\(\int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6)+e^6 (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8)+e^3 (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10})}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx\) [2215]

Optimal result

Integrand size = 175, antiderivative size = 35 \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=\left (x+5 x^2\right )^2 \left (5+x+\frac {3}{\frac {e^3}{x}+x}\right )^2-\log (-x) \] Output:

(x+5+3/(x+exp(3)/x))^2*(5*x^2+x)^2-ln(-x)
                                                                                    
                                                                                    
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(105\) vs. \(2(35)=70\).

Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.00 \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=-30 \left (-4+27 e^3\right ) x-2 \left (-278+75 e^3\right ) x^2+1070 x^3+876 x^4+260 x^5+25 x^6+\frac {9 e^6 \left (1-25 e^3+10 x\right )}{\left (e^3+x^2\right )^2}-\frac {3 e^3 \left (6+50 e^6+70 x-3 e^3 (109+90 x)\right )}{e^3+x^2}-\log (x) \] Input:

Integrate[(-x^6 + 120*x^7 + 1112*x^8 + 3210*x^9 + 3504*x^10 + 1300*x^11 + 
150*x^12 + E^9*(-1 + 50*x^2 + 780*x^3 + 2904*x^4 + 1300*x^5 + 150*x^6) + E 
^6*(-3*x^2 + 90*x^3 + 1374*x^4 + 6390*x^5 + 9612*x^6 + 3900*x^7 + 450*x^8) 
 + E^3*(33*x^4 + 570*x^5 + 3336*x^6 + 8820*x^7 + 10212*x^8 + 3900*x^9 + 45 
0*x^10))/(E^9*x + 3*E^6*x^3 + 3*E^3*x^5 + x^7),x]
 

Output:

-30*(-4 + 27*E^3)*x - 2*(-278 + 75*E^3)*x^2 + 1070*x^3 + 876*x^4 + 260*x^5 
 + 25*x^6 + (9*E^6*(1 - 25*E^3 + 10*x))/(E^3 + x^2)^2 - (3*E^3*(6 + 50*E^6 
 + 70*x - 3*E^3*(109 + 90*x)))/(E^3 + x^2) - Log[x]
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(138\) vs. \(2(35)=70\).

Time = 2.03 (sec) , antiderivative size = 138, normalized size of antiderivative = 3.94, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {2026, 2070, 2336, 27, 2336, 27, 2019, 2010, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {150 x^{12}+1300 x^{11}+3504 x^{10}+3210 x^9+1112 x^8+120 x^7-x^6+e^9 \left (150 x^6+1300 x^5+2904 x^4+780 x^3+50 x^2-1\right )+e^3 \left (450 x^{10}+3900 x^9+10212 x^8+8820 x^7+3336 x^6+570 x^5+33 x^4\right )+e^6 \left (450 x^8+3900 x^7+9612 x^6+6390 x^5+1374 x^4+90 x^3-3 x^2\right )}{x^7+3 e^3 x^5+3 e^6 x^3+e^9 x} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {150 x^{12}+1300 x^{11}+3504 x^{10}+3210 x^9+1112 x^8+120 x^7-x^6+e^9 \left (150 x^6+1300 x^5+2904 x^4+780 x^3+50 x^2-1\right )+e^3 \left (450 x^{10}+3900 x^9+10212 x^8+8820 x^7+3336 x^6+570 x^5+33 x^4\right )+e^6 \left (450 x^8+3900 x^7+9612 x^6+6390 x^5+1374 x^4+90 x^3-3 x^2\right )}{x \left (x^6+3 e^3 x^4+3 e^6 x^2+e^9\right )}dx\)

\(\Big \downarrow \) 2070

\(\displaystyle \int \frac {150 x^{12}+1300 x^{11}+3504 x^{10}+3210 x^9+1112 x^8+120 x^7-x^6+e^9 \left (150 x^6+1300 x^5+2904 x^4+780 x^3+50 x^2-1\right )+e^3 \left (450 x^{10}+3900 x^9+10212 x^8+8820 x^7+3336 x^6+570 x^5+33 x^4\right )+e^6 \left (450 x^8+3900 x^7+9612 x^6+6390 x^5+1374 x^4+90 x^3-3 x^2\right )}{x \left (x^2+e^3\right )^3}dx\)

\(\Big \downarrow \) 2336

\(\displaystyle \frac {9 e^6 \left (10 x-25 e^3+1\right )}{\left (x^2+e^3\right )^2}-\frac {\int \frac {4 \left (-150 e^3 x^{10}-1300 e^3 x^9-12 e^3 \left (292+25 e^3\right ) x^8-10 e^3 \left (321+260 e^3\right ) x^7-2 e^3 \left (556+3354 e^3+75 e^6\right ) x^6-10 e^3 \left (12+561 e^3+130 e^6\right ) x^5+e^3 \left (1-2224 e^3-2904 e^6\right ) x^4-30 e^6 \left (15+26 e^3\right ) x^3-34 e^6 \left (1-25 e^3\right ) x^2+90 e^9 x+e^9\right )}{x \left (x^2+e^3\right )^2}dx}{4 e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {9 e^6 \left (10 x-25 e^3+1\right )}{\left (x^2+e^3\right )^2}-\frac {\int \frac {-150 e^3 x^{10}-1300 e^3 x^9-12 e^3 \left (292+25 e^3\right ) x^8-10 e^3 \left (321+260 e^3\right ) x^7-2 e^3 \left (556+3354 e^3+75 e^6\right ) x^6-10 e^3 \left (12+561 e^3+130 e^6\right ) x^5+e^3 \left (1-2224 e^3-2904 e^6\right ) x^4-30 e^6 \left (15+26 e^3\right ) x^3-34 e^6 \left (1-25 e^3\right ) x^2+90 e^9 x+e^9}{x \left (x^2+e^3\right )^2}dx}{e^3}\)

\(\Big \downarrow \) 2336

\(\displaystyle \frac {9 e^6 \left (10 x-25 e^3+1\right )}{\left (x^2+e^3\right )^2}-\frac {\frac {3 e^6 \left (10 \left (7-27 e^3\right ) x+50 e^6-327 e^3+6\right )}{x^2+e^3}-\frac {\int -\frac {2 \left (-150 e^6 x^8-1300 e^6 x^7-6 e^6 \left (584+25 e^3\right ) x^6-10 e^6 \left (321+130 e^3\right ) x^5-4 e^6 \left (278+801 e^3\right ) x^4-120 e^6 \left (1+20 e^3\right ) x^3+e^6 \left (1-1112 e^3+300 e^6\right ) x^2-30 e^9 \left (4-27 e^3\right ) x+e^9\right )}{x \left (x^2+e^3\right )}dx}{2 e^3}}{e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {9 e^6 \left (10 x-25 e^3+1\right )}{\left (x^2+e^3\right )^2}-\frac {\frac {\int \frac {-150 e^6 x^8-1300 e^6 x^7-6 e^6 \left (584+25 e^3\right ) x^6-10 e^6 \left (321+130 e^3\right ) x^5-4 e^6 \left (278+801 e^3\right ) x^4-120 e^6 \left (1+20 e^3\right ) x^3+e^6 \left (1-1112 e^3+300 e^6\right ) x^2-30 e^9 \left (4-27 e^3\right ) x+e^9}{x \left (x^2+e^3\right )}dx}{e^3}+\frac {3 e^6 \left (10 \left (7-27 e^3\right ) x+50 e^6-327 e^3+6\right )}{x^2+e^3}}{e^3}\)

\(\Big \downarrow \) 2019

\(\displaystyle \frac {9 e^6 \left (10 x-25 e^3+1\right )}{\left (x^2+e^3\right )^2}-\frac {\frac {\int \frac {-150 e^6 x^6-1300 e^6 x^5-3504 e^6 x^4-3210 e^6 x^3+\left (-1112 e^6+300 e^9\right ) x^2+\left (-120 e^6+810 e^9\right ) x+e^6}{x}dx}{e^3}+\frac {3 e^6 \left (10 \left (7-27 e^3\right ) x+50 e^6-327 e^3+6\right )}{x^2+e^3}}{e^3}\)

\(\Big \downarrow \) 2010

\(\displaystyle \frac {9 e^6 \left (10 x-25 e^3+1\right )}{\left (x^2+e^3\right )^2}-\frac {\frac {\int \left (-150 e^6 x^5-1300 e^6 x^4-3504 e^6 x^3-3210 e^6 x^2-4 e^6 \left (278-75 e^3\right ) x-30 e^6 \left (4-27 e^3\right )+\frac {e^6}{x}\right )dx}{e^3}+\frac {3 e^6 \left (10 \left (7-27 e^3\right ) x+50 e^6-327 e^3+6\right )}{x^2+e^3}}{e^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {9 e^6 \left (10 x-25 e^3+1\right )}{\left (x^2+e^3\right )^2}-\frac {\frac {3 e^6 \left (10 \left (7-27 e^3\right ) x+50 e^6-327 e^3+6\right )}{x^2+e^3}+\frac {-25 e^6 x^6-260 e^6 x^5-876 e^6 x^4-1070 e^6 x^3-2 e^6 \left (278-75 e^3\right ) x^2-30 e^6 \left (4-27 e^3\right ) x+e^6 \log (x)}{e^3}}{e^3}\)

Input:

Int[(-x^6 + 120*x^7 + 1112*x^8 + 3210*x^9 + 3504*x^10 + 1300*x^11 + 150*x^ 
12 + E^9*(-1 + 50*x^2 + 780*x^3 + 2904*x^4 + 1300*x^5 + 150*x^6) + E^6*(-3 
*x^2 + 90*x^3 + 1374*x^4 + 6390*x^5 + 9612*x^6 + 3900*x^7 + 450*x^8) + E^3 
*(33*x^4 + 570*x^5 + 3336*x^6 + 8820*x^7 + 10212*x^8 + 3900*x^9 + 450*x^10 
))/(E^9*x + 3*E^6*x^3 + 3*E^3*x^5 + x^7),x]
 

Output:

(9*E^6*(1 - 25*E^3 + 10*x))/(E^3 + x^2)^2 - ((3*E^6*(6 - 327*E^3 + 50*E^6 
+ 10*(7 - 27*E^3)*x))/(E^3 + x^2) + (-30*E^6*(4 - 27*E^3)*x - 2*E^6*(278 - 
 75*E^3)*x^2 - 1070*E^6*x^3 - 876*E^6*x^4 - 260*E^6*x^5 - 25*E^6*x^6 + E^6 
*Log[x])/E^3)/E^3
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

rule 2019

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px 
, Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && 
 EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
 

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 2070

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x^2, 0], Expon[Px 
, x^2]], b = Rt[Coeff[Px, x^2, Expon[Px, x^2]], Expon[Px, x^2]]}, Int[u*(a 
+ b*x^2)^(Expon[Px, x^2]*p), x] /; EqQ[Px, (a + b*x^2)^Expon[Px, x^2]]] /; 
IntegerQ[p] && PolyQ[Px, x^2] && GtQ[Expon[Px, x^2], 1] && NeQ[Coeff[Px, x^ 
2, 0], 0]
 

rule 2336

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema 
inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) 
^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* 
b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex 
pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F 
reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(115\) vs. \(2(34)=68\).

Time = 0.65 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.31

method	result	size
risch	\(25 x^{6}+260 x^{5}+876 x^{4}-150 x^{2} {\mathrm e}^{3}+1070 x^{3}-810 x \,{\mathrm e}^{3}+556 x^{2}+120 x +\frac {30 \,{\mathrm e}^{3} \left (27 \,{\mathrm e}^{3}-7\right ) x^{3}+\left (-150 \,{\mathrm e}^{9}+981 \,{\mathrm e}^{6}-18 \,{\mathrm e}^{3}\right ) x^{2}+\left (810 \,{\mathrm e}^{9}-120 \,{\mathrm e}^{6}\right ) x -150 \,{\mathrm e}^{12}+756 \,{\mathrm e}^{9}-9 \,{\mathrm e}^{6}}{x^{4}+2 x^{2} {\mathrm e}^{3}+{\mathrm e}^{6}}-\ln \left (x \right )\)	\(116\)
norman	\(\frac {\left (876+50 \,{\mathrm e}^{3}\right ) x^{8}+\left (1070+520 \,{\mathrm e}^{3}\right ) x^{7}+\left (260 \,{\mathrm e}^{6}+30 \,{\mathrm e}^{3}\right ) x^{3}+\left (25 \,{\mathrm e}^{6}+1602 \,{\mathrm e}^{3}+556\right ) x^{6}+\left (260 \,{\mathrm e}^{6}+1330 \,{\mathrm e}^{3}+120\right ) x^{5}+\left (-1452 \,{\mathrm e}^{9}-687 \,{\mathrm e}^{6}-18 \,{\mathrm e}^{3}\right ) x^{2}+260 x^{9}+25 x^{10}-726 \,{\mathrm e}^{12}-356 \,{\mathrm e}^{9}-9 \,{\mathrm e}^{6}}{\left (x^{2}+{\mathrm e}^{3}\right )^{2}}-\ln \left (x \right )\)	\(132\)
parallelrisch	\(-\frac {-260 x^{5} {\mathrm e}^{6}-25 \,{\mathrm e}^{6} x^{6}+x^{4} \ln \left (x \right )-260 x^{3} {\mathrm e}^{6}-30 x^{3} {\mathrm e}^{3}+687 x^{2} {\mathrm e}^{6}-1330 x^{5} {\mathrm e}^{3}+18 x^{2} {\mathrm e}^{3}+726 \,{\mathrm e}^{12}+9 \,{\mathrm e}^{6}-1070 x^{7}-876 x^{8}-25 x^{10}-260 x^{9}-556 x^{6}-120 x^{5}+356 \,{\mathrm e}^{9}+1452 \,{\mathrm e}^{9} x^{2}+2 \ln \left (x \right ) {\mathrm e}^{3} x^{2}+\ln \left (x \right ) {\mathrm e}^{6}-1602 \,{\mathrm e}^{3} x^{6}-50 \,{\mathrm e}^{3} x^{8}-520 \,{\mathrm e}^{3} x^{7}}{x^{4}+2 x^{2} {\mathrm e}^{3}+{\mathrm e}^{6}}\)	\(178\)

Input:

int(((150*x^6+1300*x^5+2904*x^4+780*x^3+50*x^2-1)*exp(3)^3+(450*x^8+3900*x 
^7+9612*x^6+6390*x^5+1374*x^4+90*x^3-3*x^2)*exp(3)^2+(450*x^10+3900*x^9+10 
212*x^8+8820*x^7+3336*x^6+570*x^5+33*x^4)*exp(3)+150*x^12+1300*x^11+3504*x 
^10+3210*x^9+1112*x^8+120*x^7-x^6)/(x*exp(3)^3+3*x^3*exp(3)^2+3*x^5*exp(3) 
+x^7),x,method=_RETURNVERBOSE)
 

Output:

25*x^6+260*x^5+876*x^4-150*x^2*exp(3)+1070*x^3-810*x*exp(3)+556*x^2+120*x+ 
(30*exp(3)*(27*exp(3)-7)*x^3+(-150*exp(9)+981*exp(6)-18*exp(3))*x^2+(810*e 
xp(9)-120*exp(6))*x-150*exp(12)+756*exp(9)-9*exp(6))/(x^4+2*x^2*exp(3)+exp 
(6))-ln(x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (34) = 68\).

Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 4.26 \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=\frac {25 \, x^{10} + 260 \, x^{9} + 876 \, x^{8} + 1070 \, x^{7} + 556 \, x^{6} + 120 \, x^{5} - 12 \, {\left (25 \, x^{2} - 63\right )} e^{9} + {\left (25 \, x^{6} + 260 \, x^{5} + 576 \, x^{4} + 260 \, x^{3} + 1537 \, x^{2} - 9\right )} e^{6} + 2 \, {\left (25 \, x^{8} + 260 \, x^{7} + 801 \, x^{6} + 665 \, x^{5} + 556 \, x^{4} + 15 \, x^{3} - 9 \, x^{2}\right )} e^{3} - {\left (x^{4} + 2 \, x^{2} e^{3} + e^{6}\right )} \log \left (x\right ) - 150 \, e^{12}}{x^{4} + 2 \, x^{2} e^{3} + e^{6}} \] Input:

integrate(((150*x^6+1300*x^5+2904*x^4+780*x^3+50*x^2-1)*exp(3)^3+(450*x^8+ 
3900*x^7+9612*x^6+6390*x^5+1374*x^4+90*x^3-3*x^2)*exp(3)^2+(450*x^10+3900* 
x^9+10212*x^8+8820*x^7+3336*x^6+570*x^5+33*x^4)*exp(3)+150*x^12+1300*x^11+ 
3504*x^10+3210*x^9+1112*x^8+120*x^7-x^6)/(x*exp(3)^3+3*x^3*exp(3)^2+3*x^5* 
exp(3)+x^7),x, algorithm="fricas")
 

Output:

(25*x^10 + 260*x^9 + 876*x^8 + 1070*x^7 + 556*x^6 + 120*x^5 - 12*(25*x^2 - 
 63)*e^9 + (25*x^6 + 260*x^5 + 576*x^4 + 260*x^3 + 1537*x^2 - 9)*e^6 + 2*( 
25*x^8 + 260*x^7 + 801*x^6 + 665*x^5 + 556*x^4 + 15*x^3 - 9*x^2)*e^3 - (x^ 
4 + 2*x^2*e^3 + e^6)*log(x) - 150*e^12)/(x^4 + 2*x^2*e^3 + e^6)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (26) = 52\).

Time = 2.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.31 \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=25 x^{6} + 260 x^{5} + 876 x^{4} + 1070 x^{3} + x^{2} \cdot \left (556 - 150 e^{3}\right ) + x \left (120 - 810 e^{3}\right ) - \log {\left (x \right )} + \frac {x^{3} \left (- 210 e^{3} + 810 e^{6}\right ) + x^{2} \left (- 150 e^{9} - 18 e^{3} + 981 e^{6}\right ) + x \left (- 120 e^{6} + 810 e^{9}\right ) - 150 e^{12} - 9 e^{6} + 756 e^{9}}{x^{4} + 2 x^{2} e^{3} + e^{6}} \] Input:

integrate(((150*x**6+1300*x**5+2904*x**4+780*x**3+50*x**2-1)*exp(3)**3+(45 
0*x**8+3900*x**7+9612*x**6+6390*x**5+1374*x**4+90*x**3-3*x**2)*exp(3)**2+( 
450*x**10+3900*x**9+10212*x**8+8820*x**7+3336*x**6+570*x**5+33*x**4)*exp(3 
)+150*x**12+1300*x**11+3504*x**10+3210*x**9+1112*x**8+120*x**7-x**6)/(x*ex 
p(3)**3+3*x**3*exp(3)**2+3*x**5*exp(3)+x**7),x)
 

Output:

25*x**6 + 260*x**5 + 876*x**4 + 1070*x**3 + x**2*(556 - 150*exp(3)) + x*(1 
20 - 810*exp(3)) - log(x) + (x**3*(-210*exp(3) + 810*exp(6)) + x**2*(-150* 
exp(9) - 18*exp(3) + 981*exp(6)) + x*(-120*exp(6) + 810*exp(9)) - 150*exp( 
12) - 9*exp(6) + 756*exp(9))/(x**4 + 2*x**2*exp(3) + exp(6))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (34) = 68\).

Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.40 \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=25 \, x^{6} + 260 \, x^{5} + 876 \, x^{4} + 1070 \, x^{3} - 2 \, x^{2} {\left (75 \, e^{3} - 278\right )} - 30 \, x {\left (27 \, e^{3} - 4\right )} + \frac {3 \, {\left (10 \, x^{3} {\left (27 \, e^{6} - 7 \, e^{3}\right )} - x^{2} {\left (50 \, e^{9} - 327 \, e^{6} + 6 \, e^{3}\right )} + 10 \, x {\left (27 \, e^{9} - 4 \, e^{6}\right )} - 50 \, e^{12} + 252 \, e^{9} - 3 \, e^{6}\right )}}{x^{4} + 2 \, x^{2} e^{3} + e^{6}} - \log \left (x\right ) \] Input:

integrate(((150*x^6+1300*x^5+2904*x^4+780*x^3+50*x^2-1)*exp(3)^3+(450*x^8+ 
3900*x^7+9612*x^6+6390*x^5+1374*x^4+90*x^3-3*x^2)*exp(3)^2+(450*x^10+3900* 
x^9+10212*x^8+8820*x^7+3336*x^6+570*x^5+33*x^4)*exp(3)+150*x^12+1300*x^11+ 
3504*x^10+3210*x^9+1112*x^8+120*x^7-x^6)/(x*exp(3)^3+3*x^3*exp(3)^2+3*x^5* 
exp(3)+x^7),x, algorithm="maxima")
 

Output:

25*x^6 + 260*x^5 + 876*x^4 + 1070*x^3 - 2*x^2*(75*e^3 - 278) - 30*x*(27*e^ 
3 - 4) + 3*(10*x^3*(27*e^6 - 7*e^3) - x^2*(50*e^9 - 327*e^6 + 6*e^3) + 10* 
x*(27*e^9 - 4*e^6) - 50*e^12 + 252*e^9 - 3*e^6)/(x^4 + 2*x^2*e^3 + e^6) - 
log(x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (34) = 68\).

Time = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.26 \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=25 \, x^{6} + 260 \, x^{5} + 876 \, x^{4} + 1070 \, x^{3} - 150 \, x^{2} e^{3} + 556 \, x^{2} - 810 \, x e^{3} + 120 \, x + \frac {3 \, {\left (270 \, x^{3} e^{6} - 70 \, x^{3} e^{3} - 50 \, x^{2} e^{9} + 327 \, x^{2} e^{6} - 6 \, x^{2} e^{3} + 270 \, x e^{9} - 40 \, x e^{6} - 50 \, e^{12} + 252 \, e^{9} - 3 \, e^{6}\right )}}{{\left (x^{2} + e^{3}\right )}^{2}} - \log \left ({\left | x \right |}\right ) \] Input:

integrate(((150*x^6+1300*x^5+2904*x^4+780*x^3+50*x^2-1)*exp(3)^3+(450*x^8+ 
3900*x^7+9612*x^6+6390*x^5+1374*x^4+90*x^3-3*x^2)*exp(3)^2+(450*x^10+3900* 
x^9+10212*x^8+8820*x^7+3336*x^6+570*x^5+33*x^4)*exp(3)+150*x^12+1300*x^11+ 
3504*x^10+3210*x^9+1112*x^8+120*x^7-x^6)/(x*exp(3)^3+3*x^3*exp(3)^2+3*x^5* 
exp(3)+x^7),x, algorithm="giac")
 

Output:

25*x^6 + 260*x^5 + 876*x^4 + 1070*x^3 - 150*x^2*e^3 + 556*x^2 - 810*x*e^3 
+ 120*x + 3*(270*x^3*e^6 - 70*x^3*e^3 - 50*x^2*e^9 + 327*x^2*e^6 - 6*x^2*e 
^3 + 270*x*e^9 - 40*x*e^6 - 50*e^12 + 252*e^9 - 3*e^6)/(x^2 + e^3)^2 - log 
(abs(x))
 

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.31 \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=1070\,x^3-x^2\,\left (150\,{\mathrm {e}}^3-556\right )-\frac {\left (210\,{\mathrm {e}}^3-810\,{\mathrm {e}}^6\right )\,x^3+\left (18\,{\mathrm {e}}^3-981\,{\mathrm {e}}^6+150\,{\mathrm {e}}^9\right )\,x^2+\left (120\,{\mathrm {e}}^6-810\,{\mathrm {e}}^9\right )\,x+9\,{\mathrm {e}}^6-756\,{\mathrm {e}}^9+150\,{\mathrm {e}}^{12}}{x^4+2\,{\mathrm {e}}^3\,x^2+{\mathrm {e}}^6}-\ln \left (x\right )+876\,x^4+260\,x^5+25\,x^6-x\,\left (810\,{\mathrm {e}}^3-120\right ) \] Input:

int((exp(9)*(50*x^2 + 780*x^3 + 2904*x^4 + 1300*x^5 + 150*x^6 - 1) + exp(6 
)*(90*x^3 - 3*x^2 + 1374*x^4 + 6390*x^5 + 9612*x^6 + 3900*x^7 + 450*x^8) + 
 exp(3)*(33*x^4 + 570*x^5 + 3336*x^6 + 8820*x^7 + 10212*x^8 + 3900*x^9 + 4 
50*x^10) - x^6 + 120*x^7 + 1112*x^8 + 3210*x^9 + 3504*x^10 + 1300*x^11 + 1 
50*x^12)/(x*exp(9) + 3*x^5*exp(3) + 3*x^3*exp(6) + x^7),x)
 

Output:

1070*x^3 - x^2*(150*exp(3) - 556) - (9*exp(6) - 756*exp(9) + 150*exp(12) + 
 x^3*(210*exp(3) - 810*exp(6)) + x*(120*exp(6) - 810*exp(9)) + x^2*(18*exp 
(3) - 981*exp(6) + 150*exp(9)))/(exp(6) + 2*x^2*exp(3) + x^4) - log(x) + 8 
76*x^4 + 260*x^5 + 25*x^6 - x*(810*exp(3) - 120)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 167, normalized size of antiderivative = 4.77 \[ \int \frac {-x^6+120 x^7+1112 x^8+3210 x^9+3504 x^{10}+1300 x^{11}+150 x^{12}+e^9 \left (-1+50 x^2+780 x^3+2904 x^4+1300 x^5+150 x^6\right )+e^6 \left (-3 x^2+90 x^3+1374 x^4+6390 x^5+9612 x^6+3900 x^7+450 x^8\right )+e^3 \left (33 x^4+570 x^5+3336 x^6+8820 x^7+10212 x^8+3900 x^9+450 x^{10}\right )}{e^9 x+3 e^6 x^3+3 e^3 x^5+x^7} \, dx=\frac {-2 \,\mathrm {log}\left (x \right ) e^{6}-4 \,\mathrm {log}\left (x \right ) e^{3} x^{2}-2 \,\mathrm {log}\left (x \right ) x^{4}-25 e^{9}+50 e^{6} x^{6}+520 e^{6} x^{5}+1452 e^{6} x^{4}+520 e^{6} x^{3}+100 e^{3} x^{8}+1040 e^{3} x^{7}+3204 e^{3} x^{6}+2660 e^{3} x^{5}+687 e^{3} x^{4}+60 e^{3} x^{3}+50 x^{10}+520 x^{9}+1752 x^{8}+2140 x^{7}+1112 x^{6}+240 x^{5}+18 x^{4}}{2 e^{6}+4 e^{3} x^{2}+2 x^{4}} \] Input:

int(((150*x^6+1300*x^5+2904*x^4+780*x^3+50*x^2-1)*exp(3)^3+(450*x^8+3900*x 
^7+9612*x^6+6390*x^5+1374*x^4+90*x^3-3*x^2)*exp(3)^2+(450*x^10+3900*x^9+10 
212*x^8+8820*x^7+3336*x^6+570*x^5+33*x^4)*exp(3)+150*x^12+1300*x^11+3504*x 
^10+3210*x^9+1112*x^8+120*x^7-x^6)/(x*exp(3)^3+3*x^3*exp(3)^2+3*x^5*exp(3) 
+x^7),x)
 

Output:

( - 2*log(x)*e**6 - 4*log(x)*e**3*x**2 - 2*log(x)*x**4 - 25*e**9 + 50*e**6 
*x**6 + 520*e**6*x**5 + 1452*e**6*x**4 + 520*e**6*x**3 + 100*e**3*x**8 + 1 
040*e**3*x**7 + 3204*e**3*x**6 + 2660*e**3*x**5 + 687*e**3*x**4 + 60*e**3* 
x**3 + 50*x**10 + 520*x**9 + 1752*x**8 + 2140*x**7 + 1112*x**6 + 240*x**5 
+ 18*x**4)/(2*(e**6 + 2*e**3*x**2 + x**4))