### 3.92 $$\int \frac{x^8 (4+x^2+3 x^4+5 x^6)}{(2+3 x^2+x^4)^3} \, dx$$

Optimal. Leaf size=80 $\frac{5 x^3}{3}+\frac{\left (24-409 x^2\right ) x}{8 \left (x^4+3 x^2+2\right )}-\frac{\left (207 x^2+206\right ) x}{4 \left (x^4+3 x^2+2\right )^2}-42 x-\frac{449}{8} \tan ^{-1}(x)+\frac{219 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}}$

[Out]

-42*x + (5*x^3)/3 - (x*(206 + 207*x^2))/(4*(2 + 3*x^2 + x^4)^2) + (x*(24 - 409*x^2))/(8*(2 + 3*x^2 + x^4)) - (
449*ArcTan[x])/8 + (219*ArcTan[x/Sqrt[2]])/Sqrt[2]

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Rubi [A]  time = 0.100481, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.161, Rules used = {1668, 1678, 1676, 1166, 203} $\frac{5 x^3}{3}+\frac{\left (24-409 x^2\right ) x}{8 \left (x^4+3 x^2+2\right )}-\frac{\left (207 x^2+206\right ) x}{4 \left (x^4+3 x^2+2\right )^2}-42 x-\frac{449}{8} \tan ^{-1}(x)+\frac{219 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-42*x + (5*x^3)/3 - (x*(206 + 207*x^2))/(4*(2 + 3*x^2 + x^4)^2) + (x*(24 - 409*x^2))/(8*(2 + 3*x^2 + x^4)) - (
449*ArcTan[x])/8 + (219*ArcTan[x/Sqrt[2]])/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 1678

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(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)*PolynomialQuot
ient[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] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

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 )^3} \, dx &=-\frac{x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac{1}{8} \int \frac{-412+1230 x^2+424 x^4-216 x^6+96 x^8-40 x^{10}}{\left (2+3 x^2+x^4\right )^2} \, dx\\ &=-\frac{x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \frac{728+1500 x^2-864 x^4+160 x^6}{2+3 x^2+x^4} \, dx\\ &=-\frac{x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \left (-1344+160 x^2+\frac{4 \left (854+1303 x^2\right )}{2+3 x^2+x^4}\right ) \, dx\\ &=-42 x+\frac{5 x^3}{3}-\frac{x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{1}{8} \int \frac{854+1303 x^2}{2+3 x^2+x^4} \, dx\\ &=-42 x+\frac{5 x^3}{3}-\frac{x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac{449}{8} \int \frac{1}{1+x^2} \, dx+219 \int \frac{1}{2+x^2} \, dx\\ &=-42 x+\frac{5 x^3}{3}-\frac{x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac{449}{8} \tan ^{-1}(x)+\frac{219 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0552934, size = 66, normalized size = 0.82 $\frac{x \left (40 x^{10}-768 x^8-6755 x^6-16233 x^4-15416 x^2-5124\right )}{24 \left (x^4+3 x^2+2\right )^2}-\frac{449}{8} \tan ^{-1}(x)+\frac{219 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*(-5124 - 15416*x^2 - 16233*x^4 - 6755*x^6 - 768*x^8 + 40*x^10))/(24*(2 + 3*x^2 + x^4)^2) - (449*ArcTan[x])/
8 + (219*ArcTan[x/Sqrt[2]])/Sqrt[2]

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Maple [A]  time = 0.011, size = 62, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{3}}{3}}-42\,x+16\,{\frac{1}{ \left ({x}^{2}+2 \right ) ^{2}} \left ( -{\frac{53\,{x}^{3}}{16}}-{\frac{27\,x}{8}} \right ) }+{\frac{219\,\sqrt{2}}{2}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }-{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{15\,{x}^{3}}{8}}-{\frac{17\,x}{8}} \right ) }-{\frac{449\,\arctan \left ( x \right ) }{8}} \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)^3,x)

[Out]

5/3*x^3-42*x+16*(-53/16*x^3-27/8*x)/(x^2+2)^2+219/2*arctan(1/2*x*2^(1/2))*2^(1/2)-(-15/8*x^3-17/8*x)/(x^2+1)^2
-449/8*arctan(x)

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Maxima [A]  time = 1.49531, size = 92, normalized size = 1.15 \begin{align*} \frac{5}{3} \, x^{3} + \frac{219}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 42 \, x - \frac{409 \, x^{7} + 1203 \, x^{5} + 1160 \, x^{3} + 364 \, x}{8 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} - \frac{449}{8} \, \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)^3,x, algorithm="maxima")

[Out]

5/3*x^3 + 219/2*sqrt(2)*arctan(1/2*sqrt(2)*x) - 42*x - 1/8*(409*x^7 + 1203*x^5 + 1160*x^3 + 364*x)/(x^8 + 6*x^
6 + 13*x^4 + 12*x^2 + 4) - 449/8*arctan(x)

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Fricas [A]  time = 1.53826, size = 313, normalized size = 3.91 \begin{align*} \frac{40 \, x^{11} - 768 \, x^{9} - 6755 \, x^{7} - 16233 \, x^{5} - 15416 \, x^{3} + 2628 \, \sqrt{2}{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 1347 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) - 5124 \, x}{24 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\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)^3,x, algorithm="fricas")

[Out]

1/24*(40*x^11 - 768*x^9 - 6755*x^7 - 16233*x^5 - 15416*x^3 + 2628*sqrt(2)*(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)*
arctan(1/2*sqrt(2)*x) - 1347*(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)*arctan(x) - 5124*x)/(x^8 + 6*x^6 + 13*x^4 + 1
2*x^2 + 4)

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Sympy [A]  time = 0.238198, size = 75, normalized size = 0.94 \begin{align*} \frac{5 x^{3}}{3} - 42 x - \frac{409 x^{7} + 1203 x^{5} + 1160 x^{3} + 364 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} - \frac{449 \operatorname{atan}{\left (x \right )}}{8} + \frac{219 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{2} \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)**3,x)

[Out]

5*x**3/3 - 42*x - (409*x**7 + 1203*x**5 + 1160*x**3 + 364*x)/(8*x**8 + 48*x**6 + 104*x**4 + 96*x**2 + 32) - 44
9*atan(x)/8 + 219*sqrt(2)*atan(sqrt(2)*x/2)/2

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Giac [A]  time = 1.11295, size = 78, normalized size = 0.98 \begin{align*} \frac{5}{3} \, x^{3} + \frac{219}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 42 \, x - \frac{409 \, x^{7} + 1203 \, x^{5} + 1160 \, x^{3} + 364 \, x}{8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac{449}{8} \, \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)^3,x, algorithm="giac")

[Out]

5/3*x^3 + 219/2*sqrt(2)*arctan(1/2*sqrt(2)*x) - 42*x - 1/8*(409*x^7 + 1203*x^5 + 1160*x^3 + 364*x)/(x^4 + 3*x^
2 + 2)^2 - 449/8*arctan(x)