\(\int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} (18-\log (4 x+2 x^2))}-x} (-6 x-3 x^2+e^{\frac {1}{6} (18-\log (4 x+2 x^2))} (1+x))}{6 x+3 x^2} \, dx\) [736]

Optimal result

Integrand size = 84, antiderivative size = 27 \[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=e^{5-x-\frac {e^3}{\sqrt [6]{x (4+2 x)}}}-x \]

[Out]

exp(5-exp(3-1/6*ln(x*(4+2*x)))-x)-x

Rubi [F]

\[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=\int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx \]

[In]

Int[(-6*x - 3*x^2 + E^(5 - E^((18 - Log[4*x + 2*x^2])/6) - x)*(-6*x - 3*x^2 + E^((18 - Log[4*x + 2*x^2])/6)*(1
+ x)))/(6*x + 3*x^2),x]

[Out]

-x - Defer[Int][E^(5 - x - E^3/(2^(1/6)*(x*(2 + x))^(1/6))), x] + (2^(5/6)*x^(1/6)*(2 + x)^(1/6)*Defer[Subst][
Defer[Int][E^(8 - x^6 - E^3/(2^(1/6)*(x^6*(2 + x^6))^(1/6)))/(x^2*(2 + x^6)^(7/6)), x], x, x^(1/6)])/(2*x + x^
2)^(1/6) + (2^(5/6)*x^(1/6)*(2 + x)^(1/6)*Defer[Subst][Defer[Int][(E^(8 - x^6 - E^3/(2^(1/6)*(x^6*(2 + x^6))^(
1/6)))*x^4)/(2 + x^6)^(7/6), x], x, x^(1/6)])/(2*x + x^2)^(1/6)

Rubi steps \begin{align*} \text {integral}& = \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{x (6+3 x)} \, dx \\ & = \int \left (-e^{-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} \left (e^5+e^{x+\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}}\right )+\frac {e^{8-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} (1+x)}{3 \sqrt [6]{2} x (2+x) \sqrt [6]{2 x+x^2}}\right ) \, dx \\ & = \frac {\int \frac {e^{8-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} (1+x)}{x (2+x) \sqrt [6]{2 x+x^2}} \, dx}{3 \sqrt [6]{2}}-\int e^{-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} \left (e^5+e^{x+\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}}\right ) \, dx \\ & = \frac {\int \frac {e^{8-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} (1+x)}{\left (2 x+x^2\right )^{7/6}} \, dx}{3 \sqrt [6]{2}}-\int \left (1+e^{5-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}}\right ) \, dx \\ & = -x+\frac {\left (\sqrt [6]{x} \sqrt [6]{2+x}\right ) \int \frac {e^{8-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} (1+x)}{x^{7/6} (2+x)^{7/6}} \, dx}{3 \sqrt [6]{2} \sqrt [6]{2 x+x^2}}-\int e^{5-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} \, dx \\ & = -x+\frac {\left (2^{5/6} \sqrt [6]{x} \sqrt [6]{2+x}\right ) \text {Subst}\left (\int \frac {e^{8-x^6-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x^6 \left (2+x^6\right )}}} \left (1+x^6\right )}{x^2 \left (2+x^6\right )^{7/6}} \, dx,x,\sqrt [6]{x}\right )}{\sqrt [6]{2 x+x^2}}-\int e^{5-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} \, dx \\ & = -x+\frac {\left (2^{5/6} \sqrt [6]{x} \sqrt [6]{2+x}\right ) \text {Subst}\left (\int \left (\frac {e^{8-x^6-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x^6 \left (2+x^6\right )}}}}{x^2 \left (2+x^6\right )^{7/6}}+\frac {e^{8-x^6-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x^6 \left (2+x^6\right )}}} x^4}{\left (2+x^6\right )^{7/6}}\right ) \, dx,x,\sqrt [6]{x}\right )}{\sqrt [6]{2 x+x^2}}-\int e^{5-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} \, dx \\ & = -x+\frac {\left (2^{5/6} \sqrt [6]{x} \sqrt [6]{2+x}\right ) \text {Subst}\left (\int \frac {e^{8-x^6-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x^6 \left (2+x^6\right )}}}}{x^2 \left (2+x^6\right )^{7/6}} \, dx,x,\sqrt [6]{x}\right )}{\sqrt [6]{2 x+x^2}}+\frac {\left (2^{5/6} \sqrt [6]{x} \sqrt [6]{2+x}\right ) \text {Subst}\left (\int \frac {e^{8-x^6-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x^6 \left (2+x^6\right )}}} x^4}{\left (2+x^6\right )^{7/6}} \, dx,x,\sqrt [6]{x}\right )}{\sqrt [6]{2 x+x^2}}-\int e^{5-x-\frac {e^3}{\sqrt [6]{2} \sqrt [6]{x (2+x)}}} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=\int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx \]

[In]

Integrate[(-6*x - 3*x^2 + E^(5 - E^((18 - Log[4*x + 2*x^2])/6) - x)*(-6*x - 3*x^2 + E^((18 - Log[4*x + 2*x^2])
/6)*(1 + x)))/(6*x + 3*x^2),x]

[Out]

Integrate[(-6*x - 3*x^2 + E^(5 - E^((18 - Log[4*x + 2*x^2])/6) - x)*(-6*x - 3*x^2 + E^((18 - Log[4*x + 2*x^2])
/6)*(1 + x)))/(6*x + 3*x^2), x]

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07

method	result	size
parallelrisch	\(1-x +{\mathrm e}^{-{\mathrm e}^{-\frac {\ln \left (2 x^{2}+4 x \right )}{6}+3}+5-x}\)	\(29\)

[In]

int((((1+x)*exp(-1/6*ln(2*x^2+4*x)+3)-3*x^2-6*x)*exp(-exp(-1/6*ln(2*x^2+4*x)+3)+5-x)-3*x^2-6*x)/(3*x^2+6*x),x,
method=_RETURNVERBOSE)

[Out]

1-x+exp(-exp(-1/6*ln(2*x^2+4*x)+3)+5-x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=-x + e^{\left (-\frac {2 \, x^{3} - 6 \, x^{2} + {\left (2 \, x^{2} + 4 \, x\right )}^{\frac {5}{6}} e^{3} - 20 \, x}{2 \, {\left (x^{2} + 2 \, x\right )}}\right )} \]

[In]

integrate((((1+x)*exp(-1/6*log(2*x^2+4*x)+3)-3*x^2-6*x)*exp(-exp(-1/6*log(2*x^2+4*x)+3)+5-x)-3*x^2-6*x)/(3*x^2
+6*x),x, algorithm="fricas")

[Out]

-x + e^(-1/2*(2*x^3 - 6*x^2 + (2*x^2 + 4*x)^(5/6)*e^3 - 20*x)/(x^2 + 2*x))

Sympy [F(-1)]

Timed out. \[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=\text {Timed out} \]

[In]

integrate((((1+x)*exp(-1/6*ln(2*x**2+4*x)+3)-3*x**2-6*x)*exp(-exp(-1/6*ln(2*x**2+4*x)+3)+5-x)-3*x**2-6*x)/(3*x
**2+6*x),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=-x + e^{\left (-\frac {2^{\frac {5}{6}} e^{3}}{2 \, {\left (x + 2\right )}^{\frac {1}{6}} x^{\frac {1}{6}}} - x + 5\right )} \]

[In]

integrate((((1+x)*exp(-1/6*log(2*x^2+4*x)+3)-3*x^2-6*x)*exp(-exp(-1/6*log(2*x^2+4*x)+3)+5-x)-3*x^2-6*x)/(3*x^2
+6*x),x, algorithm="maxima")

[Out]

-x + e^(-1/2*2^(5/6)*e^3/((x + 2)^(1/6)*x^(1/6)) - x + 5)

Giac [F]

\[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=\int { -\frac {3 \, x^{2} + {\left (3 \, x^{2} - {\left (x + 1\right )} e^{\left (-\frac {1}{6} \, \log \left (2 \, x^{2} + 4 \, x\right ) + 3\right )} + 6 \, x\right )} e^{\left (-x - e^{\left (-\frac {1}{6} \, \log \left (2 \, x^{2} + 4 \, x\right ) + 3\right )} + 5\right )} + 6 \, x}{3 \, {\left (x^{2} + 2 \, x\right )}} \,d x } \]

[In]

integrate((((1+x)*exp(-1/6*log(2*x^2+4*x)+3)-3*x^2-6*x)*exp(-exp(-1/6*log(2*x^2+4*x)+3)+5-x)-3*x^2-6*x)/(3*x^2
+6*x),x, algorithm="giac")

[Out]

integrate(-1/3*(3*x^2 + (3*x^2 - (x + 1)*e^(-1/6*log(2*x^2 + 4*x) + 3) + 6*x)*e^(-x - e^(-1/6*log(2*x^2 + 4*x)
+ 3) + 5) + 6*x)/(x^2 + 2*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-6 x-3 x^2+e^{5-e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )}-x} \left (-6 x-3 x^2+e^{\frac {1}{6} \left (18-\log \left (4 x+2 x^2\right )\right )} (1+x)\right )}{6 x+3 x^2} \, dx=\int -\frac {6\,x+3\,x^2+{\mathrm {e}}^{5-{\mathrm {e}}^{3-\frac {\ln \left (2\,x^2+4\,x\right )}{6}}-x}\,\left (6\,x-{\mathrm {e}}^{3-\frac {\ln \left (2\,x^2+4\,x\right )}{6}}\,\left (x+1\right )+3\,x^2\right )}{3\,x^2+6\,x} \,d x \]

[In]

int(-(6*x + 3*x^2 + exp(5 - exp(3 - log(4*x + 2*x^2)/6) - x)*(6*x - exp(3 - log(4*x + 2*x^2)/6)*(x + 1) + 3*x^
2))/(6*x + 3*x^2),x)

[Out]

int(-(6*x + 3*x^2 + exp(5 - exp(3 - log(4*x + 2*x^2)/6) - x)*(6*x - exp(3 - log(4*x + 2*x^2)/6)*(x + 1) + 3*x^
2))/(6*x + 3*x^2), x)