\(\int \frac {75+10 e^{3 x}-5 e^{4 x}+e^x (-25-25 x)+(25+e^{4 x} (-5-20 x)+30 e^{2 x} x+e^{3 x} (20+20 x-10 x^2)) \log (x)+(10 e^{2 x} x+3 e^{4 x} x^2-2 e^{6 x} x^2+e^{3 x} (5+15 x)+e^{5 x} (x+x^2-x^3)) \log ^2(x)+(e^{4 x} x^2+e^{5 x} x^2) \log ^3(x)}{25+10 e^{2 x} x \log (x)+e^{4 x} x^2 \log ^2(x)} \, dx\) [7851]

Optimal result

Integrand size = 178, antiderivative size = 32 \[ \int \frac {75+10 e^{3 x}-5 e^{4 x}+e^x (-25-25 x)+\left (25+e^{4 x} (-5-20 x)+30 e^{2 x} x+e^{3 x} \left (20+20 x-10 x^2\right )\right ) \log (x)+\left (10 e^{2 x} x+3 e^{4 x} x^2-2 e^{6 x} x^2+e^{3 x} (5+15 x)+e^{5 x} \left (x+x^2-x^3\right )\right ) \log ^2(x)+\left (e^{4 x} x^2+e^{5 x} x^2\right ) \log ^3(x)}{25+10 e^{2 x} x \log (x)+e^{4 x} x^2 \log ^2(x)} \, dx=\left (x+\frac {e^x x}{x+\frac {5 e^{-2 x}}{\log (x)}}\right ) \left (2-e^x+\log (x)\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {75+10 e^{3 x}-5 e^{4 x}+e^x (-25-25 x)+\left (25+e^{4 x} (-5-20 x)+30 e^{2 x} x+e^{3 x} \left (20+20 x-10 x^2\right )\right ) \log (x)+\left (10 e^{2 x} x+3 e^{4 x} x^2-2 e^{6 x} x^2+e^{3 x} (5+15 x)+e^{5 x} \left (x+x^2-x^3\right )\right ) \log ^2(x)+\left (e^{4 x} x^2+e^{5 x} x^2\right ) \log ^3(x)}{25+10 e^{2 x} x \log (x)+e^{4 x} x^2 \log ^2(x)} \, dx=\int \frac {75+10 e^{3 x}-5 e^{4 x}+e^x (-25-25 x)+\left (25+e^{4 x} (-5-20 x)+30 e^{2 x} x+e^{3 x} \left (20+20 x-10 x^2\right )\right ) \log (x)+\left (10 e^{2 x} x+3 e^{4 x} x^2-2 e^{6 x} x^2+e^{3 x} (5+15 x)+e^{5 x} \left (x+x^2-x^3\right )\right ) \log ^2(x)+\left (e^{4 x} x^2+e^{5 x} x^2\right ) \log ^3(x)}{25+10 e^{2 x} x \log (x)+e^{4 x} x^2 \log ^2(x)} \, dx \]

[In]

Int[(75 + 10*E^(3*x) - 5*E^(4*x) + E^x*(-25 - 25*x) + (25 + E^(4*x)*(-5 - 20*x) + 30*E^(2*x)*x + E^(3*x)*(20 +
20*x - 10*x^2))*Log[x] + (10*E^(2*x)*x + 3*E^(4*x)*x^2 - 2*E^(6*x)*x^2 + E^(3*x)*(5 + 15*x) + E^(5*x)*(x + x^
2 - x^3))*Log[x]^2 + (E^(4*x)*x^2 + E^(5*x)*x^2)*Log[x]^3)/(25 + 10*E^(2*x)*x*Log[x] + E^(4*x)*x^2*Log[x]^2),x
]

[Out]

2*E^x - E^(2*x) + 2*x - E^x*x + 5/(x*Log[x]) + E^x*Log[x] + x*Log[x] - 100*Defer[Int][E^x/(5 + E^(2*x)*x*Log[x
])^2, x] - 75*Defer[Int][E^x/(x*(5 + E^(2*x)*x*Log[x])^2), x] - 125*Defer[Int][1/(x^2*Log[x]^2*(5 + E^(2*x)*x*
Log[x])^2), x] - 125*Defer[Int][1/(x^2*Log[x]*(5 + E^(2*x)*x*Log[x])^2), x] - 250*Defer[Int][1/(x*Log[x]*(5 +
E^(2*x)*x*Log[x])^2), x] - 50*Defer[Int][E^x/(x*Log[x]*(5 + E^(2*x)*x*Log[x])^2), x] - 50*Defer[Int][(E^x*Log[
x])/(5 + E^(2*x)*x*Log[x])^2, x] - 25*Defer[Int][(E^x*Log[x])/(x*(5 + E^(2*x)*x*Log[x])^2), x] + 10*Defer[Int]
[E^x/(5 + E^(2*x)*x*Log[x]), x] + 10*Defer[Int][E^x/(x*(5 + E^(2*x)*x*Log[x])), x] + 50*Defer[Int][1/(x^2*Log[
x]^2*(5 + E^(2*x)*x*Log[x])), x] + 50*Defer[Int][1/(x^2*Log[x]*(5 + E^(2*x)*x*Log[x])), x] + 50*Defer[Int][1/(
x*Log[x]*(5 + E^(2*x)*x*Log[x])), x] + 10*Defer[Int][E^x/(x*Log[x]*(5 + E^(2*x)*x*Log[x])), x] + 5*Defer[Int][
(E^x*Log[x])/(5 + E^(2*x)*x*Log[x]), x] + 5*Defer[Int][(E^x*Log[x])/(x*(5 + E^(2*x)*x*Log[x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {75+10 e^{3 x}-5 e^{4 x}+e^x (-25-25 x)+\left (25+e^{4 x} (-5-20 x)+30 e^{2 x} x+e^{3 x} \left (20+20 x-10 x^2\right )\right ) \log (x)+\left (10 e^{2 x} x+3 e^{4 x} x^2-2 e^{6 x} x^2+e^{3 x} (5+15 x)+e^{5 x} \left (x+x^2-x^3\right )\right ) \log ^2(x)+\left (e^{4 x} x^2+e^{5 x} x^2\right ) \log ^3(x)}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx \\ & = \int \left (-2 e^{2 x}-\frac {e^x \left (-1-x+x^2-x \log (x)\right )}{x}-\frac {25 (1+\log (x)+2 x \log (x)) \left (5+2 e^x x \log (x)+e^x x \log ^2(x)\right )}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2}+\frac {-5-5 \log (x)+3 x^2 \log ^2(x)+x^2 \log ^3(x)}{x^2 \log ^2(x)}+\frac {5 \left (10+10 \log (x)+10 x \log (x)+2 e^x x \log (x)+2 e^x x \log ^2(x)+2 e^x x^2 \log ^2(x)+e^x x \log ^3(x)+e^x x^2 \log ^3(x)\right )}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )}\right ) \, dx \\ & = -\left (2 \int e^{2 x} \, dx\right )+5 \int \frac {10+10 \log (x)+10 x \log (x)+2 e^x x \log (x)+2 e^x x \log ^2(x)+2 e^x x^2 \log ^2(x)+e^x x \log ^3(x)+e^x x^2 \log ^3(x)}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )} \, dx-25 \int \frac {(1+\log (x)+2 x \log (x)) \left (5+2 e^x x \log (x)+e^x x \log ^2(x)\right )}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-\int \frac {e^x \left (-1-x+x^2-x \log (x)\right )}{x} \, dx+\int \frac {-5-5 \log (x)+3 x^2 \log ^2(x)+x^2 \log ^3(x)}{x^2 \log ^2(x)} \, dx \\ & = -e^{2 x}+5 \int \frac {10+2 \left (5+\left (5+e^x\right ) x\right ) \log (x)+2 e^x x (1+x) \log ^2(x)+e^x x (1+x) \log ^3(x)}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )} \, dx-25 \int \left (\frac {4 e^x}{\left (5+e^{2 x} x \log (x)\right )^2}+\frac {3 e^x}{x \left (5+e^{2 x} x \log (x)\right )^2}+\frac {5}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2}+\frac {5}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )^2}+\frac {10}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2}+\frac {2 e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2}+\frac {2 e^x \log (x)}{\left (5+e^{2 x} x \log (x)\right )^2}+\frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )^2}\right ) \, dx+\int \left (3-\frac {5}{x^2 \log ^2(x)}-\frac {5}{x^2 \log (x)}+\log (x)\right ) \, dx-\int \left (\frac {e^x \left (-1-x+x^2\right )}{x}-e^x \log (x)\right ) \, dx \\ & = -e^{2 x}+3 x-5 \int \frac {1}{x^2 \log ^2(x)} \, dx-5 \int \frac {1}{x^2 \log (x)} \, dx+5 \int \left (\frac {2 e^x}{5+e^{2 x} x \log (x)}+\frac {2 e^x}{x \left (5+e^{2 x} x \log (x)\right )}+\frac {10}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )}+\frac {10}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )}+\frac {10}{x \log (x) \left (5+e^{2 x} x \log (x)\right )}+\frac {2 e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )}+\frac {e^x \log (x)}{5+e^{2 x} x \log (x)}+\frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )}\right ) \, dx-25 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x \log (x)}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx-75 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-100 \int \frac {e^x}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-250 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-\int \frac {e^x \left (-1-x+x^2\right )}{x} \, dx+\int \log (x) \, dx+\int e^x \log (x) \, dx \\ & = -e^{2 x}+2 x+\frac {5}{x \log (x)}+e^x \log (x)+x \log (x)+5 \int \frac {1}{x^2 \log (x)} \, dx+5 \int \frac {e^x \log (x)}{5+e^{2 x} x \log (x)} \, dx+5 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )} \, dx-5 \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )+10 \int \frac {e^x}{5+e^{2 x} x \log (x)} \, dx+10 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )} \, dx+10 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-25 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x \log (x)}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx+50 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-75 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-100 \int \frac {e^x}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-250 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-\int \frac {e^x}{x} \, dx-\int \left (-e^x-\frac {e^x}{x}+e^x x\right ) \, dx \\ & = -e^{2 x}+2 x-\operatorname {ExpIntegralEi}(x)-5 \operatorname {ExpIntegralEi}(-\log (x))+\frac {5}{x \log (x)}+e^x \log (x)+x \log (x)+5 \int \frac {e^x \log (x)}{5+e^{2 x} x \log (x)} \, dx+5 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )} \, dx+5 \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )+10 \int \frac {e^x}{5+e^{2 x} x \log (x)} \, dx+10 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )} \, dx+10 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-25 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x \log (x)}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx+50 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-75 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-100 \int \frac {e^x}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-250 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx+\int e^x \, dx+\int \frac {e^x}{x} \, dx-\int e^x x \, dx \\ & = e^x-e^{2 x}+2 x-e^x x+\frac {5}{x \log (x)}+e^x \log (x)+x \log (x)+5 \int \frac {e^x \log (x)}{5+e^{2 x} x \log (x)} \, dx+5 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )} \, dx+10 \int \frac {e^x}{5+e^{2 x} x \log (x)} \, dx+10 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )} \, dx+10 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-25 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x \log (x)}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx+50 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-75 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-100 \int \frac {e^x}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-250 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx+\int e^x \, dx \\ & = 2 e^x-e^{2 x}+2 x-e^x x+\frac {5}{x \log (x)}+e^x \log (x)+x \log (x)+5 \int \frac {e^x \log (x)}{5+e^{2 x} x \log (x)} \, dx+5 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )} \, dx+10 \int \frac {e^x}{5+e^{2 x} x \log (x)} \, dx+10 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )} \, dx+10 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-25 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x \log (x)}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx+50 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-75 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-100 \int \frac {e^x}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-250 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \frac {75+10 e^{3 x}-5 e^{4 x}+e^x (-25-25 x)+\left (25+e^{4 x} (-5-20 x)+30 e^{2 x} x+e^{3 x} \left (20+20 x-10 x^2\right )\right ) \log (x)+\left (10 e^{2 x} x+3 e^{4 x} x^2-2 e^{6 x} x^2+e^{3 x} (5+15 x)+e^{5 x} \left (x+x^2-x^3\right )\right ) \log ^2(x)+\left (e^{4 x} x^2+e^{5 x} x^2\right ) \log ^3(x)}{25+10 e^{2 x} x \log (x)+e^{4 x} x^2 \log ^2(x)} \, dx=-\frac {x \left (-2+e^x-\log (x)\right ) \left (5+e^{2 x} \left (e^x+x\right ) \log (x)\right )}{5+e^{2 x} x \log (x)} \]

[In]

Integrate[(75 + 10*E^(3*x) - 5*E^(4*x) + E^x*(-25 - 25*x) + (25 + E^(4*x)*(-5 - 20*x) + 30*E^(2*x)*x + E^(3*x)
*(20 + 20*x - 10*x^2))*Log[x] + (10*E^(2*x)*x + 3*E^(4*x)*x^2 - 2*E^(6*x)*x^2 + E^(3*x)*(5 + 15*x) + E^(5*x)*(
x + x^2 - x^3))*Log[x]^2 + (E^(4*x)*x^2 + E^(5*x)*x^2)*Log[x]^3)/(25 + 10*E^(2*x)*x*Log[x] + E^(4*x)*x^2*Log[x
]^2),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(29)=58\).

Time = 0.55 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.62

method	result	size
risch	\(\left ({\mathrm e}^{x}+x \right ) \ln \left (x \right )-\frac {\left ({\mathrm e}^{2 x} x^{2}+x \,{\mathrm e}^{3 x}-2 \,{\mathrm e}^{x} x^{2}-2 x \,{\mathrm e}^{2 x}+5\right ) {\mathrm e}^{-x}}{x}+\frac {5 \left (x \,{\mathrm e}^{3 x}-2 x \,{\mathrm e}^{2 x}+5\right ) {\mathrm e}^{-x}}{x \left (\ln \left (x \right ) {\mathrm e}^{2 x} x +5\right )}\)	\(84\)
parallelrisch	\(-\frac {-5 \ln \left (x \right )^{2} {\mathrm e}^{2 x} x^{2}+5 \ln \left (x \right ) {\mathrm e}^{3 x} x^{2}-5 \ln \left (x \right )^{2} {\mathrm e}^{3 x} x +5 \ln \left (x \right ) {\mathrm e}^{4 x} x -10 x^{2} \ln \left (x \right ) {\mathrm e}^{2 x}-10 \ln \left (x \right ) {\mathrm e}^{3 x} x -25 x \ln \left (x \right )+25 \,{\mathrm e}^{x} x -50 x}{5 \left (\ln \left (x \right ) {\mathrm e}^{2 x} x +5\right )}\)	\(93\)

[In]

int(((x^2*exp(x)^5+x^2*exp(x)^4)*ln(x)^3+(-2*x^2*exp(x)^6+(-x^3+x^2+x)*exp(x)^5+3*x^2*exp(x)^4+(15*x+5)*exp(x)
^3+10*x*exp(x)^2)*ln(x)^2+((-20*x-5)*exp(x)^4+(-10*x^2+20*x+20)*exp(x)^3+30*x*exp(x)^2+25)*ln(x)-5*exp(x)^4+10
*exp(x)^3+(-25*x-25)*exp(x)+75)/(x^2*exp(x)^4*ln(x)^2+10*x*exp(x)^2*ln(x)+25),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (30) = 60\).

Time = 0.24 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.44 \[ \int \frac {75+10 e^{3 x}-5 e^{4 x}+e^x (-25-25 x)+\left (25+e^{4 x} (-5-20 x)+30 e^{2 x} x+e^{3 x} \left (20+20 x-10 x^2\right )\right ) \log (x)+\left (10 e^{2 x} x+3 e^{4 x} x^2-2 e^{6 x} x^2+e^{3 x} (5+15 x)+e^{5 x} \left (x+x^2-x^3\right )\right ) \log ^2(x)+\left (e^{4 x} x^2+e^{5 x} x^2\right ) \log ^3(x)}{25+10 e^{2 x} x \log (x)+e^{4 x} x^2 \log ^2(x)} \, dx=\frac {{\left (x^{2} e^{\left (2 \, x\right )} + x e^{\left (3 \, x\right )}\right )} \log \left (x\right )^{2} - 5 \, x e^{x} + {\left (2 \, x^{2} e^{\left (2 \, x\right )} - x e^{\left (4 \, x\right )} - {\left (x^{2} - 2 \, x\right )} e^{\left (3 \, x\right )} + 5 \, x\right )} \log \left (x\right ) + 10 \, x}{x e^{\left (2 \, x\right )} \log \left (x\right ) + 5} \]

[In]

integrate(((x^2*exp(x)^5+x^2*exp(x)^4)*log(x)^3+(-2*x^2*exp(x)^6+(-x^3+x^2+x)*exp(x)^5+3*x^2*exp(x)^4+(15*x+5)
*exp(x)^3+10*x*exp(x)^2)*log(x)^2+((-20*x-5)*exp(x)^4+(-10*x^2+20*x+20)*exp(x)^3+30*x*exp(x)^2+25)*log(x)-5*ex
p(x)^4+10*exp(x)^3+(-25*x-25)*exp(x)+75)/(x^2*exp(x)^4*log(x)^2+10*x*exp(x)^2*log(x)+25),x, algorithm="fricas"
)

[Out]

((x^2*e^(2*x) + x*e^(3*x))*log(x)^2 - 5*x*e^x + (2*x^2*e^(2*x) - x*e^(4*x) - (x^2 - 2*x)*e^(3*x) + 5*x)*log(x)
+ 10*x)/(x*e^(2*x)*log(x) + 5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (26) = 52\).

Time = 0.22 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.22 \[ \int \frac {75+10 e^{3 x}-5 e^{4 x}+e^x (-25-25 x)+\left (25+e^{4 x} (-5-20 x)+30 e^{2 x} x+e^{3 x} \left (20+20 x-10 x^2\right )\right ) \log (x)+\left (10 e^{2 x} x+3 e^{4 x} x^2-2 e^{6 x} x^2+e^{3 x} (5+15 x)+e^{5 x} \left (x+x^2-x^3\right )\right ) \log ^2(x)+\left (e^{4 x} x^2+e^{5 x} x^2\right ) \log ^3(x)}{25+10 e^{2 x} x \log (x)+e^{4 x} x^2 \log ^2(x)} \, dx=x \log {\left (x \right )} + 2 x + \frac {\left (- 5 x \log {\left (x \right )}^{2} - 10 x \log {\left (x \right )}\right ) e^{x} - 25}{x^{2} e^{2 x} \log {\left (x \right )}^{2} + 5 x \log {\left (x \right )}} + \left (- x + \log {\left (x \right )} + 2\right ) e^{x} - e^{2 x} + \frac {5}{x \log {\left (x \right )}} \]

[In]

integrate(((x**2*exp(x)**5+x**2*exp(x)**4)*ln(x)**3+(-2*x**2*exp(x)**6+(-x**3+x**2+x)*exp(x)**5+3*x**2*exp(x)*
*4+(15*x+5)*exp(x)**3+10*x*exp(x)**2)*ln(x)**2+((-20*x-5)*exp(x)**4+(-10*x**2+20*x+20)*exp(x)**3+30*x*exp(x)**
2+25)*ln(x)-5*exp(x)**4+10*exp(x)**3+(-25*x-25)*exp(x)+75)/(x**2*exp(x)**4*ln(x)**2+10*x*exp(x)**2*ln(x)+25),x
)

[Out]

x*log(x) + 2*x + ((-5*x*log(x)**2 - 10*x*log(x))*exp(x) - 25)/(x**2*exp(2*x)*log(x)**2 + 5*x*log(x)) + (-x + l
og(x) + 2)*exp(x) - exp(2*x) + 5/(x*log(x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (30) = 60\).

Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.56 \[ \int \frac {75+10 e^{3 x}-5 e^{4 x}+e^x (-25-25 x)+\left (25+e^{4 x} (-5-20 x)+30 e^{2 x} x+e^{3 x} \left (20+20 x-10 x^2\right )\right ) \log (x)+\left (10 e^{2 x} x+3 e^{4 x} x^2-2 e^{6 x} x^2+e^{3 x} (5+15 x)+e^{5 x} \left (x+x^2-x^3\right )\right ) \log ^2(x)+\left (e^{4 x} x^2+e^{5 x} x^2\right ) \log ^3(x)}{25+10 e^{2 x} x \log (x)+e^{4 x} x^2 \log ^2(x)} \, dx=-\frac {x e^{\left (4 \, x\right )} \log \left (x\right ) - {\left (x \log \left (x\right )^{2} - {\left (x^{2} - 2 \, x\right )} \log \left (x\right )\right )} e^{\left (3 \, x\right )} - {\left (x^{2} \log \left (x\right )^{2} + 2 \, x^{2} \log \left (x\right )\right )} e^{\left (2 \, x\right )} + 5 \, x e^{x} - 5 \, x \log \left (x\right ) - 10 \, x}{x e^{\left (2 \, x\right )} \log \left (x\right ) + 5} \]

[In]

integrate(((x^2*exp(x)^5+x^2*exp(x)^4)*log(x)^3+(-2*x^2*exp(x)^6+(-x^3+x^2+x)*exp(x)^5+3*x^2*exp(x)^4+(15*x+5)
*exp(x)^3+10*x*exp(x)^2)*log(x)^2+((-20*x-5)*exp(x)^4+(-10*x^2+20*x+20)*exp(x)^3+30*x*exp(x)^2+25)*log(x)-5*ex
p(x)^4+10*exp(x)^3+(-25*x-25)*exp(x)+75)/(x^2*exp(x)^4*log(x)^2+10*x*exp(x)^2*log(x)+25),x, algorithm="maxima"
)

[Out]

-(x*e^(4*x)*log(x) - (x*log(x)^2 - (x^2 - 2*x)*log(x))*e^(3*x) - (x^2*log(x)^2 + 2*x^2*log(x))*e^(2*x) + 5*x*e
^x - 5*x*log(x) - 10*x)/(x*e^(2*x)*log(x) + 5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (30) = 60\).

Time = 0.35 (sec) , antiderivative size = 204, normalized size of antiderivative = 6.38 \[ \int \frac {75+10 e^{3 x}-5 e^{4 x}+e^x (-25-25 x)+\left (25+e^{4 x} (-5-20 x)+30 e^{2 x} x+e^{3 x} \left (20+20 x-10 x^2\right )\right ) \log (x)+\left (10 e^{2 x} x+3 e^{4 x} x^2-2 e^{6 x} x^2+e^{3 x} (5+15 x)+e^{5 x} \left (x+x^2-x^3\right )\right ) \log ^2(x)+\left (e^{4 x} x^2+e^{5 x} x^2\right ) \log ^3(x)}{25+10 e^{2 x} x \log (x)+e^{4 x} x^2 \log ^2(x)} \, dx=\frac {x^{4} e^{\left (5 \, x\right )} \log \left (x\right )^{3} - x^{4} e^{\left (6 \, x\right )} \log \left (x\right )^{2} + 2 \, x^{4} e^{\left (5 \, x\right )} \log \left (x\right )^{2} + x^{3} e^{\left (6 \, x\right )} \log \left (x\right )^{3} - x^{3} e^{\left (7 \, x\right )} \log \left (x\right )^{2} + 2 \, x^{3} e^{\left (6 \, x\right )} \log \left (x\right )^{2} + 10 \, x^{3} e^{\left (3 \, x\right )} \log \left (x\right )^{2} - 10 \, x^{3} e^{\left (4 \, x\right )} \log \left (x\right ) + 20 \, x^{3} e^{\left (3 \, x\right )} \log \left (x\right ) + 5 \, x^{2} e^{\left (4 \, x\right )} \log \left (x\right )^{2} - 5 \, x^{2} e^{\left (5 \, x\right )} \log \left (x\right ) + 10 \, x^{2} e^{\left (4 \, x\right )} \log \left (x\right ) + 25 \, x^{2} e^{x} \log \left (x\right ) - 25 \, x^{2} e^{\left (2 \, x\right )} + 50 \, x^{2} e^{x}}{x^{3} e^{\left (5 \, x\right )} \log \left (x\right )^{2} + 10 \, x^{2} e^{\left (3 \, x\right )} \log \left (x\right ) + 25 \, x e^{x}} \]

[In]

integrate(((x^2*exp(x)^5+x^2*exp(x)^4)*log(x)^3+(-2*x^2*exp(x)^6+(-x^3+x^2+x)*exp(x)^5+3*x^2*exp(x)^4+(15*x+5)
*exp(x)^3+10*x*exp(x)^2)*log(x)^2+((-20*x-5)*exp(x)^4+(-10*x^2+20*x+20)*exp(x)^3+30*x*exp(x)^2+25)*log(x)-5*ex
p(x)^4+10*exp(x)^3+(-25*x-25)*exp(x)+75)/(x^2*exp(x)^4*log(x)^2+10*x*exp(x)^2*log(x)+25),x, algorithm="giac")

[Out]

(x^4*e^(5*x)*log(x)^3 - x^4*e^(6*x)*log(x)^2 + 2*x^4*e^(5*x)*log(x)^2 + x^3*e^(6*x)*log(x)^3 - x^3*e^(7*x)*log
(x)^2 + 2*x^3*e^(6*x)*log(x)^2 + 10*x^3*e^(3*x)*log(x)^2 - 10*x^3*e^(4*x)*log(x) + 20*x^3*e^(3*x)*log(x) + 5*x
^2*e^(4*x)*log(x)^2 - 5*x^2*e^(5*x)*log(x) + 10*x^2*e^(4*x)*log(x) + 25*x^2*e^x*log(x) - 25*x^2*e^(2*x) + 50*x
^2*e^x)/(x^3*e^(5*x)*log(x)^2 + 10*x^2*e^(3*x)*log(x) + 25*x*e^x)

Mupad [B] (verification not implemented)

Time = 13.79 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.16 \[ \int \frac {75+10 e^{3 x}-5 e^{4 x}+e^x (-25-25 x)+\left (25+e^{4 x} (-5-20 x)+30 e^{2 x} x+e^{3 x} \left (20+20 x-10 x^2\right )\right ) \log (x)+\left (10 e^{2 x} x+3 e^{4 x} x^2-2 e^{6 x} x^2+e^{3 x} (5+15 x)+e^{5 x} \left (x+x^2-x^3\right )\right ) \log ^2(x)+\left (e^{4 x} x^2+e^{5 x} x^2\right ) \log ^3(x)}{25+10 e^{2 x} x \log (x)+e^{4 x} x^2 \log ^2(x)} \, dx=2\,x-{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (x-2\right )-\frac {5\,{\mathrm {e}}^{-x}}{x}+\ln \left (x\right )\,\left (x+{\mathrm {e}}^x\right )+\frac {5\,\left (25\,{\mathrm {e}}^x-15\,x\,{\mathrm {e}}^{3\,x}+5\,x\,{\mathrm {e}}^{4\,x}-20\,x^2\,{\mathrm {e}}^{3\,x}+10\,x^2\,{\mathrm {e}}^{4\,x}+2\,x^2\,{\mathrm {e}}^{5\,x}-x^2\,{\mathrm {e}}^{6\,x}+50\,x\,{\mathrm {e}}^x\right )}{x\,\left (x\,{\mathrm {e}}^{2\,x}\,\ln \left (x\right )+5\right )\,\left (5\,{\mathrm {e}}^{2\,x}+10\,x\,{\mathrm {e}}^{2\,x}-x\,{\mathrm {e}}^{4\,x}\right )} \]

[In]

int((10*exp(3*x) - 5*exp(4*x) + log(x)^2*(10*x*exp(2*x) + 3*x^2*exp(4*x) - 2*x^2*exp(6*x) + exp(5*x)*(x + x^2
- x^3) + exp(3*x)*(15*x + 5)) - exp(x)*(25*x + 25) + log(x)*(exp(3*x)*(20*x - 10*x^2 + 20) + 30*x*exp(2*x) - e
xp(4*x)*(20*x + 5) + 25) + log(x)^3*(x^2*exp(4*x) + x^2*exp(5*x)) + 75)/(x^2*exp(4*x)*log(x)^2 + 10*x*exp(2*x)
*log(x) + 25),x)

[Out]

2*x - exp(2*x) - exp(x)*(x - 2) - (5*exp(-x))/x + log(x)*(x + exp(x)) + (5*(25*exp(x) - 15*x*exp(3*x) + 5*x*ex
p(4*x) - 20*x^2*exp(3*x) + 10*x^2*exp(4*x) + 2*x^2*exp(5*x) - x^2*exp(6*x) + 50*x*exp(x)))/(x*(x*exp(2*x)*log(
x) + 5)*(5*exp(2*x) + 10*x*exp(2*x) - x*exp(4*x)))