3.101.72 \(\int \frac {-700+e^{\frac {1}{25} (-125+145 x-16 x^2+25 x \log (3 x))} (25-170 x+32 x^2-25 x \log (3 x))}{19600-1400 e^{\frac {1}{25} (-125+145 x-16 x^2+25 x \log (3 x))}+25 e^{\frac {2}{25} (-125+145 x-16 x^2+25 x \log (3 x))}} \, dx\)

Optimal. Leaf size=28 \[ \frac {x}{-28+e^{4-\left (-3+\frac {4 x}{5}\right )^2+x+x \log (3 x)}} \]

________________________________________________________________________________________

Rubi [F]  time = 3.93, 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 {-700+e^{\frac {1}{25} \left (-125+145 x-16 x^2+25 x \log (3 x)\right )} \left (25-170 x+32 x^2-25 x \log (3 x)\right )}{19600-1400 e^{\frac {1}{25} \left (-125+145 x-16 x^2+25 x \log (3 x)\right )}+25 e^{\frac {2}{25} \left (-125+145 x-16 x^2+25 x \log (3 x)\right )}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-700 + E^((-125 + 145*x - 16*x^2 + 25*x*Log[3*x])/25)*(25 - 170*x + 32*x^2 - 25*x*Log[3*x]))/(19600 - 140
0*E^((-125 + 145*x - 16*x^2 + 25*x*Log[3*x])/25) + 25*E^((2*(-125 + 145*x - 16*x^2 + 25*x*Log[3*x]))/25)),x]

[Out]

-28*Defer[Int][E^(10 + (32*x^2)/25)/(-28*E^(5 + (16*x^2)/25) + 3^x*E^((29*x)/5)*x^x)^2, x] + Defer[Int][(E^(5
+ (16*x^2)/25 + (x*(29 + Log[243]))/5)*x^x)/(-28*E^(5 + (16*x^2)/25) + 3^x*E^((29*x)/5)*x^x)^2, x] - (34*Defer
[Int][(E^(5 + (16*x^2)/25 + (x*(29 + Log[243]))/5)*x^(1 + x))/(-28*E^(5 + (16*x^2)/25) + 3^x*E^((29*x)/5)*x^x)
^2, x])/5 - Log[3*x]*Defer[Int][(E^(5 + (16*x^2)/25 + (x*(29 + Log[243]))/5)*x^(1 + x))/(-28*E^(5 + (16*x^2)/2
5) + 3^x*E^((29*x)/5)*x^x)^2, x] + (32*Defer[Int][(E^(5 + (16*x^2)/25 + (x*(29 + Log[243]))/5)*x^(2 + x))/(-28
*E^(5 + (16*x^2)/25) + 3^x*E^((29*x)/5)*x^x)^2, x])/25 + Defer[Int][Defer[Int][(3^x*E^(5 + (29*x)/5 + (16*x^2)
/25)*x^(1 + x))/(-28*E^(5 + (16*x^2)/25) + 3^x*E^((29*x)/5)*x^x)^2, x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{10+\frac {32 x^2}{25}} \left (-700+e^{\frac {1}{25} \left (-125+145 x-16 x^2+25 x \log (3 x)\right )} \left (25-170 x+32 x^2-25 x \log (3 x)\right )\right )}{25 \left (28 e^{5+\frac {16 x^2}{25}}-3^x e^{29 x/5} x^x\right )^2} \, dx\\ &=\frac {1}{25} \int \frac {e^{10+\frac {32 x^2}{25}} \left (-700+e^{\frac {1}{25} \left (-125+145 x-16 x^2+25 x \log (3 x)\right )} \left (25-170 x+32 x^2-25 x \log (3 x)\right )\right )}{\left (28 e^{5+\frac {16 x^2}{25}}-3^x e^{29 x/5} x^x\right )^2} \, dx\\ &=\frac {1}{25} \int \left (-\frac {700 e^{10+\frac {32 x^2}{25}}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2}+\frac {25\ 3^x e^{5+\frac {29 x}{5}+\frac {16 x^2}{25}} x^x}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2}-\frac {170\ 3^x e^{5+\frac {29 x}{5}+\frac {16 x^2}{25}} x^{1+x}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2}+\frac {32\ 3^x e^{5+\frac {29 x}{5}+\frac {16 x^2}{25}} x^{2+x}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2}-\frac {25\ 3^x e^{5+\frac {29 x}{5}+\frac {16 x^2}{25}} x^{1+x} \log (3 x)}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2}\right ) \, dx\\ &=\frac {32}{25} \int \frac {3^x e^{5+\frac {29 x}{5}+\frac {16 x^2}{25}} x^{2+x}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx-\frac {34}{5} \int \frac {3^x e^{5+\frac {29 x}{5}+\frac {16 x^2}{25}} x^{1+x}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx-28 \int \frac {e^{10+\frac {32 x^2}{25}}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx+\int \frac {3^x e^{5+\frac {29 x}{5}+\frac {16 x^2}{25}} x^x}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx-\int \frac {3^x e^{5+\frac {29 x}{5}+\frac {16 x^2}{25}} x^{1+x} \log (3 x)}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx\\ &=\frac {32}{25} \int \frac {e^{5+\frac {16 x^2}{25}+\frac {1}{5} x (29+\log (243))} x^{2+x}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx-\frac {34}{5} \int \frac {e^{5+\frac {16 x^2}{25}+\frac {1}{5} x (29+\log (243))} x^{1+x}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx-28 \int \frac {e^{10+\frac {32 x^2}{25}}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx+\int \frac {e^{5+\frac {16 x^2}{25}+\frac {1}{5} x (29+\log (243))} x^x}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx-\int \frac {e^{5+\frac {16 x^2}{25}+\frac {1}{5} x (29+\log (243))} x^{1+x} \log (3 x)}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx\\ &=\frac {32}{25} \int \frac {e^{5+\frac {16 x^2}{25}+\frac {1}{5} x (29+\log (243))} x^{2+x}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx-\frac {34}{5} \int \frac {e^{5+\frac {16 x^2}{25}+\frac {1}{5} x (29+\log (243))} x^{1+x}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx-28 \int \frac {e^{10+\frac {32 x^2}{25}}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx-\log (3 x) \int \frac {e^{5+\frac {16 x^2}{25}+\frac {1}{5} x (29+\log (243))} x^{1+x}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx+\int \frac {e^{5+\frac {16 x^2}{25}+\frac {1}{5} x (29+\log (243))} x^x}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx+\int \frac {\int \frac {3^x e^{5+\frac {29 x}{5}+\frac {16 x^2}{25}} x^{1+x}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx}{x} \, dx\\ &=\frac {32}{25} \int \frac {e^{5+\frac {16 x^2}{25}+\frac {1}{5} x (29+\log (243))} x^{2+x}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx-\frac {34}{5} \int \frac {e^{5+\frac {16 x^2}{25}+\frac {1}{5} x (29+\log (243))} x^{1+x}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx-28 \int \frac {e^{10+\frac {32 x^2}{25}}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx-\log (3 x) \int \frac {e^{5+\frac {16 x^2}{25}+\frac {1}{5} x (29+\log (243))} x^{1+x}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx+\int \frac {e^{5+\frac {16 x^2}{25}+\frac {1}{5} x (29+\log (243))} x^x}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx+\int \frac {\int \frac {e^{5+\frac {16 x^2}{25}+\frac {1}{5} x (29+\log (243))} x^{1+x}}{\left (-28 e^{5+\frac {16 x^2}{25}}+3^x e^{29 x/5} x^x\right )^2} \, dx}{x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.17, size = 45, normalized size = 1.61 \begin {gather*} -\frac {e^{5+\frac {16 x^2}{25}} x}{28 e^{5+\frac {16 x^2}{25}}-3^x e^{29 x/5} x^x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-700 + E^((-125 + 145*x - 16*x^2 + 25*x*Log[3*x])/25)*(25 - 170*x + 32*x^2 - 25*x*Log[3*x]))/(19600
- 1400*E^((-125 + 145*x - 16*x^2 + 25*x*Log[3*x])/25) + 25*E^((2*(-125 + 145*x - 16*x^2 + 25*x*Log[3*x]))/25)
),x]

[Out]

-((E^(5 + (16*x^2)/25)*x)/(28*E^(5 + (16*x^2)/25) - 3^x*E^((29*x)/5)*x^x))

________________________________________________________________________________________

fricas [A]  time = 2.44, size = 23, normalized size = 0.82 \begin {gather*} \frac {x}{e^{\left (-\frac {16}{25} \, x^{2} + x \log \left (3 \, x\right ) + \frac {29}{5} \, x - 5\right )} - 28} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x*log(3*x)+32*x^2-170*x+25)*exp(x*log(3*x)-16/25*x^2+29/5*x-5)-700)/(25*exp(x*log(3*x)-16/25*x
^2+29/5*x-5)^2-1400*exp(x*log(3*x)-16/25*x^2+29/5*x-5)+19600),x, algorithm="fricas")

[Out]

x/(e^(-16/25*x^2 + x*log(3*x) + 29/5*x - 5) - 28)

________________________________________________________________________________________

giac [A]  time = 0.44, size = 30, normalized size = 1.07 \begin {gather*} -\frac {x e^{5}}{28 \, e^{5} - e^{\left (-\frac {16}{25} \, x^{2} + x \log \left (3 \, x\right ) + \frac {29}{5} \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x*log(3*x)+32*x^2-170*x+25)*exp(x*log(3*x)-16/25*x^2+29/5*x-5)-700)/(25*exp(x*log(3*x)-16/25*x
^2+29/5*x-5)^2-1400*exp(x*log(3*x)-16/25*x^2+29/5*x-5)+19600),x, algorithm="giac")

[Out]

-x*e^5/(28*e^5 - e^(-16/25*x^2 + x*log(3*x) + 29/5*x))

________________________________________________________________________________________

maple [A]  time = 0.06, size = 24, normalized size = 0.86

method	result	size
norman	\(\frac {x}{{\mathrm e}^{x \ln \left (3 x \right )-\frac {16 x^{2}}{25}+\frac {29 x}{5}-5}-28}\)	\(24\)
risch	\(\frac {x}{\left (3 x \right )^{x} {\mathrm e}^{-5-\frac {16}{25} x^{2}+\frac {29}{5} x}-28}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-25*x*ln(3*x)+32*x^2-170*x+25)*exp(x*ln(3*x)-16/25*x^2+29/5*x-5)-700)/(25*exp(x*ln(3*x)-16/25*x^2+29/5*x
-5)^2-1400*exp(x*ln(3*x)-16/25*x^2+29/5*x-5)+19600),x,method=_RETURNVERBOSE)

[Out]

x/(exp(x*ln(3*x)-16/25*x^2+29/5*x-5)-28)

________________________________________________________________________________________

maxima [A]  time = 0.74, size = 39, normalized size = 1.39 \begin {gather*} -\frac {x e^{\left (\frac {16}{25} \, x^{2} + 5\right )}}{28 \, e^{\left (\frac {16}{25} \, x^{2} + 5\right )} - e^{\left (x \log \relax (3) + x \log \relax (x) + \frac {29}{5} \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x*log(3*x)+32*x^2-170*x+25)*exp(x*log(3*x)-16/25*x^2+29/5*x-5)-700)/(25*exp(x*log(3*x)-16/25*x
^2+29/5*x-5)^2-1400*exp(x*log(3*x)-16/25*x^2+29/5*x-5)+19600),x, algorithm="maxima")

[Out]

-x*e^(16/25*x^2 + 5)/(28*e^(16/25*x^2 + 5) - e^(x*log(3) + x*log(x) + 29/5*x))

________________________________________________________________________________________

mupad [B]  time = 8.08, size = 51, normalized size = 1.82 \begin {gather*} \frac {170\,x+25\,x\,\ln \left (3\,x\right )-32\,x^2}{\left ({\mathrm {e}}^{-\frac {16\,x^2}{25}+\frac {29\,x}{5}-5}\,{\left (3\,x\right )}^x-28\right )\,\left (25\,\ln \left (3\,x\right )-32\,x+170\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((29*x)/5 + x*log(3*x) - (16*x^2)/25 - 5)*(170*x + 25*x*log(3*x) - 32*x^2 - 25) + 700)/(25*exp((58*x)
/5 + 2*x*log(3*x) - (32*x^2)/25 - 10) - 1400*exp((29*x)/5 + x*log(3*x) - (16*x^2)/25 - 5) + 19600),x)

[Out]

(170*x + 25*x*log(3*x) - 32*x^2)/((exp((29*x)/5 - (16*x^2)/25 - 5)*(3*x)^x - 28)*(25*log(3*x) - 32*x + 170))

________________________________________________________________________________________

sympy [A]  time = 0.40, size = 24, normalized size = 0.86 \begin {gather*} \frac {x}{e^{- \frac {16 x^{2}}{25} + x \log {\left (3 x \right )} + \frac {29 x}{5} - 5} - 28} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x*ln(3*x)+32*x**2-170*x+25)*exp(x*ln(3*x)-16/25*x**2+29/5*x-5)-700)/(25*exp(x*ln(3*x)-16/25*x*
*2+29/5*x-5)**2-1400*exp(x*ln(3*x)-16/25*x**2+29/5*x-5)+19600),x)

[Out]

x/(exp(-16*x**2/25 + x*log(3*x) + 29*x/5 - 5) - 28)

________________________________________________________________________________________