\(\int \frac {e^{\frac {2+30 x^2+15 x^3+x^5}{15 x^2}} (-4-30 x^2+15 x^3+3 x^5)}{15 x^5} \, dx\) [5583]

Optimal result

Integrand size = 48, antiderivative size = 21 \[ \int \frac {e^{\frac {2+30 x^2+15 x^3+x^5}{15 x^2}} \left (-4-30 x^2+15 x^3+3 x^5\right )}{15 x^5} \, dx=\frac {e^{2+x+\frac {2+x^5}{15 x^2}}}{x^2} \]

[Out]

exp(2+1/15*(x^5+2)/x^2+x)/x^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(21)=42\).

Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.86, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {12, 2326} \[ \int \frac {e^{\frac {2+30 x^2+15 x^3+x^5}{15 x^2}} \left (-4-30 x^2+15 x^3+3 x^5\right )}{15 x^5} \, dx=-\frac {e^{\frac {x^5+15 x^3+30 x^2+2}{15 x^2}} \left (-3 x^5-15 x^3+4\right )}{x^5 \left (\frac {5 \left (x^4+9 x^2+12 x\right )}{x^2}-\frac {2 \left (x^5+15 x^3+30 x^2+2\right )}{x^3}\right )} \]

[In]

Int[(E^((2 + 30*x^2 + 15*x^3 + x^5)/(15*x^2))*(-4 - 30*x^2 + 15*x^3 + 3*x^5))/(15*x^5),x]

[Out]

-((E^((2 + 30*x^2 + 15*x^3 + x^5)/(15*x^2))*(4 - 15*x^3 - 3*x^5))/(x^5*((5*(12*x + 9*x^2 + x^4))/x^2 - (2*(2 +
30*x^2 + 15*x^3 + x^5))/x^3)))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{15} \int \frac {e^{\frac {2+30 x^2+15 x^3+x^5}{15 x^2}} \left (-4-30 x^2+15 x^3+3 x^5\right )}{x^5} \, dx \\ & = -\frac {e^{\frac {2+30 x^2+15 x^3+x^5}{15 x^2}} \left (4-15 x^3-3 x^5\right )}{x^5 \left (\frac {5 \left (12 x+9 x^2+x^4\right )}{x^2}-\frac {2 \left (2+30 x^2+15 x^3+x^5\right )}{x^3}\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {e^{\frac {2+30 x^2+15 x^3+x^5}{15 x^2}} \left (-4-30 x^2+15 x^3+3 x^5\right )}{15 x^5} \, dx=\frac {e^{2+\frac {2}{15 x^2}+x+\frac {x^3}{15}}}{x^2} \]

[In]

Integrate[(E^((2 + 30*x^2 + 15*x^3 + x^5)/(15*x^2))*(-4 - 30*x^2 + 15*x^3 + 3*x^5))/(15*x^5),x]

[Out]

E^(2 + 2/(15*x^2) + x + x^3/15)/x^2

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24

method	result	size
gosper	\(\frac {{\mathrm e}^{\frac {x^{5}+15 x^{3}+30 x^{2}+2}{15 x^{2}}}}{x^{2}}\)	\(26\)
norman	\(\frac {{\mathrm e}^{\frac {x^{5}+15 x^{3}+30 x^{2}+2}{15 x^{2}}}}{x^{2}}\)	\(26\)
risch	\(\frac {{\mathrm e}^{\frac {x^{5}+15 x^{3}+30 x^{2}+2}{15 x^{2}}}}{x^{2}}\)	\(26\)
parallelrisch	\(\frac {{\mathrm e}^{\frac {x^{5}+15 x^{3}+30 x^{2}+2}{15 x^{2}}}}{x^{2}}\)	\(26\)

[In]

int(1/15*(3*x^5+15*x^3-30*x^2-4)*exp(1/15*(x^5+15*x^3+30*x^2+2)/x^2)/x^5,x,method=_RETURNVERBOSE)

[Out]

1/x^2*exp(1/15*(x^5+15*x^3+30*x^2+2)/x^2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\frac {2+30 x^2+15 x^3+x^5}{15 x^2}} \left (-4-30 x^2+15 x^3+3 x^5\right )}{15 x^5} \, dx=\frac {e^{\left (\frac {x^{5} + 15 \, x^{3} + 30 \, x^{2} + 2}{15 \, x^{2}}\right )}}{x^{2}} \]

[In]

integrate(1/15*(3*x^5+15*x^3-30*x^2-4)*exp(1/15*(x^5+15*x^3+30*x^2+2)/x^2)/x^5,x, algorithm="fricas")

[Out]

e^(1/15*(x^5 + 15*x^3 + 30*x^2 + 2)/x^2)/x^2

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {e^{\frac {2+30 x^2+15 x^3+x^5}{15 x^2}} \left (-4-30 x^2+15 x^3+3 x^5\right )}{15 x^5} \, dx=\frac {e^{\frac {\frac {x^{5}}{15} + x^{3} + 2 x^{2} + \frac {2}{15}}{x^{2}}}}{x^{2}} \]

[In]

integrate(1/15*(3*x**5+15*x**3-30*x**2-4)*exp(1/15*(x**5+15*x**3+30*x**2+2)/x**2)/x**5,x)

[Out]

exp((x**5/15 + x**3 + 2*x**2 + 2/15)/x**2)/x**2

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\frac {2+30 x^2+15 x^3+x^5}{15 x^2}} \left (-4-30 x^2+15 x^3+3 x^5\right )}{15 x^5} \, dx=\frac {e^{\left (\frac {1}{15} \, x^{3} + x + \frac {2}{15 \, x^{2}} + 2\right )}}{x^{2}} \]

[In]

integrate(1/15*(3*x^5+15*x^3-30*x^2-4)*exp(1/15*(x^5+15*x^3+30*x^2+2)/x^2)/x^5,x, algorithm="maxima")

[Out]

e^(1/15*x^3 + x + 2/15/x^2 + 2)/x^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\frac {2+30 x^2+15 x^3+x^5}{15 x^2}} \left (-4-30 x^2+15 x^3+3 x^5\right )}{15 x^5} \, dx=\frac {e^{\left (\frac {x^{5} + 15 \, x^{3} + 30 \, x^{2} + 2}{15 \, x^{2}}\right )}}{x^{2}} \]

[In]

integrate(1/15*(3*x^5+15*x^3-30*x^2-4)*exp(1/15*(x^5+15*x^3+30*x^2+2)/x^2)/x^5,x, algorithm="giac")

[Out]

e^(1/15*(x^5 + 15*x^3 + 30*x^2 + 2)/x^2)/x^2

Mupad [B] (verification not implemented)

Time = 10.79 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\frac {2+30 x^2+15 x^3+x^5}{15 x^2}} \left (-4-30 x^2+15 x^3+3 x^5\right )}{15 x^5} \, dx=\frac {{\mathrm {e}}^{x+\frac {2}{15\,x^2}+\frac {x^3}{15}+2}}{x^2} \]

[In]

int(-(exp((2*x^2 + x^3 + x^5/15 + 2/15)/x^2)*(30*x^2 - 15*x^3 - 3*x^5 + 4))/(15*x^5),x)

[Out]

exp(x + 2/(15*x^2) + x^3/15 + 2)/x^2