\(\int \frac {e^x (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6)+e^x (-32+2 x+32 x^2-x^3-8 x^4) \log (\frac {-4+2 x^2}{-80+5 x+40 x^2})}{32-2 x-32 x^2+x^3+8 x^4} \, dx\) [820]

Optimal result

Integrand size = 101, antiderivative size = 28 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=e^x \left (x^2-\log \left (\frac {2}{5 \left (8+\frac {x}{-2+x^2}\right )}\right )\right ) \] Output:

(x^2-ln(2/(40+5*x/(x^2-2))))*exp(x)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=e^x \left (x^2-\log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )\right ) \] Input:

Integrate[(E^x*(-2 + 64*x + 27*x^2 - 66*x^3 - 30*x^4 + 17*x^5 + 8*x^6) + E 
^x*(-32 + 2*x + 32*x^2 - x^3 - 8*x^4)*Log[(-4 + 2*x^2)/(-80 + 5*x + 40*x^2 
)])/(32 - 2*x - 32*x^2 + x^3 + 8*x^4),x]
 

Output:

E^x*(x^2 - Log[(2*(-2 + x^2))/(5*(-16 + x + 8*x^2))])
 

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 14.47 (sec) , antiderivative size = 723, normalized size of antiderivative = 25.82, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2463, 7239, 7276, 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.

\(\displaystyle \int \frac {e^x \left (-8 x^4-x^3+32 x^2+2 x-32\right ) \log \left (\frac {2 x^2-4}{40 x^2+5 x-80}\right )+e^x \left (8 x^6+17 x^5-30 x^4-66 x^3+27 x^2+64 x-2\right )}{8 x^4+x^3-32 x^2-2 x+32} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {x \left (e^x \left (-8 x^4-x^3+32 x^2+2 x-32\right ) \log \left (\frac {2 x^2-4}{40 x^2+5 x-80}\right )+e^x \left (8 x^6+17 x^5-30 x^4-66 x^3+27 x^2+64 x-2\right )\right )}{2 \left (x^2-2\right )}+\frac {(-8 x-1) \left (e^x \left (-8 x^4-x^3+32 x^2+2 x-32\right ) \log \left (\frac {2 x^2-4}{40 x^2+5 x-80}\right )+e^x \left (8 x^6+17 x^5-30 x^4-66 x^3+27 x^2+64 x-2\right )\right )}{2 \left (8 x^2+x-16\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^x \left (8 x^6+17 x^5-30 x^4-66 x^3+27 x^2-\left (8 x^4+x^3-32 x^2-2 x+32\right ) \log \left (\frac {2 \left (x^2-2\right )}{5 \left (8 x^2+x-16\right )}\right )+64 x-2\right )}{\left (-8 x^2-x+16\right ) \left (2-x^2\right )}dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {27 e^x x^2}{\left (x^2-2\right ) \left (8 x^2+x-16\right )}+\frac {64 e^x x}{\left (x^2-2\right ) \left (8 x^2+x-16\right )}-\frac {2 e^x}{\left (x^2-2\right ) \left (8 x^2+x-16\right )}-e^x \log \left (\frac {2 \left (x^2-2\right )}{5 \left (8 x^2+x-16\right )}\right )+\frac {8 e^x x^6}{\left (x^2-2\right ) \left (8 x^2+x-16\right )}+\frac {17 e^x x^5}{\left (x^2-2\right ) \left (8 x^2+x-16\right )}-\frac {30 e^x x^4}{\left (x^2-2\right ) \left (8 x^2+x-16\right )}-\frac {66 e^x x^3}{\left (x^2-2\right ) \left (8 x^2+x-16\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{342} \left (171+\sqrt {57}\right ) e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )-\frac {3}{38} \left (171-\sqrt {57}\right ) e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )-\frac {11}{456} \left (171-257 \sqrt {57}\right ) e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )+\frac {5 \left (22059-385 \sqrt {57}\right ) e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )}{3648}+\frac {17 \left (43947-33281 \sqrt {57}\right ) e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )}{175104}-\frac {\left (2867499-82561 \sqrt {57}\right ) e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )}{175104}-\frac {512 e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )}{3 \sqrt {57}}-e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )-\frac {\left (2867499+82561 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )}{175104}+\frac {17 \left (43947+33281 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )}{175104}+\frac {5 \left (22059+385 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )}{3648}-\frac {11}{456} \left (171+257 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )-\frac {3}{38} \left (171+\sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )+\frac {1}{342} \left (171-\sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )+\frac {512 e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )}{3 \sqrt {57}}-e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )+e^x x^2-e^x \log \left (\frac {2 \left (2-x^2\right )}{5 \left (-8 x^2-x+16\right )}\right )\)

Input:

Int[(E^x*(-2 + 64*x + 27*x^2 - 66*x^3 - 30*x^4 + 17*x^5 + 8*x^6) + E^x*(-3 
2 + 2*x + 32*x^2 - x^3 - 8*x^4)*Log[(-4 + 2*x^2)/(-80 + 5*x + 40*x^2)])/(3 
2 - 2*x - 32*x^2 + x^3 + 8*x^4),x]
 

Output:

E^x*x^2 - E^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/1 
6] - (512*E^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/1 
6])/(3*Sqrt[57]) - ((2867499 - 82561*Sqrt[57])*E^((-1 + 3*Sqrt[57])/16)*Ex 
pIntegralEi[(1 - 3*Sqrt[57] + 16*x)/16])/175104 + (17*(43947 - 33281*Sqrt[ 
57])*E^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/16])/1 
75104 + (5*(22059 - 385*Sqrt[57])*E^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[( 
1 - 3*Sqrt[57] + 16*x)/16])/3648 - (11*(171 - 257*Sqrt[57])*E^((-1 + 3*Sqr 
t[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/16])/456 - (3*(171 - Sqrt 
[57])*E^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/16])/ 
38 + ((171 + Sqrt[57])*E^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[ 
57] + 16*x)/16])/342 - E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[ 
57] + 16*x)/16] + (512*E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[ 
57] + 16*x)/16])/(3*Sqrt[57]) + ((171 - Sqrt[57])*E^((-1 - 3*Sqrt[57])/16) 
*ExpIntegralEi[(1 + 3*Sqrt[57] + 16*x)/16])/342 - (3*(171 + Sqrt[57])*E^(( 
-1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[57] + 16*x)/16])/38 - (11*( 
171 + 257*Sqrt[57])*E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[57] 
 + 16*x)/16])/456 + (5*(22059 + 385*Sqrt[57])*E^((-1 - 3*Sqrt[57])/16)*Exp 
IntegralEi[(1 + 3*Sqrt[57] + 16*x)/16])/3648 + (17*(43947 + 33281*Sqrt[57] 
)*E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[57] + 16*x)/16])/1751 
04 - ((2867499 + 82561*Sqrt[57])*E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi...
 

Defintions of rubi rules used

rule 2009

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

rule 2463

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u, Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt 
Q[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 
0]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [A] (verified)

Time = 1.49 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07

method	result	size
parallelrisch	\({\mathrm e}^{x} x^{2}-{\mathrm e}^{x} \ln \left (\frac {\frac {2 x^{2}}{5}-\frac {4}{5}}{8 x^{2}+x -16}\right )\)	\(30\)
risch	\({\mathrm e}^{x} \ln \left (x^{2}+\frac {1}{8} x -2\right )-{\mathrm e}^{x} \ln \left (x^{2}-2\right )+\frac {i {\mathrm e}^{x} \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}+\frac {1}{8} x -2}\right ) \operatorname {csgn}\left (i \left (x^{2}-2\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )}{x^{2}+\frac {1}{8} x -2}\right )}{2}-\frac {i {\mathrm e}^{x} \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}+\frac {1}{8} x -2}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )}{x^{2}+\frac {1}{8} x -2}\right )}^{2}}{2}-\frac {i {\mathrm e}^{x} \pi \,\operatorname {csgn}\left (i \left (x^{2}-2\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )}{x^{2}+\frac {1}{8} x -2}\right )}^{2}}{2}+\frac {i {\mathrm e}^{x} \pi {\operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )}{x^{2}+\frac {1}{8} x -2}\right )}^{3}}{2}+{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x} \ln \left (2\right )+{\mathrm e}^{x} \ln \left (5\right )\)	\(193\)
orering	\(\frac {\left (256 x^{9}+192 x^{8}-2012 x^{7}-1422 x^{6}+5950 x^{5}+3865 x^{4}-7728 x^{3}-4528 x^{2}+3656 x +1916\right ) \left (\left (-8 x^{4}-x^{3}+32 x^{2}+2 x -32\right ) {\mathrm e}^{x} \ln \left (\frac {2 x^{2}-4}{40 x^{2}+5 x -80}\right )+\left (8 x^{6}+17 x^{5}-30 x^{4}-66 x^{3}+27 x^{2}+64 x -2\right ) {\mathrm e}^{x}\right )}{\left (128 x^{9}+160 x^{8}-990 x^{7}-1222 x^{6}+2887 x^{5}+3465 x^{4}-3640 x^{3}-4304 x^{2}+1604 x +1980\right ) \left (8 x^{4}+x^{3}-32 x^{2}-2 x +32\right )}-\frac {\left (16 x^{5}+2 x^{4}-64 x^{3}-5 x^{2}+64 x -2\right ) \left (x^{2}-2\right ) \left (8 x^{2}+x -16\right ) \left (\frac {\left (-32 x^{3}-3 x^{2}+64 x +2\right ) {\mathrm e}^{x} \ln \left (\frac {2 x^{2}-4}{40 x^{2}+5 x -80}\right )+\left (-8 x^{4}-x^{3}+32 x^{2}+2 x -32\right ) {\mathrm e}^{x} \ln \left (\frac {2 x^{2}-4}{40 x^{2}+5 x -80}\right )+\frac {\left (-8 x^{4}-x^{3}+32 x^{2}+2 x -32\right ) {\mathrm e}^{x} \left (\frac {4 x}{40 x^{2}+5 x -80}-\frac {\left (2 x^{2}-4\right ) \left (80 x +5\right )}{\left (40 x^{2}+5 x -80\right )^{2}}\right ) \left (40 x^{2}+5 x -80\right )}{2 x^{2}-4}+\left (48 x^{5}+85 x^{4}-120 x^{3}-198 x^{2}+54 x +64\right ) {\mathrm e}^{x}+\left (8 x^{6}+17 x^{5}-30 x^{4}-66 x^{3}+27 x^{2}+64 x -2\right ) {\mathrm e}^{x}}{8 x^{4}+x^{3}-32 x^{2}-2 x +32}-\frac {\left (\left (-8 x^{4}-x^{3}+32 x^{2}+2 x -32\right ) {\mathrm e}^{x} \ln \left (\frac {2 x^{2}-4}{40 x^{2}+5 x -80}\right )+\left (8 x^{6}+17 x^{5}-30 x^{4}-66 x^{3}+27 x^{2}+64 x -2\right ) {\mathrm e}^{x}\right ) \left (32 x^{3}+3 x^{2}-64 x -2\right )}{\left (8 x^{4}+x^{3}-32 x^{2}-2 x +32\right )^{2}}\right )}{128 x^{9}+160 x^{8}-990 x^{7}-1222 x^{6}+2887 x^{5}+3465 x^{4}-3640 x^{3}-4304 x^{2}+1604 x +1980}\)	\(646\)

Input:

int(((-8*x^4-x^3+32*x^2+2*x-32)*exp(x)*ln((2*x^2-4)/(40*x^2+5*x-80))+(8*x^ 
6+17*x^5-30*x^4-66*x^3+27*x^2+64*x-2)*exp(x))/(8*x^4+x^3-32*x^2-2*x+32),x, 
method=_RETURNVERBOSE)
 

Output:

exp(x)*x^2-exp(x)*ln(2/5*(x^2-2)/(8*x^2+x-16))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=x^{2} e^{x} - e^{x} \log \left (\frac {2 \, {\left (x^{2} - 2\right )}}{5 \, {\left (8 \, x^{2} + x - 16\right )}}\right ) \] Input:

integrate(((-8*x^4-x^3+32*x^2+2*x-32)*exp(x)*log((2*x^2-4)/(40*x^2+5*x-80) 
)+(8*x^6+17*x^5-30*x^4-66*x^3+27*x^2+64*x-2)*exp(x))/(8*x^4+x^3-32*x^2-2*x 
+32),x, algorithm="fricas")
 

Output:

x^2*e^x - e^x*log(2/5*(x^2 - 2)/(8*x^2 + x - 16))
 

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=\left (x^{2} - \log {\left (\frac {2 x^{2} - 4}{40 x^{2} + 5 x - 80} \right )}\right ) e^{x} \] Input:

integrate(((-8*x**4-x**3+32*x**2+2*x-32)*exp(x)*ln((2*x**2-4)/(40*x**2+5*x 
-80))+(8*x**6+17*x**5-30*x**4-66*x**3+27*x**2+64*x-2)*exp(x))/(8*x**4+x**3 
-32*x**2-2*x+32),x)
 

Output:

(x**2 - log((2*x**2 - 4)/(40*x**2 + 5*x - 80)))*exp(x)
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx={\left (x^{2} + \log \left (5\right ) - \log \left (2\right )\right )} e^{x} + e^{x} \log \left (8 \, x^{2} + x - 16\right ) - e^{x} \log \left (x^{2} - 2\right ) \] Input:

integrate(((-8*x^4-x^3+32*x^2+2*x-32)*exp(x)*log((2*x^2-4)/(40*x^2+5*x-80) 
)+(8*x^6+17*x^5-30*x^4-66*x^3+27*x^2+64*x-2)*exp(x))/(8*x^4+x^3-32*x^2-2*x 
+32),x, algorithm="maxima")
 

Output:

(x^2 + log(5) - log(2))*e^x + e^x*log(8*x^2 + x - 16) - e^x*log(x^2 - 2)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=x^{2} e^{x} - e^{x} \log \left (\frac {2 \, {\left (x^{2} - 2\right )}}{5 \, {\left (8 \, x^{2} + x - 16\right )}}\right ) \] Input:

integrate(((-8*x^4-x^3+32*x^2+2*x-32)*exp(x)*log((2*x^2-4)/(40*x^2+5*x-80) 
)+(8*x^6+17*x^5-30*x^4-66*x^3+27*x^2+64*x-2)*exp(x))/(8*x^4+x^3-32*x^2-2*x 
+32),x, algorithm="giac")
 

Output:

x^2*e^x - e^x*log(2/5*(x^2 - 2)/(8*x^2 + x - 16))
 

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=-{\mathrm {e}}^x\,\left (\ln \left (\frac {2\,x^2-4}{40\,x^2+5\,x-80}\right )-x^2\right ) \] Input:

int((exp(x)*(64*x + 27*x^2 - 66*x^3 - 30*x^4 + 17*x^5 + 8*x^6 - 2) - log(( 
2*x^2 - 4)/(5*x + 40*x^2 - 80))*exp(x)*(x^3 - 32*x^2 - 2*x + 8*x^4 + 32))/ 
(x^3 - 32*x^2 - 2*x + 8*x^4 + 32),x)
 

Output:

-exp(x)*(log((2*x^2 - 4)/(5*x + 40*x^2 - 80)) - x^2)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=e^{x} \left (-\mathrm {log}\left (\frac {2 x^{2}-4}{40 x^{2}+5 x -80}\right )+x^{2}\right ) \] Input:

int(((-8*x^4-x^3+32*x^2+2*x-32)*exp(x)*log((2*x^2-4)/(40*x^2+5*x-80))+(8*x 
^6+17*x^5-30*x^4-66*x^3+27*x^2+64*x-2)*exp(x))/(8*x^4+x^3-32*x^2-2*x+32),x 
)
 

Output:

e**x*( - log((2*x**2 - 4)/(40*x**2 + 5*x - 80)) + x**2)