\(\int \frac {36+27 x^2+3 x^4}{80+40 x^2+5 x^4} \, dx\) [5476]

Optimal result

Integrand size = 27, antiderivative size = 30 \[ \int \frac {36+27 x^2+3 x^4}{80+40 x^2+5 x^4} \, dx=\frac {3 \left (3+\left (1+\frac {-x+x^2}{x}\right )^2\right )}{5 \left (\frac {4}{x}+x\right )} \]

[Out]

3*(3+((x^2-x)/x+1)^2)/(5*x+20/x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.60, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {28, 1171, 21, 8} \[ \int \frac {36+27 x^2+3 x^4}{80+40 x^2+5 x^4} \, dx=\frac {3 x}{5}-\frac {3 x}{5 \left (x^2+4\right )} \]

[In]

Int[(36 + 27*x^2 + 3*x^4)/(80 + 40*x^2 + 5*x^4),x]

[Out]

(3*x)/5 - (3*x)/(5*(4 + x^2))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rubi steps \begin{align*} \text {integral}& = 5 \int \frac {36+27 x^2+3 x^4}{\left (20+5 x^2\right )^2} \, dx \\ & = -\frac {3 x}{5 \left (4+x^2\right )}-\frac {1}{8} \int \frac {-96-24 x^2}{20+5 x^2} \, dx \\ & = -\frac {3 x}{5 \left (4+x^2\right )}+\frac {3 \int 1 \, dx}{5} \\ & = \frac {3 x}{5}-\frac {3 x}{5 \left (4+x^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.53 \[ \int \frac {36+27 x^2+3 x^4}{80+40 x^2+5 x^4} \, dx=\frac {3}{5} \left (x-\frac {x}{4+x^2}\right ) \]

[In]

Integrate[(36 + 27*x^2 + 3*x^4)/(80 + 40*x^2 + 5*x^4),x]

[Out]

(3*(x - x/(4 + x^2)))/5

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.50

method	result	size
default	\(\frac {3 x}{5}-\frac {3 x}{5 \left (x^{2}+4\right )}\)	\(15\)
risch	\(\frac {3 x}{5}-\frac {3 x}{5 \left (x^{2}+4\right )}\)	\(15\)
gosper	\(\frac {3 \left (x^{2}+3\right ) x}{5 \left (x^{2}+4\right )}\)	\(16\)
norman	\(\frac {\frac {9}{5} x +\frac {3}{5} x^{3}}{x^{2}+4}\)	\(18\)
parallelrisch	\(\frac {3 x^{3}+9 x}{5 x^{2}+20}\)	\(19\)

[In]

int((3*x^4+27*x^2+36)/(5*x^4+40*x^2+80),x,method=_RETURNVERBOSE)

[Out]

3/5*x-3/5*x/(x^2+4)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.53 \[ \int \frac {36+27 x^2+3 x^4}{80+40 x^2+5 x^4} \, dx=\frac {3 \, {\left (x^{3} + 3 \, x\right )}}{5 \, {\left (x^{2} + 4\right )}} \]

[In]

integrate((3*x^4+27*x^2+36)/(5*x^4+40*x^2+80),x, algorithm="fricas")

[Out]

3/5*(x^3 + 3*x)/(x^2 + 4)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.47 \[ \int \frac {36+27 x^2+3 x^4}{80+40 x^2+5 x^4} \, dx=\frac {3 x}{5} - \frac {3 x}{5 x^{2} + 20} \]

[In]

integrate((3*x**4+27*x**2+36)/(5*x**4+40*x**2+80),x)

[Out]

3*x/5 - 3*x/(5*x**2 + 20)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.47 \[ \int \frac {36+27 x^2+3 x^4}{80+40 x^2+5 x^4} \, dx=\frac {3}{5} \, x - \frac {3 \, x}{5 \, {\left (x^{2} + 4\right )}} \]

[In]

integrate((3*x^4+27*x^2+36)/(5*x^4+40*x^2+80),x, algorithm="maxima")

[Out]

3/5*x - 3/5*x/(x^2 + 4)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.47 \[ \int \frac {36+27 x^2+3 x^4}{80+40 x^2+5 x^4} \, dx=\frac {3}{5} \, x - \frac {3 \, x}{5 \, {\left (x^{2} + 4\right )}} \]

[In]

integrate((3*x^4+27*x^2+36)/(5*x^4+40*x^2+80),x, algorithm="giac")

[Out]

3/5*x - 3/5*x/(x^2 + 4)

Mupad [B] (verification not implemented)

Time = 10.96 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.57 \[ \int \frac {36+27 x^2+3 x^4}{80+40 x^2+5 x^4} \, dx=\frac {3\,x\,\left (x^2+3\right )}{5\,\left (x^2+4\right )} \]

[In]

int((27*x^2 + 3*x^4 + 36)/(40*x^2 + 5*x^4 + 80),x)

[Out]

(3*x*(x^2 + 3))/(5*(x^2 + 4))