\(\int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+(1218 x+2 e^2 x+10600 x^2+3750 x^3) \log (x)) \, dx\) [7789]

Optimal result

Integrand size = 72, antiderivative size = 19 \[ \int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx=2 x^{(4+x) \left (e^2+(3+25 x)^2\right )} \]

[Out]

2*exp(((3+25*x)^2+exp(2))*(4+x)*ln(x))

Rubi [F]

\[ \int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx=\int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx \]

[In]

Int[x^(35 + 609*x + 2650*x^2 + 625*x^3 + E^2*(4 + x))*(72 + 1218*x + 5300*x^2 + 1250*x^3 + E^2*(8 + 2*x) + (12
18*x + 2*E^2*x + 10600*x^2 + 3750*x^3)*Log[x]),x]

[Out]

2*(609 + E^2)*Defer[Int][x^((4 + x)*(9 + E^2 + 150*x + 625*x^2)), x] + 2*(609 + E^2)*Log[x]*Defer[Int][x^((4 +
x)*(9 + E^2 + 150*x + 625*x^2)), x] + 8*(9 + E^2)*Defer[Int][x^(35 + 4*E^2 + (609 + E^2)*x + 2650*x^2 + 625*x
^3), x] + 5300*Defer[Int][x^(37 + 4*E^2 + (609 + E^2)*x + 2650*x^2 + 625*x^3), x] + 10600*Log[x]*Defer[Int][x^
(37 + 4*E^2 + (609 + E^2)*x + 2650*x^2 + 625*x^3), x] + 1250*Defer[Int][x^(2*(19 + 2*E^2) + (609 + E^2)*x + 26
50*x^2 + 625*x^3), x] + 3750*Log[x]*Defer[Int][x^(2*(19 + 2*E^2) + (609 + E^2)*x + 2650*x^2 + 625*x^3), x] - 2
*(609 + E^2)*Defer[Int][Defer[Int][x^((4 + x)*(9 + E^2 + 150*x + 625*x^2)), x]/x, x] - 10600*Defer[Int][Defer[
Int][x^(37 + 4*E^2 + (609 + E^2)*x + 2650*x^2 + 625*x^3), x]/x, x] - 3750*Defer[Int][Defer[Int][x^(2*(19 + 2*E
^2) + (609 + E^2)*x + 2650*x^2 + 625*x^3), x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int x^{35+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx \\ & = \int 2 x^{35+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \left ((4+x) \left (e^2+(3+25 x)^2\right )+x \left (609+e^2+5300 x+1875 x^2\right ) \log (x)\right ) \, dx \\ & = 2 \int x^{35+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \left ((4+x) \left (e^2+(3+25 x)^2\right )+x \left (609+e^2+5300 x+1875 x^2\right ) \log (x)\right ) \, dx \\ & = 2 \int \left (x^{35+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} (4+x) \left (9+e^2+150 x+625 x^2\right )+x^{36+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \left (609+e^2+5300 x+1875 x^2\right ) \log (x)\right ) \, dx \\ & = 2 \int x^{35+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} (4+x) \left (9+e^2+150 x+625 x^2\right ) \, dx+2 \int x^{36+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \left (609+e^2+5300 x+1875 x^2\right ) \log (x) \, dx \\ & = 2 \int \left (4 \left (9+e^2\right ) x^{35+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3}+\left (609+e^2\right ) x^{36+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3}+2650 x^{37+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3}+625 x^{38+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3}\right ) \, dx-2 \int \frac {5300 \int x^{37+609 x+2650 x^2+625 x^3+e^2 (4+x)} \, dx+1875 \int x^{38+609 x+2650 x^2+625 x^3+e^2 (4+x)} \, dx+\left (609+e^2\right ) \int x^{(4+x) \left (e^2+(3+25 x)^2\right )} \, dx}{x} \, dx+(3750 \log (x)) \int x^{2 \left (19+2 e^2\right )+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+(10600 \log (x)) \int x^{37+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+\left (2 \left (609+e^2\right ) \log (x)\right ) \int x^{(4+x) \left (9+e^2+150 x+625 x^2\right )} \, dx \\ & = -\left (2 \int \left (\frac {1875 \int x^{2 \left (19+2 e^2\right )+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx}{x}+\frac {5300 \int x^{37+609 x+2650 x^2+625 x^3+e^2 (4+x)} \, dx+609 \left (1+\frac {e^2}{609}\right ) \int x^{(4+x) \left (e^2+(3+25 x)^2\right )} \, dx}{x}\right ) \, dx\right )+1250 \int x^{38+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+5300 \int x^{37+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+\left (8 \left (9+e^2\right )\right ) \int x^{35+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+\left (2 \left (609+e^2\right )\right ) \int x^{36+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+(3750 \log (x)) \int x^{2 \left (19+2 e^2\right )+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+(10600 \log (x)) \int x^{37+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+\left (2 \left (609+e^2\right ) \log (x)\right ) \int x^{(4+x) \left (9+e^2+150 x+625 x^2\right )} \, dx \\ & = -\left (2 \int \frac {609 \left (1+\frac {e^2}{609}\right ) \int x^{(4+x) \left (9+e^2+150 x+625 x^2\right )} \, dx+5300 \int x^{37+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx}{x} \, dx\right )+1250 \int x^{2 \left (19+2 e^2\right )+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx-3750 \int \frac {\int x^{2 \left (19+2 e^2\right )+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx}{x} \, dx+5300 \int x^{37+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+\left (8 \left (9+e^2\right )\right ) \int x^{35+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+\left (2 \left (609+e^2\right )\right ) \int x^{(4+x) \left (9+e^2+150 x+625 x^2\right )} \, dx+(3750 \log (x)) \int x^{2 \left (19+2 e^2\right )+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+(10600 \log (x)) \int x^{37+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+\left (2 \left (609+e^2\right ) \log (x)\right ) \int x^{(4+x) \left (9+e^2+150 x+625 x^2\right )} \, dx \\ & = -\left (2 \int \left (\frac {\left (609+e^2\right ) \int x^{(4+x) \left (9+e^2+150 x+625 x^2\right )} \, dx}{x}+\frac {5300 \int x^{37+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx}{x}\right ) \, dx\right )+1250 \int x^{2 \left (19+2 e^2\right )+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx-3750 \int \frac {\int x^{2 \left (19+2 e^2\right )+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx}{x} \, dx+5300 \int x^{37+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+\left (8 \left (9+e^2\right )\right ) \int x^{35+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+\left (2 \left (609+e^2\right )\right ) \int x^{(4+x) \left (9+e^2+150 x+625 x^2\right )} \, dx+(3750 \log (x)) \int x^{2 \left (19+2 e^2\right )+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+(10600 \log (x)) \int x^{37+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+\left (2 \left (609+e^2\right ) \log (x)\right ) \int x^{(4+x) \left (9+e^2+150 x+625 x^2\right )} \, dx \\ & = 1250 \int x^{2 \left (19+2 e^2\right )+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx-3750 \int \frac {\int x^{2 \left (19+2 e^2\right )+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx}{x} \, dx+5300 \int x^{37+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx-10600 \int \frac {\int x^{37+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx}{x} \, dx+\left (8 \left (9+e^2\right )\right ) \int x^{35+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+\left (2 \left (609+e^2\right )\right ) \int x^{(4+x) \left (9+e^2+150 x+625 x^2\right )} \, dx-\left (2 \left (609+e^2\right )\right ) \int \frac {\int x^{(4+x) \left (9+e^2+150 x+625 x^2\right )} \, dx}{x} \, dx+(3750 \log (x)) \int x^{2 \left (19+2 e^2\right )+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+(10600 \log (x)) \int x^{37+4 e^2+\left (609+e^2\right ) x+2650 x^2+625 x^3} \, dx+\left (2 \left (609+e^2\right ) \log (x)\right ) \int x^{(4+x) \left (9+e^2+150 x+625 x^2\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx=2 x^{(4+x) \left (e^2+(3+25 x)^2\right )} \]

[In]

Integrate[x^(35 + 609*x + 2650*x^2 + 625*x^3 + E^2*(4 + x))*(72 + 1218*x + 5300*x^2 + 1250*x^3 + E^2*(8 + 2*x)
+ (1218*x + 2*E^2*x + 10600*x^2 + 3750*x^3)*Log[x]),x]

[Out]

2*x^((4 + x)*(E^2 + (3 + 25*x)^2))

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11

method	result	size
risch	\(2 x^{\left (625 x^{2}+{\mathrm e}^{2}+150 x +9\right ) \left (4+x \right )}\)	\(21\)
norman	\(2 \,{\mathrm e}^{\left (\left (4+x \right ) {\mathrm e}^{2}+625 x^{3}+2650 x^{2}+609 x +36\right ) \ln \left (x \right )}\)	\(28\)
parallelrisch	\(2 \,{\mathrm e}^{\left (\left (4+x \right ) {\mathrm e}^{2}+625 x^{3}+2650 x^{2}+609 x +36\right ) \ln \left (x \right )}\)	\(28\)

[In]

int(((2*exp(2)*x+3750*x^3+10600*x^2+1218*x)*ln(x)+(2*x+8)*exp(2)+1250*x^3+5300*x^2+1218*x+72)*exp(((4+x)*exp(2
)+625*x^3+2650*x^2+609*x+36)*ln(x))/x,x,method=_RETURNVERBOSE)

[Out]

2*x^((625*x^2+exp(2)+150*x+9)*(4+x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32 \[ \int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx=2 \, x^{625 \, x^{3} + 2650 \, x^{2} + {\left (x + 4\right )} e^{2} + 609 \, x + 36} \]

[In]

integrate(((2*exp(2)*x+3750*x^3+10600*x^2+1218*x)*log(x)+(2*x+8)*exp(2)+1250*x^3+5300*x^2+1218*x+72)*exp(((4+x
)*exp(2)+625*x^3+2650*x^2+609*x+36)*log(x))/x,x, algorithm="fricas")

[Out]

2*x^(625*x^3 + 2650*x^2 + (x + 4)*e^2 + 609*x + 36)

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx=2 e^{\left (625 x^{3} + 2650 x^{2} + 609 x + \left (x + 4\right ) e^{2} + 36\right ) \log {\left (x \right )}} \]

[In]

integrate(((2*exp(2)*x+3750*x**3+10600*x**2+1218*x)*ln(x)+(2*x+8)*exp(2)+1250*x**3+5300*x**2+1218*x+72)*exp(((
4+x)*exp(2)+625*x**3+2650*x**2+609*x+36)*ln(x))/x,x)

[Out]

2*exp((625*x**3 + 2650*x**2 + 609*x + (x + 4)*exp(2) + 36)*log(x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (18) = 36\).

Time = 0.37 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.00 \[ \int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx=2 \, x^{36} e^{\left (625 \, x^{3} \log \left (x\right ) + 2650 \, x^{2} \log \left (x\right ) + x e^{2} \log \left (x\right ) + 609 \, x \log \left (x\right ) + 4 \, e^{2} \log \left (x\right )\right )} \]

[In]

integrate(((2*exp(2)*x+3750*x^3+10600*x^2+1218*x)*log(x)+(2*x+8)*exp(2)+1250*x^3+5300*x^2+1218*x+72)*exp(((4+x
)*exp(2)+625*x^3+2650*x^2+609*x+36)*log(x))/x,x, algorithm="maxima")

[Out]

2*x^36*e^(625*x^3*log(x) + 2650*x^2*log(x) + x*e^2*log(x) + 609*x*log(x) + 4*e^2*log(x))

Giac [F]

\[ \int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx=\int { \frac {2 \, {\left (625 \, x^{3} + 2650 \, x^{2} + {\left (x + 4\right )} e^{2} + {\left (1875 \, x^{3} + 5300 \, x^{2} + x e^{2} + 609 \, x\right )} \log \left (x\right ) + 609 \, x + 36\right )} x^{625 \, x^{3} + 2650 \, x^{2} + {\left (x + 4\right )} e^{2} + 609 \, x + 36}}{x} \,d x } \]

[In]

integrate(((2*exp(2)*x+3750*x^3+10600*x^2+1218*x)*log(x)+(2*x+8)*exp(2)+1250*x^3+5300*x^2+1218*x+72)*exp(((4+x
)*exp(2)+625*x^3+2650*x^2+609*x+36)*log(x))/x,x, algorithm="giac")

[Out]

integrate(2*(625*x^3 + 2650*x^2 + (x + 4)*e^2 + (1875*x^3 + 5300*x^2 + x*e^2 + 609*x)*log(x) + 609*x + 36)*x^(
625*x^3 + 2650*x^2 + (x + 4)*e^2 + 609*x + 36)/x, x)

Mupad [B] (verification not implemented)

Time = 14.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx=2\,x^{625\,x^3}\,x^{2650\,x^2}\,x^{x\,{\mathrm {e}}^2}\,x^{4\,{\mathrm {e}}^2}\,x^{609\,x}\,x^{36} \]

[In]

int((exp(log(x)*(609*x + exp(2)*(x + 4) + 2650*x^2 + 625*x^3 + 36))*(1218*x + log(x)*(1218*x + 2*x*exp(2) + 10
600*x^2 + 3750*x^3) + 5300*x^2 + 1250*x^3 + exp(2)*(2*x + 8) + 72))/x,x)

[Out]

2*x^(625*x^3)*x^(2650*x^2)*x^(x*exp(2))*x^(4*exp(2))*x^(609*x)*x^36