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

Optimal. Leaf size=30 \[ \frac {3 \left (3+\left (1+\frac {-x+x^2}{x}\right )^2\right )}{5 \left (\frac {4}{x}+x\right )} \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 0.60, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {28, 1157, 21, 8} \begin {gather*} \frac {3 x}{5}-\frac {3 x}{5 \left (x^2+4\right )} \end {gather*}

Antiderivative was successfully verified.

[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 1157

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] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.53 \begin {gather*} \frac {3}{5} \left (x-\frac {x}{4+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.74, size = 16, normalized size = 0.53 \begin {gather*} \frac {3 \, {\left (x^{3} + 3 \, x\right )}}{5 \, {\left (x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.15, size = 14, normalized size = 0.47 \begin {gather*} \frac {3}{5} \, x - \frac {3 \, x}{5 \, {\left (x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.02, size = 15, normalized size = 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	\(\frac {\frac {9}{5} x +\frac {3}{5} x^{3}}{x^{2}+4}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.37, size = 14, normalized size = 0.47 \begin {gather*} \frac {3}{5} \, x - \frac {3 \, x}{5 \, {\left (x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 3.44, size = 17, normalized size = 0.57 \begin {gather*} \frac {3\,x\,\left (x^2+3\right )}{5\,\left (x^2+4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.08, size = 14, normalized size = 0.47 \begin {gather*} \frac {3 x}{5} - \frac {3 x}{5 x^{2} + 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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