\(\int \frac {468512 \log (5 x)+e^{2-2 x} x^2 (702768 \log (5 x)+(702768-702768 x) \log ^2(5 x))+e^{1-x} x (937024 \log (5 x)+(468512-468512 x) \log ^2(5 x))+e^{3-3 x} x^3 (234256 \log (5 x)+(351384-351384 x) \log ^2(5 x))+e^{4-4 x} x^4 (29282 \log (5 x)+(58564-58564 x) \log ^2(5 x))}{x} \, dx\) [1038]

Optimal result

Integrand size = 129, antiderivative size = 21 \[ \int \frac {468512 \log (5 x)+e^{2-2 x} x^2 \left (702768 \log (5 x)+(702768-702768 x) \log ^2(5 x)\right )+e^{1-x} x \left (937024 \log (5 x)+(468512-468512 x) \log ^2(5 x)\right )+e^{3-3 x} x^3 \left (234256 \log (5 x)+(351384-351384 x) \log ^2(5 x)\right )+e^{4-4 x} x^4 \left (29282 \log (5 x)+(58564-58564 x) \log ^2(5 x)\right )}{x} \, dx=14641 \left (2+e^{1-x} x\right )^4 \log ^2(5 x) \] Output:

121*(exp(1+ln(x)-x)+2)^2*(11*exp(1+ln(x)-x)+22)^2*ln(5*x)^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {468512 \log (5 x)+e^{2-2 x} x^2 \left (702768 \log (5 x)+(702768-702768 x) \log ^2(5 x)\right )+e^{1-x} x \left (937024 \log (5 x)+(468512-468512 x) \log ^2(5 x)\right )+e^{3-3 x} x^3 \left (234256 \log (5 x)+(351384-351384 x) \log ^2(5 x)\right )+e^{4-4 x} x^4 \left (29282 \log (5 x)+(58564-58564 x) \log ^2(5 x)\right )}{x} \, dx=14641 e^{-4 x} \left (2 e^x+e x\right )^4 \log ^2(5 x) \] Input:

Integrate[(468512*Log[5*x] + E^(2 - 2*x)*x^2*(702768*Log[5*x] + (702768 - 
702768*x)*Log[5*x]^2) + E^(1 - x)*x*(937024*Log[5*x] + (468512 - 468512*x) 
*Log[5*x]^2) + E^(3 - 3*x)*x^3*(234256*Log[5*x] + (351384 - 351384*x)*Log[ 
5*x]^2) + E^(4 - 4*x)*x^4*(29282*Log[5*x] + (58564 - 58564*x)*Log[5*x]^2)) 
/x,x]
 

Output:

(14641*(2*E^x + E*x)^4*Log[5*x]^2)/E^(4*x)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(79\) vs. \(2(21)=42\).

Time = 0.51 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.76, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {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.

\(\displaystyle \int \frac {e^{4-4 x} x^4 \left ((58564-58564 x) \log ^2(5 x)+29282 \log (5 x)\right )+e^{3-3 x} x^3 \left ((351384-351384 x) \log ^2(5 x)+234256 \log (5 x)\right )+e^{2-2 x} x^2 \left ((702768-702768 x) \log ^2(5 x)+702768 \log (5 x)\right )+e^{1-x} x \left ((468512-468512 x) \log ^2(5 x)+937024 \log (5 x)\right )+468512 \log (5 x)}{x} \, dx\)

\(\Big \downarrow \) 2010

\(\displaystyle \int \left (-29282 e^{4-4 x} x^3 \log (5 x) (2 x \log (5 x)-2 \log (5 x)-1)-117128 e^{3-3 x} x^2 \log (5 x) (3 x \log (5 x)-3 \log (5 x)-2)-702768 e^{2-2 x} x \log (5 x) (x \log (5 x)-\log (5 x)-1)-468512 e^{1-x} \log (5 x) (x \log (5 x)-\log (5 x)-2)+\frac {468512 \log (5 x)}{x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 14641 e^{4-4 x} x^4 \log ^2(5 x)+117128 e^{3-3 x} x^3 \log ^2(5 x)+351384 e^{2-2 x} x^2 \log ^2(5 x)+468512 e^{1-x} x \log ^2(5 x)+234256 \log ^2(5 x)\)

Input:

Int[(468512*Log[5*x] + E^(2 - 2*x)*x^2*(702768*Log[5*x] + (702768 - 702768 
*x)*Log[5*x]^2) + E^(1 - x)*x*(937024*Log[5*x] + (468512 - 468512*x)*Log[5 
*x]^2) + E^(3 - 3*x)*x^3*(234256*Log[5*x] + (351384 - 351384*x)*Log[5*x]^2 
) + E^(4 - 4*x)*x^4*(29282*Log[5*x] + (58564 - 58564*x)*Log[5*x]^2))/x,x]
 

Output:

234256*Log[5*x]^2 + 468512*E^(1 - x)*x*Log[5*x]^2 + 351384*E^(2 - 2*x)*x^2 
*Log[5*x]^2 + 117128*E^(3 - 3*x)*x^3*Log[5*x]^2 + 14641*E^(4 - 4*x)*x^4*Lo 
g[5*x]^2
 

Defintions of rubi rules used

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]]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(127\) vs. \(2(34)=68\).

Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 6.10

\[234256 \ln \left (x \right )^{2}+468512 \ln \left (5\right ) \ln \left (x \right )+\left (14641 \ln \left (5\right )^{2}+29282 \ln \left (5\right ) \ln \left (x \right )+14641 \ln \left (x \right )^{2}\right ) x^{4} {\mathrm e}^{4-4 x}+\left (117128 \ln \left (5\right )^{2}+234256 \ln \left (5\right ) \ln \left (x \right )+117128 \ln \left (x \right )^{2}\right ) x^{3} {\mathrm e}^{-3 x +3}+\left (351384 \ln \left (5\right )^{2}+702768 \ln \left (5\right ) \ln \left (x \right )+351384 \ln \left (x \right )^{2}\right ) x^{2} {\mathrm e}^{2-2 x}+\left (468512 \ln \left (5\right )^{2}+937024 \ln \left (5\right ) \ln \left (x \right )+468512 \ln \left (x \right )^{2}\right ) x \,{\mathrm e}^{1-x}\]

Input:

int((((-58564*x+58564)*ln(5*x)^2+29282*ln(5*x))*exp(1+ln(x)-x)^4+((-351384 
*x+351384)*ln(5*x)^2+234256*ln(5*x))*exp(1+ln(x)-x)^3+((-702768*x+702768)* 
ln(5*x)^2+702768*ln(5*x))*exp(1+ln(x)-x)^2+((-468512*x+468512)*ln(5*x)^2+9 
37024*ln(5*x))*exp(1+ln(x)-x)+468512*ln(5*x))/x,x)
 

Output:

234256*ln(x)^2+468512*ln(5)*ln(x)+(14641*ln(5)^2+29282*ln(5)*ln(x)+14641*l 
n(x)^2)*x^4*exp(4-4*x)+(117128*ln(5)^2+234256*ln(5)*ln(x)+117128*ln(x)^2)* 
x^3*exp(-3*x+3)+(351384*ln(5)^2+702768*ln(5)*ln(x)+351384*ln(x)^2)*x^2*exp 
(2-2*x)+(468512*ln(5)^2+937024*ln(5)*ln(x)+468512*ln(x)^2)*x*exp(1-x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (20) = 40\).

Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 5.67 \[ \int \frac {468512 \log (5 x)+e^{2-2 x} x^2 \left (702768 \log (5 x)+(702768-702768 x) \log ^2(5 x)\right )+e^{1-x} x \left (937024 \log (5 x)+(468512-468512 x) \log ^2(5 x)\right )+e^{3-3 x} x^3 \left (234256 \log (5 x)+(351384-351384 x) \log ^2(5 x)\right )+e^{4-4 x} x^4 \left (29282 \log (5 x)+(58564-58564 x) \log ^2(5 x)\right )}{x} \, dx=468512 \, {\left (\log \left (5\right )^{2} + 2 \, \log \left (5\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )} e^{\left (-x + \log \left (x\right ) + 1\right )} + 351384 \, {\left (\log \left (5\right )^{2} + 2 \, \log \left (5\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )} e^{\left (-2 \, x + 2 \, \log \left (x\right ) + 2\right )} + 117128 \, {\left (\log \left (5\right )^{2} + 2 \, \log \left (5\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )} e^{\left (-3 \, x + 3 \, \log \left (x\right ) + 3\right )} + 14641 \, {\left (\log \left (5\right )^{2} + 2 \, \log \left (5\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )} e^{\left (-4 \, x + 4 \, \log \left (x\right ) + 4\right )} + 468512 \, \log \left (5\right ) \log \left (x\right ) + 234256 \, \log \left (x\right )^{2} \] Input:

integrate((((-58564*x+58564)*log(5*x)^2+29282*log(5*x))*exp(1+log(x)-x)^4+ 
((-351384*x+351384)*log(5*x)^2+234256*log(5*x))*exp(1+log(x)-x)^3+((-70276 
8*x+702768)*log(5*x)^2+702768*log(5*x))*exp(1+log(x)-x)^2+((-468512*x+4685 
12)*log(5*x)^2+937024*log(5*x))*exp(1+log(x)-x)+468512*log(5*x))/x,x, algo 
rithm="fricas")
 

Output:

468512*(log(5)^2 + 2*log(5)*log(x) + log(x)^2)*e^(-x + log(x) + 1) + 35138 
4*(log(5)^2 + 2*log(5)*log(x) + log(x)^2)*e^(-2*x + 2*log(x) + 2) + 117128 
*(log(5)^2 + 2*log(5)*log(x) + log(x)^2)*e^(-3*x + 3*log(x) + 3) + 14641*( 
log(5)^2 + 2*log(5)*log(x) + log(x)^2)*e^(-4*x + 4*log(x) + 4) + 468512*lo 
g(5)*log(x) + 234256*log(x)^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (29) = 58\).

Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.71 \[ \int \frac {468512 \log (5 x)+e^{2-2 x} x^2 \left (702768 \log (5 x)+(702768-702768 x) \log ^2(5 x)\right )+e^{1-x} x \left (937024 \log (5 x)+(468512-468512 x) \log ^2(5 x)\right )+e^{3-3 x} x^3 \left (234256 \log (5 x)+(351384-351384 x) \log ^2(5 x)\right )+e^{4-4 x} x^4 \left (29282 \log (5 x)+(58564-58564 x) \log ^2(5 x)\right )}{x} \, dx=14641 x^{4} e^{4 - 4 x} \log {\left (5 x \right )}^{2} + 117128 x^{3} e^{3 - 3 x} \log {\left (5 x \right )}^{2} + 351384 x^{2} e^{2 - 2 x} \log {\left (5 x \right )}^{2} + 468512 x e^{1 - x} \log {\left (5 x \right )}^{2} + 234256 \log {\left (5 x \right )}^{2} \] Input:

integrate((((-58564*x+58564)*ln(5*x)**2+29282*ln(5*x))*exp(1+ln(x)-x)**4+( 
(-351384*x+351384)*ln(5*x)**2+234256*ln(5*x))*exp(1+ln(x)-x)**3+((-702768* 
x+702768)*ln(5*x)**2+702768*ln(5*x))*exp(1+ln(x)-x)**2+((-468512*x+468512) 
*ln(5*x)**2+937024*ln(5*x))*exp(1+ln(x)-x)+468512*ln(5*x))/x,x)
 

Output:

14641*x**4*exp(4 - 4*x)*log(5*x)**2 + 117128*x**3*exp(3 - 3*x)*log(5*x)**2 
 + 351384*x**2*exp(2 - 2*x)*log(5*x)**2 + 468512*x*exp(1 - x)*log(5*x)**2 
+ 234256*log(5*x)**2
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (20) = 40\).

Time = 0.20 (sec) , antiderivative size = 155, normalized size of antiderivative = 7.38 \[ \int \frac {468512 \log (5 x)+e^{2-2 x} x^2 \left (702768 \log (5 x)+(702768-702768 x) \log ^2(5 x)\right )+e^{1-x} x \left (937024 \log (5 x)+(468512-468512 x) \log ^2(5 x)\right )+e^{3-3 x} x^3 \left (234256 \log (5 x)+(351384-351384 x) \log ^2(5 x)\right )+e^{4-4 x} x^4 \left (29282 \log (5 x)+(58564-58564 x) \log ^2(5 x)\right )}{x} \, dx=468512 \, {\left (x e \log \left (5\right )^{2} + 2 \, x e \log \left (5\right ) \log \left (x\right ) + x e \log \left (x\right )^{2}\right )} e^{\left (-x\right )} + 351384 \, {\left (x^{2} e^{2} \log \left (5\right )^{2} + 2 \, x^{2} e^{2} \log \left (5\right ) \log \left (x\right ) + x^{2} e^{2} \log \left (x\right )^{2}\right )} e^{\left (-2 \, x\right )} + 117128 \, {\left (x^{3} e^{3} \log \left (5\right )^{2} + 2 \, x^{3} e^{3} \log \left (5\right ) \log \left (x\right ) + x^{3} e^{3} \log \left (x\right )^{2}\right )} e^{\left (-3 \, x\right )} + 14641 \, {\left (x^{4} e^{4} \log \left (5\right )^{2} + 2 \, x^{4} e^{4} \log \left (5\right ) \log \left (x\right ) + x^{4} e^{4} \log \left (x\right )^{2}\right )} e^{\left (-4 \, x\right )} + 234256 \, \log \left (5 \, x\right )^{2} \] Input:

integrate((((-58564*x+58564)*log(5*x)^2+29282*log(5*x))*exp(1+log(x)-x)^4+ 
((-351384*x+351384)*log(5*x)^2+234256*log(5*x))*exp(1+log(x)-x)^3+((-70276 
8*x+702768)*log(5*x)^2+702768*log(5*x))*exp(1+log(x)-x)^2+((-468512*x+4685 
12)*log(5*x)^2+937024*log(5*x))*exp(1+log(x)-x)+468512*log(5*x))/x,x, algo 
rithm="maxima")
 

Output:

468512*(x*e*log(5)^2 + 2*x*e*log(5)*log(x) + x*e*log(x)^2)*e^(-x) + 351384 
*(x^2*e^2*log(5)^2 + 2*x^2*e^2*log(5)*log(x) + x^2*e^2*log(x)^2)*e^(-2*x) 
+ 117128*(x^3*e^3*log(5)^2 + 2*x^3*e^3*log(5)*log(x) + x^3*e^3*log(x)^2)*e 
^(-3*x) + 14641*(x^4*e^4*log(5)^2 + 2*x^4*e^4*log(5)*log(x) + x^4*e^4*log( 
x)^2)*e^(-4*x) + 234256*log(5*x)^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (20) = 40\).

Time = 0.14 (sec) , antiderivative size = 187, normalized size of antiderivative = 8.90 \[ \int \frac {468512 \log (5 x)+e^{2-2 x} x^2 \left (702768 \log (5 x)+(702768-702768 x) \log ^2(5 x)\right )+e^{1-x} x \left (937024 \log (5 x)+(468512-468512 x) \log ^2(5 x)\right )+e^{3-3 x} x^3 \left (234256 \log (5 x)+(351384-351384 x) \log ^2(5 x)\right )+e^{4-4 x} x^4 \left (29282 \log (5 x)+(58564-58564 x) \log ^2(5 x)\right )}{x} \, dx=14641 \, x^{4} e^{\left (-4 \, x + 4\right )} \log \left (5\right )^{2} + 29282 \, x^{4} e^{\left (-4 \, x + 4\right )} \log \left (5\right ) \log \left (x\right ) + 14641 \, x^{4} e^{\left (-4 \, x + 4\right )} \log \left (x\right )^{2} + 117128 \, x^{3} e^{\left (-3 \, x + 3\right )} \log \left (5\right )^{2} + 234256 \, x^{3} e^{\left (-3 \, x + 3\right )} \log \left (5\right ) \log \left (x\right ) + 117128 \, x^{3} e^{\left (-3 \, x + 3\right )} \log \left (x\right )^{2} + 351384 \, x^{2} e^{\left (-2 \, x + 2\right )} \log \left (5\right )^{2} + 702768 \, x^{2} e^{\left (-2 \, x + 2\right )} \log \left (5\right ) \log \left (x\right ) + 351384 \, x^{2} e^{\left (-2 \, x + 2\right )} \log \left (x\right )^{2} + 468512 \, x e^{\left (-x + 1\right )} \log \left (5\right )^{2} + 937024 \, x e^{\left (-x + 1\right )} \log \left (5\right ) \log \left (x\right ) + 468512 \, x e^{\left (-x + 1\right )} \log \left (x\right )^{2} + 468512 \, \log \left (5\right ) \log \left (x\right ) + 234256 \, \log \left (x\right )^{2} \] Input:

integrate((((-58564*x+58564)*log(5*x)^2+29282*log(5*x))*exp(1+log(x)-x)^4+ 
((-351384*x+351384)*log(5*x)^2+234256*log(5*x))*exp(1+log(x)-x)^3+((-70276 
8*x+702768)*log(5*x)^2+702768*log(5*x))*exp(1+log(x)-x)^2+((-468512*x+4685 
12)*log(5*x)^2+937024*log(5*x))*exp(1+log(x)-x)+468512*log(5*x))/x,x, algo 
rithm="giac")
 

Output:

14641*x^4*e^(-4*x + 4)*log(5)^2 + 29282*x^4*e^(-4*x + 4)*log(5)*log(x) + 1 
4641*x^4*e^(-4*x + 4)*log(x)^2 + 117128*x^3*e^(-3*x + 3)*log(5)^2 + 234256 
*x^3*e^(-3*x + 3)*log(5)*log(x) + 117128*x^3*e^(-3*x + 3)*log(x)^2 + 35138 
4*x^2*e^(-2*x + 2)*log(5)^2 + 702768*x^2*e^(-2*x + 2)*log(5)*log(x) + 3513 
84*x^2*e^(-2*x + 2)*log(x)^2 + 468512*x*e^(-x + 1)*log(5)^2 + 937024*x*e^( 
-x + 1)*log(5)*log(x) + 468512*x*e^(-x + 1)*log(x)^2 + 468512*log(5)*log(x 
) + 234256*log(x)^2
 

Mupad [B] (verification not implemented)

Time = 4.35 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.57 \[ \int \frac {468512 \log (5 x)+e^{2-2 x} x^2 \left (702768 \log (5 x)+(702768-702768 x) \log ^2(5 x)\right )+e^{1-x} x \left (937024 \log (5 x)+(468512-468512 x) \log ^2(5 x)\right )+e^{3-3 x} x^3 \left (234256 \log (5 x)+(351384-351384 x) \log ^2(5 x)\right )+e^{4-4 x} x^4 \left (29282 \log (5 x)+(58564-58564 x) \log ^2(5 x)\right )}{x} \, dx=234256\,{\ln \left (5\,x\right )}^2+468512\,x\,{\ln \left (5\,x\right )}^2\,{\mathrm {e}}^{1-x}+351384\,x^2\,{\ln \left (5\,x\right )}^2\,{\mathrm {e}}^{2-2\,x}+117128\,x^3\,{\ln \left (5\,x\right )}^2\,{\mathrm {e}}^{3-3\,x}+14641\,x^4\,{\ln \left (5\,x\right )}^2\,{\mathrm {e}}^{4-4\,x} \] Input:

int((468512*log(5*x) + exp(log(x) - x + 1)*(937024*log(5*x) - log(5*x)^2*( 
468512*x - 468512)) + exp(4*log(x) - 4*x + 4)*(29282*log(5*x) - log(5*x)^2 
*(58564*x - 58564)) + exp(3*log(x) - 3*x + 3)*(234256*log(5*x) - log(5*x)^ 
2*(351384*x - 351384)) + exp(2*log(x) - 2*x + 2)*(702768*log(5*x) - log(5* 
x)^2*(702768*x - 702768)))/x,x)
 

Output:

234256*log(5*x)^2 + 468512*x*log(5*x)^2*exp(1 - x) + 351384*x^2*log(5*x)^2 
*exp(2 - 2*x) + 117128*x^3*log(5*x)^2*exp(3 - 3*x) + 14641*x^4*log(5*x)^2* 
exp(4 - 4*x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.00 \[ \int \frac {468512 \log (5 x)+e^{2-2 x} x^2 \left (702768 \log (5 x)+(702768-702768 x) \log ^2(5 x)\right )+e^{1-x} x \left (937024 \log (5 x)+(468512-468512 x) \log ^2(5 x)\right )+e^{3-3 x} x^3 \left (234256 \log (5 x)+(351384-351384 x) \log ^2(5 x)\right )+e^{4-4 x} x^4 \left (29282 \log (5 x)+(58564-58564 x) \log ^2(5 x)\right )}{x} \, dx=\frac {14641 \mathrm {log}\left (5 x \right )^{2} \left (16 e^{4 x}+32 e^{3 x} e x +24 e^{2 x} e^{2} x^{2}+8 e^{x} e^{3} x^{3}+e^{4} x^{4}\right )}{e^{4 x}} \] Input:

int((((-58564*x+58564)*log(5*x)^2+29282*log(5*x))*exp(1+log(x)-x)^4+((-351 
384*x+351384)*log(5*x)^2+234256*log(5*x))*exp(1+log(x)-x)^3+((-702768*x+70 
2768)*log(5*x)^2+702768*log(5*x))*exp(1+log(x)-x)^2+((-468512*x+468512)*lo 
g(5*x)^2+937024*log(5*x))*exp(1+log(x)-x)+468512*log(5*x))/x,x)
 

Output:

(14641*log(5*x)**2*(16*e**(4*x) + 32*e**(3*x)*e*x + 24*e**(2*x)*e**2*x**2 
+ 8*e**x*e**3*x**3 + e**4*x**4))/e**(4*x)