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 = 0.07, antiderivative size = 103, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}
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*}
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Mathematica [A] time = 0.03, size = 79, normalized size = 0.77 \begin {gather*} -\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 (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )}{60 e^4 (d+e x)^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a+b x+c x^2\right )}{(d+e x)^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 132, normalized size = 1.28 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 80, normalized size = 0.78 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 93, normalized size = 0.90 \begin {gather*} -\frac {c}{2 \left (e x +d \right )^{2} e^{4}}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d}{5 \left (e x +d \right )^{5} e^{4}}-\frac {a \,e^{2}-2 b d e +3 c \,d^{2}}{4 \left (e x +d \right )^{4} e^{4}}-\frac {b e -3 c d}{3 \left (e x +d \right )^{3} e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 132, normalized size = 1.28 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 128, normalized size = 1.24 \begin {gather*} -\frac {\frac {c\,x^3}{2\,e}+\frac {d\,\left (3\,c\,d^2+2\,b\,d\,e+3\,a\,e^2\right )}{60\,e^4}+\frac {x\,\left (3\,c\,d^2+2\,b\,d\,e+3\,a\,e^2\right )}{12\,e^3}+\frac {x^2\,\left (2\,b\,e+3\,c\,d\right )}{6\,e^2}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.66, size = 141, normalized size = 1.37 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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