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

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

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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 = {} $$\begin{array}{c}\int \frac{30+20x+e^{{\left(\frac{3+2x}{x}\right)}^{5x}}(15+10x+{\left(\frac{3+2x}{x}\right)}^{5x}(75x+(-75x-50x^2)\log \left(\frac{3+2x}{x}\right)))}{6x^2+4x^3}dx\end{array}$$

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{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{30+20x+e^{{\left(\frac{3+2x}{x}\right)}^{5x}}(15+10x+{\left(\frac{3+2x}{x}\right)}^{5x}(75x+(-75x-50x^2)\log \left(\frac{3+2x}{x}\right)))}{x^2(6+4x)}dx\\ & =\int (\frac{5(2+e^{{(2+\frac{3}{x})}^{5x}})}{2x^2}-\frac{25e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}(-3+3\log (2+\frac{3}{x})+2x\log (2+\frac{3}{x}))}{2x(3+2x)})dx\\ & =\frac{5}{2}\int \frac{2+e^{{(2+\frac{3}{x})}^{5x}}}{x^2}dx-\frac{25}{2}\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}(-3+3\log (2+\frac{3}{x})+2x\log (2+\frac{3}{x}))}{x(3+2x)}dx\\ & =\frac{5}{2}\int (\frac{2}{x^2}+\frac{e^{{(2+\frac{3}{x})}^{5x}}}{x^2})dx-\frac{25}{2}\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}(-3+(3+2x)\log (2+\frac{3}{x}))}{x(3+2x)}dx\\ & =-\frac{5}{x}+\frac{5}{2}\int \frac{e^{{(2+\frac{3}{x})}^{5x}}}{x^2}dx-\frac{25}{2}\int (-\frac{3e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{x(3+2x)}+\frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}\log (2+\frac{3}{x})}{x})dx\\ & =-\frac{5}{x}+\frac{5}{2}\int \frac{e^{{(2+\frac{3}{x})}^{5x}}}{x^2}dx-\frac{25}{2}\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}\log (2+\frac{3}{x})}{x}dx+\frac{75}{2}\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{x(3+2x)}dx\\ & =-\frac{5}{x}+\frac{5}{2}\int \frac{e^{{(2+\frac{3}{x})}^{5x}}}{x^2}dx+\frac{25}{2}\int \frac{3\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{x}dx}{(-3-2x)x}dx+\frac{75}{2}\int (\frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{3x}-\frac{2e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{3(3+2x)})dx-\frac{1}{2}(25\log (2+\frac{3}{x}))\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{x}dx\\ & =-\frac{5}{x}+\frac{5}{2}\int \frac{e^{{(2+\frac{3}{x})}^{5x}}}{x^2}dx+\frac{25}{2}\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{x}dx-25\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{3+2x}dx+\frac{75}{2}\int \frac{\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{x}dx}{(-3-2x)x}dx-\frac{1}{2}(25\log (2+\frac{3}{x}))\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{x}dx\\ & =-\frac{5}{x}+\frac{5}{2}\int \frac{e^{{(2+\frac{3}{x})}^{5x}}}{x^2}dx+\frac{25}{2}\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{x}dx-25\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{3+2x}dx+\frac{75}{2}\int (-\frac{\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{x}dx}{3x}+\frac{2\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{x}dx}{3(3+2x)})dx-\frac{1}{2}(25\log (2+\frac{3}{x}))\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{x}dx\\ & =-\frac{5}{x}+\frac{5}{2}\int \frac{e^{{(2+\frac{3}{x})}^{5x}}}{x^2}dx+\frac{25}{2}\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{x}dx-\frac{25}{2}\int \frac{\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{x}dx}{x}dx-25\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{3+2x}dx+25\int \frac{\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{x}dx}{3+2x}dx-\frac{1}{2}(25\log (2+\frac{3}{x}))\int \frac{e^{{(2+\frac{3}{x})}^{5x}}{(2+\frac{3}{x})}^{5x}}{x}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.81, size = 22, normalized size = 0.81 $$\begin{array}{c}-\frac{5(2+e^{{(2+\frac{3}{x})}^{5x}})}{2x}\end{array}$$

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)

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

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\int -\frac{5((5((2x^2+3x)\log \left(\frac{2x+3}{x}\right)-3x){\left(\frac{2x+3}{x}\right)}^{5x}-2x-3)e^{\left({\left(\frac{2x+3}{x}\right)}^{5x}\right)}-4x-6)}{2(2x^3+3x^2)}dx\end{array}$$

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)

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

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

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))

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

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

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

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)

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

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

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