3.1.15 \(\int \frac {40+15 x-7 x^2+e^{4+x^2} (-24+12 x^2-12 x^3-4 x^5-6 x^6-2 x^7)}{1024+3840 x+7040 x^2+8160 x^3+6580 x^4+3843 x^5+1645 x^6+510 x^7+110 x^8+15 x^9+x^{10}} \, dx\)
Optimal. Leaf size=29 \[ \frac {-3+x-e^{4+x^2} \left (-2+x^4\right )}{\left (x-(2+x)^2\right )^4} \]
________________________________________________________________________________________
Rubi [F] time = 12.70, 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 {40+15 x-7 x^2+e^{4+x^2} \left (-24+12 x^2-12 x^3-4 x^5-6 x^6-2 x^7\right )}{1024+3840 x+7040 x^2+8160 x^3+6580 x^4+3843 x^5+1645 x^6+510 x^7+110 x^8+15 x^9+x^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(40 + 15*x - 7*x^2 + E^(4 + x^2)*(-24 + 12*x^2 - 12*x^3 - 4*x^5 - 6*x^6 - 2*x^7))/(1024 + 3840*x + 7040*x^
2 + 8160*x^3 + 6580*x^4 + 3843*x^5 + 1645*x^6 + 510*x^7 + 110*x^8 + 15*x^9 + x^10),x]
[Out]
(((-96*I)/7)*E^(4 + x^2))/(Sqrt[7]*(3 - I*Sqrt[7] + 2*x)^4) + (40*(7 + (3*I)*Sqrt[7])*E^(4 + x^2))/(49*(3 - I*
Sqrt[7] + 2*x)^4) + (32*E^(4 + x^2))/(49*(3 - I*Sqrt[7] + 2*x)^3) + (344*(3 - I*Sqrt[7])*E^(4 + x^2))/(147*(3
- I*Sqrt[7] + 2*x)^3) - (80*(5 - I*Sqrt[7])*E^(4 + x^2))/(49*(3 - I*Sqrt[7] + 2*x)^3) - (40*(21 + I*Sqrt[7])*E
^(4 + x^2))/(147*(3 - I*Sqrt[7] + 2*x)^3) + (16*(7 + (3*I)*Sqrt[7])*E^(4 + x^2))/(49*(3 - I*Sqrt[7] + 2*x)^3)
+ (((612*I)/49)*E^(4 + x^2))/(Sqrt[7]*(3 - I*Sqrt[7] + 2*x)^2) + (160*(1 - I*Sqrt[7])*E^(4 + x^2))/(49*(3 - I*
Sqrt[7] + 2*x)^2) - (8*(3 - I*Sqrt[7])*E^(4 + x^2))/(49*(3 - I*Sqrt[7] + 2*x)^2) - (8*(21 + I*Sqrt[7])*E^(4 +
x^2))/(49*(3 - I*Sqrt[7] + 2*x)^2) - (24*(7 + (3*I)*Sqrt[7])*E^(4 + x^2))/(49*(3 - I*Sqrt[7] + 2*x)^2) + (172*
(7 + (6*I)*Sqrt[7])*E^(4 + x^2))/(343*(3 - I*Sqrt[7] + 2*x)^2) - (100*(7 + (9*I)*Sqrt[7])*E^(4 + x^2))/(343*(3
- I*Sqrt[7] + 2*x)^2) + (86*(3*I + Sqrt[7])^2*E^(4 + x^2))/(147*(3 - I*Sqrt[7] + 2*x)^2) - (5*(3*I + Sqrt[7])
^3*E^(4 + x^2))/(21*Sqrt[7]*(3 - I*Sqrt[7] + 2*x)^2) - (1004*E^(4 + x^2))/(343*(3 - I*Sqrt[7] + 2*x)) + (214*(
3 - I*Sqrt[7])*E^(4 + x^2))/(147*(3 - I*Sqrt[7] + 2*x)) + (43*(3 - I*Sqrt[7])^3*E^(4 + x^2))/(147*(3 - I*Sqrt[
7] + 2*x)) - (40*(5 - I*Sqrt[7])*E^(4 + x^2))/(49*(3 - I*Sqrt[7] + 2*x)) + (34*(9 - I*Sqrt[7])*E^(4 + x^2))/(4
9*(3 - I*Sqrt[7] + 2*x)) - (172*(15 - I*Sqrt[7])*E^(4 + x^2))/(343*(3 - I*Sqrt[7] + 2*x)) + (100*(21 - I*Sqrt[
7])*E^(4 + x^2))/(343*(3 - I*Sqrt[7] + 2*x)) + (320*(1 + I*Sqrt[7])*E^(4 + x^2))/(49*(3 - I*Sqrt[7] + 2*x)) +
(52*(21 + I*Sqrt[7])*E^(4 + x^2))/(147*(3 - I*Sqrt[7] + 2*x)) - (250*(7 + (3*I)*Sqrt[7])*E^(4 + x^2))/(343*(3
- I*Sqrt[7] + 2*x)) + (200*(21 + (5*I)*Sqrt[7])*E^(4 + x^2))/(343*(3 - I*Sqrt[7] + 2*x)) + (8*(35 - (9*I)*Sqrt
[7])*E^(4 + x^2))/(49*(3 - I*Sqrt[7] + 2*x)) - (86*(63 + (11*I)*Sqrt[7])*E^(4 + x^2))/(343*(3 - I*Sqrt[7] + 2*
x)) - (20*(21 - (31*I)*Sqrt[7])*E^(4 + x^2))/(147*(3 - I*Sqrt[7] + 2*x)) - (4*(3*I + Sqrt[7])^2*E^(4 + x^2))/(
49*(3 - I*Sqrt[7] + 2*x)) + (((96*I)/7)*E^(4 + x^2))/(Sqrt[7]*(3 + I*Sqrt[7] + 2*x)^4) + (40*(7 - (3*I)*Sqrt[7
])*E^(4 + x^2))/(49*(3 + I*Sqrt[7] + 2*x)^4) + (32*E^(4 + x^2))/(49*(3 + I*Sqrt[7] + 2*x)^3) - (40*(21 - I*Sqr
t[7])*E^(4 + x^2))/(147*(3 + I*Sqrt[7] + 2*x)^3) + (344*(3 + I*Sqrt[7])*E^(4 + x^2))/(147*(3 + I*Sqrt[7] + 2*x
)^3) - (80*(5 + I*Sqrt[7])*E^(4 + x^2))/(49*(3 + I*Sqrt[7] + 2*x)^3) + (16*(7 - (3*I)*Sqrt[7])*E^(4 + x^2))/(4
9*(3 + I*Sqrt[7] + 2*x)^3) - (((612*I)/49)*E^(4 + x^2))/(Sqrt[7]*(3 + I*Sqrt[7] + 2*x)^2) + (86*(3*I - Sqrt[7]
)^2*E^(4 + x^2))/(147*(3 + I*Sqrt[7] + 2*x)^2) - (8*(21 - I*Sqrt[7])*E^(4 + x^2))/(49*(3 + I*Sqrt[7] + 2*x)^2)
+ (160*(1 + I*Sqrt[7])*E^(4 + x^2))/(49*(3 + I*Sqrt[7] + 2*x)^2) - (8*(3 + I*Sqrt[7])*E^(4 + x^2))/(49*(3 + I
*Sqrt[7] + 2*x)^2) - (24*(7 - (3*I)*Sqrt[7])*E^(4 + x^2))/(49*(3 + I*Sqrt[7] + 2*x)^2) + (172*(7 - (6*I)*Sqrt[
7])*E^(4 + x^2))/(343*(3 + I*Sqrt[7] + 2*x)^2) - (100*(7 - (9*I)*Sqrt[7])*E^(4 + x^2))/(343*(3 + I*Sqrt[7] + 2
*x)^2) + (20*(35 + (9*I)*Sqrt[7])*E^(4 + x^2))/(147*(3 + I*Sqrt[7] + 2*x)^2) - (1004*E^(4 + x^2))/(343*(3 + I*
Sqrt[7] + 2*x)) - (4*(3*I - Sqrt[7])^2*E^(4 + x^2))/(49*(3 + I*Sqrt[7] + 2*x)) + (320*(1 - I*Sqrt[7])*E^(4 + x
^2))/(49*(3 + I*Sqrt[7] + 2*x)) + (52*(21 - I*Sqrt[7])*E^(4 + x^2))/(147*(3 + I*Sqrt[7] + 2*x)) + (214*(3 + I*
Sqrt[7])*E^(4 + x^2))/(147*(3 + I*Sqrt[7] + 2*x)) - (40*(5 + I*Sqrt[7])*E^(4 + x^2))/(49*(3 + I*Sqrt[7] + 2*x)
) + (34*(9 + I*Sqrt[7])*E^(4 + x^2))/(49*(3 + I*Sqrt[7] + 2*x)) - (172*(15 + I*Sqrt[7])*E^(4 + x^2))/(343*(3 +
I*Sqrt[7] + 2*x)) + (100*(21 + I*Sqrt[7])*E^(4 + x^2))/(343*(3 + I*Sqrt[7] + 2*x)) - (250*(7 - (3*I)*Sqrt[7])
*E^(4 + x^2))/(343*(3 + I*Sqrt[7] + 2*x)) - (172*(9 - (5*I)*Sqrt[7])*E^(4 + x^2))/(147*(3 + I*Sqrt[7] + 2*x))
+ (200*(21 - (5*I)*Sqrt[7])*E^(4 + x^2))/(343*(3 + I*Sqrt[7] + 2*x)) + (8*(35 + (9*I)*Sqrt[7])*E^(4 + x^2))/(4
9*(3 + I*Sqrt[7] + 2*x)) - (86*(63 - (11*I)*Sqrt[7])*E^(4 + x^2))/(343*(3 + I*Sqrt[7] + 2*x)) - (20*(21 + (31*
I)*Sqrt[7])*E^(4 + x^2))/(147*(3 + I*Sqrt[7] + 2*x)) + (10*(3 + 2*x))/(7*(4 + 3*x + x^2)^4) - (15*(8 + 3*x))/(
28*(4 + 3*x + x^2)^4) + (x*(8 + 3*x))/(4*(4 + 3*x + x^2)^4) + (5*(3 + 2*x))/(12*(4 + 3*x + x^2)^3) - (12 + 5*x
)/(6*(4 + 3*x + x^2)^3) - (43*(11*I - 9*Sqrt[7])*E^4*Sqrt[Pi/7]*Erfi[x])/49 + (4*(9*I - 5*Sqrt[7])*E^4*Sqrt[Pi
/7]*Erfi[x])/7 + (332*(I - 3*Sqrt[7])*E^4*Sqrt[Pi/7]*Erfi[x])/147 + (100*(5*I - 3*Sqrt[7])*E^4*Sqrt[Pi/7]*Erfi
[x])/49 - (10*(31*I - 3*Sqrt[7])*E^4*Sqrt[Pi/7]*Erfi[x])/21 - (125*(3*I - Sqrt[7])*E^4*Sqrt[Pi/7]*Erfi[x])/49
+ (125*(3*I + Sqrt[7])*E^4*Sqrt[Pi/7]*Erfi[x])/49 - (332*(I + 3*Sqrt[7])*E^4*Sqrt[Pi/7]*Erfi[x])/147 - (100*(5
*I + 3*Sqrt[7])*E^4*Sqrt[Pi/7]*Erfi[x])/49 + (10*(31*I + 3*Sqrt[7])*E^4*Sqrt[Pi/7]*Erfi[x])/21 - (4*(9*I + 5*S
qrt[7])*E^4*Sqrt[Pi/7]*Erfi[x])/7 + (43*(11*I + 9*Sqrt[7])*E^4*Sqrt[Pi/7]*Erfi[x])/49 + (1004*E^4*Sqrt[Pi]*Erf
i[x])/343 + (2*(3*I - Sqrt[7])^2*E^4*Sqrt[Pi]*Erfi[x])/49 - (160*(1 - I*Sqrt[7])*E^4*Sqrt[Pi]*Erfi[x])/49 - (1
07*(3 - I*Sqrt[7])*E^4*Sqrt[Pi]*Erfi[x])/147 - (43*(3 - I*Sqrt[7])^3*E^4*Sqrt[Pi]*Erfi[x])/294 + (20*(5 - I*Sq
rt[7])*E^4*Sqrt[Pi]*Erfi[x])/49 - (17*(9 - I*Sqrt[7])*E^4*Sqrt[Pi]*Erfi[x])/49 + (86*(15 - I*Sqrt[7])*E^4*Sqrt
[Pi]*Erfi[x])/343 - (107*(3 + I*Sqrt[7])*E^4*Sqrt[Pi]*Erfi[x])/147 + (20*(5 + I*Sqrt[7])*E^4*Sqrt[Pi]*Erfi[x])
/49 - (17*(9 + I*Sqrt[7])*E^4*Sqrt[Pi]*Erfi[x])/49 + (86*(15 + I*Sqrt[7])*E^4*Sqrt[Pi]*Erfi[x])/343 + (86*(9 -
(5*I)*Sqrt[7])*E^4*Sqrt[Pi]*Erfi[x])/147 + (2*(3*I + Sqrt[7])^2*E^4*Sqrt[Pi]*Erfi[x])/49 - (10*(5*I + Sqrt[7]
)*(I + 3*Sqrt[7])*E^4*Sqrt[Pi]*Erfi[x])/49 + (((852*I)/49)*Defer[Int][E^(4 + x^2)/(-3 + I*Sqrt[7] - 2*x), x])/
Sqrt[7] + (480*(1 - I*Sqrt[7])*Defer[Int][E^(4 + x^2)/(-3 + I*Sqrt[7] - 2*x), x])/49 + (948*(3 - I*Sqrt[7])*De
fer[Int][E^(4 + x^2)/(-3 + I*Sqrt[7] - 2*x), x])/343 - (640*(5 + I*Sqrt[7])*Defer[Int][E^(4 + x^2)/(-3 + I*Sqr
t[7] - 2*x), x])/49 + (444*(21 + I*Sqrt[7])*Defer[Int][E^(4 + x^2)/(-3 + I*Sqrt[7] - 2*x), x])/343 - (136*(5 -
(3*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(-3 + I*Sqrt[7] - 2*x), x])/49 - (800*(7 - (3*I)*Sqrt[7])*Defer[Int][E^
(4 + x^2)/(-3 + I*Sqrt[7] - 2*x), x])/343 - (400*(49 - (3*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(-3 + I*Sqrt[7] -
2*x), x])/343 - (24*(7 + (3*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(-3 + I*Sqrt[7] - 2*x), x])/49 + (344*(31 + (3
*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(-3 + I*Sqrt[7] - 2*x), x])/147 + (16*(9 + (5*I)*Sqrt[7])*Defer[Int][E^(4
+ x^2)/(-3 + I*Sqrt[7] - 2*x), x])/49 + (172*(7 + (6*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(-3 + I*Sqrt[7] - 2*x)
, x])/343 + (344*(19 - (9*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(-3 + I*Sqrt[7] - 2*x), x])/343 - (100*(7 + (9*I)
*Sqrt[7])*Defer[Int][E^(4 + x^2)/(-3 + I*Sqrt[7] - 2*x), x])/343 + (172*(133 - (15*I)*Sqrt[7])*Defer[Int][E^(4
+ x^2)/(-3 + I*Sqrt[7] - 2*x), x])/343 - (16*(21 - (31*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(-3 + I*Sqrt[7] - 2
*x), x])/49 - (40*(77 + (57*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(-3 + I*Sqrt[7] - 2*x), x])/147 + (100*(3*I + S
qrt[7])^2*Defer[Int][E^(4 + x^2)/(-3 + I*Sqrt[7] - 2*x), x])/49 + ((3*I + Sqrt[7])^3*Defer[Int][E^(4 + x^2)/(-
3 + I*Sqrt[7] - 2*x), x])/Sqrt[7] + (((852*I)/49)*Defer[Int][E^(4 + x^2)/(3 + I*Sqrt[7] + 2*x), x])/Sqrt[7] +
(640*(5 - I*Sqrt[7])*Defer[Int][E^(4 + x^2)/(3 + I*Sqrt[7] + 2*x), x])/49 - (444*(21 - I*Sqrt[7])*Defer[Int][E
^(4 + x^2)/(3 + I*Sqrt[7] + 2*x), x])/343 - (480*(1 + I*Sqrt[7])*Defer[Int][E^(4 + x^2)/(3 + I*Sqrt[7] + 2*x),
x])/49 - (948*(3 + I*Sqrt[7])*Defer[Int][E^(4 + x^2)/(3 + I*Sqrt[7] + 2*x), x])/343 + (24*(7 - (3*I)*Sqrt[7])
*Defer[Int][E^(4 + x^2)/(3 + I*Sqrt[7] + 2*x), x])/49 - (344*(31 - (3*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(3 +
I*Sqrt[7] + 2*x), x])/147 + (200*(1 + (3*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(3 + I*Sqrt[7] + 2*x), x])/49 + (1
36*(5 + (3*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(3 + I*Sqrt[7] + 2*x), x])/49 + (800*(7 + (3*I)*Sqrt[7])*Defer[I
nt][E^(4 + x^2)/(3 + I*Sqrt[7] + 2*x), x])/343 + (400*(49 + (3*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(3 + I*Sqrt[
7] + 2*x), x])/343 - (16*(9 - (5*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(3 + I*Sqrt[7] + 2*x), x])/49 - (172*(7 -
(6*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(3 + I*Sqrt[7] + 2*x), x])/343 + (100*(7 - (9*I)*Sqrt[7])*Defer[Int][E^(
4 + x^2)/(3 + I*Sqrt[7] + 2*x), x])/343 - (344*(19 + (9*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(3 + I*Sqrt[7] + 2*
x), x])/343 + (4*(35 + (9*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(3 + I*Sqrt[7] + 2*x), x])/7 - (172*(133 + (15*I)
*Sqrt[7])*Defer[Int][E^(4 + x^2)/(3 + I*Sqrt[7] + 2*x), x])/343 + (16*(21 + (31*I)*Sqrt[7])*Defer[Int][E^(4 +
x^2)/(3 + I*Sqrt[7] + 2*x), x])/49 + (40*(77 - (57*I)*Sqrt[7])*Defer[Int][E^(4 + x^2)/(3 + I*Sqrt[7] + 2*x), x
])/147
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {40+15 x-7 x^2-2 e^{4+x^2} \left (12-6 x^2+6 x^3+2 x^5+3 x^6+x^7\right )}{\left (4+3 x+x^2\right )^5} \, dx\\ &=\int \left (\frac {40}{\left (4+3 x+x^2\right )^5}+\frac {15 x}{\left (4+3 x+x^2\right )^5}-\frac {7 x^2}{\left (4+3 x+x^2\right )^5}-\frac {2 e^{4+x^2} (1+x) \left (12-12 x+6 x^2+2 x^5+x^6\right )}{\left (4+3 x+x^2\right )^5}\right ) \, dx\\ &=-\left (2 \int \frac {e^{4+x^2} (1+x) \left (12-12 x+6 x^2+2 x^5+x^6\right )}{\left (4+3 x+x^2\right )^5} \, dx\right )-7 \int \frac {x^2}{\left (4+3 x+x^2\right )^5} \, dx+15 \int \frac {x}{\left (4+3 x+x^2\right )^5} \, dx+40 \int \frac {1}{\left (4+3 x+x^2\right )^5} \, dx\\ &=\frac {10 (3+2 x)}{7 \left (4+3 x+x^2\right )^4}-\frac {15 (8+3 x)}{28 \left (4+3 x+x^2\right )^4}+\frac {x (8+3 x)}{4 \left (4+3 x+x^2\right )^4}-\frac {1}{4} \int \frac {8-18 x}{\left (4+3 x+x^2\right )^4} \, dx-2 \int \left (\frac {14 e^{4+x^2} (6+5 x)}{\left (4+3 x+x^2\right )^5}+\frac {e^{4+x^2} (-54-43 x)}{\left (4+3 x+x^2\right )^4}+\frac {e^{4+x^2} (33+17 x)}{\left (4+3 x+x^2\right )^3}+\frac {e^{4+x^2} (-6+x)}{\left (4+3 x+x^2\right )^2}\right ) \, dx-\frac {45}{4} \int \frac {1}{\left (4+3 x+x^2\right )^4} \, dx+20 \int \frac {1}{\left (4+3 x+x^2\right )^4} \, dx\\ &=\frac {10 (3+2 x)}{7 \left (4+3 x+x^2\right )^4}-\frac {15 (8+3 x)}{28 \left (4+3 x+x^2\right )^4}+\frac {x (8+3 x)}{4 \left (4+3 x+x^2\right )^4}+\frac {5 (3+2 x)}{12 \left (4+3 x+x^2\right )^3}-\frac {12+5 x}{6 \left (4+3 x+x^2\right )^3}-2 \int \frac {e^{4+x^2} (-54-43 x)}{\left (4+3 x+x^2\right )^4} \, dx-2 \int \frac {e^{4+x^2} (33+17 x)}{\left (4+3 x+x^2\right )^3} \, dx-2 \int \frac {e^{4+x^2} (-6+x)}{\left (4+3 x+x^2\right )^2} \, dx-\frac {25}{6} \int \frac {1}{\left (4+3 x+x^2\right )^3} \, dx-\frac {75}{14} \int \frac {1}{\left (4+3 x+x^2\right )^3} \, dx+\frac {200}{21} \int \frac {1}{\left (4+3 x+x^2\right )^3} \, dx-28 \int \frac {e^{4+x^2} (6+5 x)}{\left (4+3 x+x^2\right )^5} \, dx\\ &=\frac {10 (3+2 x)}{7 \left (4+3 x+x^2\right )^4}-\frac {15 (8+3 x)}{28 \left (4+3 x+x^2\right )^4}+\frac {x (8+3 x)}{4 \left (4+3 x+x^2\right )^4}+\frac {5 (3+2 x)}{12 \left (4+3 x+x^2\right )^3}-\frac {12+5 x}{6 \left (4+3 x+x^2\right )^3}-\frac {25}{14} \int \frac {1}{\left (4+3 x+x^2\right )^2} \, dx-2 \int \left (-\frac {54 e^{4+x^2}}{\left (4+3 x+x^2\right )^4}-\frac {43 e^{4+x^2} x}{\left (4+3 x+x^2\right )^4}\right ) \, dx-2 \int \left (\frac {33 e^{4+x^2}}{\left (4+3 x+x^2\right )^3}+\frac {17 e^{4+x^2} x}{\left (4+3 x+x^2\right )^3}\right ) \, dx-2 \int \left (-\frac {6 e^{4+x^2}}{\left (4+3 x+x^2\right )^2}+\frac {e^{4+x^2} x}{\left (4+3 x+x^2\right )^2}\right ) \, dx-\frac {225}{98} \int \frac {1}{\left (4+3 x+x^2\right )^2} \, dx+\frac {200}{49} \int \frac {1}{\left (4+3 x+x^2\right )^2} \, dx-28 \int \left (\frac {6 e^{4+x^2}}{\left (4+3 x+x^2\right )^5}+\frac {5 e^{4+x^2} x}{\left (4+3 x+x^2\right )^5}\right ) \, dx\\ &=\frac {10 (3+2 x)}{7 \left (4+3 x+x^2\right )^4}-\frac {15 (8+3 x)}{28 \left (4+3 x+x^2\right )^4}+\frac {x (8+3 x)}{4 \left (4+3 x+x^2\right )^4}+\frac {5 (3+2 x)}{12 \left (4+3 x+x^2\right )^3}-\frac {12+5 x}{6 \left (4+3 x+x^2\right )^3}-\frac {25}{49} \int \frac {1}{4+3 x+x^2} \, dx-\frac {225}{343} \int \frac {1}{4+3 x+x^2} \, dx+\frac {400}{343} \int \frac {1}{4+3 x+x^2} \, dx-2 \int \frac {e^{4+x^2} x}{\left (4+3 x+x^2\right )^2} \, dx+12 \int \frac {e^{4+x^2}}{\left (4+3 x+x^2\right )^2} \, dx-34 \int \frac {e^{4+x^2} x}{\left (4+3 x+x^2\right )^3} \, dx-66 \int \frac {e^{4+x^2}}{\left (4+3 x+x^2\right )^3} \, dx+86 \int \frac {e^{4+x^2} x}{\left (4+3 x+x^2\right )^4} \, dx+108 \int \frac {e^{4+x^2}}{\left (4+3 x+x^2\right )^4} \, dx-140 \int \frac {e^{4+x^2} x}{\left (4+3 x+x^2\right )^5} \, dx-168 \int \frac {e^{4+x^2}}{\left (4+3 x+x^2\right )^5} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.96, size = 28, normalized size = 0.97 \begin {gather*} \frac {-3+x-e^{4+x^2} \left (-2+x^4\right )}{\left (4+3 x+x^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(40 + 15*x - 7*x^2 + E^(4 + x^2)*(-24 + 12*x^2 - 12*x^3 - 4*x^5 - 6*x^6 - 2*x^7))/(1024 + 3840*x + 7
040*x^2 + 8160*x^3 + 6580*x^4 + 3843*x^5 + 1645*x^6 + 510*x^7 + 110*x^8 + 15*x^9 + x^10),x]
[Out]
(-3 + x - E^(4 + x^2)*(-2 + x^4))/(4 + 3*x + x^2)^4
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fricas [A] time = 0.69, size = 59, normalized size = 2.03 \begin {gather*} -\frac {{\left (x^{4} - 2\right )} e^{\left (x^{2} + 4\right )} - x + 3}{x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-2*x^7-6*x^6-4*x^5-12*x^3+12*x^2-24)*exp(x^2+4)-7*x^2+15*x+40)/(x^10+15*x^9+110*x^8+510*x^7+1645*x
^6+3843*x^5+6580*x^4+8160*x^3+7040*x^2+3840*x+1024),x, algorithm="fricas")
[Out]
-((x^4 - 2)*e^(x^2 + 4) - x + 3)/(x^8 + 12*x^7 + 70*x^6 + 252*x^5 + 609*x^4 + 1008*x^3 + 1120*x^2 + 768*x + 25
6)
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giac [B] time = 0.20, size = 65, normalized size = 2.24 \begin {gather*} -\frac {x^{4} e^{\left (x^{2} + 4\right )} - x - 2 \, e^{\left (x^{2} + 4\right )} + 3}{x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-2*x^7-6*x^6-4*x^5-12*x^3+12*x^2-24)*exp(x^2+4)-7*x^2+15*x+40)/(x^10+15*x^9+110*x^8+510*x^7+1645*x
^6+3843*x^5+6580*x^4+8160*x^3+7040*x^2+3840*x+1024),x, algorithm="giac")
[Out]
-(x^4*e^(x^2 + 4) - x - 2*e^(x^2 + 4) + 3)/(x^8 + 12*x^7 + 70*x^6 + 252*x^5 + 609*x^4 + 1008*x^3 + 1120*x^2 +
768*x + 256)
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maple [A] time = 0.34, size = 34, normalized size = 1.17
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| method |
result |
size |
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| norman |
\(\frac {x -3-{\mathrm e}^{x^{2}+4} x^{4}+2 \,{\mathrm e}^{x^{2}+4}}{\left (x^{2}+3 x +4\right )^{4}}\) |
\(34\) |
| risch |
\(\frac {x -3}{x^{8}+12 x^{7}+70 x^{6}+252 x^{5}+609 x^{4}+1008 x^{3}+1120 x^{2}+768 x +256}-\frac {\left (x^{4}-2\right ) {\mathrm e}^{x^{2}+4}}{\left (x^{2}+3 x +4\right )^{4}}\) |
\(69\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((-2*x^7-6*x^6-4*x^5-12*x^3+12*x^2-24)*exp(x^2+4)-7*x^2+15*x+40)/(x^10+15*x^9+110*x^8+510*x^7+1645*x^6+384
3*x^5+6580*x^4+8160*x^3+7040*x^2+3840*x+1024),x,method=_RETURNVERBOSE)
[Out]
(x-3-exp(x^2+4)*x^4+2*exp(x^2+4))/(x^2+3*x+4)^4
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maxima [B] time = 0.70, size = 289, normalized size = 9.97 \begin {gather*} -\frac {{\left (x^{4} e^{4} - 2 \, e^{4}\right )} e^{\left (x^{2}\right )}}{x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256} - \frac {300 \, x^{7} + 3150 \, x^{6} + 16100 \, x^{5} + 49875 \, x^{4} + 100940 \, x^{3} + 132930 \, x^{2} + 106512 \, x + 41904}{588 \, {\left (x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256\right )}} + \frac {10 \, {\left (120 \, x^{7} + 1260 \, x^{6} + 6440 \, x^{5} + 19950 \, x^{4} + 40376 \, x^{3} + 53172 \, x^{2} + 42840 \, x + 16497\right )}}{1029 \, {\left (x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256\right )}} - \frac {15 \, {\left (60 \, x^{7} + 630 \, x^{6} + 3220 \, x^{5} + 9975 \, x^{4} + 20188 \, x^{3} + 26586 \, x^{2} + 21420 \, x + 8420\right )}}{1372 \, {\left (x^{8} + 12 \, x^{7} + 70 \, x^{6} + 252 \, x^{5} + 609 \, x^{4} + 1008 \, x^{3} + 1120 \, x^{2} + 768 \, x + 256\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-2*x^7-6*x^6-4*x^5-12*x^3+12*x^2-24)*exp(x^2+4)-7*x^2+15*x+40)/(x^10+15*x^9+110*x^8+510*x^7+1645*x
^6+3843*x^5+6580*x^4+8160*x^3+7040*x^2+3840*x+1024),x, algorithm="maxima")
[Out]
-(x^4*e^4 - 2*e^4)*e^(x^2)/(x^8 + 12*x^7 + 70*x^6 + 252*x^5 + 609*x^4 + 1008*x^3 + 1120*x^2 + 768*x + 256) - 1
/588*(300*x^7 + 3150*x^6 + 16100*x^5 + 49875*x^4 + 100940*x^3 + 132930*x^2 + 106512*x + 41904)/(x^8 + 12*x^7 +
70*x^6 + 252*x^5 + 609*x^4 + 1008*x^3 + 1120*x^2 + 768*x + 256) + 10/1029*(120*x^7 + 1260*x^6 + 6440*x^5 + 19
950*x^4 + 40376*x^3 + 53172*x^2 + 42840*x + 16497)/(x^8 + 12*x^7 + 70*x^6 + 252*x^5 + 609*x^4 + 1008*x^3 + 112
0*x^2 + 768*x + 256) - 15/1372*(60*x^7 + 630*x^6 + 3220*x^5 + 9975*x^4 + 20188*x^3 + 26586*x^2 + 21420*x + 842
0)/(x^8 + 12*x^7 + 70*x^6 + 252*x^5 + 609*x^4 + 1008*x^3 + 1120*x^2 + 768*x + 256)
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mupad [B] time = 0.38, size = 33, normalized size = 1.14 \begin {gather*} \frac {x+2\,{\mathrm {e}}^{x^2+4}-x^4\,{\mathrm {e}}^{x^2+4}-3}{{\left (x^2+3\,x+4\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((15*x - exp(x^2 + 4)*(12*x^3 - 12*x^2 + 4*x^5 + 6*x^6 + 2*x^7 + 24) - 7*x^2 + 40)/(3840*x + 7040*x^2 + 816
0*x^3 + 6580*x^4 + 3843*x^5 + 1645*x^6 + 510*x^7 + 110*x^8 + 15*x^9 + x^10 + 1024),x)
[Out]
(x + 2*exp(x^2 + 4) - x^4*exp(x^2 + 4) - 3)/(3*x + x^2 + 4)^4
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sympy [B] time = 0.20, size = 92, normalized size = 3.17 \begin {gather*} \frac {\left (2 - x^{4}\right ) e^{x^{2} + 4}}{x^{8} + 12 x^{7} + 70 x^{6} + 252 x^{5} + 609 x^{4} + 1008 x^{3} + 1120 x^{2} + 768 x + 256} - \frac {3 - x}{x^{8} + 12 x^{7} + 70 x^{6} + 252 x^{5} + 609 x^{4} + 1008 x^{3} + 1120 x^{2} + 768 x + 256} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-2*x**7-6*x**6-4*x**5-12*x**3+12*x**2-24)*exp(x**2+4)-7*x**2+15*x+40)/(x**10+15*x**9+110*x**8+510*
x**7+1645*x**6+3843*x**5+6580*x**4+8160*x**3+7040*x**2+3840*x+1024),x)
[Out]
(2 - x**4)*exp(x**2 + 4)/(x**8 + 12*x**7 + 70*x**6 + 252*x**5 + 609*x**4 + 1008*x**3 + 1120*x**2 + 768*x + 256
) - (3 - x)/(x**8 + 12*x**7 + 70*x**6 + 252*x**5 + 609*x**4 + 1008*x**3 + 1120*x**2 + 768*x + 256)
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