Optimal. Leaf size=151 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{3 e^5 (d+e x)^3}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^5}+\frac{c (2 c d-b e)}{e^5 (d+e x)^2}-\frac{c^2}{e^5 (d+e x)} \]
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Rubi [A] time = 0.111422, antiderivative size = 151, 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} \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{3 e^5 (d+e x)^3}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^5}+\frac{c (2 c d-b e)}{e^5 (d+e x)^2}-\frac{c^2}{e^5 (d+e x)} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^2}{e^4 (d+e x)^6}+\frac{2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)^5}+\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^4 (d+e x)^4}-\frac{2 c (2 c d-b e)}{e^4 (d+e x)^3}+\frac{c^2}{e^4 (d+e x)^2}\right ) \, dx\\ &=-\frac{\left (c d^2-b d e+a e^2\right )^2}{5 e^5 (d+e x)^5}+\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{2 e^5 (d+e x)^4}-\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{3 e^5 (d+e x)^3}+\frac{c (2 c d-b e)}{e^5 (d+e x)^2}-\frac{c^2}{e^5 (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.0722244, size = 160, normalized size = 1.06 \[ -\frac{e^2 \left (6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )\right )+c e \left (2 a e \left (d^2+5 d e x+10 e^2 x^2\right )+3 b \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )\right )+6 c^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )}{30 e^5 (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 195, normalized size = 1.3 \begin{align*} -{\frac{2\,ab{e}^{3}-4\,ad{e}^{2}c-2\,{b}^{2}d{e}^{2}+6\,{d}^{2}ebc-4\,{c}^{2}{d}^{3}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{2\,ac{e}^{2}+{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{c \left ( be-2\,cd \right ) }{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}-{\frac{{a}^{2}{e}^{4}-2\,d{e}^{3}ab+2\,ac{d}^{2}{e}^{2}+{b}^{2}{d}^{2}{e}^{2}-2\,{d}^{3}ebc+{c}^{2}{d}^{4}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987873, size = 296, normalized size = 1.96 \begin{align*} -\frac{30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + 3 \, a b d e^{3} + 6 \, a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 30 \,{\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \,{\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 5 \,{\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 3 \, a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{30 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03934, size = 466, normalized size = 3.09 \begin{align*} -\frac{30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + 3 \, a b d e^{3} + 6 \, a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 30 \,{\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \,{\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 5 \,{\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 3 \, a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{30 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 38.7147, size = 250, normalized size = 1.66 \begin{align*} - \frac{6 a^{2} e^{4} + 3 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} + 3 b c d^{3} e + 6 c^{2} d^{4} + 30 c^{2} e^{4} x^{4} + x^{3} \left (30 b c e^{4} + 60 c^{2} d e^{3}\right ) + x^{2} \left (20 a c e^{4} + 10 b^{2} e^{4} + 30 b c d e^{3} + 60 c^{2} d^{2} e^{2}\right ) + x \left (15 a b e^{4} + 10 a c d e^{3} + 5 b^{2} d e^{3} + 15 b c d^{2} e^{2} + 30 c^{2} d^{3} e\right )}{30 d^{5} e^{5} + 150 d^{4} e^{6} x + 300 d^{3} e^{7} x^{2} + 300 d^{2} e^{8} x^{3} + 150 d e^{9} x^{4} + 30 e^{10} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11779, size = 242, normalized size = 1.6 \begin{align*} -\frac{{\left (30 \, c^{2} x^{4} e^{4} + 60 \, c^{2} d x^{3} e^{3} + 60 \, c^{2} d^{2} x^{2} e^{2} + 30 \, c^{2} d^{3} x e + 6 \, c^{2} d^{4} + 30 \, b c x^{3} e^{4} + 30 \, b c d x^{2} e^{3} + 15 \, b c d^{2} x e^{2} + 3 \, b c d^{3} e + 10 \, b^{2} x^{2} e^{4} + 20 \, a c x^{2} e^{4} + 5 \, b^{2} d x e^{3} + 10 \, a c d x e^{3} + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + 15 \, a b x e^{4} + 3 \, a b d e^{3} + 6 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{30 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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