Optimal. Leaf size=103 \[ -\frac{3 c d^2-e (2 b d-a e)}{4 e^4 (d+e x)^4}+\frac{d \left (a e^2-b d e+c d^2\right )}{5 e^4 (d+e x)^5}+\frac{3 c d-b e}{3 e^4 (d+e x)^3}-\frac{c}{2 e^4 (d+e x)^2} \]
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Rubi [A] time = 0.0749924, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {771} \[ -\frac{3 c d^2-e (2 b d-a e)}{4 e^4 (d+e x)^4}+\frac{d \left (a e^2-b d e+c d^2\right )}{5 e^4 (d+e x)^5}+\frac{3 c d-b e}{3 e^4 (d+e x)^3}-\frac{c}{2 e^4 (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{x \left (a+b x+c x^2\right )}{(d+e x)^6} \, dx &=\int \left (-\frac{d \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^6}+\frac{3 c d^2-e (2 b d-a e)}{e^3 (d+e x)^5}+\frac{-3 c d+b e}{e^3 (d+e x)^4}+\frac{c}{e^3 (d+e x)^3}\right ) \, dx\\ &=\frac{d \left (c d^2-b d e+a e^2\right )}{5 e^4 (d+e x)^5}-\frac{3 c d^2-e (2 b d-a e)}{4 e^4 (d+e x)^4}+\frac{3 c d-b e}{3 e^4 (d+e x)^3}-\frac{c}{2 e^4 (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.0327718, size = 79, normalized size = 0.77 \[ -\frac{e \left (3 a e (d+5 e x)+2 b \left (d^2+5 d e x+10 e^2 x^2\right )\right )+3 c \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )}{60 e^4 (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 93, normalized size = 0.9 \begin{align*}{\frac{d \left ( a{e}^{2}-bde+c{d}^{2} \right ) }{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{be-3\,cd}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{c}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{a{e}^{2}-2\,bde+3\,c{d}^{2}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.2618, size = 178, normalized size = 1.73 \begin{align*} -\frac{30 \, c e^{3} x^{3} + 3 \, c d^{3} + 2 \, b d^{2} e + 3 \, a d e^{2} + 10 \,{\left (3 \, c d e^{2} + 2 \, b e^{3}\right )} x^{2} + 5 \,{\left (3 \, c d^{2} e + 2 \, b d e^{2} + 3 \, a e^{3}\right )} x}{60 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35966, size = 282, normalized size = 2.74 \begin{align*} -\frac{30 \, c e^{3} x^{3} + 3 \, c d^{3} + 2 \, b d^{2} e + 3 \, a d e^{2} + 10 \,{\left (3 \, c d e^{2} + 2 \, b e^{3}\right )} x^{2} + 5 \,{\left (3 \, c d^{2} e + 2 \, b d e^{2} + 3 \, a e^{3}\right )} x}{60 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.0645, size = 139, normalized size = 1.35 \begin{align*} - \frac{3 a d e^{2} + 2 b d^{2} e + 3 c d^{3} + 30 c e^{3} x^{3} + x^{2} \left (20 b e^{3} + 30 c d e^{2}\right ) + x \left (15 a e^{3} + 10 b d e^{2} + 15 c d^{2} e\right )}{60 d^{5} e^{4} + 300 d^{4} e^{5} x + 600 d^{3} e^{6} x^{2} + 600 d^{2} e^{7} x^{3} + 300 d e^{8} x^{4} + 60 e^{9} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10845, size = 108, normalized size = 1.05 \begin{align*} -\frac{{\left (30 \, c x^{3} e^{3} + 30 \, c d x^{2} e^{2} + 15 \, c d^{2} x e + 3 \, c d^{3} + 20 \, b x^{2} e^{3} + 10 \, b d x e^{2} + 2 \, b d^{2} e + 15 \, a x e^{3} + 3 \, a d e^{2}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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