\(\int \frac {e^{256+x} (342 x+171 x^2-81 x^4+81 x^5)}{361+342 x^3+81 x^6} \, dx\) [4182]

Optimal result

Integrand size = 39, antiderivative size = 17 \[ \int \frac {e^{256+x} \left (342 x+171 x^2-81 x^4+81 x^5\right )}{361+342 x^3+81 x^6} \, dx=\frac {e^{256+x}}{\frac {19}{9 x^2}+x} \]

[Out]

exp(256+x)/(x+19/9/x^2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.59, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {28, 2326} \[ \int \frac {e^{256+x} \left (342 x+171 x^2-81 x^4+81 x^5\right )}{361+342 x^3+81 x^6} \, dx=\frac {9 e^{x+256} \left (9 x^5+19 x^2\right )}{\left (9 x^3+19\right )^2} \]

[In]

Int[(E^(256 + x)*(342*x + 171*x^2 - 81*x^4 + 81*x^5))/(361 + 342*x^3 + 81*x^6),x]

[Out]

(9*E^(256 + x)*(19*x^2 + 9*x^5))/(19 + 9*x^3)^2

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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}& = 81 \int \frac {e^{256+x} \left (342 x+171 x^2-81 x^4+81 x^5\right )}{\left (171+81 x^3\right )^2} \, dx \\ & = \frac {9 e^{256+x} \left (19 x^2+9 x^5\right )}{\left (19+9 x^3\right )^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {e^{256+x} \left (342 x+171 x^2-81 x^4+81 x^5\right )}{361+342 x^3+81 x^6} \, dx=\frac {9 e^{256+x} x^2}{19+9 x^3} \]

[In]

Integrate[(E^(256 + x)*(342*x + 171*x^2 - 81*x^4 + 81*x^5))/(361 + 342*x^3 + 81*x^6),x]

[Out]

(9*E^(256 + x)*x^2)/(19 + 9*x^3)

Maple [A] (verified)

Time = 1.62 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12

method	result	size
gosper	\(\frac {9 x^{2} {\mathrm e}^{256+x}}{9 x^{3}+19}\)	\(19\)
norman	\(\frac {9 x^{2} {\mathrm e}^{256+x}}{9 x^{3}+19}\)	\(19\)
risch	\(\frac {9 x^{2} {\mathrm e}^{256+x}}{9 x^{3}+19}\)	\(19\)
parallelrisch	\(\frac {9 x^{2} {\mathrm e}^{256+x}}{9 x^{3}+19}\)	\(19\)
derivativedivides	\(\text {Expression too large to display}\)	\(520\)
default	\(\text {Expression too large to display}\)	\(520\)

[In]

int((81*x^5-81*x^4+171*x^2+342*x)*exp(256+x)/(81*x^6+342*x^3+361),x,method=_RETURNVERBOSE)

[Out]

9*x^2*exp(256+x)/(9*x^3+19)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{256+x} \left (342 x+171 x^2-81 x^4+81 x^5\right )}{361+342 x^3+81 x^6} \, dx=\frac {9 \, x^{2} e^{\left (x + 256\right )}}{9 \, x^{3} + 19} \]

[In]

integrate((81*x^5-81*x^4+171*x^2+342*x)*exp(256+x)/(81*x^6+342*x^3+361),x, algorithm="fricas")

[Out]

9*x^2*e^(x + 256)/(9*x^3 + 19)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {e^{256+x} \left (342 x+171 x^2-81 x^4+81 x^5\right )}{361+342 x^3+81 x^6} \, dx=\frac {9 x^{2} e^{x + 256}}{9 x^{3} + 19} \]

[In]

integrate((81*x**5-81*x**4+171*x**2+342*x)*exp(256+x)/(81*x**6+342*x**3+361),x)

[Out]

9*x**2*exp(x + 256)/(9*x**3 + 19)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{256+x} \left (342 x+171 x^2-81 x^4+81 x^5\right )}{361+342 x^3+81 x^6} \, dx=\frac {9 \, x^{2} e^{\left (x + 256\right )}}{9 \, x^{3} + 19} \]

[In]

integrate((81*x^5-81*x^4+171*x^2+342*x)*exp(256+x)/(81*x^6+342*x^3+361),x, algorithm="maxima")

[Out]

9*x^2*e^(x + 256)/(9*x^3 + 19)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{256+x} \left (342 x+171 x^2-81 x^4+81 x^5\right )}{361+342 x^3+81 x^6} \, dx=\frac {9 \, x^{2} e^{\left (x + 256\right )}}{9 \, x^{3} + 19} \]

[In]

integrate((81*x^5-81*x^4+171*x^2+342*x)*exp(256+x)/(81*x^6+342*x^3+361),x, algorithm="giac")

[Out]

9*x^2*e^(x + 256)/(9*x^3 + 19)

Mupad [B] (verification not implemented)

Time = 10.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{256+x} \left (342 x+171 x^2-81 x^4+81 x^5\right )}{361+342 x^3+81 x^6} \, dx=\frac {9\,x^2\,{\mathrm {e}}^{256}\,{\mathrm {e}}^x}{9\,x^3+19} \]

[In]

int((exp(x + 256)*(342*x + 171*x^2 - 81*x^4 + 81*x^5))/(342*x^3 + 81*x^6 + 361),x)

[Out]

(9*x^2*exp(256)*exp(x))/(9*x^3 + 19)