Integrand size = 79, antiderivative size = 29 \[ \int \frac {140+70 x-77 x^2+35 x^3+77 x^4+21 x^5+e^x \left (28 x+28 x^2+7 x^3\right )+\left (-56 x^2-56 x^3-14 x^4\right ) \log (5 x)}{4 x+4 x^2+x^3} \, dx=7 \left (e^x+\left (5-x^2\right ) \left (-x-\frac {2}{2+x}+\log (5 x)\right )\right ) \] Output:
7*exp(x)+7*(-x^2+5)*(ln(5*x)-2/(2+x)-x)
Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {140+70 x-77 x^2+35 x^3+77 x^4+21 x^5+e^x \left (28 x+28 x^2+7 x^3\right )+\left (-56 x^2-56 x^3-14 x^4\right ) \log (5 x)}{4 x+4 x^2+x^3} \, dx=7 \left (e^x-3 x+x^3-\frac {2}{2+x}+5 \log (x)-x^2 \log (5 x)\right ) \] Input:
Integrate[(140 + 70*x - 77*x^2 + 35*x^3 + 77*x^4 + 21*x^5 + E^x*(28*x + 28 *x^2 + 7*x^3) + (-56*x^2 - 56*x^3 - 14*x^4)*Log[5*x])/(4*x + 4*x^2 + x^3), x]
Output:
7*(E^x - 3*x + x^3 - 2/(2 + x) + 5*Log[x] - x^2*Log[5*x])
Time = 0.76 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {2026, 2007, 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.
\(\displaystyle \int \frac {21 x^5+77 x^4+35 x^3-77 x^2+e^x \left (7 x^3+28 x^2+28 x\right )+\left (-14 x^4-56 x^3-56 x^2\right ) \log (5 x)+70 x+140}{x^3+4 x^2+4 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {21 x^5+77 x^4+35 x^3-77 x^2+e^x \left (7 x^3+28 x^2+28 x\right )+\left (-14 x^4-56 x^3-56 x^2\right ) \log (5 x)+70 x+140}{x \left (x^2+4 x+4\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {21 x^5+77 x^4+35 x^3-77 x^2+e^x \left (7 x^3+28 x^2+28 x\right )+\left (-14 x^4-56 x^3-56 x^2\right ) \log (5 x)+70 x+140}{x (x+2)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {21 x^4}{(x+2)^2}+\frac {77 x^3}{(x+2)^2}+\frac {35 x^2}{(x+2)^2}-\frac {77 x}{(x+2)^2}+7 e^x+\frac {70}{(x+2)^2}+\frac {140}{(x+2)^2 x}-14 x \log (5 x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 7 x^3-7 x^2 \log (5 x)-21 x+7 e^x-\frac {14}{x+2}+35 \log (x)\) |
Input:
Int[(140 + 70*x - 77*x^2 + 35*x^3 + 77*x^4 + 21*x^5 + E^x*(28*x + 28*x^2 + 7*x^3) + (-56*x^2 - 56*x^3 - 14*x^4)*Log[5*x])/(4*x + 4*x^2 + x^3),x]
Output:
7*E^x - 21*x + 7*x^3 - 14/(2 + x) + 35*Log[x] - 7*x^2*Log[5*x]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
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])
Time = 0.98 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17
method | result | size |
default | \(-7 \ln \left (5 x \right ) x^{2}+7 x^{3}-21 x -\frac {14}{2+x}+35 \ln \left (x \right )+7 \,{\mathrm e}^{x}\) | \(34\) |
parts | \(-7 \ln \left (5 x \right ) x^{2}+7 x^{3}-21 x -\frac {14}{2+x}+35 \ln \left (x \right )+7 \,{\mathrm e}^{x}\) | \(34\) |
risch | \(-7 \ln \left (5 x \right ) x^{2}+\frac {7 x^{4}+14 x^{3}+35 x \ln \left (x \right )-21 x^{2}+7 \,{\mathrm e}^{x} x +70 \ln \left (x \right )-42 x +14 \,{\mathrm e}^{x}-14}{2+x}\) | \(53\) |
parallelrisch | \(-\frac {-14 x^{4}+14 \ln \left (5 x \right ) x^{3}-140-28 x^{3}+28 \ln \left (5 x \right ) x^{2}-70 x \ln \left (x \right )+42 x^{2}-14 \,{\mathrm e}^{x} x -140 \ln \left (x \right )-28 \,{\mathrm e}^{x}}{2 \left (2+x \right )}\) | \(61\) |
orering | \(\frac {\left (4 x^{10}+23 x^{9}+2 x^{8}-172 x^{7}-40 x^{6}+1197 x^{5}+2354 x^{4}+2020 x^{3}+2096 x^{2}+2640 x +1120\right ) \left (\left (-14 x^{4}-56 x^{3}-56 x^{2}\right ) \ln \left (5 x \right )+\left (7 x^{3}+28 x^{2}+28 x \right ) {\mathrm e}^{x}+21 x^{5}+77 x^{4}+35 x^{3}-77 x^{2}+70 x +140\right )}{2 \left (3 x^{9}+16 x^{8}+19 x^{7}-20 x^{6}-56 x^{5}-162 x^{4}-364 x^{3}-272 x^{2}+160 x +160\right ) \left (x^{3}+4 x^{2}+4 x \right )}-\frac {\left (x^{11}+7 x^{10}-2 x^{9}-72 x^{8}+111 x^{7}+1259 x^{6}+2438 x^{5}+1482 x^{4}+116 x^{3}+1120 x^{2}+1920 x +800\right ) \left (\frac {\left (-56 x^{3}-168 x^{2}-112 x \right ) \ln \left (5 x \right )+\frac {-14 x^{4}-56 x^{3}-56 x^{2}}{x}+\left (21 x^{2}+56 x +28\right ) {\mathrm e}^{x}+\left (7 x^{3}+28 x^{2}+28 x \right ) {\mathrm e}^{x}+105 x^{4}+308 x^{3}+105 x^{2}-154 x +70}{x^{3}+4 x^{2}+4 x}-\frac {\left (\left (-14 x^{4}-56 x^{3}-56 x^{2}\right ) \ln \left (5 x \right )+\left (7 x^{3}+28 x^{2}+28 x \right ) {\mathrm e}^{x}+21 x^{5}+77 x^{4}+35 x^{3}-77 x^{2}+70 x +140\right ) \left (3 x^{2}+8 x +4\right )}{\left (x^{3}+4 x^{2}+4 x \right )^{2}}\right )}{2 \left (3 x^{9}+16 x^{8}+19 x^{7}-20 x^{6}-56 x^{5}-162 x^{4}-364 x^{3}-272 x^{2}+160 x +160\right )}+\frac {\left (x^{9}+x^{8}-21 x^{7}+321 x^{5}+617 x^{4}-105 x^{3}-986 x^{2}-660 x -200\right ) x \left (2+x \right ) \left (\frac {\left (-168 x^{2}-336 x -112\right ) \ln \left (5 x \right )+\frac {-112 x^{3}-336 x^{2}-224 x}{x}-\frac {-14 x^{4}-56 x^{3}-56 x^{2}}{x^{2}}+\left (42 x +56\right ) {\mathrm e}^{x}+2 \left (21 x^{2}+56 x +28\right ) {\mathrm e}^{x}+\left (7 x^{3}+28 x^{2}+28 x \right ) {\mathrm e}^{x}+420 x^{3}+924 x^{2}+210 x -154}{x^{3}+4 x^{2}+4 x}-\frac {2 \left (\left (-56 x^{3}-168 x^{2}-112 x \right ) \ln \left (5 x \right )+\frac {-14 x^{4}-56 x^{3}-56 x^{2}}{x}+\left (21 x^{2}+56 x +28\right ) {\mathrm e}^{x}+\left (7 x^{3}+28 x^{2}+28 x \right ) {\mathrm e}^{x}+105 x^{4}+308 x^{3}+105 x^{2}-154 x +70\right ) \left (3 x^{2}+8 x +4\right )}{\left (x^{3}+4 x^{2}+4 x \right )^{2}}+\frac {2 \left (\left (-14 x^{4}-56 x^{3}-56 x^{2}\right ) \ln \left (5 x \right )+\left (7 x^{3}+28 x^{2}+28 x \right ) {\mathrm e}^{x}+21 x^{5}+77 x^{4}+35 x^{3}-77 x^{2}+70 x +140\right ) \left (3 x^{2}+8 x +4\right )^{2}}{\left (x^{3}+4 x^{2}+4 x \right )^{3}}-\frac {\left (\left (-14 x^{4}-56 x^{3}-56 x^{2}\right ) \ln \left (5 x \right )+\left (7 x^{3}+28 x^{2}+28 x \right ) {\mathrm e}^{x}+21 x^{5}+77 x^{4}+35 x^{3}-77 x^{2}+70 x +140\right ) \left (6 x +8\right )}{\left (x^{3}+4 x^{2}+4 x \right )^{2}}\right )}{6 x^{9}+32 x^{8}+38 x^{7}-40 x^{6}-112 x^{5}-324 x^{4}-728 x^{3}-544 x^{2}+320 x +320}\) | \(978\) |
Input:
int(((-14*x^4-56*x^3-56*x^2)*ln(5*x)+(7*x^3+28*x^2+28*x)*exp(x)+21*x^5+77* x^4+35*x^3-77*x^2+70*x+140)/(x^3+4*x^2+4*x),x,method=_RETURNVERBOSE)
Output:
-7*ln(5*x)*x^2+7*x^3-21*x-14/(2+x)+35*ln(x)+7*exp(x)
Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {140+70 x-77 x^2+35 x^3+77 x^4+21 x^5+e^x \left (28 x+28 x^2+7 x^3\right )+\left (-56 x^2-56 x^3-14 x^4\right ) \log (5 x)}{4 x+4 x^2+x^3} \, dx=\frac {7 \, {\left (x^{4} + 2 \, x^{3} - 3 \, x^{2} + {\left (x + 2\right )} e^{x} - {\left (x^{3} + 2 \, x^{2} - 5 \, x - 10\right )} \log \left (5 \, x\right ) - 6 \, x - 2\right )}}{x + 2} \] Input:
integrate(((-14*x^4-56*x^3-56*x^2)*log(5*x)+(7*x^3+28*x^2+28*x)*exp(x)+21* x^5+77*x^4+35*x^3-77*x^2+70*x+140)/(x^3+4*x^2+4*x),x, algorithm="fricas")
Output:
7*(x^4 + 2*x^3 - 3*x^2 + (x + 2)*e^x - (x^3 + 2*x^2 - 5*x - 10)*log(5*x) - 6*x - 2)/(x + 2)
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {140+70 x-77 x^2+35 x^3+77 x^4+21 x^5+e^x \left (28 x+28 x^2+7 x^3\right )+\left (-56 x^2-56 x^3-14 x^4\right ) \log (5 x)}{4 x+4 x^2+x^3} \, dx=7 x^{3} - 7 x^{2} \log {\left (5 x \right )} - 21 x + 7 e^{x} + 35 \log {\left (x \right )} - \frac {14}{x + 2} \] Input:
integrate(((-14*x**4-56*x**3-56*x**2)*ln(5*x)+(7*x**3+28*x**2+28*x)*exp(x) +21*x**5+77*x**4+35*x**3-77*x**2+70*x+140)/(x**3+4*x**2+4*x),x)
Output:
7*x**3 - 7*x**2*log(5*x) - 21*x + 7*exp(x) + 35*log(x) - 14/(x + 2)
\[ \int \frac {140+70 x-77 x^2+35 x^3+77 x^4+21 x^5+e^x \left (28 x+28 x^2+7 x^3\right )+\left (-56 x^2-56 x^3-14 x^4\right ) \log (5 x)}{4 x+4 x^2+x^3} \, dx=\int { \frac {7 \, {\left (3 \, x^{5} + 11 \, x^{4} + 5 \, x^{3} - 11 \, x^{2} + {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} e^{x} - 2 \, {\left (x^{4} + 4 \, x^{3} + 4 \, x^{2}\right )} \log \left (5 \, x\right ) + 10 \, x + 20\right )}}{x^{3} + 4 \, x^{2} + 4 \, x} \,d x } \] Input:
integrate(((-14*x^4-56*x^3-56*x^2)*log(5*x)+(7*x^3+28*x^2+28*x)*exp(x)+21* x^5+77*x^4+35*x^3-77*x^2+70*x+140)/(x^3+4*x^2+4*x),x, algorithm="maxima")
Output:
7*x^3 - 7/2*x^2*(2*log(5) - 1) - 7*x^2*log(x) - 7/2*x^2 - 21*x - 28*e^(-2) *exp_integral_e(2, -x - 2)/(x + 2) - 14/(x + 2) + 7*integrate((x^2 + 4*x)* e^x/(x^2 + 4*x + 4), x) + 35*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (27) = 54\).
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07 \[ \int \frac {140+70 x-77 x^2+35 x^3+77 x^4+21 x^5+e^x \left (28 x+28 x^2+7 x^3\right )+\left (-56 x^2-56 x^3-14 x^4\right ) \log (5 x)}{4 x+4 x^2+x^3} \, dx=\frac {7 \, {\left (x^{4} - x^{3} \log \left (5 \, x\right ) + 2 \, x^{3} - 2 \, x^{2} \log \left (5 \, x\right ) - 3 \, x^{2} + x e^{x} + 5 \, x \log \left (x\right ) - 6 \, x + 2 \, e^{x} + 10 \, \log \left (x\right ) - 2\right )}}{x + 2} \] Input:
integrate(((-14*x^4-56*x^3-56*x^2)*log(5*x)+(7*x^3+28*x^2+28*x)*exp(x)+21* x^5+77*x^4+35*x^3-77*x^2+70*x+140)/(x^3+4*x^2+4*x),x, algorithm="giac")
Output:
7*(x^4 - x^3*log(5*x) + 2*x^3 - 2*x^2*log(5*x) - 3*x^2 + x*e^x + 5*x*log(x ) - 6*x + 2*e^x + 10*log(x) - 2)/(x + 2)
Time = 3.97 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {140+70 x-77 x^2+35 x^3+77 x^4+21 x^5+e^x \left (28 x+28 x^2+7 x^3\right )+\left (-56 x^2-56 x^3-14 x^4\right ) \log (5 x)}{4 x+4 x^2+x^3} \, dx=7\,{\mathrm {e}}^x-21\,x+35\,\ln \left (x\right )-7\,x^2\,\ln \left (x\right )-\frac {14}{x+2}-7\,x^2\,\ln \left (5\right )+7\,x^3 \] Input:
int((70*x - log(5*x)*(56*x^2 + 56*x^3 + 14*x^4) - 77*x^2 + 35*x^3 + 77*x^4 + 21*x^5 + exp(x)*(28*x + 28*x^2 + 7*x^3) + 140)/(4*x + 4*x^2 + x^3),x)
Output:
7*exp(x) - 21*x + 35*log(x) - 7*x^2*log(x) - 14/(x + 2) - 7*x^2*log(5) + 7 *x^3
Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.10 \[ \int \frac {140+70 x-77 x^2+35 x^3+77 x^4+21 x^5+e^x \left (28 x+28 x^2+7 x^3\right )+\left (-56 x^2-56 x^3-14 x^4\right ) \log (5 x)}{4 x+4 x^2+x^3} \, dx=\frac {7 e^{x} x +14 e^{x}-7 \,\mathrm {log}\left (5 x \right ) x^{3}-14 \,\mathrm {log}\left (5 x \right ) x^{2}+35 \,\mathrm {log}\left (x \right ) x +70 \,\mathrm {log}\left (x \right )+7 x^{4}+14 x^{3}-21 x^{2}-35 x}{x +2} \] Input:
int(((-14*x^4-56*x^3-56*x^2)*log(5*x)+(7*x^3+28*x^2+28*x)*exp(x)+21*x^5+77 *x^4+35*x^3-77*x^2+70*x+140)/(x^3+4*x^2+4*x),x)
Output:
(7*(e**x*x + 2*e**x - log(5*x)*x**3 - 2*log(5*x)*x**2 + 5*log(x)*x + 10*lo g(x) + x**4 + 2*x**3 - 3*x**2 - 5*x))/(x + 2)