Optimal. Leaf size=256 \[ \frac{c (d+e x)^3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{(d+e x)^2 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7}+\frac{3 x \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}-\frac{\left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)}-\frac{3 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^7}-\frac{3 c^2 (d+e x)^4 (2 c d-b e)}{4 e^7}+\frac{c^3 (d+e x)^5}{5 e^7} \]
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Rubi [A] time = 0.320798, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {698} \[ \frac{c (d+e x)^3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{(d+e x)^2 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7}+\frac{3 x \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}-\frac{\left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)}-\frac{3 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^7}-\frac{3 c^2 (d+e x)^4 (2 c d-b e)}{4 e^7}+\frac{c^3 (d+e x)^5}{5 e^7} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx &=\int \left (\frac{3 \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^6}+\frac{\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^2}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{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 ) (d+e x)}{e^6}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^2}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^3}{e^6}+\frac{c^3 (d+e x)^4}{e^6}\right ) \, dx\\ &=\frac{3 \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 ) x}{e^6}-\frac{\left (c d^2-b d e+a e^2\right )^3}{e^7 (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 ) (d+e x)^2}{2 e^7}+\frac{c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^3}{e^7}-\frac{3 c^2 (2 c d-b e) (d+e x)^4}{4 e^7}+\frac{c^3 (d+e x)^5}{5 e^7}-\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 \log (d+e x)}{e^7}\\ \end{align*}
Mathematica [A] time = 0.103753, size = 255, normalized size = 1. \[ \frac{20 e x \left (3 c e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )+b^2 e^3 (3 a e-2 b d)+3 c^2 d^2 e (3 a e-4 b d)+5 c^3 d^4\right )+20 c e^3 x^3 \left (c e (a e-2 b d)+b^2 e^2+c^2 d^2\right )+10 e^2 x^2 (b e-c d) \left (c e (6 a e-5 b d)+b^2 e^2+4 c^2 d^2\right )-\frac{20 \left (e (a e-b d)+c d^2\right )^3}{d+e x}-60 (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^2+5 c^2 e^4 x^4 (3 b e-2 c d)+4 c^3 e^5 x^5}{20 e^7} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 585, normalized size = 2.3 \begin{align*} 18\,{\frac{\ln \left ( ex+d \right ) abc{d}^{2}}{{e}^{4}}}-3\,{\frac{a{x}^{2}{c}^{2}d}{{e}^{3}}}-3\,{\frac{{b}^{2}{x}^{2}cd}{{e}^{3}}}+{\frac{{c}^{3}{x}^{5}}{5\,{e}^{2}}}+9\,{\frac{a{c}^{2}{d}^{2}x}{{e}^{4}}}+9\,{\frac{c{b}^{2}{d}^{2}x}{{e}^{4}}}-12\,{\frac{{d}^{3}b{c}^{2}x}{{e}^{5}}}+3\,{\frac{ab{x}^{2}c}{{e}^{2}}}-6\,{\frac{dc\ln \left ( ex+d \right ){a}^{2}}{{e}^{3}}}-6\,{\frac{\ln \left ( ex+d \right ) a{b}^{2}d}{{e}^{3}}}-12\,{\frac{\ln \left ( ex+d \right ) a{c}^{2}{d}^{3}}{{e}^{5}}}-12\,{\frac{\ln \left ( ex+d \right ){b}^{2}c{d}^{3}}{{e}^{5}}}+15\,{\frac{\ln \left ( ex+d \right ) b{c}^{2}{d}^{4}}{{e}^{6}}}+3\,{\frac{{a}^{2}db}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{{a}^{2}c{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-3\,{\frac{a{b}^{2}{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-3\,{\frac{a{c}^{2}{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}+3\,{\frac{b{c}^{2}{d}^{5}}{{e}^{6} \left ( ex+d \right ) }}-2\,{\frac{b{x}^{3}{c}^{2}d}{{e}^{3}}}-{\frac{{c}^{3}{d}^{6}}{{e}^{7} \left ( ex+d \right ) }}+3\,{\frac{\ln \left ( ex+d \right ){a}^{2}b}{{e}^{2}}}+3\,{\frac{\ln \left ( ex+d \right ){b}^{3}{d}^{2}}{{e}^{4}}}-6\,{\frac{\ln \left ( ex+d \right ){c}^{3}{d}^{5}}{{e}^{7}}}+{\frac{{b}^{3}{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}+3\,{\frac{{b}^{2}ax}{{e}^{2}}}-2\,{\frac{{b}^{3}dx}{{e}^{3}}}+5\,{\frac{{c}^{3}{d}^{4}x}{{e}^{6}}}+{\frac{3\,b{x}^{4}{c}^{2}}{4\,{e}^{2}}}+{\frac{a{x}^{3}{c}^{2}}{{e}^{2}}}+{\frac{{b}^{2}c{x}^{3}}{{e}^{2}}}+{\frac{{x}^{3}{c}^{3}{d}^{2}}{{e}^{4}}}-2\,{\frac{{x}^{2}{c}^{3}{d}^{3}}{{e}^{5}}}+3\,{\frac{{a}^{2}cx}{{e}^{2}}}-{\frac{{a}^{3}}{e \left ( ex+d \right ) }}-3\,{\frac{{b}^{2}c{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}+{\frac{{x}^{2}{b}^{3}}{2\,{e}^{2}}}+{\frac{9\,b{x}^{2}{c}^{2}{d}^{2}}{2\,{e}^{4}}}+6\,{\frac{abc{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-12\,{\frac{cabdx}{{e}^{3}}}-{\frac{{c}^{3}d{x}^{4}}{2\,{e}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989355, size = 554, normalized size = 2.16 \begin{align*} -\frac{c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}}{e^{8} x + d e^{7}} + \frac{4 \, c^{3} e^{4} x^{5} - 5 \,{\left (2 \, c^{3} d e^{3} - 3 \, b c^{2} e^{4}\right )} x^{4} + 20 \,{\left (c^{3} d^{2} e^{2} - 2 \, b c^{2} d e^{3} +{\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{3} - 10 \,{\left (4 \, c^{3} d^{3} e - 9 \, b c^{2} d^{2} e^{2} + 6 \,{\left (b^{2} c + a c^{2}\right )} d e^{3} -{\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{2} + 20 \,{\left (5 \, c^{3} d^{4} - 12 \, b c^{2} d^{3} e + 9 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - 2 \,{\left (b^{3} + 6 \, a b c\right )} d e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x}{20 \, e^{6}} - \frac{3 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d 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 = 2.11092, size = 1197, normalized size = 4.68 \begin{align*} \frac{4 \, c^{3} e^{6} x^{6} - 20 \, c^{3} d^{6} + 60 \, b c^{2} d^{5} e + 60 \, a^{2} b d e^{5} - 20 \, a^{3} e^{6} - 60 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 20 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 60 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 3 \,{\left (2 \, c^{3} d e^{5} - 5 \, b c^{2} e^{6}\right )} x^{5} + 5 \,{\left (2 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 10 \,{\left (2 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} + 4 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} -{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 30 \,{\left (2 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 2 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 20 \,{\left (5 \, c^{3} d^{5} e - 12 \, b c^{2} d^{4} e^{2} + 9 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 2 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 3 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x - 60 \,{\left (2 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e - a^{2} b d e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} +{\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{20 \,{\left (e^{8} x + d e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.64275, size = 403, normalized size = 1.57 \begin{align*} \frac{c^{3} x^{5}}{5 e^{2}} - \frac{a^{3} e^{6} - 3 a^{2} b d e^{5} + 3 a^{2} c d^{2} e^{4} + 3 a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 3 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}}{d e^{7} + e^{8} x} + \frac{x^{4} \left (3 b c^{2} e - 2 c^{3} d\right )}{4 e^{3}} + \frac{x^{3} \left (a c^{2} e^{2} + b^{2} c e^{2} - 2 b c^{2} d e + c^{3} d^{2}\right )}{e^{4}} + \frac{x^{2} \left (6 a b c e^{3} - 6 a c^{2} d e^{2} + b^{3} e^{3} - 6 b^{2} c d e^{2} + 9 b c^{2} d^{2} e - 4 c^{3} d^{3}\right )}{2 e^{5}} + \frac{x \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 12 a b c d e^{3} + 9 a c^{2} d^{2} e^{2} - 2 b^{3} d e^{3} + 9 b^{2} c d^{2} e^{2} - 12 b c^{2} d^{3} e + 5 c^{3} d^{4}\right )}{e^{6}} + \frac{3 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2} \log{\left (d + e x \right )}}{e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1272, size = 730, normalized size = 2.85 \begin{align*} \frac{1}{20} \,{\left (4 \, c^{3} - \frac{15 \,{\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{20 \,{\left (5 \, c^{3} d^{2} e^{2} - 5 \, b c^{2} d e^{3} + b^{2} c e^{4} + a c^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{10 \,{\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 12 \, a c^{2} d e^{5} - b^{3} e^{6} - 6 \, a b c e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac{60 \,{\left (5 \, c^{3} d^{4} e^{4} - 10 \, b c^{2} d^{3} e^{5} + 6 \, b^{2} c d^{2} e^{6} + 6 \, a c^{2} d^{2} e^{6} - b^{3} d e^{7} - 6 \, a b c d e^{7} + a b^{2} e^{8} + a^{2} c e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}}\right )}{\left (x e + d\right )}^{5} e^{\left (-7\right )} + 3 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, 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}\right )} e^{\left (-7\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{c^{3} d^{6} e^{5}}{x e + d} - \frac{3 \, b c^{2} d^{5} e^{6}}{x e + d} + \frac{3 \, b^{2} c d^{4} e^{7}}{x e + d} + \frac{3 \, a c^{2} d^{4} e^{7}}{x e + d} - \frac{b^{3} d^{3} e^{8}}{x e + d} - \frac{6 \, a b c d^{3} e^{8}}{x e + d} + \frac{3 \, a b^{2} d^{2} e^{9}}{x e + d} + \frac{3 \, a^{2} c d^{2} e^{9}}{x e + d} - \frac{3 \, a^{2} b d e^{10}}{x e + d} + \frac{a^{3} e^{11}}{x e + d}\right )} e^{\left (-12\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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