\(\int \frac {2+50 e^6 x^2+600 e^3 x^4+1350 x^6}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx\) [113]

Optimal result

Integrand size = 56, antiderivative size = 23 \[ \int \frac {2+50 e^6 x^2+600 e^3 x^4+1350 x^6}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx=\log \left (3+25 x^2 \left (e^3+3 x^2\right )^2+2 \log (x)\right ) \]

[Out]

ln(2*ln(x)+25*x^2*(3*x^2+exp(3))^2+3)

Rubi [F]

\[ \int \frac {2+50 e^6 x^2+600 e^3 x^4+1350 x^6}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx=\int \frac {2+50 e^6 x^2+600 e^3 x^4+1350 x^6}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx \]

[In]

Int[(2 + 50*E^6*x^2 + 600*E^3*x^4 + 1350*x^6)/(3*x + 25*E^6*x^3 + 150*E^3*x^5 + 225*x^7 + 2*x*Log[x]),x]

[Out]

2*Defer[Int][1/(x*(3 + 25*E^6*x^2 + 150*E^3*x^4 + 225*x^6 + 2*Log[x])), x] + 50*E^6*Defer[Int][x/(3 + 25*E^6*x
^2 + 150*E^3*x^4 + 225*x^6 + 2*Log[x]), x] + 600*E^3*Defer[Int][x^3/(3 + 25*E^6*x^2 + 150*E^3*x^4 + 225*x^6 +
2*Log[x]), x] + 1350*Defer[Int][x^5/(3 + 25*E^6*x^2 + 150*E^3*x^4 + 225*x^6 + 2*Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (1+25 e^6 x^2+300 e^3 x^4+675 x^6\right )}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx \\ & = 2 \int \frac {1+25 e^6 x^2+300 e^3 x^4+675 x^6}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx \\ & = 2 \int \left (\frac {1}{x \left (3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)\right )}+\frac {25 e^6 x}{3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)}+\frac {300 e^3 x^3}{3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)}+\frac {675 x^5}{3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)}\right ) \, dx \\ & = 2 \int \frac {1}{x \left (3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)\right )} \, dx+1350 \int \frac {x^5}{3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)} \, dx+\left (600 e^3\right ) \int \frac {x^3}{3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)} \, dx+\left (50 e^6\right ) \int \frac {x}{3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {2+50 e^6 x^2+600 e^3 x^4+1350 x^6}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx=\log \left (3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)\right ) \]

[In]

Integrate[(2 + 50*E^6*x^2 + 600*E^3*x^4 + 1350*x^6)/(3*x + 25*E^6*x^3 + 150*E^3*x^5 + 225*x^7 + 2*x*Log[x]),x]

[Out]

Log[3 + 25*E^6*x^2 + 150*E^3*x^4 + 225*x^6 + 2*Log[x]]

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09

method	result	size
risch	\(\ln \left (\frac {225 x^{6}}{2}+75 x^{4} {\mathrm e}^{3}+\frac {25 x^{2} {\mathrm e}^{6}}{2}+\ln \left (x \right )+\frac {3}{2}\right )\)	\(25\)
parallelrisch	\(\ln \left (x^{6}+\frac {2 x^{4} {\mathrm e}^{3}}{3}+\frac {x^{2} {\mathrm e}^{6}}{9}+\frac {2 \ln \left (x \right )}{225}+\frac {1}{75}\right )\)	\(27\)
default	\(\ln \left (225 x^{6}+150 x^{4} {\mathrm e}^{3}+25 x^{2} {\mathrm e}^{6}+2 \ln \left (x \right )+3\right )\)	\(29\)
norman	\(\ln \left (225 x^{6}+150 x^{4} {\mathrm e}^{3}+25 x^{2} {\mathrm e}^{6}+2 \ln \left (x \right )+3\right )\)	\(29\)

[In]

int((50*x^2*exp(3)^2+600*x^4*exp(3)+1350*x^6+2)/(2*x*ln(x)+25*x^3*exp(3)^2+150*x^5*exp(3)+225*x^7+3*x),x,metho
d=_RETURNVERBOSE)

[Out]

ln(225/2*x^6+75*x^4*exp(3)+25/2*x^2*exp(6)+ln(x)+3/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {2+50 e^6 x^2+600 e^3 x^4+1350 x^6}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx=\log \left (225 \, x^{6} + 150 \, x^{4} e^{3} + 25 \, x^{2} e^{6} + 2 \, \log \left (x\right ) + 3\right ) \]

[In]

integrate((50*x^2*exp(3)^2+600*x^4*exp(3)+1350*x^6+2)/(2*x*log(x)+25*x^3*exp(3)^2+150*x^5*exp(3)+225*x^7+3*x),
x, algorithm="fricas")

[Out]

log(225*x^6 + 150*x^4*e^3 + 25*x^2*e^6 + 2*log(x) + 3)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {2+50 e^6 x^2+600 e^3 x^4+1350 x^6}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx=\log {\left (\frac {225 x^{6}}{2} + 75 x^{4} e^{3} + \frac {25 x^{2} e^{6}}{2} + \log {\left (x \right )} + \frac {3}{2} \right )} \]

[In]

integrate((50*x**2*exp(3)**2+600*x**4*exp(3)+1350*x**6+2)/(2*x*ln(x)+25*x**3*exp(3)**2+150*x**5*exp(3)+225*x**
7+3*x),x)

[Out]

log(225*x**6/2 + 75*x**4*exp(3) + 25*x**2*exp(6)/2 + log(x) + 3/2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {2+50 e^6 x^2+600 e^3 x^4+1350 x^6}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx=\log \left (\frac {225}{2} \, x^{6} + 75 \, x^{4} e^{3} + \frac {25}{2} \, x^{2} e^{6} + \log \left (x\right ) + \frac {3}{2}\right ) \]

[In]

integrate((50*x^2*exp(3)^2+600*x^4*exp(3)+1350*x^6+2)/(2*x*log(x)+25*x^3*exp(3)^2+150*x^5*exp(3)+225*x^7+3*x),
x, algorithm="maxima")

[Out]

log(225/2*x^6 + 75*x^4*e^3 + 25/2*x^2*e^6 + log(x) + 3/2)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {2+50 e^6 x^2+600 e^3 x^4+1350 x^6}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx=\log \left (225 \, x^{6} + 150 \, x^{4} e^{3} + 25 \, x^{2} e^{6} + 2 \, \log \left (x\right ) + 3\right ) \]

[In]

integrate((50*x^2*exp(3)^2+600*x^4*exp(3)+1350*x^6+2)/(2*x*log(x)+25*x^3*exp(3)^2+150*x^5*exp(3)+225*x^7+3*x),
x, algorithm="giac")

[Out]

log(225*x^6 + 150*x^4*e^3 + 25*x^2*e^6 + 2*log(x) + 3)

Mupad [B] (verification not implemented)

Time = 8.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {2+50 e^6 x^2+600 e^3 x^4+1350 x^6}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx=\ln \left (\ln \left (x\right )+75\,x^4\,{\mathrm {e}}^3+\frac {25\,x^2\,{\mathrm {e}}^6}{2}+\frac {225\,x^6}{2}+\frac {3}{2}\right ) \]

[In]

int((600*x^4*exp(3) + 50*x^2*exp(6) + 1350*x^6 + 2)/(3*x + 150*x^5*exp(3) + 25*x^3*exp(6) + 2*x*log(x) + 225*x
^7),x)

[Out]

log(log(x) + 75*x^4*exp(3) + (25*x^2*exp(6))/2 + (225*x^6)/2 + 3/2)