### 3.82 $$\int \frac{x^8 (4+x^2+3 x^4+5 x^6)}{(2+3 x^2+x^4)^2} \, dx$$

Optimal. Leaf size=70 $\frac{5 x^7}{7}-\frac{27 x^5}{5}+\frac{98 x^3}{3}-\frac{\left (207 x^2+206\right ) x}{2 \left (x^4+3 x^2+2\right )}-293 x+\frac{9}{2} \tan ^{-1}(x)+340 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )$

[Out]

-293*x + (98*x^3)/3 - (27*x^5)/5 + (5*x^7)/7 - (x*(206 + 207*x^2))/(2*(2 + 3*x^2 + x^4)) + (9*ArcTan[x])/2 + 3
40*Sqrt[2]*ArcTan[x/Sqrt[2]]

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Rubi [A]  time = 0.0845314, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.129, Rules used = {1668, 1676, 1166, 203} $\frac{5 x^7}{7}-\frac{27 x^5}{5}+\frac{98 x^3}{3}-\frac{\left (207 x^2+206\right ) x}{2 \left (x^4+3 x^2+2\right )}-293 x+\frac{9}{2} \tan ^{-1}(x)+340 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]

[Out]

-293*x + (98*x^3)/3 - (27*x^5)/5 + (5*x^7)/7 - (x*(206 + 207*x^2))/(2*(2 + 3*x^2 + x^4)) + (9*ArcTan[x])/2 + 3
40*Sqrt[2]*ArcTan[x/Sqrt[2]]

Rule 1668

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainde
r[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, S
imp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)
), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a
*c)*PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p +
7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x^2], 1] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[m/2, 0]

Rule 1676

Int[(Pq_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^2 + c*x^4), x], x
] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx &=-\frac{x \left (206+207 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \frac{-412-6 x^2+212 x^4-108 x^6+48 x^8-20 x^{10}}{2+3 x^2+x^4} \, dx\\ &=-\frac{x \left (206+207 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \left (1172-392 x^2+108 x^4-20 x^6-\frac{2 \left (1378+1369 x^2\right )}{2+3 x^2+x^4}\right ) \, dx\\ &=-293 x+\frac{98 x^3}{3}-\frac{27 x^5}{5}+\frac{5 x^7}{7}-\frac{x \left (206+207 x^2\right )}{2 \left (2+3 x^2+x^4\right )}+\frac{1}{2} \int \frac{1378+1369 x^2}{2+3 x^2+x^4} \, dx\\ &=-293 x+\frac{98 x^3}{3}-\frac{27 x^5}{5}+\frac{5 x^7}{7}-\frac{x \left (206+207 x^2\right )}{2 \left (2+3 x^2+x^4\right )}+\frac{9}{2} \int \frac{1}{1+x^2} \, dx+680 \int \frac{1}{2+x^2} \, dx\\ &=-293 x+\frac{98 x^3}{3}-\frac{27 x^5}{5}+\frac{5 x^7}{7}-\frac{x \left (206+207 x^2\right )}{2 \left (2+3 x^2+x^4\right )}+\frac{9}{2} \tan ^{-1}(x)+340 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.045937, size = 71, normalized size = 1.01 $\frac{5 x^7}{7}-\frac{27 x^5}{5}+\frac{98 x^3}{3}+\frac{-207 x^3-206 x}{2 \left (x^4+3 x^2+2\right )}-293 x+\frac{9}{2} \tan ^{-1}(x)+340 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]

[Out]

-293*x + (98*x^3)/3 - (27*x^5)/5 + (5*x^7)/7 + (-206*x - 207*x^3)/(2*(2 + 3*x^2 + x^4)) + (9*ArcTan[x])/2 + 34
0*Sqrt[2]*ArcTan[x/Sqrt[2]]

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Maple [A]  time = 0.013, size = 56, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{7}}{7}}-{\frac{27\,{x}^{5}}{5}}+{\frac{98\,{x}^{3}}{3}}-293\,x-104\,{\frac{x}{{x}^{2}+2}}+340\,\arctan \left ( 1/2\,x\sqrt{2} \right ) \sqrt{2}+{\frac{x}{2\,{x}^{2}+2}}+{\frac{9\,\arctan \left ( x \right ) }{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(5*x^6+3*x^4+x^2+4)/(x^4+3*x^2+2)^2,x)

[Out]

5/7*x^7-27/5*x^5+98/3*x^3-293*x-104*x/(x^2+2)+340*arctan(1/2*x*2^(1/2))*2^(1/2)+1/2*x/(x^2+1)+9/2*arctan(x)

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Maxima [A]  time = 1.46827, size = 78, normalized size = 1.11 \begin{align*} \frac{5}{7} \, x^{7} - \frac{27}{5} \, x^{5} + \frac{98}{3} \, x^{3} + 340 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 293 \, x - \frac{207 \, x^{3} + 206 \, x}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac{9}{2} \, \arctan \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(5*x^6+3*x^4+x^2+4)/(x^4+3*x^2+2)^2,x, algorithm="maxima")

[Out]

5/7*x^7 - 27/5*x^5 + 98/3*x^3 + 340*sqrt(2)*arctan(1/2*sqrt(2)*x) - 293*x - 1/2*(207*x^3 + 206*x)/(x^4 + 3*x^2
+ 2) + 9/2*arctan(x)

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Fricas [A]  time = 1.84306, size = 247, normalized size = 3.53 \begin{align*} \frac{150 \, x^{11} - 684 \, x^{9} + 3758 \, x^{7} - 43218 \, x^{5} - 192605 \, x^{3} + 71400 \, \sqrt{2}{\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 945 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (x\right ) - 144690 \, x}{210 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(5*x^6+3*x^4+x^2+4)/(x^4+3*x^2+2)^2,x, algorithm="fricas")

[Out]

1/210*(150*x^11 - 684*x^9 + 3758*x^7 - 43218*x^5 - 192605*x^3 + 71400*sqrt(2)*(x^4 + 3*x^2 + 2)*arctan(1/2*sqr
t(2)*x) + 945*(x^4 + 3*x^2 + 2)*arctan(x) - 144690*x)/(x^4 + 3*x^2 + 2)

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Sympy [A]  time = 0.186366, size = 66, normalized size = 0.94 \begin{align*} \frac{5 x^{7}}{7} - \frac{27 x^{5}}{5} + \frac{98 x^{3}}{3} - 293 x - \frac{207 x^{3} + 206 x}{2 x^{4} + 6 x^{2} + 4} + \frac{9 \operatorname{atan}{\left (x \right )}}{2} + 340 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(5*x**6+3*x**4+x**2+4)/(x**4+3*x**2+2)**2,x)

[Out]

5*x**7/7 - 27*x**5/5 + 98*x**3/3 - 293*x - (207*x**3 + 206*x)/(2*x**4 + 6*x**2 + 4) + 9*atan(x)/2 + 340*sqrt(2
)*atan(sqrt(2)*x/2)

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Giac [A]  time = 1.08255, size = 78, normalized size = 1.11 \begin{align*} \frac{5}{7} \, x^{7} - \frac{27}{5} \, x^{5} + \frac{98}{3} \, x^{3} + 340 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 293 \, x - \frac{207 \, x^{3} + 206 \, x}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac{9}{2} \, \arctan \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(5*x^6+3*x^4+x^2+4)/(x^4+3*x^2+2)^2,x, algorithm="giac")

[Out]

5/7*x^7 - 27/5*x^5 + 98/3*x^3 + 340*sqrt(2)*arctan(1/2*sqrt(2)*x) - 293*x - 1/2*(207*x^3 + 206*x)/(x^4 + 3*x^2
+ 2) + 9/2*arctan(x)