### 3.91 $$\int \frac{x^{10} (4+x^2+3 x^4+5 x^6)}{(2+3 x^2+x^4)^3} \, dx$$

Optimal. Leaf size=81 $x^5-14 x^3+\frac{\left (1669 x^2+824\right ) x}{8 \left (x^4+3 x^2+2\right )}+\frac{\left (415 x^2+414\right ) x}{4 \left (x^4+3 x^2+2\right )^2}+214 x+\frac{477}{8} \tan ^{-1}(x)-351 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )$

[Out]

214*x - 14*x^3 + x^5 + (x*(414 + 415*x^2))/(4*(2 + 3*x^2 + x^4)^2) + (x*(824 + 1669*x^2))/(8*(2 + 3*x^2 + x^4)
) + (477*ArcTan[x])/8 - 351*Sqrt[2]*ArcTan[x/Sqrt[2]]

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Rubi [A]  time = 0.112492, antiderivative size = 81, 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} $x^5-14 x^3+\frac{\left (1669 x^2+824\right ) x}{8 \left (x^4+3 x^2+2\right )}+\frac{\left (415 x^2+414\right ) x}{4 \left (x^4+3 x^2+2\right )^2}+214 x+\frac{477}{8} \tan ^{-1}(x)-351 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(x^10*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^3,x]

[Out]

214*x - 14*x^3 + x^5 + (x*(414 + 415*x^2))/(4*(2 + 3*x^2 + x^4)^2) + (x*(824 + 1669*x^2))/(8*(2 + 3*x^2 + x^4)
) + (477*ArcTan[x])/8 - 351*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 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^{10} \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx &=\frac{x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac{1}{8} \int \frac{828-2478 x^2-840 x^4+424 x^6-216 x^8+96 x^{10}-40 x^{12}}{\left (2+3 x^2+x^4\right )^2} \, dx\\ &=\frac{x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \frac{-4952-2700 x^2+3136 x^4-864 x^6+160 x^8}{2+3 x^2+x^4} \, dx\\ &=\frac{x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \left (6848-1344 x^2+160 x^4-\frac{36 \left (518+571 x^2\right )}{2+3 x^2+x^4}\right ) \, dx\\ &=214 x-14 x^3+x^5+\frac{x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac{9}{8} \int \frac{518+571 x^2}{2+3 x^2+x^4} \, dx\\ &=214 x-14 x^3+x^5+\frac{x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{477}{8} \int \frac{1}{1+x^2} \, dx-702 \int \frac{1}{2+x^2} \, dx\\ &=214 x-14 x^3+x^5+\frac{x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{477}{8} \tan ^{-1}(x)-351 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0603729, size = 71, normalized size = 0.88 $\frac{x \left (8 x^{12}-64 x^{10}+1144 x^8+10581 x^6+26775 x^4+26736 x^2+9324\right )}{8 \left (x^4+3 x^2+2\right )^2}+\frac{477}{8} \tan ^{-1}(x)-351 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x^10*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^3,x]

[Out]

(x*(9324 + 26736*x^2 + 26775*x^4 + 10581*x^6 + 1144*x^8 - 64*x^10 + 8*x^12))/(8*(2 + 3*x^2 + x^4)^2) + (477*Ar
cTan[x])/8 - 351*Sqrt[2]*ArcTan[x/Sqrt[2]]

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Maple [A]  time = 0.013, size = 64, normalized size = 0.8 \begin{align*}{x}^{5}-14\,{x}^{3}+214\,x-16\,{\frac{1}{ \left ({x}^{2}+2 \right ) ^{2}} \left ( -{\frac{105\,{x}^{3}}{8}}-{\frac{79\,x}{4}} \right ) }-351\,\arctan \left ( 1/2\,x\sqrt{2} \right ) \sqrt{2}+{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{11\,{x}^{3}}{8}}-{\frac{13\,x}{8}} \right ) }+{\frac{477\,\arctan \left ( x \right ) }{8}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^10*(5*x^6+3*x^4+x^2+4)/(x^4+3*x^2+2)^3,x)

[Out]

x^5-14*x^3+214*x-16*(-105/8*x^3-79/4*x)/(x^2+2)^2-351*arctan(1/2*x*2^(1/2))*2^(1/2)+(-11/8*x^3-13/8*x)/(x^2+1)
^2+477/8*arctan(x)

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Maxima [A]  time = 1.49385, size = 96, normalized size = 1.19 \begin{align*} x^{5} - 14 \, x^{3} - 351 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 214 \, x + \frac{1669 \, x^{7} + 5831 \, x^{5} + 6640 \, x^{3} + 2476 \, x}{8 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} + \frac{477}{8} \, \arctan \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(5*x^6+3*x^4+x^2+4)/(x^4+3*x^2+2)^3,x, algorithm="maxima")

[Out]

x^5 - 14*x^3 - 351*sqrt(2)*arctan(1/2*sqrt(2)*x) + 214*x + 1/8*(1669*x^7 + 5831*x^5 + 6640*x^3 + 2476*x)/(x^8
+ 6*x^6 + 13*x^4 + 12*x^2 + 4) + 477/8*arctan(x)

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Fricas [A]  time = 1.58518, size = 325, normalized size = 4.01 \begin{align*} \frac{8 \, x^{13} - 64 \, x^{11} + 1144 \, x^{9} + 10581 \, x^{7} + 26775 \, x^{5} + 26736 \, x^{3} - 2808 \, \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 ) + 477 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) + 9324 \, x}{8 \,{\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^10*(5*x^6+3*x^4+x^2+4)/(x^4+3*x^2+2)^3,x, algorithm="fricas")

[Out]

1/8*(8*x^13 - 64*x^11 + 1144*x^9 + 10581*x^7 + 26775*x^5 + 26736*x^3 - 2808*sqrt(2)*(x^8 + 6*x^6 + 13*x^4 + 12
*x^2 + 4)*arctan(1/2*sqrt(2)*x) + 477*(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)*arctan(x) + 9324*x)/(x^8 + 6*x^6 + 1
3*x^4 + 12*x^2 + 4)

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Sympy [A]  time = 0.240526, size = 75, normalized size = 0.93 \begin{align*} x^{5} - 14 x^{3} + 214 x + \frac{1669 x^{7} + 5831 x^{5} + 6640 x^{3} + 2476 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} + \frac{477 \operatorname{atan}{\left (x \right )}}{8} - 351 \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**10*(5*x**6+3*x**4+x**2+4)/(x**4+3*x**2+2)**3,x)

[Out]

x**5 - 14*x**3 + 214*x + (1669*x**7 + 5831*x**5 + 6640*x**3 + 2476*x)/(8*x**8 + 48*x**6 + 104*x**4 + 96*x**2 +
32) + 477*atan(x)/8 - 351*sqrt(2)*atan(sqrt(2)*x/2)

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Giac [A]  time = 1.09603, size = 82, normalized size = 1.01 \begin{align*} x^{5} - 14 \, x^{3} - 351 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 214 \, x + \frac{1669 \, x^{7} + 5831 \, x^{5} + 6640 \, x^{3} + 2476 \, x}{8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} + \frac{477}{8} \, \arctan \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10*(5*x^6+3*x^4+x^2+4)/(x^4+3*x^2+2)^3,x, algorithm="giac")

[Out]

x^5 - 14*x^3 - 351*sqrt(2)*arctan(1/2*sqrt(2)*x) + 214*x + 1/8*(1669*x^7 + 5831*x^5 + 6640*x^3 + 2476*x)/(x^4
+ 3*x^2 + 2)^2 + 477/8*arctan(x)