, b^{6} d^{6} - 108 \, a b^{5} d^{5} e + 210 \, a^{2} b^{4} d^{4} e^{2} - 200 \, a^{3} b^{3} d^{3} e^{3} + 90 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 2 \,{\left (b^{6} d e^{5} - 4 \, a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (b^{6} d^{2} e^{4} - 4 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \,{\left (b^{6} d^{3} e^{3} - 4 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} - 4 \, a^{3} b^{3} e^{6}\right )} x^{3} - 2 \,{\left (34 \, b^{6} d^{4} e^{2} - 126 \, a b^{5} d^{3} e^{3} + 165 \, a^{2} b^{4} d^{2} e^{4} - 80 \, a^{3} b^{3} d e^{5}\right )} x^{2} - 4 \,{\left (4 \, b^{6} d^{5} e - 6 \, a b^{5} d^{4} e^{2} - 15 \, a^{2} b^{4} d^{3} e^{3} + 40 \, a^{3} b^{3} d^{2} e^{4} - 30 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x + 60 \,{\left (b^{6} d^{6} - 4 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} - 4 \, a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} +{\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, 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} + 2 \,{\left (b^{6} d^{5} e - 4 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} - 4 \, a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.16562, size = 335, normalized size = 2.12 \begin{align*} \frac{b^{6} x^{4}}{4 e^{3}} + \frac{15 b^{2} \left (a e - b d\right )^{4} \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{6} e^{6} + 6 a^{5} b d e^{5} - 45 a^{4} b^{2} d^{2} e^{4} + 100 a^{3} b^{3} d^{3} e^{3} - 105 a^{2} b^{4} d^{4} e^{2} + 54 a b^{5} d^{5} e - 11 b^{6} d^{6} + x \left (12 a^{5} b e^{6} - 60 a^{4} b^{2} d e^{5} + 120 a^{3} b^{3} d^{2} e^{4} - 120 a^{2} b^{4} d^{3} e^{3} + 60 a b^{5} d^{4} e^{2} - 12 b^{6} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac{x^{3} \left (2 a b^{5} e - b^{6} d\right )}{e^{4}} + \frac{x^{2} \left (15 a^{2} b^{4} e^{2} - 18 a b^{5} d e + 6 b^{6} d^{2}\right )}{2 e^{5}} + \frac{x \left (20 a^{3} b^{3} e^{3} - 45 a^{2} b^{4} d e^{2} + 36 a b^{5} d^{2} e - 10 b^{6} d^{3}\right )}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14652, size = 460, normalized size = 2.91 \begin{align*} 15 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{4} \,{\left (b^{6} x^{4} e^{9} - 4 \, b^{6} d x^{3} e^{8} + 12 \, b^{6} d^{2} x^{2} e^{7} - 40 \, b^{6} d^{3} x e^{6} + 8 \, a b^{5} x^{3} e^{9} - 36 \, a b^{5} d x^{2} e^{8} + 144 \, a b^{5} d^{2} x e^{7} + 30 \, a^{2} b^{4} x^{2} e^{9} - 180 \, a^{2} b^{4} d x e^{8} + 80 \, a^{3} b^{3} x e^{9}\right )} e^{\left (-12\right )} + \frac{{\left (11 \, b^{6} d^{6} - 54 \, a b^{5} d^{5} e + 105 \, a^{2} b^{4} d^{4} e^{2} - 100 \, a^{3} b^{3} d^{3} e^{3} + 45 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} - a^{6} e^{6} + 12 \,{\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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