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

Optimal. Leaf size=27 \[ \frac {5 \left (-2-e^{\left (\frac {3+2 x}{x}\right )^{5 x}}+x\right )}{2 x} \]

________________________________________________________________________________________

Rubi [F]  time = 2.64, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.81, size = 22, normalized size = 0.81 \begin {gather*} -\frac {5 \left (2+e^{\left (2+\frac {3}{x}\right )^{5 x}}\right )}{2 x} \end {gather*}

Antiderivative was successfully verified.

[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)

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 21, normalized size = 0.78 \begin {gather*} -\frac {5 \, {\left (e^{\left (\left (\frac {2 \, x + 3}{x}\right )^{5 \, x}\right )} + 2\right )}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-50*x^2-75*x)*log((2*x+3)/x)+75*x)*exp(5*x*log((2*x+3)/x))+10*x+15)*exp(exp(5*x*log((2*x+3)/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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-50*x^2-75*x)*log((2*x+3)/x)+75*x)*exp(5*x*log((2*x+3)/x))+10*x+15)*exp(exp(5*x*log((2*x+3)/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)

________________________________________________________________________________________

maple [A]  time = 0.12, size = 26, normalized size = 0.96

method	result	size
risch	\(-\frac {5}{x}-\frac {5 \,{\mathrm e}^{\left (\frac {2 x +3}{x}\right )^{5 x}}}{2 x}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 28, normalized size = 1.04 \begin {gather*} -\frac {5 \, e^{\left (e^{\left (5 \, x \log \left (2 \, x + 3\right ) - 5 \, x \log \relax (x)\right )}\right )}}{2 \, x} - \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-50*x^2-75*x)*log((2*x+3)/x)+75*x)*exp(5*x*log((2*x+3)/x))+10*x+15)*exp(exp(5*x*log((2*x+3)/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

________________________________________________________________________________________

mupad [B]  time = 4.45, size = 19, normalized size = 0.70 \begin {gather*} -\frac {5\,\left ({\mathrm {e}}^{{\left (\frac {3}{x}+2\right )}^{5\,x}}+2\right )}{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 1.00, size = 24, normalized size = 0.89 \begin {gather*} - \frac {5 e^{e^{5 x \log {\left (\frac {2 x + 3}{x} \right )}}}}{2 x} - \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-50*x**2-75*x)*ln((2*x+3)/x)+75*x)*exp(5*x*ln((2*x+3)/x))+10*x+15)*exp(exp(5*x*ln((2*x+3)/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

________________________________________________________________________________________