∫ −24+4ex∕2+ex∕4(3−16x)+16x2+ex(24+24x−16x2+16x3+ex∕2(−4+4x)+ex∕4(−15+16x−16x2))____________________________________________________________________

________________________________________________________________________________________

Rubi [F] time = 3.65, 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 {-24+4 e^{x/2}+e^{x/4} (3-16 x)+16 x^2+e^x \left (24+24 x-16 x^2+16 x^3+e^{x/2} (-4+4 x)+e^{x/4} \left (-15+16 x-16 x^2\right )\right )}{36+48 x^2+4 e^{x/2} x^2+16 x^4+e^{x/4} \left (-24 x-16 x^3\right )} \, dx \end {gather*}

Int[(-24 + 4*E^(x/2) + E^(x/4)*(3 - 16*x) + 16*x^2 + E^x*(24 + 24*x - 16*x^2 + 16*x^3 + E^(x/2)*(-4 + 4*x) + E

^(x/4)*(-15 + 16*x - 16*x^2)))/(36 + 48*x^2 + 4*E^(x/2)*x^2 + 16*x^4 + E^(x/4)*(-24*x - 16*x^3)),x]

12*E^(x/4) + 81/x^5 + (27*E^(x/4))/x^4 + 162/x^3 + (9*E^(x/2))/x^3 + (36*E^(x/4))/x^2 + (3*E^((3*x)/4))/x^2 +

107/x + (6*E^(x/2))/x + E^x/x + 24*x + 426*Defer[Int][(-3 + E^(x/4)*x - 2*x^2)^(-2), x] - 729*Defer[Int][1/(x^

6*(3 - E^(x/4)*x + 2*x^2)^2), x] - (729*Defer[Int][1/(x^5*(3 - E^(x/4)*x + 2*x^2)^2), x])/4 - 1458*Defer[Int][

1/(x^4*(3 - E^(x/4)*x + 2*x^2)^2), x] - (1215*Defer[Int][1/(x^3*(3 - E^(x/4)*x + 2*x^2)^2), x])/2 - 639*Defer[

Int][1/(x^2*(3 - E^(x/4)*x + 2*x^2)^2), x] - (3231*Defer[Int][1/(x*(3 - E^(x/4)*x + 2*x^2)^2), x])/4 - (1077*D

efer[Int][x/(3 - E^(x/4)*x + 2*x^2)^2, x])/2 + 432*Defer[Int][x^2/(3 - E^(x/4)*x + 2*x^2)^2, x] - 180*Defer[In

t][x^3/(3 - E^(x/4)*x + 2*x^2)^2, x] + 96*Defer[Int][x^4/(3 - E^(x/4)*x + 2*x^2)^2, x] - 24*Defer[Int][x^5/(3

- E^(x/4)*x + 2*x^2)^2, x] + 1458*Defer[Int][1/(x^6*(3 - E^(x/4)*x + 2*x^2)), x] + (243*Defer[Int][1/(x^5*(3 -

E^(x/4)*x + 2*x^2)), x])/4 + 2592*Defer[Int][1/(x^4*(3 - E^(x/4)*x + 2*x^2)), x] + 162*Defer[Int][1/(x^3*(3 -

E^(x/4)*x + 2*x^2)), x] + 1290*Defer[Int][1/(x^2*(3 - E^(x/4)*x + 2*x^2)), x] + (645*Defer[Int][1/(x*(3 - E^(

x/4)*x + 2*x^2)), x])/4 + 72*Defer[Int][x/(3 - E^(x/4)*x + 2*x^2), x] - 96*Defer[Int][x^2/(3 - E^(x/4)*x + 2*x

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-24+4 e^{x/2}+e^{x/4} (3-16 x)+16 x^2+e^x \left (24+24 x-16 x^2+16 x^3+e^{x/2} (-4+4 x)+e^{x/4} \left (-15+16 x-16 x^2\right )\right )}{4 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx\\ &=\frac {1}{4} \int \frac {-24+4 e^{x/2}+e^{x/4} (3-16 x)+16 x^2+e^x \left (24+24 x-16 x^2+16 x^3+e^{x/2} (-4+4 x)+e^{x/4} \left (-15+16 x-16 x^2\right )\right )}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx\\ &=\frac {1}{4} \int \left (\frac {4 e^x (-1+x)}{x^2}+\frac {3 e^{3 x/4} (-8+3 x)}{x^3}+\frac {6 e^{x/2} \left (-18+3 x-4 x^2+2 x^3\right )}{x^4}+\frac {3 e^{x/4} \left (-144+9 x-96 x^2+12 x^3+4 x^5\right )}{x^5}+\frac {4 \left (-405-486 x^2-107 x^4+24 x^6\right )}{x^6}+\frac {3 \left (1944+81 x+3456 x^2+216 x^3+1720 x^4+215 x^5+96 x^7-128 x^8+16 x^9\right )}{x^6 \left (3-e^{x/4} x+2 x^2\right )}-\frac {3 \left (972+243 x+1944 x^2+810 x^3+852 x^4+1077 x^5-568 x^6+718 x^7-576 x^8+240 x^9-128 x^{10}+32 x^{11}\right )}{x^6 \left (3-e^{x/4} x+2 x^2\right )^2}\right ) \, dx\\ &=\frac {3}{4} \int \frac {e^{3 x/4} (-8+3 x)}{x^3} \, dx+\frac {3}{4} \int \frac {e^{x/4} \left (-144+9 x-96 x^2+12 x^3+4 x^5\right )}{x^5} \, dx+\frac {3}{4} \int \frac {1944+81 x+3456 x^2+216 x^3+1720 x^4+215 x^5+96 x^7-128 x^8+16 x^9}{x^6 \left (3-e^{x/4} x+2 x^2\right )} \, dx-\frac {3}{4} \int \frac {972+243 x+1944 x^2+810 x^3+852 x^4+1077 x^5-568 x^6+718 x^7-576 x^8+240 x^9-128 x^{10}+32 x^{11}}{x^6 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+\frac {3}{2} \int \frac {e^{x/2} \left (-18+3 x-4 x^2+2 x^3\right )}{x^4} \, dx+\int \frac {e^x (-1+x)}{x^2} \, dx+\int \frac {-405-486 x^2-107 x^4+24 x^6}{x^6} \, dx\\ &=\frac {3 e^{3 x/4}}{x^2}+\frac {e^x}{x}+\frac {3}{4} \int \left (4 e^{x/4}-\frac {144 e^{x/4}}{x^5}+\frac {9 e^{x/4}}{x^4}-\frac {96 e^{x/4}}{x^3}+\frac {12 e^{x/4}}{x^2}\right ) \, dx-\frac {3}{4} \int \left (-\frac {568}{\left (-3+e^{x/4} x-2 x^2\right )^2}+\frac {972}{x^6 \left (3-e^{x/4} x+2 x^2\right )^2}+\frac {243}{x^5 \left (3-e^{x/4} x+2 x^2\right )^2}+\frac {1944}{x^4 \left (3-e^{x/4} x+2 x^2\right )^2}+\frac {810}{x^3 \left (3-e^{x/4} x+2 x^2\right )^2}+\frac {852}{x^2 \left (3-e^{x/4} x+2 x^2\right )^2}+\frac {1077}{x \left (3-e^{x/4} x+2 x^2\right )^2}+\frac {718 x}{\left (3-e^{x/4} x+2 x^2\right )^2}-\frac {576 x^2}{\left (3-e^{x/4} x+2 x^2\right )^2}+\frac {240 x^3}{\left (3-e^{x/4} x+2 x^2\right )^2}-\frac {128 x^4}{\left (3-e^{x/4} x+2 x^2\right )^2}+\frac {32 x^5}{\left (3-e^{x/4} x+2 x^2\right )^2}\right ) \, dx+\frac {3}{4} \int \left (\frac {1944}{x^6 \left (3-e^{x/4} x+2 x^2\right )}+\frac {81}{x^5 \left (3-e^{x/4} x+2 x^2\right )}+\frac {3456}{x^4 \left (3-e^{x/4} x+2 x^2\right )}+\frac {216}{x^3 \left (3-e^{x/4} x+2 x^2\right )}+\frac {1720}{x^2 \left (3-e^{x/4} x+2 x^2\right )}+\frac {215}{x \left (3-e^{x/4} x+2 x^2\right )}+\frac {96 x}{3-e^{x/4} x+2 x^2}-\frac {128 x^2}{3-e^{x/4} x+2 x^2}+\frac {16 x^3}{3-e^{x/4} x+2 x^2}\right ) \, dx+\frac {3}{2} \int \left (-\frac {18 e^{x/2}}{x^4}+\frac {3 e^{x/2}}{x^3}-\frac {4 e^{x/2}}{x^2}+\frac {2 e^{x/2}}{x}\right ) \, dx+\int \left (24-\frac {405}{x^6}-\frac {486}{x^4}-\frac {107}{x^2}\right ) \, dx\\ &=\frac {81}{x^5}+\frac {162}{x^3}+\frac {3 e^{3 x/4}}{x^2}+\frac {107}{x}+\frac {e^x}{x}+24 x+3 \int e^{x/4} \, dx+3 \int \frac {e^{x/2}}{x} \, dx+\frac {9}{2} \int \frac {e^{x/2}}{x^3} \, dx-6 \int \frac {e^{x/2}}{x^2} \, dx+\frac {27}{4} \int \frac {e^{x/4}}{x^4} \, dx+9 \int \frac {e^{x/4}}{x^2} \, dx+12 \int \frac {x^3}{3-e^{x/4} x+2 x^2} \, dx-24 \int \frac {x^5}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-27 \int \frac {e^{x/2}}{x^4} \, dx+\frac {243}{4} \int \frac {1}{x^5 \left (3-e^{x/4} x+2 x^2\right )} \, dx-72 \int \frac {e^{x/4}}{x^3} \, dx+72 \int \frac {x}{3-e^{x/4} x+2 x^2} \, dx+96 \int \frac {x^4}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-96 \int \frac {x^2}{3-e^{x/4} x+2 x^2} \, dx-108 \int \frac {e^{x/4}}{x^5} \, dx+\frac {645}{4} \int \frac {1}{x \left (3-e^{x/4} x+2 x^2\right )} \, dx+162 \int \frac {1}{x^3 \left (3-e^{x/4} x+2 x^2\right )} \, dx-180 \int \frac {x^3}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {729}{4} \int \frac {1}{x^5 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+426 \int \frac {1}{\left (-3+e^{x/4} x-2 x^2\right )^2} \, dx+432 \int \frac {x^2}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {1077}{2} \int \frac {x}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {1215}{2} \int \frac {1}{x^3 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-639 \int \frac {1}{x^2 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-729 \int \frac {1}{x^6 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {3231}{4} \int \frac {1}{x \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+1290 \int \frac {1}{x^2 \left (3-e^{x/4} x+2 x^2\right )} \, dx-1458 \int \frac {1}{x^4 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+1458 \int \frac {1}{x^6 \left (3-e^{x/4} x+2 x^2\right )} \, dx+2592 \int \frac {1}{x^4 \left (3-e^{x/4} x+2 x^2\right )} \, dx\\ &=12 e^{x/4}+\frac {81}{x^5}+\frac {27 e^{x/4}}{x^4}+\frac {162}{x^3}-\frac {9 e^{x/4}}{4 x^3}+\frac {9 e^{x/2}}{x^3}+\frac {36 e^{x/4}}{x^2}-\frac {9 e^{x/2}}{4 x^2}+\frac {3 e^{3 x/4}}{x^2}+\frac {107}{x}-\frac {9 e^{x/4}}{x}+\frac {6 e^{x/2}}{x}+\frac {e^x}{x}+24 x+3 \text {Ei}\left (\frac {x}{2}\right )+\frac {9}{16} \int \frac {e^{x/4}}{x^3} \, dx+\frac {9}{8} \int \frac {e^{x/2}}{x^2} \, dx+\frac {9}{4} \int \frac {e^{x/4}}{x} \, dx-3 \int \frac {e^{x/2}}{x} \, dx-\frac {9}{2} \int \frac {e^{x/2}}{x^3} \, dx-\frac {27}{4} \int \frac {e^{x/4}}{x^4} \, dx-9 \int \frac {e^{x/4}}{x^2} \, dx+12 \int \frac {x^3}{3-e^{x/4} x+2 x^2} \, dx-24 \int \frac {x^5}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx+\frac {243}{4} \int \frac {1}{x^5 \left (3-e^{x/4} x+2 x^2\right )} \, dx+72 \int \frac {x}{3-e^{x/4} x+2 x^2} \, dx+96 \int \frac {x^4}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-96 \int \frac {x^2}{3-e^{x/4} x+2 x^2} \, dx+\frac {645}{4} \int \frac {1}{x \left (3-e^{x/4} x+2 x^2\right )} \, dx+162 \int \frac {1}{x^3 \left (3-e^{x/4} x+2 x^2\right )} \, dx-180 \int \frac {x^3}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {729}{4} \int \frac {1}{x^5 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+426 \int \frac {1}{\left (-3+e^{x/4} x-2 x^2\right )^2} \, dx+432 \int \frac {x^2}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {1077}{2} \int \frac {x}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {1215}{2} \int \frac {1}{x^3 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-639 \int \frac {1}{x^2 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-729 \int \frac {1}{x^6 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {3231}{4} \int \frac {1}{x \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+1290 \int \frac {1}{x^2 \left (3-e^{x/4} x+2 x^2\right )} \, dx-1458 \int \frac {1}{x^4 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+1458 \int \frac {1}{x^6 \left (3-e^{x/4} x+2 x^2\right )} \, dx+2592 \int \frac {1}{x^4 \left (3-e^{x/4} x+2 x^2\right )} \, dx\\ &=12 e^{x/4}+\frac {81}{x^5}+\frac {27 e^{x/4}}{x^4}+\frac {162}{x^3}+\frac {9 e^{x/2}}{x^3}+\frac {1143 e^{x/4}}{32 x^2}+\frac {3 e^{3 x/4}}{x^2}+\frac {107}{x}+\frac {39 e^{x/2}}{8 x}+\frac {e^x}{x}+24 x+\frac {9 \text {Ei}\left (\frac {x}{4}\right )}{4}+\frac {9}{128} \int \frac {e^{x/4}}{x^2} \, dx-\frac {9}{16} \int \frac {e^{x/4}}{x^3} \, dx+\frac {9}{16} \int \frac {e^{x/2}}{x} \, dx-\frac {9}{8} \int \frac {e^{x/2}}{x^2} \, dx-\frac {9}{4} \int \frac {e^{x/4}}{x} \, dx+12 \int \frac {x^3}{3-e^{x/4} x+2 x^2} \, dx-24 \int \frac {x^5}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx+\frac {243}{4} \int \frac {1}{x^5 \left (3-e^{x/4} x+2 x^2\right )} \, dx+72 \int \frac {x}{3-e^{x/4} x+2 x^2} \, dx+96 \int \frac {x^4}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-96 \int \frac {x^2}{3-e^{x/4} x+2 x^2} \, dx+\frac {645}{4} \int \frac {1}{x \left (3-e^{x/4} x+2 x^2\right )} \, dx+162 \int \frac {1}{x^3 \left (3-e^{x/4} x+2 x^2\right )} \, dx-180 \int \frac {x^3}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {729}{4} \int \frac {1}{x^5 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+426 \int \frac {1}{\left (-3+e^{x/4} x-2 x^2\right )^2} \, dx+432 \int \frac {x^2}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {1077}{2} \int \frac {x}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {1215}{2} \int \frac {1}{x^3 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-639 \int \frac {1}{x^2 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-729 \int \frac {1}{x^6 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {3231}{4} \int \frac {1}{x \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+1290 \int \frac {1}{x^2 \left (3-e^{x/4} x+2 x^2\right )} \, dx-1458 \int \frac {1}{x^4 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+1458 \int \frac {1}{x^6 \left (3-e^{x/4} x+2 x^2\right )} \, dx+2592 \int \frac {1}{x^4 \left (3-e^{x/4} x+2 x^2\right )} \, dx\\ &=12 e^{x/4}+\frac {81}{x^5}+\frac {27 e^{x/4}}{x^4}+\frac {162}{x^3}+\frac {9 e^{x/2}}{x^3}+\frac {36 e^{x/4}}{x^2}+\frac {3 e^{3 x/4}}{x^2}+\frac {107}{x}-\frac {9 e^{x/4}}{128 x}+\frac {6 e^{x/2}}{x}+\frac {e^x}{x}+24 x+\frac {9 \text {Ei}\left (\frac {x}{2}\right )}{16}+\frac {9}{512} \int \frac {e^{x/4}}{x} \, dx-\frac {9}{128} \int \frac {e^{x/4}}{x^2} \, dx-\frac {9}{16} \int \frac {e^{x/2}}{x} \, dx+12 \int \frac {x^3}{3-e^{x/4} x+2 x^2} \, dx-24 \int \frac {x^5}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx+\frac {243}{4} \int \frac {1}{x^5 \left (3-e^{x/4} x+2 x^2\right )} \, dx+72 \int \frac {x}{3-e^{x/4} x+2 x^2} \, dx+96 \int \frac {x^4}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-96 \int \frac {x^2}{3-e^{x/4} x+2 x^2} \, dx+\frac {645}{4} \int \frac {1}{x \left (3-e^{x/4} x+2 x^2\right )} \, dx+162 \int \frac {1}{x^3 \left (3-e^{x/4} x+2 x^2\right )} \, dx-180 \int \frac {x^3}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {729}{4} \int \frac {1}{x^5 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+426 \int \frac {1}{\left (-3+e^{x/4} x-2 x^2\right )^2} \, dx+432 \int \frac {x^2}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {1077}{2} \int \frac {x}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {1215}{2} \int \frac {1}{x^3 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-639 \int \frac {1}{x^2 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-729 \int \frac {1}{x^6 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {3231}{4} \int \frac {1}{x \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+1290 \int \frac {1}{x^2 \left (3-e^{x/4} x+2 x^2\right )} \, dx-1458 \int \frac {1}{x^4 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+1458 \int \frac {1}{x^6 \left (3-e^{x/4} x+2 x^2\right )} \, dx+2592 \int \frac {1}{x^4 \left (3-e^{x/4} x+2 x^2\right )} \, dx\\ &=12 e^{x/4}+\frac {81}{x^5}+\frac {27 e^{x/4}}{x^4}+\frac {162}{x^3}+\frac {9 e^{x/2}}{x^3}+\frac {36 e^{x/4}}{x^2}+\frac {3 e^{3 x/4}}{x^2}+\frac {107}{x}+\frac {6 e^{x/2}}{x}+\frac {e^x}{x}+24 x+\frac {9 \text {Ei}\left (\frac {x}{4}\right )}{512}-\frac {9}{512} \int \frac {e^{x/4}}{x} \, dx+12 \int \frac {x^3}{3-e^{x/4} x+2 x^2} \, dx-24 \int \frac {x^5}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx+\frac {243}{4} \int \frac {1}{x^5 \left (3-e^{x/4} x+2 x^2\right )} \, dx+72 \int \frac {x}{3-e^{x/4} x+2 x^2} \, dx+96 \int \frac {x^4}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-96 \int \frac {x^2}{3-e^{x/4} x+2 x^2} \, dx+\frac {645}{4} \int \frac {1}{x \left (3-e^{x/4} x+2 x^2\right )} \, dx+162 \int \frac {1}{x^3 \left (3-e^{x/4} x+2 x^2\right )} \, dx-180 \int \frac {x^3}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {729}{4} \int \frac {1}{x^5 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+426 \int \frac {1}{\left (-3+e^{x/4} x-2 x^2\right )^2} \, dx+432 \int \frac {x^2}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {1077}{2} \int \frac {x}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {1215}{2} \int \frac {1}{x^3 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-639 \int \frac {1}{x^2 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-729 \int \frac {1}{x^6 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {3231}{4} \int \frac {1}{x \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+1290 \int \frac {1}{x^2 \left (3-e^{x/4} x+2 x^2\right )} \, dx-1458 \int \frac {1}{x^4 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+1458 \int \frac {1}{x^6 \left (3-e^{x/4} x+2 x^2\right )} \, dx+2592 \int \frac {1}{x^4 \left (3-e^{x/4} x+2 x^2\right )} \, dx\\ &=12 e^{x/4}+\frac {81}{x^5}+\frac {27 e^{x/4}}{x^4}+\frac {162}{x^3}+\frac {9 e^{x/2}}{x^3}+\frac {36 e^{x/4}}{x^2}+\frac {3 e^{3 x/4}}{x^2}+\frac {107}{x}+\frac {6 e^{x/2}}{x}+\frac {e^x}{x}+24 x+12 \int \frac {x^3}{3-e^{x/4} x+2 x^2} \, dx-24 \int \frac {x^5}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx+\frac {243}{4} \int \frac {1}{x^5 \left (3-e^{x/4} x+2 x^2\right )} \, dx+72 \int \frac {x}{3-e^{x/4} x+2 x^2} \, dx+96 \int \frac {x^4}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-96 \int \frac {x^2}{3-e^{x/4} x+2 x^2} \, dx+\frac {645}{4} \int \frac {1}{x \left (3-e^{x/4} x+2 x^2\right )} \, dx+162 \int \frac {1}{x^3 \left (3-e^{x/4} x+2 x^2\right )} \, dx-180 \int \frac {x^3}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {729}{4} \int \frac {1}{x^5 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+426 \int \frac {1}{\left (-3+e^{x/4} x-2 x^2\right )^2} \, dx+432 \int \frac {x^2}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {1077}{2} \int \frac {x}{\left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {1215}{2} \int \frac {1}{x^3 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-639 \int \frac {1}{x^2 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-729 \int \frac {1}{x^6 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx-\frac {3231}{4} \int \frac {1}{x \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+1290 \int \frac {1}{x^2 \left (3-e^{x/4} x+2 x^2\right )} \, dx-1458 \int \frac {1}{x^4 \left (3-e^{x/4} x+2 x^2\right )^2} \, dx+1458 \int \frac {1}{x^6 \left (3-e^{x/4} x+2 x^2\right )} \, dx+2592 \int \frac {1}{x^4 \left (3-e^{x/4} x+2 x^2\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A] time = 0.06, size = 35, normalized size = 1.40 \begin {gather*} \frac {\left (-1+e^x\right ) \left (e^{x/4}-2 x\right )}{-3+e^{x/4} x-2 x^2} \end {gather*}

Integrate[(-24 + 4*E^(x/2) + E^(x/4)*(3 - 16*x) + 16*x^2 + E^x*(24 + 24*x - 16*x^2 + 16*x^3 + E^(x/2)*(-4 + 4*

x) + E^(x/4)*(-15 + 16*x - 16*x^2)))/(36 + 48*x^2 + 4*E^(x/2)*x^2 + 16*x^4 + E^(x/4)*(-24*x - 16*x^3)),x]

________________________________________________________________________________________

fricas [A] time = 0.58, size = 36, normalized size = 1.44 \begin {gather*} \frac {2 \, x e^{x} - 2 \, x - e^{\left (\frac {5}{4} \, x\right )} + e^{\left (\frac {1}{4} \, x\right )}}{2 \, x^{2} - x e^{\left (\frac {1}{4} \, x\right )} + 3} \end {gather*}

integrate((((4*x-4)*exp(1/4*x)^2+(-16*x^2+16*x-15)*exp(1/4*x)+16*x^3-16*x^2+24*x+24)*exp(x)+4*exp(1/4*x)^2+(3-

16*x)*exp(1/4*x)+16*x^2-24)/(4*x^2*exp(1/4*x)^2+(-16*x^3-24*x)*exp(1/4*x)+16*x^4+48*x^2+36),x, algorithm="fric

________________________________________________________________________________________

giac [B] time = 0.18, size = 53, normalized size = 2.12 \begin {gather*} \frac {2 \, x^{4} e^{x} - 2 \, x^{4} - x^{3} e^{\left (\frac {5}{4} \, x\right )} + x^{3} e^{\left (\frac {1}{4} \, x\right )}}{2 \, x^{5} - x^{4} e^{\left (\frac {1}{4} \, x\right )} + 3 \, x^{3}} \end {gather*}

integrate((((4*x-4)*exp(1/4*x)^2+(-16*x^2+16*x-15)*exp(1/4*x)+16*x^3-16*x^2+24*x+24)*exp(x)+4*exp(1/4*x)^2+(3-

16*x)*exp(1/4*x)+16*x^2-24)/(4*x^2*exp(1/4*x)^2+(-16*x^3-24*x)*exp(1/4*x)+16*x^4+48*x^2+36),x, algorithm="giac

________________________________________________________________________________________


method	result	size

risch	\(24 x +\frac {107 x^{4}+162 x^{2}+81}{x^{5}}+\frac {{\mathrm e}^{x}}{x}+\frac {3 \,{\mathrm e}^{\frac {3 x}{4}}}{x^{2}}+\frac {3 \left (2 x^{2}+3\right ) {\mathrm e}^{\frac {x}{2}}}{x^{3}}+\frac {3 \left (4 x^{4}+12 x^{2}+9\right ) {\mathrm e}^{\frac {x}{4}}}{x^{4}}-\frac {3 \left (16 x^{8}+96 x^{6}+215 x^{4}+216 x^{2}+81\right )}{x^{5} \left (2 x^{2}-x \,{\mathrm e}^{\frac {x}{4}}+3\right )}\)	\(116\)

int((((4*x-4)*exp(1/4*x)^2+(-16*x^2+16*x-15)*exp(1/4*x)+16*x^3-16*x^2+24*x+24)*exp(x)+4*exp(1/4*x)^2+(3-16*x)*

exp(1/4*x)+16*x^2-24)/(4*x^2*exp(1/4*x)^2+(-16*x^3-24*x)*exp(1/4*x)+16*x^4+48*x^2+36),x,method=_RETURNVERBOSE)

24*x+(107*x^4+162*x^2+81)/x^5+exp(x)/x+3/x^2*exp(3/4*x)+3*(2*x^2+3)/x^3*exp(1/2*x)+3*(4*x^4+12*x^2+9)/x^4*exp(

________________________________________________________________________________________

maxima [A] time = 0.38, size = 36, normalized size = 1.44 \begin {gather*} \frac {2 \, x e^{x} - 2 \, x - e^{\left (\frac {5}{4} \, x\right )} + e^{\left (\frac {1}{4} \, x\right )}}{2 \, x^{2} - x e^{\left (\frac {1}{4} \, x\right )} + 3} \end {gather*}

integrate((((4*x-4)*exp(1/4*x)^2+(-16*x^2+16*x-15)*exp(1/4*x)+16*x^3-16*x^2+24*x+24)*exp(x)+4*exp(1/4*x)^2+(3-

16*x)*exp(1/4*x)+16*x^2-24)/(4*x^2*exp(1/4*x)^2+(-16*x^3-24*x)*exp(1/4*x)+16*x^4+48*x^2+36),x, algorithm="maxi

________________________________________________________________________________________

mupad [B] time = 3.50, size = 163, normalized size = 6.52 \begin {gather*} 24\,x+\frac {107\,x^4+162\,x^2+81}{x^5}+\frac {{\mathrm {e}}^x}{x}+\frac {3\,{\mathrm {e}}^{\frac {3\,x}{4}}}{x^2}+\frac {{\mathrm {e}}^{x/2}\,\left (6\,x^2+9\right )}{x^3}+\frac {{\mathrm {e}}^{x/4}\,\left (12\,x^4+36\,x^2+27\right )}{x^4}-\frac {3\,\left (32\,x^{11}-128\,x^{10}+240\,x^9-576\,x^8+718\,x^7-568\,x^6+1077\,x^5+852\,x^4+810\,x^3+1944\,x^2+243\,x+972\right )}{x^5\,\left (2\,x^2-x\,{\mathrm {e}}^{x/4}+3\right )\,\left (2\,x^3-8\,x^2+3\,x+12\right )} \end {gather*}

int((4*exp(x/2) + exp(x)*(24*x - exp(x/4)*(16*x^2 - 16*x + 15) + exp(x/2)*(4*x - 4) - 16*x^2 + 16*x^3 + 24) -

exp(x/4)*(16*x - 3) + 16*x^2 - 24)/(4*x^2*exp(x/2) - exp(x/4)*(24*x + 16*x^3) + 48*x^2 + 16*x^4 + 36),x)

24*x + (162*x^2 + 107*x^4 + 81)/x^5 + exp(x)/x + (3*exp((3*x)/4))/x^2 + (exp(x/2)*(6*x^2 + 9))/x^3 + (exp(x/4)

*(36*x^2 + 12*x^4 + 27))/x^4 - (3*(243*x + 1944*x^2 + 810*x^3 + 852*x^4 + 1077*x^5 - 568*x^6 + 718*x^7 - 576*x

^8 + 240*x^9 - 128*x^10 + 32*x^11 + 972))/(x^5*(2*x^2 - x*exp(x/4) + 3)*(3*x - 8*x^2 + 2*x^3 + 12))

________________________________________________________________________________________

sympy [B] time = 0.34, size = 116, normalized size = 4.64 \begin {gather*} 24 x + \frac {48 x^{8} + 288 x^{6} + 645 x^{4} + 648 x^{2} + 243}{- 2 x^{7} + x^{6} e^{\frac {x}{4}} - 3 x^{5}} + \frac {107 x^{4} + 162 x^{2} + 81}{x^{5}} + \frac {x^{9} e^{x} + 3 x^{8} e^{\frac {3 x}{4}} + \left (6 x^{9} + 9 x^{7}\right ) e^{\frac {x}{2}} + \left (12 x^{10} + 36 x^{8} + 27 x^{6}\right ) e^{\frac {x}{4}}}{x^{10}} \end {gather*}

integrate((((4*x-4)*exp(1/4*x)**2+(-16*x**2+16*x-15)*exp(1/4*x)+16*x**3-16*x**2+24*x+24)*exp(x)+4*exp(1/4*x)**

2+(3-16*x)*exp(1/4*x)+16*x**2-24)/(4*x**2*exp(1/4*x)**2+(-16*x**3-24*x)*exp(1/4*x)+16*x**4+48*x**2+36),x)

24*x + (48*x**8 + 288*x**6 + 645*x**4 + 648*x**2 + 243)/(-2*x**7 + x**6*exp(x/4) - 3*x**5) + (107*x**4 + 162*x

**2 + 81)/x**5 + (x**9*exp(x) + 3*x**8*exp(3*x/4) + (6*x**9 + 9*x**7)*exp(x/2) + (12*x**10 + 36*x**8 + 27*x**6

________________________________________________________________________________________

3.51.92 \(\int \frac {-24+4 e^{x/2}+e^{x/4} (3-16 x)+16 x^2+e^x (24+24 x-16 x^2+16 x^3+e^{x/2} (-4+4 x)+e^{x/4} (-15+16 x-16 x^2))}{36+48 x^2+4 e^{x/2} x^2+16 x^4+e^{x/4} (-24 x-16 x^3)} \, dx\)