Optimal. Leaf size=103 \[ -\frac {3 c d^2-e (2 b d-a e)}{5 e^4 (d+e x)^5}+\frac {d \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac {3 c d-b e}{4 e^4 (d+e x)^4}-\frac {c}{3 e^4 (d+e x)^3} \]
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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)}{5 e^4 (d+e x)^5}+\frac {d \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac {3 c d-b e}{4 e^4 (d+e x)^4}-\frac {c}{3 e^4 (d+e x)^3} \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)^7} \, dx &=\int \left (-\frac {d \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^7}+\frac {3 c d^2-e (2 b d-a e)}{e^3 (d+e x)^6}+\frac {-3 c d+b e}{e^3 (d+e x)^5}+\frac {c}{e^3 (d+e x)^4}\right ) \, dx\\ &=\frac {d \left (c d^2-b d e+a e^2\right )}{6 e^4 (d+e x)^6}-\frac {3 c d^2-e (2 b d-a e)}{5 e^4 (d+e x)^5}+\frac {3 c d-b e}{4 e^4 (d+e x)^4}-\frac {c}{3 e^4 (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 77, normalized size = 0.75 \begin {gather*} -\frac {e \left (2 a e (d+6 e x)+b \left (d^2+6 d e x+15 e^2 x^2\right )\right )+c \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )}{60 e^4 (d+e x)^6} \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)^7} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.37, size = 137, normalized size = 1.33 \begin {gather*} -\frac {20 \, c e^{3} x^{3} + c d^{3} + b d^{2} e + 2 \, a d e^{2} + 15 \, {\left (c d e^{2} + b e^{3}\right )} x^{2} + 6 \, {\left (c d^{2} e + b d e^{2} + 2 \, a e^{3}\right )} x}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 78, normalized size = 0.76 \begin {gather*} -\frac {{\left (20 \, c x^{3} e^{3} + 15 \, c d x^{2} e^{2} + 6 \, c d^{2} x e + c d^{3} + 15 \, b x^{2} e^{3} + 6 \, b d x e^{2} + b d^{2} e + 12 \, a x e^{3} + 2 \, a d e^{2}\right )} e^{\left (-4\right )}}{60 \, {\left (x e + d\right )}^{6}} \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}{3 \left (e x +d \right )^{3} e^{4}}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d}{6 \left (e x +d \right )^{6} e^{4}}-\frac {b e -3 c d}{4 \left (e x +d \right )^{4} e^{4}}-\frac {a \,e^{2}-2 b d e +3 c \,d^{2}}{5 \left (e x +d \right )^{5} e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 137, normalized size = 1.33 \begin {gather*} -\frac {20 \, c e^{3} x^{3} + c d^{3} + b d^{2} e + 2 \, a d e^{2} + 15 \, {\left (c d e^{2} + b e^{3}\right )} x^{2} + 6 \, {\left (c d^{2} e + b d e^{2} + 2 \, a e^{3}\right )} x}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.35, size = 133, normalized size = 1.29 \begin {gather*} -\frac {\frac {c\,x^3}{3\,e}+\frac {d\,\left (c\,d^2+b\,d\,e+2\,a\,e^2\right )}{60\,e^4}+\frac {x\,\left (c\,d^2+b\,d\,e+2\,a\,e^2\right )}{10\,e^3}+\frac {x^2\,\left (b\,e+c\,d\right )}{4\,e^2}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.83, size = 150, normalized size = 1.46 \begin {gather*} \frac {- 2 a d e^{2} - b d^{2} e - c d^{3} - 20 c e^{3} x^{3} + x^{2} \left (- 15 b e^{3} - 15 c d e^{2}\right ) + x \left (- 12 a e^{3} - 6 b d e^{2} - 6 c d^{2} e\right )}{60 d^{6} e^{4} + 360 d^{5} e^{5} x + 900 d^{4} e^{6} x^{2} + 1200 d^{3} e^{7} x^{3} + 900 d^{2} e^{8} x^{4} + 360 d e^{9} x^{5} + 60 e^{10} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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