Optimal. Leaf size=256 \[ -\frac{3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^7 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac{3 c^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac{c^3 x}{e^6} \]
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Rubi [A] time = 0.244684, 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} \[ -\frac{3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^7 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac{3 c^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac{c^3 x}{e^6} \]
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)^6} \, dx &=\int \left (\frac{c^3}{e^6}+\frac{\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^6}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^5}+\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)^4}+\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)^3}+\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)^2}-\frac{3 c^2 (2 c d-b e)}{e^6 (d+e x)}\right ) \, dx\\ &=\frac{c^3 x}{e^6}-\frac{\left (c d^2-b d e+a e^2\right )^3}{5 e^7 (d+e x)^5}+\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{4 e^7 (d+e x)^4}-\frac{\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^7 (d+e x)^3}+\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 )}{2 e^7 (d+e x)^2}-\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)}-\frac{3 c^2 (2 c d-b e) \log (d+e x)}{e^7}\\ \end{align*}
Mathematica [A] time = 0.255258, size = 396, normalized size = 1.55 \[ -\frac{2 c e^2 \left (a^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 a b e \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )+6 b^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )\right )+e^3 \left (3 a^2 b e^2 (d+5 e x)+4 a^3 e^3+2 a b^2 e \left (d^2+5 d e x+10 e^2 x^2\right )+b^3 \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )\right )+c^2 e \left (12 a e \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )-b d \left (1100 d^2 e^2 x^2+625 d^3 e x+137 d^4+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 c^2 (d+e x)^5 (2 c d-b e) \log (d+e x)+2 c^3 \left (600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4+375 d^5 e x+87 d^6-50 d e^5 x^5-10 e^6 x^6\right )}{20 e^7 (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.051, size = 688, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06975, size = 606, normalized size = 2.37 \begin{align*} -\frac{174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 3 \, a^{2} b d e^{5} + 4 \, a^{3} e^{6} + 12 \,{\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} + 60 \,{\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} +{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 10 \,{\left (100 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} +{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 10 \,{\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \,{\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} + 5 \,{\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} + 12 \,{\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}{20 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} + \frac{c^{3} x}{e^{6}} - \frac{3 \,{\left (2 \, c^{3} d - b c^{2} e\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 = 2.09305, size = 1247, normalized size = 4.87 \begin{align*} \frac{20 \, c^{3} e^{6} x^{6} + 100 \, c^{3} d e^{5} x^{5} - 174 \, c^{3} d^{6} + 137 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} - 4 \, a^{3} e^{6} - 12 \,{\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} - 20 \,{\left (5 \, c^{3} d^{2} e^{4} - 15 \, b c^{2} d e^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 10 \,{\left (80 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} +{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 10 \,{\left (120 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \,{\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} - 5 \,{\left (150 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} + 12 \,{\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 - 60 \,{\left (2 \, c^{3} d^{6} - b c^{2} d^{5} e +{\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 5 \,{\left (2 \, c^{3} d^{2} e^{4} - b c^{2} d e^{5}\right )} x^{4} + 10 \,{\left (2 \, c^{3} d^{3} e^{3} - b c^{2} d^{2} e^{4}\right )} x^{3} + 10 \,{\left (2 \, c^{3} d^{4} e^{2} - b c^{2} d^{3} e^{3}\right )} x^{2} + 5 \,{\left (2 \, c^{3} d^{5} e - b c^{2} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{20 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} 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 = 1.13207, size = 560, normalized size = 2.19 \begin{align*} c^{3} x e^{\left (-6\right )} - 3 \,{\left (2 \, c^{3} d - b c^{2} e\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 12 \, a c^{2} d^{4} e^{2} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 2 \, a b^{2} d^{2} e^{4} + 2 \, a^{2} c d^{2} e^{4} + 60 \,{\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + b^{2} c e^{6} + a c^{2} e^{6}\right )} x^{4} + 3 \, a^{2} b d e^{5} + 10 \,{\left (100 \, c^{3} d^{3} e^{3} - 90 \, 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 )} x^{3} + 4 \, a^{3} e^{6} + 10 \,{\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + 12 \, a c^{2} d^{2} e^{4} + b^{3} d e^{5} + 6 \, a b c d e^{5} + 2 \, a b^{2} e^{6} + 2 \, a^{2} c e^{6}\right )} x^{2} + 5 \,{\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + 12 \, a c^{2} d^{3} e^{3} + b^{3} d^{2} e^{4} + 6 \, a b c d^{2} e^{4} + 2 \, a b^{2} d e^{5} + 2 \, a^{2} c d e^{5} + 3 \, a^{2} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{20 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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