Integrand size = 67, antiderivative size = 25 \[ \int \frac {e^x \left (-7+x+9 x^2+x^3-x^4\right )+e^x \left (7-9 x-2 x^2+x^3\right ) \log (x)}{49 x^2-14 x^3-13 x^4+2 x^5+x^6} \, dx=\frac {e^x (x-\log (x))}{x \left (8+x-(1+x)^2\right )} \]
Time = 1.81 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {e^x \left (-7+x+9 x^2+x^3-x^4\right )+e^x \left (7-9 x-2 x^2+x^3\right ) \log (x)}{49 x^2-14 x^3-13 x^4+2 x^5+x^6} \, dx=\frac {e^x (-x+\log (x))}{x \left (-7+x+x^2\right )} \]
Integrate[(E^x*(-7 + x + 9*x^2 + x^3 - x^4) + E^x*(7 - 9*x - 2*x^2 + x^3)* Log[x])/(49*x^2 - 14*x^3 - 13*x^4 + 2*x^5 + x^6),x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 14.35 (sec) , antiderivative size = 1024, normalized size of antiderivative = 40.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2026, 2463, 7239, 7293, 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.
\(\displaystyle \int \frac {e^x \left (x^3-2 x^2-9 x+7\right ) \log (x)+e^x \left (-x^4+x^3+9 x^2+x-7\right )}{x^6+2 x^5-13 x^4-14 x^3+49 x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {e^x \left (x^3-2 x^2-9 x+7\right ) \log (x)+e^x \left (-x^4+x^3+9 x^2+x-7\right )}{x^2 \left (x^4+2 x^3-13 x^2-14 x+49\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {4 \left (e^x \left (x^3-2 x^2-9 x+7\right ) \log (x)+e^x \left (-x^4+x^3+9 x^2+x-7\right )\right )}{29 \sqrt {29} x^2 \left (2 x+\sqrt {29}+1\right )}+\frac {4 \left (e^x \left (x^3-2 x^2-9 x+7\right ) \log (x)+e^x \left (-x^4+x^3+9 x^2+x-7\right )\right )}{29 \sqrt {29} \left (-2 x+\sqrt {29}-1\right ) x^2}+\frac {4 \left (e^x \left (x^3-2 x^2-9 x+7\right ) \log (x)+e^x \left (-x^4+x^3+9 x^2+x-7\right )\right )}{29 \left (-2 x+\sqrt {29}-1\right )^2 x^2}+\frac {4 \left (e^x \left (x^3-2 x^2-9 x+7\right ) \log (x)+e^x \left (-x^4+x^3+9 x^2+x-7\right )\right )}{29 x^2 \left (2 x+\sqrt {29}+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^x \left (-x^4+x^3+9 x^2+\left (x^3-2 x^2-9 x+7\right ) \log (x)+x-7\right )}{x^2 \left (-x^2-x+7\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {e^x x^2}{\left (x^2+x-7\right )^2}+\frac {e^x x}{\left (x^2+x-7\right )^2}+\frac {9 e^x}{\left (x^2+x-7\right )^2}+\frac {e^x}{\left (x^2+x-7\right )^2 x}-\frac {7 e^x}{\left (x^2+x-7\right )^2 x^2}+\frac {e^x \left (x^3-2 x^2-9 x+7\right ) \log (x)}{\left (x^2+x-7\right )^2 x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{406} \left (29-\sqrt {29}\right ) e^{\frac {1}{2} \left (-1+\sqrt {29}\right )} \log (x) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-\sqrt {29}+1\right )\right )+\frac {1}{406} \left (1-\sqrt {29}\right ) e^{\frac {1}{2} \left (-1+\sqrt {29}\right )} \log (x) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-\sqrt {29}+1\right )\right )+\frac {e^{\frac {1}{2} \left (-1+\sqrt {29}\right )} \log (x) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-\sqrt {29}+1\right )\right )}{7 \sqrt {29}}-\frac {15}{203} e^{\frac {1}{2} \left (-1+\sqrt {29}\right )} \log (x) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-\sqrt {29}+1\right )\right )-\frac {\left (29+15 \sqrt {29}\right ) e^{\frac {1}{2} \left (-1+\sqrt {29}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-\sqrt {29}+1\right )\right )}{2842}+\frac {\left (29+8 \sqrt {29}\right ) e^{\frac {1}{2} \left (-1+\sqrt {29}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-\sqrt {29}+1\right )\right )}{1421}-\frac {\left (29+\sqrt {29}\right ) e^{\frac {1}{2} \left (-1+\sqrt {29}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-\sqrt {29}+1\right )\right )}{2842}-\frac {1}{29} \left (1-\sqrt {29}\right ) e^{\frac {1}{2} \left (-1+\sqrt {29}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-\sqrt {29}+1\right )\right )-\frac {e^{\frac {1}{2} \left (-1+\sqrt {29}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-\sqrt {29}+1\right )\right )}{\sqrt {29}}+\frac {1}{29} e^{\frac {1}{2} \left (-1+\sqrt {29}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-\sqrt {29}+1\right )\right )-\frac {1}{29} \left (1+\sqrt {29}\right ) e^{-\frac {1}{2}-\frac {\sqrt {29}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+\sqrt {29}+1\right )\right )-\frac {\left (29-\sqrt {29}\right ) e^{-\frac {1}{2}-\frac {\sqrt {29}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+\sqrt {29}+1\right )\right )}{2842}+\frac {\left (29-8 \sqrt {29}\right ) e^{-\frac {1}{2}-\frac {\sqrt {29}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+\sqrt {29}+1\right )\right )}{1421}-\frac {\left (29-15 \sqrt {29}\right ) e^{-\frac {1}{2}-\frac {\sqrt {29}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+\sqrt {29}+1\right )\right )}{2842}+\frac {e^{-\frac {1}{2}-\frac {\sqrt {29}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+\sqrt {29}+1\right )\right )}{\sqrt {29}}+\frac {1}{29} e^{-\frac {1}{2}-\frac {\sqrt {29}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+\sqrt {29}+1\right )\right )+\frac {1}{406} \left (29+\sqrt {29}\right ) e^{-\frac {1}{2}-\frac {\sqrt {29}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+\sqrt {29}+1\right )\right ) \log (x)+\frac {1}{406} \left (1+\sqrt {29}\right ) e^{-\frac {1}{2}-\frac {\sqrt {29}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+\sqrt {29}+1\right )\right ) \log (x)-\frac {e^{-\frac {1}{2}-\frac {\sqrt {29}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+\sqrt {29}+1\right )\right ) \log (x)}{7 \sqrt {29}}-\frac {15}{203} e^{-\frac {1}{2}-\frac {\sqrt {29}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+\sqrt {29}+1\right )\right ) \log (x)-\frac {e^x \log (x)}{7 x}-\frac {\left (1-\sqrt {29}\right ) e^x \log (x)}{203 \left (2 x-\sqrt {29}+1\right )}+\frac {30 e^x \log (x)}{203 \left (2 x-\sqrt {29}+1\right )}-\frac {\left (1+\sqrt {29}\right ) e^x \log (x)}{203 \left (2 x+\sqrt {29}+1\right )}+\frac {30 e^x \log (x)}{203 \left (2 x+\sqrt {29}+1\right )}+\frac {2 \left (1-\sqrt {29}\right ) e^x}{29 \left (2 x-\sqrt {29}+1\right )}-\frac {2 e^x}{29 \left (2 x-\sqrt {29}+1\right )}+\frac {2 \left (1+\sqrt {29}\right ) e^x}{29 \left (2 x+\sqrt {29}+1\right )}-\frac {2 e^x}{29 \left (2 x+\sqrt {29}+1\right )}\) |
Int[(E^x*(-7 + x + 9*x^2 + x^3 - x^4) + E^x*(7 - 9*x - 2*x^2 + x^3)*Log[x] )/(49*x^2 - 14*x^3 - 13*x^4 + 2*x^5 + x^6),x]
(-2*E^x)/(29*(1 - Sqrt[29] + 2*x)) + (2*(1 - Sqrt[29])*E^x)/(29*(1 - Sqrt[ 29] + 2*x)) - (2*E^x)/(29*(1 + Sqrt[29] + 2*x)) + (2*(1 + Sqrt[29])*E^x)/( 29*(1 + Sqrt[29] + 2*x)) + (E^((-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqrt[ 29] + 2*x)/2])/29 - (E^((-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqrt[29] + 2 *x)/2])/Sqrt[29] - ((1 - Sqrt[29])*E^((-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqrt[29] + 2*x)/2])/29 - ((29 + Sqrt[29])*E^((-1 + Sqrt[29])/2)*ExpInteg ralEi[(1 - Sqrt[29] + 2*x)/2])/2842 + ((29 + 8*Sqrt[29])*E^((-1 + Sqrt[29] )/2)*ExpIntegralEi[(1 - Sqrt[29] + 2*x)/2])/1421 - ((29 + 15*Sqrt[29])*E^( (-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqrt[29] + 2*x)/2])/2842 + (E^(-1/2 - Sqrt[29]/2)*ExpIntegralEi[(1 + Sqrt[29] + 2*x)/2])/29 + (E^(-1/2 - Sqrt[ 29]/2)*ExpIntegralEi[(1 + Sqrt[29] + 2*x)/2])/Sqrt[29] - ((29 - 15*Sqrt[29 ])*E^(-1/2 - Sqrt[29]/2)*ExpIntegralEi[(1 + Sqrt[29] + 2*x)/2])/2842 + ((2 9 - 8*Sqrt[29])*E^(-1/2 - Sqrt[29]/2)*ExpIntegralEi[(1 + Sqrt[29] + 2*x)/2 ])/1421 - ((29 - Sqrt[29])*E^(-1/2 - Sqrt[29]/2)*ExpIntegralEi[(1 + Sqrt[2 9] + 2*x)/2])/2842 - ((1 + Sqrt[29])*E^(-1/2 - Sqrt[29]/2)*ExpIntegralEi[( 1 + Sqrt[29] + 2*x)/2])/29 - (E^x*Log[x])/(7*x) + (30*E^x*Log[x])/(203*(1 - Sqrt[29] + 2*x)) - ((1 - Sqrt[29])*E^x*Log[x])/(203*(1 - Sqrt[29] + 2*x) ) + (30*E^x*Log[x])/(203*(1 + Sqrt[29] + 2*x)) - ((1 + Sqrt[29])*E^x*Log[x ])/(203*(1 + Sqrt[29] + 2*x)) - (15*E^((-1 + Sqrt[29])/2)*ExpIntegralEi[(1 - Sqrt[29] + 2*x)/2]*Log[x])/203 + (E^((-1 + Sqrt[29])/2)*ExpIntegralE...
3.20.55.3.1 Defintions of rubi rules used
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])
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]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(-\frac {{\mathrm e}^{x} x -{\mathrm e}^{x} \ln \left (x \right )}{x \left (x^{2}+x -7\right )}\) | \(25\) |
risch | \(\frac {{\mathrm e}^{x} \ln \left (x \right )}{x \left (x^{2}+x -7\right )}-\frac {{\mathrm e}^{x}}{x^{2}+x -7}\) | \(30\) |
int(((x^3-2*x^2-9*x+7)*exp(x)*ln(x)+(-x^4+x^3+9*x^2+x-7)*exp(x))/(x^6+2*x^ 5-13*x^4-14*x^3+49*x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (-7+x+9 x^2+x^3-x^4\right )+e^x \left (7-9 x-2 x^2+x^3\right ) \log (x)}{49 x^2-14 x^3-13 x^4+2 x^5+x^6} \, dx=-\frac {x e^{x} - e^{x} \log \left (x\right )}{x^{3} + x^{2} - 7 \, x} \]
integrate(((x^3-2*x^2-9*x+7)*exp(x)*log(x)+(-x^4+x^3+9*x^2+x-7)*exp(x))/(x ^6+2*x^5-13*x^4-14*x^3+49*x^2),x, algorithm=\
Time = 0.15 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {e^x \left (-7+x+9 x^2+x^3-x^4\right )+e^x \left (7-9 x-2 x^2+x^3\right ) \log (x)}{49 x^2-14 x^3-13 x^4+2 x^5+x^6} \, dx=\frac {\left (- x + \log {\left (x \right )}\right ) e^{x}}{x^{3} + x^{2} - 7 x} \]
integrate(((x**3-2*x**2-9*x+7)*exp(x)*ln(x)+(-x**4+x**3+9*x**2+x-7)*exp(x) )/(x**6+2*x**5-13*x**4-14*x**3+49*x**2),x)
Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {e^x \left (-7+x+9 x^2+x^3-x^4\right )+e^x \left (7-9 x-2 x^2+x^3\right ) \log (x)}{49 x^2-14 x^3-13 x^4+2 x^5+x^6} \, dx=-\frac {{\left (x - \log \left (x\right )\right )} e^{x}}{x^{3} + x^{2} - 7 \, x} \]
integrate(((x^3-2*x^2-9*x+7)*exp(x)*log(x)+(-x^4+x^3+9*x^2+x-7)*exp(x))/(x ^6+2*x^5-13*x^4-14*x^3+49*x^2),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (-7+x+9 x^2+x^3-x^4\right )+e^x \left (7-9 x-2 x^2+x^3\right ) \log (x)}{49 x^2-14 x^3-13 x^4+2 x^5+x^6} \, dx=-\frac {x e^{x} - e^{x} \log \left (x\right )}{x^{3} + x^{2} - 7 \, x} \]
integrate(((x^3-2*x^2-9*x+7)*exp(x)*log(x)+(-x^4+x^3+9*x^2+x-7)*exp(x))/(x ^6+2*x^5-13*x^4-14*x^3+49*x^2),x, algorithm=\
Timed out. \[ \int \frac {e^x \left (-7+x+9 x^2+x^3-x^4\right )+e^x \left (7-9 x-2 x^2+x^3\right ) \log (x)}{49 x^2-14 x^3-13 x^4+2 x^5+x^6} \, dx=\int \frac {{\mathrm {e}}^x\,\left (-x^4+x^3+9\,x^2+x-7\right )-{\mathrm {e}}^x\,\ln \left (x\right )\,\left (-x^3+2\,x^2+9\,x-7\right )}{x^6+2\,x^5-13\,x^4-14\,x^3+49\,x^2} \,d x \]
int((exp(x)*(x + 9*x^2 + x^3 - x^4 - 7) - exp(x)*log(x)*(9*x + 2*x^2 - x^3 - 7))/(49*x^2 - 14*x^3 - 13*x^4 + 2*x^5 + x^6),x)