\(\int \frac {30+20 x+e^{(\frac {3+2 x}{x})^{5 x}} (15+10 x+(\frac {3+2 x}{x})^{5 x} (75 x+(-75 x-50 x^2) \log (\frac {3+2 x}{x})))}{6 x^2+4 x^3} \, dx\) [6815]

Optimal result

Integrand size = 78, antiderivative size = 27 \[ \int \frac {30+20 x+e^{\left (\frac {3+2 x}{x}\right )^{5 x}} \left (15+10 x+\left (\frac {3+2 x}{x}\right )^{5 x} \left (75 x+\left (-75 x-50 x^2\right ) \log \left (\frac {3+2 x}{x}\right )\right )\right )}{6 x^2+4 x^3} \, dx=\frac {5 \left (-2-e^{\left (\frac {3+2 x}{x}\right )^{5 x}}+x\right )}{2 x} \]

[Out]

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

Rubi [F]

\[ \int \frac {30+20 x+e^{\left (\frac {3+2 x}{x}\right )^{5 x}} \left (15+10 x+\left (\frac {3+2 x}{x}\right )^{5 x} \left (75 x+\left (-75 x-50 x^2\right ) \log \left (\frac {3+2 x}{x}\right )\right )\right )}{6 x^2+4 x^3} \, dx=\int \frac {30+20 x+e^{\left (\frac {3+2 x}{x}\right )^{5 x}} \left (15+10 x+\left (\frac {3+2 x}{x}\right )^{5 x} \left (75 x+\left (-75 x-50 x^2\right ) \log \left (\frac {3+2 x}{x}\right )\right )\right )}{6 x^2+4 x^3} \, dx \]

[In]

Int[(30 + 20*x + E^((3 + 2*x)/x)^(5*x)*(15 + 10*x + ((3 + 2*x)/x)^(5*x)*(75*x + (-75*x - 50*x^2)*Log[(3 + 2*x)
/x])))/(6*x^2 + 4*x^3),x]

[Out]

-5/x + (5*Defer[Int][E^(2 + 3/x)^(5*x)/x^2, x])/2 + (25*Defer[Int][(E^(2 + 3/x)^(5*x)*(2 + 3/x)^(5*x))/x, x])/
2 - (25*Log[2 + 3/x]*Defer[Int][(E^(2 + 3/x)^(5*x)*(2 + 3/x)^(5*x))/x, x])/2 - 25*Defer[Int][(E^(2 + 3/x)^(5*x
)*(2 + 3/x)^(5*x))/(3 + 2*x), x] - (25*Defer[Int][Defer[Int][(E^(2 + 3/x)^(5*x)*(2 + 3/x)^(5*x))/x, x]/x, x])/
2 + 25*Defer[Int][Defer[Int][(E^(2 + 3/x)^(5*x)*(2 + 3/x)^(5*x))/x, x]/(3 + 2*x), x]

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

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {30+20 x+e^{\left (\frac {3+2 x}{x}\right )^{5 x}} \left (15+10 x+\left (\frac {3+2 x}{x}\right )^{5 x} \left (75 x+\left (-75 x-50 x^2\right ) \log \left (\frac {3+2 x}{x}\right )\right )\right )}{6 x^2+4 x^3} \, dx=-\frac {5 \left (2+e^{\left (2+\frac {3}{x}\right )^{5 x}}\right )}{2 x} \]

[In]

Integrate[(30 + 20*x + E^((3 + 2*x)/x)^(5*x)*(15 + 10*x + ((3 + 2*x)/x)^(5*x)*(75*x + (-75*x - 50*x^2)*Log[(3
+ 2*x)/x])))/(6*x^2 + 4*x^3),x]

[Out]

(-5*(2 + E^(2 + 3/x)^(5*x)))/(2*x)

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96

method	result	size
risch	\(-\frac {5}{x}-\frac {5 \,{\mathrm e}^{\left (\frac {3+2 x}{x}\right )^{5 x}}}{2 x}\)	\(26\)
parallelrisch	\(\frac {-360+120 x -180 \,{\mathrm e}^{{\mathrm e}^{5 x \ln \left (\frac {3+2 x}{x}\right )}}}{72 x}\)	\(28\)

[In]

int(((((-50*x^2-75*x)*ln((3+2*x)/x)+75*x)*exp(5*x*ln((3+2*x)/x))+10*x+15)*exp(exp(5*x*ln((3+2*x)/x)))+20*x+30)
/(4*x^3+6*x^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {30+20 x+e^{\left (\frac {3+2 x}{x}\right )^{5 x}} \left (15+10 x+\left (\frac {3+2 x}{x}\right )^{5 x} \left (75 x+\left (-75 x-50 x^2\right ) \log \left (\frac {3+2 x}{x}\right )\right )\right )}{6 x^2+4 x^3} \, dx=-\frac {5 \, {\left (e^{\left (\left (\frac {2 \, x + 3}{x}\right )^{5 \, x}\right )} + 2\right )}}{2 \, x} \]

[In]

integrate(((((-50*x^2-75*x)*log((3+2*x)/x)+75*x)*exp(5*x*log((3+2*x)/x))+10*x+15)*exp(exp(5*x*log((3+2*x)/x)))
+20*x+30)/(4*x^3+6*x^2),x, algorithm="fricas")

[Out]

-5/2*(e^(((2*x + 3)/x)^(5*x)) + 2)/x

Sympy [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {30+20 x+e^{\left (\frac {3+2 x}{x}\right )^{5 x}} \left (15+10 x+\left (\frac {3+2 x}{x}\right )^{5 x} \left (75 x+\left (-75 x-50 x^2\right ) \log \left (\frac {3+2 x}{x}\right )\right )\right )}{6 x^2+4 x^3} \, dx=- \frac {5 e^{e^{5 x \log {\left (\frac {2 x + 3}{x} \right )}}}}{2 x} - \frac {5}{x} \]

[In]

integrate(((((-50*x**2-75*x)*ln((3+2*x)/x)+75*x)*exp(5*x*ln((3+2*x)/x))+10*x+15)*exp(exp(5*x*ln((3+2*x)/x)))+2
0*x+30)/(4*x**3+6*x**2),x)

[Out]

-5*exp(exp(5*x*log((2*x + 3)/x)))/(2*x) - 5/x

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {30+20 x+e^{\left (\frac {3+2 x}{x}\right )^{5 x}} \left (15+10 x+\left (\frac {3+2 x}{x}\right )^{5 x} \left (75 x+\left (-75 x-50 x^2\right ) \log \left (\frac {3+2 x}{x}\right )\right )\right )}{6 x^2+4 x^3} \, dx=-\frac {5 \, e^{\left (e^{\left (5 \, x \log \left (2 \, x + 3\right ) - 5 \, x \log \left (x\right )\right )}\right )}}{2 \, x} - \frac {5}{x} \]

[In]

integrate(((((-50*x^2-75*x)*log((3+2*x)/x)+75*x)*exp(5*x*log((3+2*x)/x))+10*x+15)*exp(exp(5*x*log((3+2*x)/x)))
+20*x+30)/(4*x^3+6*x^2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {30+20 x+e^{\left (\frac {3+2 x}{x}\right )^{5 x}} \left (15+10 x+\left (\frac {3+2 x}{x}\right )^{5 x} \left (75 x+\left (-75 x-50 x^2\right ) \log \left (\frac {3+2 x}{x}\right )\right )\right )}{6 x^2+4 x^3} \, dx=\int { -\frac {5 \, {\left ({\left (5 \, {\left ({\left (2 \, x^{2} + 3 \, x\right )} \log \left (\frac {2 \, x + 3}{x}\right ) - 3 \, x\right )} \left (\frac {2 \, x + 3}{x}\right )^{5 \, x} - 2 \, x - 3\right )} e^{\left (\left (\frac {2 \, x + 3}{x}\right )^{5 \, x}\right )} - 4 \, x - 6\right )}}{2 \, {\left (2 \, x^{3} + 3 \, x^{2}\right )}} \,d x } \]

[In]

integrate(((((-50*x^2-75*x)*log((3+2*x)/x)+75*x)*exp(5*x*log((3+2*x)/x))+10*x+15)*exp(exp(5*x*log((3+2*x)/x)))
+20*x+30)/(4*x^3+6*x^2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {30+20 x+e^{\left (\frac {3+2 x}{x}\right )^{5 x}} \left (15+10 x+\left (\frac {3+2 x}{x}\right )^{5 x} \left (75 x+\left (-75 x-50 x^2\right ) \log \left (\frac {3+2 x}{x}\right )\right )\right )}{6 x^2+4 x^3} \, dx=-\frac {5\,\left ({\mathrm {e}}^{{\left (\frac {3}{x}+2\right )}^{5\,x}}+2\right )}{2\,x} \]

[In]

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

[Out]

-(5*(exp((3/x + 2)^(5*x)) + 2))/(2*x)