\(\int \frac {-9+x^2+4 x^3+e^{x^2} (8 x^3+8 x^5)}{9 x+9 x^2+x^3+2 x^4+4 e^{x^2} x^4} \, dx\) [6799]

Optimal result

Integrand size = 57, antiderivative size = 22 \[ \int \frac {-9+x^2+4 x^3+e^{x^2} \left (8 x^3+8 x^5\right )}{9 x+9 x^2+x^3+2 x^4+4 e^{x^2} x^4} \, dx=\log \left (9+\frac {9}{x}+x+\left (2+4 e^{x^2}\right ) x^2\right ) \]

[Out]

ln(9/x+x+9+(2+4*exp(x^2))*x^2)

Rubi [F]

\[ \int \frac {-9+x^2+4 x^3+e^{x^2} \left (8 x^3+8 x^5\right )}{9 x+9 x^2+x^3+2 x^4+4 e^{x^2} x^4} \, dx=\int \frac {-9+x^2+4 x^3+e^{x^2} \left (8 x^3+8 x^5\right )}{9 x+9 x^2+x^3+2 x^4+4 e^{x^2} x^4} \, dx \]

[In]

Int[(-9 + x^2 + 4*x^3 + E^x^2*(8*x^3 + 8*x^5))/(9*x + 9*x^2 + x^3 + 2*x^4 + 4*E^x^2*x^4),x]

[Out]

x^2 + 2*Log[x] - 18*Defer[Int][(9 + 9*x + x^2 + 2*x^3 + 4*E^x^2*x^3)^(-1), x] - 27*Defer[Int][1/(x*(9 + 9*x +
x^2 + 2*x^3 + 4*E^x^2*x^3)), x] - 19*Defer[Int][x/(9 + 9*x + x^2 + 2*x^3 + 4*E^x^2*x^3), x] - 18*Defer[Int][x^
2/(9 + 9*x + x^2 + 2*x^3 + 4*E^x^2*x^3), x] - 2*Defer[Int][x^3/(9 + 9*x + x^2 + 2*x^3 + 4*E^x^2*x^3), x] - 4*D
efer[Int][x^4/(9 + 9*x + x^2 + 2*x^3 + 4*E^x^2*x^3), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 \left (1+x^2\right )}{x}-\frac {27+18 x+19 x^2+18 x^3+2 x^4+4 x^5}{x \left (9+9 x+x^2+2 x^3+4 e^{x^2} x^3\right )}\right ) \, dx \\ & = 2 \int \frac {1+x^2}{x} \, dx-\int \frac {27+18 x+19 x^2+18 x^3+2 x^4+4 x^5}{x \left (9+9 x+x^2+2 x^3+4 e^{x^2} x^3\right )} \, dx \\ & = 2 \int \left (\frac {1}{x}+x\right ) \, dx-\int \left (\frac {18}{9+9 x+x^2+2 x^3+4 e^{x^2} x^3}+\frac {27}{x \left (9+9 x+x^2+2 x^3+4 e^{x^2} x^3\right )}+\frac {19 x}{9+9 x+x^2+2 x^3+4 e^{x^2} x^3}+\frac {18 x^2}{9+9 x+x^2+2 x^3+4 e^{x^2} x^3}+\frac {2 x^3}{9+9 x+x^2+2 x^3+4 e^{x^2} x^3}+\frac {4 x^4}{9+9 x+x^2+2 x^3+4 e^{x^2} x^3}\right ) \, dx \\ & = x^2+2 \log (x)-2 \int \frac {x^3}{9+9 x+x^2+2 x^3+4 e^{x^2} x^3} \, dx-4 \int \frac {x^4}{9+9 x+x^2+2 x^3+4 e^{x^2} x^3} \, dx-18 \int \frac {1}{9+9 x+x^2+2 x^3+4 e^{x^2} x^3} \, dx-18 \int \frac {x^2}{9+9 x+x^2+2 x^3+4 e^{x^2} x^3} \, dx-19 \int \frac {x}{9+9 x+x^2+2 x^3+4 e^{x^2} x^3} \, dx-27 \int \frac {1}{x \left (9+9 x+x^2+2 x^3+4 e^{x^2} x^3\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.59 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {-9+x^2+4 x^3+e^{x^2} \left (8 x^3+8 x^5\right )}{9 x+9 x^2+x^3+2 x^4+4 e^{x^2} x^4} \, dx=-\log (x)+\log \left (9+9 x+x^2+2 x^3+4 e^{x^2} x^3\right ) \]

[In]

Integrate[(-9 + x^2 + 4*x^3 + E^x^2*(8*x^3 + 8*x^5))/(9*x + 9*x^2 + x^3 + 2*x^4 + 4*E^x^2*x^4),x]

[Out]

-Log[x] + Log[9 + 9*x + x^2 + 2*x^3 + 4*E^x^2*x^3]

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32

method	result	size
norman	\(-\ln \left (x \right )+\ln \left (4 x^{3} {\mathrm e}^{x^{2}}+2 x^{3}+x^{2}+9 x +9\right )\)	\(29\)
risch	\(2 \ln \left (x \right )+\ln \left ({\mathrm e}^{x^{2}}+\frac {2 x^{3}+x^{2}+9 x +9}{4 x^{3}}\right )\)	\(30\)
parallelrisch	\(-\ln \left (x \right )+\ln \left (x^{3} {\mathrm e}^{x^{2}}+\frac {x^{3}}{2}+\frac {x^{2}}{4}+\frac {9 x}{4}+\frac {9}{4}\right )\)	\(30\)

[In]

int(((8*x^5+8*x^3)*exp(x^2)+4*x^3+x^2-9)/(4*x^4*exp(x^2)+2*x^4+x^3+9*x^2+9*x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+ln(4*x^3*exp(x^2)+2*x^3+x^2+9*x+9)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {-9+x^2+4 x^3+e^{x^2} \left (8 x^3+8 x^5\right )}{9 x+9 x^2+x^3+2 x^4+4 e^{x^2} x^4} \, dx=2 \, \log \left (x\right ) + \log \left (\frac {4 \, x^{3} e^{\left (x^{2}\right )} + 2 \, x^{3} + x^{2} + 9 \, x + 9}{x^{3}}\right ) \]

[In]

integrate(((8*x^5+8*x^3)*exp(x^2)+4*x^3+x^2-9)/(4*x^4*exp(x^2)+2*x^4+x^3+9*x^2+9*x),x, algorithm="fricas")

[Out]

2*log(x) + log((4*x^3*e^(x^2) + 2*x^3 + x^2 + 9*x + 9)/x^3)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {-9+x^2+4 x^3+e^{x^2} \left (8 x^3+8 x^5\right )}{9 x+9 x^2+x^3+2 x^4+4 e^{x^2} x^4} \, dx=2 \log {\left (x \right )} + \log {\left (e^{x^{2}} + \frac {2 x^{3} + x^{2} + 9 x + 9}{4 x^{3}} \right )} \]

[In]

integrate(((8*x**5+8*x**3)*exp(x**2)+4*x**3+x**2-9)/(4*x**4*exp(x**2)+2*x**4+x**3+9*x**2+9*x),x)

[Out]

2*log(x) + log(exp(x**2) + (2*x**3 + x**2 + 9*x + 9)/(4*x**3))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {-9+x^2+4 x^3+e^{x^2} \left (8 x^3+8 x^5\right )}{9 x+9 x^2+x^3+2 x^4+4 e^{x^2} x^4} \, dx=2 \, \log \left (x\right ) + \log \left (\frac {4 \, x^{3} e^{\left (x^{2}\right )} + 2 \, x^{3} + x^{2} + 9 \, x + 9}{4 \, x^{3}}\right ) \]

[In]

integrate(((8*x^5+8*x^3)*exp(x^2)+4*x^3+x^2-9)/(4*x^4*exp(x^2)+2*x^4+x^3+9*x^2+9*x),x, algorithm="maxima")

[Out]

2*log(x) + log(1/4*(4*x^3*e^(x^2) + 2*x^3 + x^2 + 9*x + 9)/x^3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {-9+x^2+4 x^3+e^{x^2} \left (8 x^3+8 x^5\right )}{9 x+9 x^2+x^3+2 x^4+4 e^{x^2} x^4} \, dx=\log \left (4 \, x^{3} e^{\left (x^{2}\right )} + 2 \, x^{3} + x^{2} + 9 \, x + 9\right ) - \log \left (x\right ) \]

[In]

integrate(((8*x^5+8*x^3)*exp(x^2)+4*x^3+x^2-9)/(4*x^4*exp(x^2)+2*x^4+x^3+9*x^2+9*x),x, algorithm="giac")

[Out]

log(4*x^3*e^(x^2) + 2*x^3 + x^2 + 9*x + 9) - log(x)

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {-9+x^2+4 x^3+e^{x^2} \left (8 x^3+8 x^5\right )}{9 x+9 x^2+x^3+2 x^4+4 e^{x^2} x^4} \, dx=\ln \left (9\,x+4\,x^3\,{\mathrm {e}}^{x^2}+x^2+2\,x^3+9\right )-\ln \left (x\right ) \]

[In]

int((exp(x^2)*(8*x^3 + 8*x^5) + x^2 + 4*x^3 - 9)/(9*x + 4*x^4*exp(x^2) + 9*x^2 + x^3 + 2*x^4),x)

[Out]

log(9*x + 4*x^3*exp(x^2) + x^2 + 2*x^3 + 9) - log(x)