3.51.62 \(\int \frac {32 x+26 x^2+6 x^4-2 x^5+e^6 (2 x+2 x^2)+e^3 (-16 x-14 x^2-2 x^4)}{256+832 x+708 x^2-140 x^3-279 x^4+40 x^5+38 x^6-12 x^7+x^8+e^{12} (1+4 x+6 x^2+4 x^3+x^4)+e^9 (-16-60 x-80 x^2-40 x^3+4 x^5)+e^6 (96+340 x+398 x^2+124 x^3-60 x^4-24 x^5+6 x^6)+e^3 (-256-864 x-872 x^2-76 x^3+248 x^4+24 x^5-32 x^6+4 x^7)} \, dx\)

Optimal. Leaf size=24 \[ \frac {x^2}{2 x+(1+x)^2 \left (-4+e^3+x\right )^2} \]

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Rubi [F]  time = 1.47, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {32 x+26 x^2+6 x^4-2 x^5+e^6 \left (2 x+2 x^2\right )+e^3 \left (-16 x-14 x^2-2 x^4\right )}{256+832 x+708 x^2-140 x^3-279 x^4+40 x^5+38 x^6-12 x^7+x^8+e^{12} \left (1+4 x+6 x^2+4 x^3+x^4\right )+e^9 \left (-16-60 x-80 x^2-40 x^3+4 x^5\right )+e^6 \left (96+340 x+398 x^2+124 x^3-60 x^4-24 x^5+6 x^6\right )+e^3 \left (-256-864 x-872 x^2-76 x^3+248 x^4+24 x^5-32 x^6+4 x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(32*x + 26*x^2 + 6*x^4 - 2*x^5 + E^6*(2*x + 2*x^2) + E^3*(-16*x - 14*x^2 - 2*x^4))/(256 + 832*x + 708*x^2
- 140*x^3 - 279*x^4 + 40*x^5 + 38*x^6 - 12*x^7 + x^8 + E^12*(1 + 4*x + 6*x^2 + 4*x^3 + x^4) + E^9*(-16 - 60*x
- 80*x^2 - 40*x^3 + 4*x^5) + E^6*(96 + 340*x + 398*x^2 + 124*x^3 - 60*x^4 - 24*x^5 + 6*x^6) + E^3*(-256 - 864*
x - 872*x^2 - 76*x^3 + 248*x^4 + 24*x^5 - 32*x^6 + 4*x^7)),x]

[Out]

(17 - 8*E^3 + E^6)/(2*((-4 + E^3)^2 + 2*(13 - 7*E^3 + E^6)*x + (1 - 4*E^3 + E^6)*x^2 - 2*(3 - E^3)*x^3 + x^4))
+ 2*(3 - E^3)*Defer[Int][(-(4 - E^3)^2 - 2*(13 - 7*E^3 + E^6)*x - (1 - 4*E^3 + E^6)*x^2 + 2*(3 - E^3)*x^3 - x
^4)^(-1), x] + 2*Defer[Int][x/(-(4 - E^3)^2 - 2*(13 - 7*E^3 + E^6)*x - (1 - 4*E^3 + E^6)*x^2 + 2*(3 - E^3)*x^3
- x^4), x] + (317 - 303*E^3 + 108*E^6 - 17*E^9 + E^12)*Defer[Int][((-4 + E^3)^2 + 2*(13 - 7*E^3 + E^6)*x + (1
- 4*E^3 + E^6)*x^2 - 2*(3 - E^3)*x^3 + x^4)^(-2), x] + (237 - 244*E^3 + 94*E^6 - 16*E^9 + E^12)*Defer[Int][x/
((-4 + E^3)^2 + 2*(13 - 7*E^3 + E^6)*x + (1 - 4*E^3 + E^6)*x^2 - 2*(3 - E^3)*x^3 + x^4)^2, x] - (69 - 55*E^3 +
13*E^6 - E^9)*Defer[Int][x^2/((-4 + E^3)^2 + 2*(13 - 7*E^3 + E^6)*x + (1 - 4*E^3 + E^6)*x^2 - 2*(3 - E^3)*x^3
+ x^4)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 \left (\left (3-e^3\right ) \left (4-e^3\right )^2+2 \left (55-42 e^3+11 e^6-e^9\right ) x+\left (42-34 e^3+10 e^6-e^9\right ) x^2-\left (17-8 e^3+e^6\right ) x^3\right )}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2}+\frac {2 \left (-3+e^3-x\right )}{\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4}\right ) \, dx\\ &=2 \int \frac {\left (3-e^3\right ) \left (4-e^3\right )^2+2 \left (55-42 e^3+11 e^6-e^9\right ) x+\left (42-34 e^3+10 e^6-e^9\right ) x^2-\left (17-8 e^3+e^6\right ) x^3}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2} \, dx+2 \int \frac {-3+e^3-x}{\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4} \, dx\\ &=\frac {17-8 e^3+e^6}{2 \left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )}+\frac {1}{2} \int \frac {2 \left (317-303 e^3+108 e^6-17 e^9+e^{12}\right )+2 \left (237-244 e^3+94 e^6-16 e^9+e^{12}\right ) x-2 \left (69-55 e^3+13 e^6-e^9\right ) x^2}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2} \, dx+2 \int \left (\frac {3 \left (1-\frac {e^3}{3}\right )}{-\left (-4+e^3\right )^2-2 \left (13-7 e^3+e^6\right ) x-\left (1-4 e^3+e^6\right ) x^2+2 \left (3-e^3\right ) x^3-x^4}+\frac {x}{-\left (-4+e^3\right )^2-2 \left (13-7 e^3+e^6\right ) x-\left (1-4 e^3+e^6\right ) x^2+2 \left (3-e^3\right ) x^3-x^4}\right ) \, dx\\ &=\frac {17-8 e^3+e^6}{2 \left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )}+\frac {1}{2} \int \left (\frac {2 \left (317-303 e^3+108 e^6-17 e^9+e^{12}\right )}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2}+\frac {2 \left (237-244 e^3+94 e^6-16 e^9+e^{12}\right ) x}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2}+\frac {2 \left (-69+55 e^3-13 e^6+e^9\right ) x^2}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2}\right ) \, dx+2 \int \frac {x}{-\left (-4+e^3\right )^2-2 \left (13-7 e^3+e^6\right ) x-\left (1-4 e^3+e^6\right ) x^2+2 \left (3-e^3\right ) x^3-x^4} \, dx+\left (2 \left (3-e^3\right )\right ) \int \frac {1}{-\left (-4+e^3\right )^2-2 \left (13-7 e^3+e^6\right ) x-\left (1-4 e^3+e^6\right ) x^2+2 \left (3-e^3\right ) x^3-x^4} \, dx\\ &=\frac {17-8 e^3+e^6}{2 \left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )}+2 \int \frac {x}{-\left (4-e^3\right )^2-2 \left (13-7 e^3+e^6\right ) x-\left (1-4 e^3+e^6\right ) x^2+2 \left (3-e^3\right ) x^3-x^4} \, dx+\left (2 \left (3-e^3\right )\right ) \int \frac {1}{-\left (4-e^3\right )^2-2 \left (13-7 e^3+e^6\right ) x-\left (1-4 e^3+e^6\right ) x^2+2 \left (3-e^3\right ) x^3-x^4} \, dx+\left (-69+55 e^3-13 e^6+e^9\right ) \int \frac {x^2}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2} \, dx+\left (317-303 e^3+108 e^6-17 e^9+e^{12}\right ) \int \frac {1}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2} \, dx+\left (237-244 e^3+94 e^6-16 e^9+e^{12}\right ) \int \frac {x}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 44, normalized size = 1.83 \begin {gather*} \frac {x^2}{16+26 x+x^2-6 x^3+x^4+e^6 (1+x)^2+2 e^3 (-4+x) (1+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32*x + 26*x^2 + 6*x^4 - 2*x^5 + E^6*(2*x + 2*x^2) + E^3*(-16*x - 14*x^2 - 2*x^4))/(256 + 832*x + 70
8*x^2 - 140*x^3 - 279*x^4 + 40*x^5 + 38*x^6 - 12*x^7 + x^8 + E^12*(1 + 4*x + 6*x^2 + 4*x^3 + x^4) + E^9*(-16 -
60*x - 80*x^2 - 40*x^3 + 4*x^5) + E^6*(96 + 340*x + 398*x^2 + 124*x^3 - 60*x^4 - 24*x^5 + 6*x^6) + E^3*(-256
- 864*x - 872*x^2 - 76*x^3 + 248*x^4 + 24*x^5 - 32*x^6 + 4*x^7)),x]

[Out]

x^2/(16 + 26*x + x^2 - 6*x^3 + x^4 + E^6*(1 + x)^2 + 2*E^3*(-4 + x)*(1 + x)^2)

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fricas [B]  time = 0.76, size = 50, normalized size = 2.08 \begin {gather*} \frac {x^{2}}{x^{4} - 6 \, x^{3} + x^{2} + {\left (x^{2} + 2 \, x + 1\right )} e^{6} + 2 \, {\left (x^{3} - 2 \, x^{2} - 7 \, x - 4\right )} e^{3} + 26 \, x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2*x)*exp(3)^2+(-2*x^4-14*x^2-16*x)*exp(3)-2*x^5+6*x^4+26*x^2+32*x)/((x^4+4*x^3+6*x^2+4*x+1)*
exp(3)^4+(4*x^5-40*x^3-80*x^2-60*x-16)*exp(3)^3+(6*x^6-24*x^5-60*x^4+124*x^3+398*x^2+340*x+96)*exp(3)^2+(4*x^7
-32*x^6+24*x^5+248*x^4-76*x^3-872*x^2-864*x-256)*exp(3)+x^8-12*x^7+38*x^6+40*x^5-279*x^4-140*x^3+708*x^2+832*x
+256),x, algorithm="fricas")

[Out]

x^2/(x^4 - 6*x^3 + x^2 + (x^2 + 2*x + 1)*e^6 + 2*(x^3 - 2*x^2 - 7*x - 4)*e^3 + 26*x + 16)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2*x)*exp(3)^2+(-2*x^4-14*x^2-16*x)*exp(3)-2*x^5+6*x^4+26*x^2+32*x)/((x^4+4*x^3+6*x^2+4*x+1)*
exp(3)^4+(4*x^5-40*x^3-80*x^2-60*x-16)*exp(3)^3+(6*x^6-24*x^5-60*x^4+124*x^3+398*x^2+340*x+96)*exp(3)^2+(4*x^7
-32*x^6+24*x^5+248*x^4-76*x^3-872*x^2-864*x-256)*exp(3)+x^8-12*x^7+38*x^6+40*x^5-279*x^4-140*x^3+708*x^2+832*x
+256),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.82, size = 59, normalized size = 2.46

method	result	size
risch	\(\frac {x^{2}}{x^{2} {\mathrm e}^{6}+2 x^{3} {\mathrm e}^{3}+x^{4}+2 x \,{\mathrm e}^{6}-4 x^{2} {\mathrm e}^{3}-6 x^{3}+{\mathrm e}^{6}-14 x \,{\mathrm e}^{3}+x^{2}-8 \,{\mathrm e}^{3}+26 x +16}\)	\(59\)
gosper	\(\frac {x^{2}}{x^{2} {\mathrm e}^{6}+2 x^{3} {\mathrm e}^{3}+x^{4}+2 x \,{\mathrm e}^{6}-4 x^{2} {\mathrm e}^{3}-6 x^{3}+{\mathrm e}^{6}-14 x \,{\mathrm e}^{3}+x^{2}-8 \,{\mathrm e}^{3}+26 x +16}\)	\(65\)
norman	\(\frac {x^{2}}{x^{2} {\mathrm e}^{6}+2 x^{3} {\mathrm e}^{3}+x^{4}+2 x \,{\mathrm e}^{6}-4 x^{2} {\mathrm e}^{3}-6 x^{3}+{\mathrm e}^{6}-14 x \,{\mathrm e}^{3}+x^{2}-8 \,{\mathrm e}^{3}+26 x +16}\)	\(65\)
default	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}+\left (4 \,{\mathrm e}^{3}-12\right ) \textit {\_Z}^{7}+\left (6 \,{\mathrm e}^{6}-32 \,{\mathrm e}^{3}+38\right ) \textit {\_Z}^{6}+\left (-24 \,{\mathrm e}^{6}+4 \,{\mathrm e}^{9}+24 \,{\mathrm e}^{3}+40\right ) \textit {\_Z}^{5}+\left (-60 \,{\mathrm e}^{6}+{\mathrm e}^{12}+248 \,{\mathrm e}^{3}-279\right ) \textit {\_Z}^{4}+\left (124 \,{\mathrm e}^{6}-40 \,{\mathrm e}^{9}+4 \,{\mathrm e}^{12}-76 \,{\mathrm e}^{3}-140\right ) \textit {\_Z}^{3}+\left (398 \,{\mathrm e}^{6}-80 \,{\mathrm e}^{9}+6 \,{\mathrm e}^{12}-872 \,{\mathrm e}^{3}+708\right ) \textit {\_Z}^{2}+\left (340 \,{\mathrm e}^{6}-60 \,{\mathrm e}^{9}+4 \,{\mathrm e}^{12}-864 \,{\mathrm e}^{3}+832\right ) \textit {\_Z} +256-16 \,{\mathrm e}^{9}+96 \,{\mathrm e}^{6}-256 \,{\mathrm e}^{3}+{\mathrm e}^{12}\right )}{\sum }\frac {\left (-\textit {\_R}^{5}+\left (3-{\mathrm e}^{3}\right ) \textit {\_R}^{4}+\left ({\mathrm e}^{6}-7 \,{\mathrm e}^{3}+13\right ) \textit {\_R}^{2}+\left (16+{\mathrm e}^{6}-8 \,{\mathrm e}^{3}\right ) \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{208+354 \textit {\_R} -15 \,{\mathrm e}^{9}+85 \,{\mathrm e}^{6}-216 \,{\mathrm e}^{3}+7 \textit {\_R}^{6} {\mathrm e}^{3}+9 \textit {\_R}^{5} {\mathrm e}^{6}-48 \textit {\_R}^{5} {\mathrm e}^{3}-30 \textit {\_R}^{4} {\mathrm e}^{6}+30 \textit {\_R}^{4} {\mathrm e}^{3}-60 \textit {\_R}^{3} {\mathrm e}^{6}+248 \textit {\_R}^{3} {\mathrm e}^{3}-57 \textit {\_R}^{2} {\mathrm e}^{3}-436 \textit {\_R} \,{\mathrm e}^{3}+{\mathrm e}^{12}+2 \textit {\_R}^{7}-21 \textit {\_R}^{6}+57 \textit {\_R}^{5}+50 \textit {\_R}^{4}-279 \textit {\_R}^{3}-105 \textit {\_R}^{2}+5 \textit {\_R}^{4} {\mathrm e}^{9}+3 \textit {\_R}^{2} {\mathrm e}^{12}+\textit {\_R}^{3} {\mathrm e}^{12}+3 \textit {\_R} \,{\mathrm e}^{12}-30 \textit {\_R}^{2} {\mathrm e}^{9}-40 \textit {\_R} \,{\mathrm e}^{9}+93 \textit {\_R}^{2} {\mathrm e}^{6}+199 \textit {\_R} \,{\mathrm e}^{6}}\right )}{2}\)	\(354\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2+2*x)*exp(3)^2+(-2*x^4-14*x^2-16*x)*exp(3)-2*x^5+6*x^4+26*x^2+32*x)/((x^4+4*x^3+6*x^2+4*x+1)*exp(3)
^4+(4*x^5-40*x^3-80*x^2-60*x-16)*exp(3)^3+(6*x^6-24*x^5-60*x^4+124*x^3+398*x^2+340*x+96)*exp(3)^2+(4*x^7-32*x^
6+24*x^5+248*x^4-76*x^3-872*x^2-864*x-256)*exp(3)+x^8-12*x^7+38*x^6+40*x^5-279*x^4-140*x^3+708*x^2+832*x+256),
x,method=_RETURNVERBOSE)

[Out]

x^2/(x^2*exp(6)+2*x^3*exp(3)+x^4+2*x*exp(6)-4*x^2*exp(3)-6*x^3+exp(6)-14*x*exp(3)+x^2-8*exp(3)+26*x+16)

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maxima [B]  time = 0.35, size = 49, normalized size = 2.04 \begin {gather*} \frac {x^{2}}{x^{4} + 2 \, x^{3} {\left (e^{3} - 3\right )} + x^{2} {\left (e^{6} - 4 \, e^{3} + 1\right )} + 2 \, x {\left (e^{6} - 7 \, e^{3} + 13\right )} + e^{6} - 8 \, e^{3} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+2*x)*exp(3)^2+(-2*x^4-14*x^2-16*x)*exp(3)-2*x^5+6*x^4+26*x^2+32*x)/((x^4+4*x^3+6*x^2+4*x+1)*
exp(3)^4+(4*x^5-40*x^3-80*x^2-60*x-16)*exp(3)^3+(6*x^6-24*x^5-60*x^4+124*x^3+398*x^2+340*x+96)*exp(3)^2+(4*x^7
-32*x^6+24*x^5+248*x^4-76*x^3-872*x^2-864*x-256)*exp(3)+x^8-12*x^7+38*x^6+40*x^5-279*x^4-140*x^3+708*x^2+832*x
+256),x, algorithm="maxima")

[Out]

x^2/(x^4 + 2*x^3*(e^3 - 3) + x^2*(e^6 - 4*e^3 + 1) + 2*x*(e^6 - 7*e^3 + 13) + e^6 - 8*e^3 + 16)

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mupad [B]  time = 4.00, size = 51, normalized size = 2.12 \begin {gather*} \frac {x^2}{x^4+\left (2\,{\mathrm {e}}^3-6\right )\,x^3+\left ({\mathrm {e}}^6-4\,{\mathrm {e}}^3+1\right )\,x^2+\left (2\,{\mathrm {e}}^6-14\,{\mathrm {e}}^3+26\right )\,x-8\,{\mathrm {e}}^3+{\mathrm {e}}^6+16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((32*x + exp(6)*(2*x + 2*x^2) - exp(3)*(16*x + 14*x^2 + 2*x^4) + 26*x^2 + 6*x^4 - 2*x^5)/(832*x + exp(6)*(3
40*x + 398*x^2 + 124*x^3 - 60*x^4 - 24*x^5 + 6*x^6 + 96) + exp(12)*(4*x + 6*x^2 + 4*x^3 + x^4 + 1) - exp(3)*(8
64*x + 872*x^2 + 76*x^3 - 248*x^4 - 24*x^5 + 32*x^6 - 4*x^7 + 256) - exp(9)*(60*x + 80*x^2 + 40*x^3 - 4*x^5 +
16) + 708*x^2 - 140*x^3 - 279*x^4 + 40*x^5 + 38*x^6 - 12*x^7 + x^8 + 256),x)

[Out]

x^2/(exp(6) - 8*exp(3) + x^3*(2*exp(3) - 6) + x*(2*exp(6) - 14*exp(3) + 26) + x^2*(exp(6) - 4*exp(3) + 1) + x^
4 + 16)

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sympy [B]  time = 4.83, size = 53, normalized size = 2.21 \begin {gather*} \frac {x^{2}}{x^{4} + x^{3} \left (-6 + 2 e^{3}\right ) + x^{2} \left (- 4 e^{3} + 1 + e^{6}\right ) + x \left (- 14 e^{3} + 26 + 2 e^{6}\right ) - 8 e^{3} + 16 + e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2+2*x)*exp(3)**2+(-2*x**4-14*x**2-16*x)*exp(3)-2*x**5+6*x**4+26*x**2+32*x)/((x**4+4*x**3+6*x*
*2+4*x+1)*exp(3)**4+(4*x**5-40*x**3-80*x**2-60*x-16)*exp(3)**3+(6*x**6-24*x**5-60*x**4+124*x**3+398*x**2+340*x
+96)*exp(3)**2+(4*x**7-32*x**6+24*x**5+248*x**4-76*x**3-872*x**2-864*x-256)*exp(3)+x**8-12*x**7+38*x**6+40*x**
5-279*x**4-140*x**3+708*x**2+832*x+256),x)

[Out]

x**2/(x**4 + x**3*(-6 + 2*exp(3)) + x**2*(-4*exp(3) + 1 + exp(6)) + x*(-14*exp(3) + 26 + 2*exp(6)) - 8*exp(3)
+ 16 + exp(6))

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