\(\int \frac {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} (-27-27 x-6 x^2-6 x^3+x^5)}{x^4+2 x^5+x^6} \, dx\) [3245]

Optimal result

Integrand size = 56, antiderivative size = 22 \[ \int \frac {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4+2 x^5+x^6} \, dx=\frac {-2+e^{\frac {\left (3+x^2\right )^2}{x^3}}+x}{1+x} \]

[Out]

(x+exp((x^2+3)^2/x^3)-2)/(1+x)

Rubi [F]

\[ \int \frac {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4+2 x^5+x^6} \, dx=\int \frac {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4+2 x^5+x^6} \, dx \]

[In]

Int[(3*x^4 + E^((9 + 6*x^2 + x^4)/x^3)*(-27 - 27*x - 6*x^2 - 6*x^3 + x^5))/(x^4 + 2*x^5 + x^6),x]

[Out]

-3/(1 + x) - 27*Defer[Int][E^((3 + x^2)^2/x^3)/x^4, x] + 27*Defer[Int][E^((3 + x^2)^2/x^3)/x^3, x] - 33*Defer[
Int][E^((3 + x^2)^2/x^3)/x^2, x] + 33*Defer[Int][E^((3 + x^2)^2/x^3)/x, x] - Defer[Int][E^((3 + x^2)^2/x^3)/(1
+ x)^2, x] - 32*Defer[Int][E^((3 + x^2)^2/x^3)/(1 + x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4 \left (1+2 x+x^2\right )} \, dx \\ & = \int \frac {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4 (1+x)^2} \, dx \\ & = \int \left (\frac {3}{(1+x)^2}+\frac {e^{\frac {\left (3+x^2\right )^2}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4 (1+x)^2}\right ) \, dx \\ & = -\frac {3}{1+x}+\int \frac {e^{\frac {\left (3+x^2\right )^2}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4 (1+x)^2} \, dx \\ & = -\frac {3}{1+x}+\int \left (-\frac {27 e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x^4}+\frac {27 e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x^3}-\frac {33 e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x^2}+\frac {33 e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x}-\frac {e^{\frac {\left (3+x^2\right )^2}{x^3}}}{(1+x)^2}-\frac {32 e^{\frac {\left (3+x^2\right )^2}{x^3}}}{1+x}\right ) \, dx \\ & = -\frac {3}{1+x}-27 \int \frac {e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x^4} \, dx+27 \int \frac {e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x^3} \, dx-32 \int \frac {e^{\frac {\left (3+x^2\right )^2}{x^3}}}{1+x} \, dx-33 \int \frac {e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x^2} \, dx+33 \int \frac {e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x} \, dx-\int \frac {e^{\frac {\left (3+x^2\right )^2}{x^3}}}{(1+x)^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4+2 x^5+x^6} \, dx=\frac {-3+e^{\frac {\left (3+x^2\right )^2}{x^3}}}{1+x} \]

[In]

Integrate[(3*x^4 + E^((9 + 6*x^2 + x^4)/x^3)*(-27 - 27*x - 6*x^2 - 6*x^3 + x^5))/(x^4 + 2*x^5 + x^6),x]

[Out]

(-3 + E^((3 + x^2)^2/x^3))/(1 + x)

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09

method	result	size
parallelrisch	\(\frac {-3+{\mathrm e}^{\frac {x^{4}+6 x^{2}+9}{x^{3}}}}{1+x}\)	\(24\)
risch	\(-\frac {3}{1+x}+\frac {{\mathrm e}^{\frac {\left (x^{2}+3\right )^{2}}{x^{3}}}}{1+x}\)	\(27\)
parts	\(-\frac {3}{1+x}+\frac {{\mathrm e}^{\frac {x^{4}+6 x^{2}+9}{x^{3}}}}{1+x}\)	\(30\)
norman	\(\frac {-3 x^{3}+{\mathrm e}^{\frac {x^{4}+6 x^{2}+9}{x^{3}}} x^{3}}{x^{3} \left (1+x \right )}\)	\(35\)

[In]

int(((x^5-6*x^3-6*x^2-27*x-27)*exp((x^4+6*x^2+9)/x^3)+3*x^4)/(x^6+2*x^5+x^4),x,method=_RETURNVERBOSE)

[Out]

(-3+exp((x^4+6*x^2+9)/x^3))/(1+x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4+2 x^5+x^6} \, dx=\frac {e^{\left (\frac {x^{4} + 6 \, x^{2} + 9}{x^{3}}\right )} - 3}{x + 1} \]

[In]

integrate(((x^5-6*x^3-6*x^2-27*x-27)*exp((x^4+6*x^2+9)/x^3)+3*x^4)/(x^6+2*x^5+x^4),x, algorithm="fricas")

[Out]

(e^((x^4 + 6*x^2 + 9)/x^3) - 3)/(x + 1)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4+2 x^5+x^6} \, dx=\frac {e^{\frac {x^{4} + 6 x^{2} + 9}{x^{3}}}}{x + 1} - \frac {3}{x + 1} \]

[In]

integrate(((x**5-6*x**3-6*x**2-27*x-27)*exp((x**4+6*x**2+9)/x**3)+3*x**4)/(x**6+2*x**5+x**4),x)

[Out]

exp((x**4 + 6*x**2 + 9)/x**3)/(x + 1) - 3/(x + 1)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4+2 x^5+x^6} \, dx=\frac {e^{\left (x + \frac {6}{x} + \frac {9}{x^{3}}\right )}}{x + 1} - \frac {3}{x + 1} \]

[In]

integrate(((x^5-6*x^3-6*x^2-27*x-27)*exp((x^4+6*x^2+9)/x^3)+3*x^4)/(x^6+2*x^5+x^4),x, algorithm="maxima")

[Out]

e^(x + 6/x + 9/x^3)/(x + 1) - 3/(x + 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4+2 x^5+x^6} \, dx=\frac {e^{\left (\frac {x^{4} + 6 \, x^{2} + 9}{x^{3}}\right )} - 3}{x + 1} \]

[In]

integrate(((x^5-6*x^3-6*x^2-27*x-27)*exp((x^4+6*x^2+9)/x^3)+3*x^4)/(x^6+2*x^5+x^4),x, algorithm="giac")

[Out]

(e^((x^4 + 6*x^2 + 9)/x^3) - 3)/(x + 1)

Mupad [B] (verification not implemented)

Time = 9.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4+2 x^5+x^6} \, dx=\frac {3\,x+{\mathrm {e}}^{6/x}\,{\mathrm {e}}^{\frac {9}{x^3}}\,{\mathrm {e}}^x}{x+1} \]

[In]

int(-(exp((6*x^2 + x^4 + 9)/x^3)*(27*x + 6*x^2 + 6*x^3 - x^5 + 27) - 3*x^4)/(x^4 + 2*x^5 + x^6),x)

[Out]

(3*x + exp(6/x)*exp(9/x^3)*exp(x))/(x + 1)