Optimal. Leaf size=101 \[ -\frac {3 c d^2-e (2 b d-a e)}{3 e^4 (d+e x)^3}+\frac {d \left (a e^2-b d e+c d^2\right )}{4 e^4 (d+e x)^4}+\frac {3 c d-b e}{2 e^4 (d+e x)^2}-\frac {c}{e^4 (d+e x)} \]
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Rubi [A] time = 0.07, antiderivative size = 101, 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)}{3 e^4 (d+e x)^3}+\frac {d \left (a e^2-b d e+c d^2\right )}{4 e^4 (d+e x)^4}+\frac {3 c d-b e}{2 e^4 (d+e x)^2}-\frac {c}{e^4 (d+e x)} \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)^5} \, dx &=\int \left (-\frac {d \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^5}+\frac {3 c d^2-e (2 b d-a e)}{e^3 (d+e x)^4}+\frac {-3 c d+b e}{e^3 (d+e x)^3}+\frac {c}{e^3 (d+e x)^2}\right ) \, dx\\ &=\frac {d \left (c d^2-b d e+a e^2\right )}{4 e^4 (d+e x)^4}-\frac {3 c d^2-e (2 b d-a e)}{3 e^4 (d+e x)^3}+\frac {3 c d-b e}{2 e^4 (d+e x)^2}-\frac {c}{e^4 (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 77, normalized size = 0.76 \begin {gather*} -\frac {e \left (a e (d+4 e x)+b \left (d^2+4 d e x+6 e^2 x^2\right )\right )+3 c \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )}{12 e^4 (d+e x)^4} \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)^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.36, size = 116, normalized size = 1.15 \begin {gather*} -\frac {12 \, c e^{3} x^{3} + 3 \, c d^{3} + b d^{2} e + a d e^{2} + 6 \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{2} + 4 \, {\left (3 \, c d^{2} e + b d e^{2} + a e^{3}\right )} x}{12 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 128, normalized size = 1.27 \begin {gather*} -\frac {1}{12} \, {\left (\frac {12 \, c e^{\left (-1\right )}}{x e + d} - \frac {18 \, c d e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}} + \frac {12 \, c d^{2} e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac {3 \, c d^{3} e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac {6 \, b}{{\left (x e + d\right )}^{2}} - \frac {8 \, b d}{{\left (x e + d\right )}^{3}} + \frac {3 \, b d^{2}}{{\left (x e + d\right )}^{4}} + \frac {4 \, a e}{{\left (x e + d\right )}^{3}} - \frac {3 \, a d e}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-3\right )} \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 = 0.92 \begin {gather*} -\frac {c}{\left (e x +d \right ) e^{4}}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d}{4 \left (e x +d \right )^{4} e^{4}}-\frac {b e -3 c d}{2 \left (e x +d \right )^{2} e^{4}}-\frac {a \,e^{2}-2 b d e +3 c \,d^{2}}{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 = 0.47, size = 116, normalized size = 1.15 \begin {gather*} -\frac {12 \, c e^{3} x^{3} + 3 \, c d^{3} + b d^{2} e + a d e^{2} + 6 \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{2} + 4 \, {\left (3 \, c d^{2} e + b d e^{2} + a e^{3}\right )} x}{12 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} 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 = 111, normalized size = 1.10 \begin {gather*} -\frac {\frac {c\,x^3}{e}+\frac {d\,\left (3\,c\,d^2+b\,d\,e+a\,e^2\right )}{12\,e^4}+\frac {x\,\left (3\,c\,d^2+b\,d\,e+a\,e^2\right )}{3\,e^3}+\frac {x^2\,\left (b\,e+3\,c\,d\right )}{2\,e^2}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.21, size = 126, normalized size = 1.25 \begin {gather*} \frac {- a d e^{2} - b d^{2} e - 3 c d^{3} - 12 c e^{3} x^{3} + x^{2} \left (- 6 b e^{3} - 18 c d e^{2}\right ) + x \left (- 4 a e^{3} - 4 b d e^{2} - 12 c d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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