Optimal. Leaf size=268 \[ -\frac{c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{4 e^7 (d+e x)^4}-\frac{3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7 (d+e x)^6}-\frac{\left (a e^2-b d e+c d^2\right )^3}{7 e^7 (d+e x)^7}+\frac{3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac{c^3}{e^7 (d+e x)} \]
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Rubi [A] time = 0.207406, antiderivative size = 268, 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} \[ -\frac{c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{4 e^7 (d+e x)^4}-\frac{3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7 (d+e x)^6}-\frac{\left (a e^2-b d e+c d^2\right )^3}{7 e^7 (d+e x)^7}+\frac{3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac{c^3}{e^7 (d+e x)} \]
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)^8} \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^8}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^7}+\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 (d+e x)^6}+\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^6 (d+e x)^5}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^6 (d+e x)^4}-\frac{3 c^2 (2 c d-b e)}{e^6 (d+e x)^3}+\frac{c^3}{e^6 (d+e x)^2}\right ) \, dx\\ &=-\frac{\left (c d^2-b d e+a e^2\right )^3}{7 e^7 (d+e x)^7}+\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{2 e^7 (d+e x)^6}-\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 )}{5 e^7 (d+e x)^5}+\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 )}{4 e^7 (d+e x)^4}-\frac{c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)^3}+\frac{3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac{c^3}{e^7 (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.142781, size = 377, normalized size = 1.41 \[ -\frac{2 c e^2 \left (2 a^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 a b e \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+2 b^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )\right )+e^3 \left (10 a^2 b e^2 (d+7 e x)+20 a^3 e^3+4 a b^2 e \left (d^2+7 d e x+21 e^2 x^2\right )+b^3 \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 c^2 e \left (2 a e \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )+5 b \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )\right )+20 c^3 \left (21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+7 d^5 e x+d^6+21 d e^5 x^5+7 e^6 x^6\right )}{140 e^7 (d+e x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 461, normalized size = 1.7 \begin{align*} -{\frac{6\,abc{e}^{3}-12\,{c}^{2}ad{e}^{2}+{b}^{3}{e}^{3}-12\,{b}^{2}cd{e}^{2}+30\,b{c}^{2}{d}^{2}e-20\,{c}^{3}{d}^{3}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{c \left ( ac{e}^{2}+{b}^{2}{e}^{2}-5\,bcde+5\,{c}^{2}{d}^{2} \right ) }{{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{3\,{c}^{2} \left ( be-2\,cd \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}-{\frac{3\,b{a}^{2}{e}^{5}-6\,{a}^{2}cd{e}^{4}-6\,a{b}^{2}d{e}^{4}+18\,{d}^{2}abc{e}^{3}-12\,a{c}^{2}{d}^{3}{e}^{2}+3\,{b}^{3}{d}^{2}{e}^{3}-12\,{d}^{3}{b}^{2}c{e}^{2}+15\,{d}^{4}b{c}^{2}e-6\,{c}^{3}{d}^{5}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{3\,{a}^{2}c{e}^{4}+3\,{b}^{2}a{e}^{4}-18\,cabd{e}^{3}+18\,a{c}^{2}{d}^{2}{e}^{2}-3\,{b}^{3}d{e}^{3}+18\,c{b}^{2}{d}^{2}{e}^{2}-30\,b{c}^{2}{d}^{3}e+15\,{c}^{3}{d}^{4}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{3}}{{e}^{7} \left ( ex+d \right ) }}-{\frac{{a}^{3}{e}^{6}-3\,b{a}^{2}d{e}^{5}+3\,{a}^{2}c{d}^{2}{e}^{4}+3\,a{b}^{2}{d}^{2}{e}^{4}-6\,{d}^{3}abc{e}^{3}+3\,a{c}^{2}{d}^{4}{e}^{2}-{b}^{3}{d}^{3}{e}^{3}+3\,{d}^{4}{b}^{2}c{e}^{2}-3\,b{c}^{2}{d}^{5}e+{c}^{3}{d}^{6}}{7\,{e}^{7} \left ( ex+d \right ) ^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18849, size = 637, normalized size = 2.38 \begin{align*} -\frac{140 \, c^{3} e^{6} x^{6} + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 10 \, a^{2} b d e^{5} + 20 \, a^{3} e^{6} + 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} + 4 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 210 \,{\left (2 \, c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \,{\left (10 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 2 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 35 \,{\left (20 \, c^{3} d^{3} e^{3} + 10 \, 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} + 21 \,{\left (20 \, c^{3} d^{4} e^{2} + 10 \, 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} + 4 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 7 \,{\left (20 \, c^{3} d^{5} e + 10 \, b c^{2} d^{4} e^{2} + 10 \, 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} + 4 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{140 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14285, size = 999, normalized size = 3.73 \begin{align*} -\frac{140 \, c^{3} e^{6} x^{6} + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 10 \, a^{2} b d e^{5} + 20 \, a^{3} e^{6} + 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} + 4 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 210 \,{\left (2 \, c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \,{\left (10 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 2 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 35 \,{\left (20 \, c^{3} d^{3} e^{3} + 10 \, 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} + 21 \,{\left (20 \, c^{3} d^{4} e^{2} + 10 \, 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} + 4 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 7 \,{\left (20 \, c^{3} d^{5} e + 10 \, b c^{2} d^{4} e^{2} + 10 \, 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} + 4 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{140 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09907, size = 618, normalized size = 2.31 \begin{align*} -\frac{{\left (140 \, c^{3} x^{6} e^{6} + 420 \, c^{3} d x^{5} e^{5} + 700 \, c^{3} d^{2} x^{4} e^{4} + 700 \, c^{3} d^{3} x^{3} e^{3} + 420 \, c^{3} d^{4} x^{2} e^{2} + 140 \, c^{3} d^{5} x e + 20 \, c^{3} d^{6} + 210 \, b c^{2} x^{5} e^{6} + 350 \, b c^{2} d x^{4} e^{5} + 350 \, b c^{2} d^{2} x^{3} e^{4} + 210 \, b c^{2} d^{3} x^{2} e^{3} + 70 \, b c^{2} d^{4} x e^{2} + 10 \, b c^{2} d^{5} e + 140 \, b^{2} c x^{4} e^{6} + 140 \, a c^{2} x^{4} e^{6} + 140 \, b^{2} c d x^{3} e^{5} + 140 \, a c^{2} d x^{3} e^{5} + 84 \, b^{2} c d^{2} x^{2} e^{4} + 84 \, a c^{2} d^{2} x^{2} e^{4} + 28 \, b^{2} c d^{3} x e^{3} + 28 \, a c^{2} d^{3} x e^{3} + 4 \, b^{2} c d^{4} e^{2} + 4 \, a c^{2} d^{4} e^{2} + 35 \, b^{3} x^{3} e^{6} + 210 \, a b c x^{3} e^{6} + 21 \, b^{3} d x^{2} e^{5} + 126 \, a b c d x^{2} e^{5} + 7 \, b^{3} d^{2} x e^{4} + 42 \, a b c d^{2} x e^{4} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 84 \, a b^{2} x^{2} e^{6} + 84 \, a^{2} c x^{2} e^{6} + 28 \, a b^{2} d x e^{5} + 28 \, a^{2} c d x e^{5} + 4 \, a b^{2} d^{2} e^{4} + 4 \, a^{2} c d^{2} e^{4} + 70 \, a^{2} b x e^{6} + 10 \, a^{2} b d e^{5} + 20 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{140 \,{\left (x e + d\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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