Optimal. Leaf size=219 \[ \frac{c x \left (-c e (15 b d-4 a e)+4 b^2 e^2+12 c^2 d^2\right )}{e^5}-\frac{2 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (d+e x)}-\frac{(2 c d-b e) \log (d+e x) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^6}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^6 (d+e x)^2}-\frac{c^2 x^2 (6 c d-5 b e)}{2 e^4}+\frac{2 c^3 x^3}{3 e^3} \]
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Rubi [A] time = 0.249652, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {771} \[ \frac{c x \left (-c e (15 b d-4 a e)+4 b^2 e^2+12 c^2 d^2\right )}{e^5}-\frac{2 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (d+e x)}-\frac{(2 c d-b e) \log (d+e x) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^6}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^6 (d+e x)^2}-\frac{c^2 x^2 (6 c d-5 b e)}{2 e^4}+\frac{2 c^3 x^3}{3 e^3} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx &=\int \left (\frac{c \left (12 c^2 d^2+4 b^2 e^2-c e (15 b d-4 a e)\right )}{e^5}-\frac{c^2 (6 c d-5 b e) x}{e^4}+\frac{2 c^3 x^2}{e^3}+\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^3}+\frac{2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^5 (d+e x)^2}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right )}{e^5 (d+e x)}\right ) \, dx\\ &=\frac{c \left (12 c^2 d^2+4 b^2 e^2-c e (15 b d-4 a e)\right ) x}{e^5}-\frac{c^2 (6 c d-5 b e) x^2}{2 e^4}+\frac{2 c^3 x^3}{3 e^3}+\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{2 e^6 (d+e x)^2}-\frac{2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^6 (d+e x)}-\frac{(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \log (d+e x)}{e^6}\\ \end{align*}
Mathematica [A] time = 0.0866134, size = 233, normalized size = 1.06 \[ \frac{-\frac{12 \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )+b^2 e^3 (a e-b d)+2 c^2 d^2 e (3 a e-5 b d)+5 c^3 d^4\right )}{d+e x}+6 c e x \left (c e (4 a e-15 b d)+4 b^2 e^2+12 c^2 d^2\right )-6 (2 c d-b e) \log (d+e x) \left (2 c e (3 a e-5 b d)+b^2 e^2+10 c^2 d^2\right )+\frac{3 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^2}-3 c^2 e^2 x^2 (6 c d-5 b e)+4 c^3 e^3 x^3}{6 e^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 471, normalized size = 2.2 \begin{align*} -15\,{\frac{bd{c}^{2}x}{{e}^{4}}}+{\frac{d{a}^{2}c}{{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{2}da}{{e}^{2} \left ( ex+d \right ) ^{2}}}+2\,{\frac{a{c}^{2}{d}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{b}^{2}{d}^{3}c}{{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{5\,b{d}^{4}{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-12\,{\frac{a{c}^{2}{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}-12\,{\frac{{b}^{2}{d}^{2}c}{{e}^{4} \left ( ex+d \right ) }}+20\,{\frac{b{d}^{3}{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}+6\,{\frac{\ln \left ( ex+d \right ) cab}{{e}^{3}}}-12\,{\frac{\ln \left ( ex+d \right ) a{c}^{2}d}{{e}^{4}}}-12\,{\frac{\ln \left ( ex+d \right ){b}^{2}cd}{{e}^{4}}}+30\,{\frac{\ln \left ( ex+d \right ) b{c}^{2}{d}^{2}}{{e}^{5}}}+{\frac{2\,{c}^{3}{x}^{3}}{3\,{e}^{3}}}+12\,{\frac{abcd}{{e}^{3} \left ( ex+d \right ) }}-3\,{\frac{{d}^{2}abc}{{e}^{3} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{c}^{3}{x}^{2}d}{{e}^{4}}}+4\,{\frac{a{c}^{2}x}{{e}^{3}}}+4\,{\frac{{b}^{2}cx}{{e}^{3}}}+12\,{\frac{{c}^{3}{d}^{2}x}{{e}^{5}}}+{\frac{{c}^{3}{d}^{5}}{{e}^{6} \left ( ex+d \right ) ^{2}}}-20\,{\frac{\ln \left ( ex+d \right ){c}^{3}{d}^{3}}{{e}^{6}}}-2\,{\frac{c{a}^{2}}{{e}^{2} \left ( ex+d \right ) }}-2\,{\frac{{b}^{2}a}{{e}^{2} \left ( ex+d \right ) }}+2\,{\frac{{b}^{3}d}{{e}^{3} \left ( ex+d \right ) }}-10\,{\frac{{c}^{3}{d}^{4}}{{e}^{6} \left ( ex+d \right ) }}+{\frac{5\,{c}^{2}{x}^{2}b}{2\,{e}^{3}}}-{\frac{{a}^{2}b}{2\,e \left ( ex+d \right ) ^{2}}}-{\frac{{d}^{2}{b}^{3}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{\ln \left ( ex+d \right ){b}^{3}}{{e}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05594, size = 428, normalized size = 1.95 \begin{align*} -\frac{18 \, c^{3} d^{5} - 35 \, b c^{2} d^{4} e + a^{2} b e^{5} + 20 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - 3 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4} + 4 \,{\left (5 \, c^{3} d^{4} e - 10 \, b c^{2} d^{3} e^{2} + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d e^{4} +{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x}{2 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac{4 \, c^{3} e^{2} x^{3} - 3 \,{\left (6 \, c^{3} d e - 5 \, b c^{2} e^{2}\right )} x^{2} + 6 \,{\left (12 \, c^{3} d^{2} - 15 \, b c^{2} d e + 4 \,{\left (b^{2} c + a c^{2}\right )} e^{2}\right )} x}{6 \, e^{5}} - \frac{{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.49383, size = 1021, normalized size = 4.66 \begin{align*} \frac{4 \, c^{3} e^{5} x^{5} - 54 \, c^{3} d^{5} + 105 \, b c^{2} d^{4} e - 3 \, a^{2} b e^{5} - 60 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} + 9 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} - 6 \,{\left (a b^{2} + a^{2} c\right )} d e^{4} - 5 \,{\left (2 \, c^{3} d e^{4} - 3 \, b c^{2} e^{5}\right )} x^{4} + 4 \,{\left (10 \, c^{3} d^{2} e^{3} - 15 \, b c^{2} d e^{4} + 6 \,{\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} + 3 \,{\left (42 \, c^{3} d^{3} e^{2} - 55 \, b c^{2} d^{2} e^{3} + 16 \,{\left (b^{2} c + a c^{2}\right )} d e^{4}\right )} x^{2} + 6 \,{\left (2 \, c^{3} d^{4} e + 5 \, b c^{2} d^{3} e^{2} - 8 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} + 2 \,{\left (b^{3} + 6 \, a b c\right )} d e^{4} - 2 \,{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x - 6 \,{\left (20 \, c^{3} d^{5} - 30 \, b c^{2} d^{4} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} +{\left (20 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} -{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 2 \,{\left (20 \, c^{3} d^{4} e - 30 \, b c^{2} d^{3} e^{2} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.20122, size = 357, normalized size = 1.63 \begin{align*} \frac{2 c^{3} x^{3}}{3 e^{3}} - \frac{a^{2} b e^{5} + 2 a^{2} c d e^{4} + 2 a b^{2} d e^{4} - 18 a b c d^{2} e^{3} + 20 a c^{2} d^{3} e^{2} - 3 b^{3} d^{2} e^{3} + 20 b^{2} c d^{3} e^{2} - 35 b c^{2} d^{4} e + 18 c^{3} d^{5} + x \left (4 a^{2} c e^{5} + 4 a b^{2} e^{5} - 24 a b c d e^{4} + 24 a c^{2} d^{2} e^{3} - 4 b^{3} d e^{4} + 24 b^{2} c d^{2} e^{3} - 40 b c^{2} d^{3} e^{2} + 20 c^{3} d^{4} e\right )}{2 d^{2} e^{6} + 4 d e^{7} x + 2 e^{8} x^{2}} + \frac{x^{2} \left (5 b c^{2} e - 6 c^{3} d\right )}{2 e^{4}} + \frac{x \left (4 a c^{2} e^{2} + 4 b^{2} c e^{2} - 15 b c^{2} d e + 12 c^{3} d^{2}\right )}{e^{5}} + \frac{\left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15305, size = 428, normalized size = 1.95 \begin{align*} -{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} + 12 \, a c^{2} d e^{2} - b^{3} e^{3} - 6 \, a b c e^{3}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (4 \, c^{3} x^{3} e^{6} - 18 \, c^{3} d x^{2} e^{5} + 72 \, c^{3} d^{2} x e^{4} + 15 \, b c^{2} x^{2} e^{6} - 90 \, b c^{2} d x e^{5} + 24 \, b^{2} c x e^{6} + 24 \, a c^{2} x e^{6}\right )} e^{\left (-9\right )} - \frac{{\left (18 \, c^{3} d^{5} - 35 \, b c^{2} d^{4} e + 20 \, b^{2} c d^{3} e^{2} + 20 \, a c^{2} d^{3} e^{2} - 3 \, b^{3} d^{2} e^{3} - 18 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} + a^{2} b e^{5} + 4 \,{\left (5 \, c^{3} d^{4} e - 10 \, b c^{2} d^{3} e^{2} + 6 \, b^{2} c d^{2} e^{3} + 6 \, a c^{2} d^{2} e^{3} - b^{3} d e^{4} - 6 \, a b c d e^{4} + a b^{2} e^{5} + a^{2} c e^{5}\right )} x\right )} e^{\left (-6\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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