3.81 \(\int \frac{4+x^2+3 x^4+5 x^6}{x^5 (2+3 x^2+x^4)^2} \, dx\)

Optimal. Leaf size=64 \[ -\frac{9 x^2+5}{8 \left (x^4+3 x^2+2\right )}+\frac{11}{8 x^2}-\frac{1}{4 x^4}-\frac{11}{2} \log \left (x^2+1\right )+\frac{21}{8} \log \left (x^2+2\right )+\frac{23 \log (x)}{4} \]

[Out]

-1/(4*x^4) + 11/(8*x^2) - (5 + 9*x^2)/(8*(2 + 3*x^2 + x^4)) + (23*Log[x])/4 - (11*Log[1 + x^2])/2 + (21*Log[2
+ x^2])/8

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Rubi [A]  time = 0.110765, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {1663, 1646, 1628} \[ -\frac{9 x^2+5}{8 \left (x^4+3 x^2+2\right )}+\frac{11}{8 x^2}-\frac{1}{4 x^4}-\frac{11}{2} \log \left (x^2+1\right )+\frac{21}{8} \log \left (x^2+2\right )+\frac{23 \log (x)}{4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(4*x^4) + 11/(8*x^2) - (5 + 9*x^2)/(8*(2 + 3*x^2 + x^4)) + (23*Log[x])/4 - (11*Log[1 + x^2])/2 + (21*Log[2
+ x^2])/8

Rule 1663

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)
*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && Inte
gerQ[(m - 1)/2]

Rule 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{x^5 \left (2+3 x^2+x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{4+x+3 x^2+5 x^3}{x^3 \left (2+3 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac{5+9 x^2}{8 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{-2+\frac{5 x}{2}-\frac{17 x^2}{4}+\frac{9 x^3}{4}}{x^3 \left (2+3 x+x^2\right )} \, dx,x,x^2\right )\\ &=-\frac{5+9 x^2}{8 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{1}{x^3}+\frac{11}{4 x^2}-\frac{23}{4 x}+\frac{11}{1+x}-\frac{21}{4 (2+x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{4 x^4}+\frac{11}{8 x^2}-\frac{5+9 x^2}{8 \left (2+3 x^2+x^4\right )}+\frac{23 \log (x)}{4}-\frac{11}{2} \log \left (1+x^2\right )+\frac{21}{8} \log \left (2+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0293778, size = 56, normalized size = 0.88 \[ \frac{1}{8} \left (-\frac{9 x^2+5}{x^4+3 x^2+2}+\frac{11}{x^2}-\frac{2}{x^4}-44 \log \left (x^2+1\right )+21 \log \left (x^2+2\right )+46 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2/x^4 + 11/x^2 - (5 + 9*x^2)/(2 + 3*x^2 + x^4) + 46*Log[x] - 44*Log[1 + x^2] + 21*Log[2 + x^2])/8

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Maple [A]  time = 0.018, size = 50, normalized size = 0.8 \begin{align*}{\frac{21\,\ln \left ({x}^{2}+2 \right ) }{8}}-{\frac{13}{8\,{x}^{2}+16}}-{\frac{11\,\ln \left ({x}^{2}+1 \right ) }{2}}+{\frac{1}{2\,{x}^{2}+2}}-{\frac{1}{4\,{x}^{4}}}+{\frac{11}{8\,{x}^{2}}}+{\frac{23\,\ln \left ( x \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21/8*ln(x^2+2)-13/8/(x^2+2)-11/2*ln(x^2+1)+1/2/(x^2+1)-1/4/x^4+11/8/x^2+23/4*ln(x)

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Maxima [A]  time = 0.952735, size = 76, normalized size = 1.19 \begin{align*} \frac{x^{6} + 13 \, x^{4} + 8 \, x^{2} - 2}{4 \,{\left (x^{8} + 3 \, x^{6} + 2 \, x^{4}\right )}} + \frac{21}{8} \, \log \left (x^{2} + 2\right ) - \frac{11}{2} \, \log \left (x^{2} + 1\right ) + \frac{23}{8} \, \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(x^6 + 13*x^4 + 8*x^2 - 2)/(x^8 + 3*x^6 + 2*x^4) + 21/8*log(x^2 + 2) - 11/2*log(x^2 + 1) + 23/8*log(x^2)

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Fricas [A]  time = 1.7272, size = 231, normalized size = 3.61 \begin{align*} \frac{2 \, x^{6} + 26 \, x^{4} + 16 \, x^{2} + 21 \,{\left (x^{8} + 3 \, x^{6} + 2 \, x^{4}\right )} \log \left (x^{2} + 2\right ) - 44 \,{\left (x^{8} + 3 \, x^{6} + 2 \, x^{4}\right )} \log \left (x^{2} + 1\right ) + 46 \,{\left (x^{8} + 3 \, x^{6} + 2 \, x^{4}\right )} \log \left (x\right ) - 4}{8 \,{\left (x^{8} + 3 \, x^{6} + 2 \, x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*x^6 + 26*x^4 + 16*x^2 + 21*(x^8 + 3*x^6 + 2*x^4)*log(x^2 + 2) - 44*(x^8 + 3*x^6 + 2*x^4)*log(x^2 + 1) +
46*(x^8 + 3*x^6 + 2*x^4)*log(x) - 4)/(x^8 + 3*x^6 + 2*x^4)

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Sympy [A]  time = 0.204195, size = 56, normalized size = 0.88 \begin{align*} \frac{23 \log{\left (x \right )}}{4} - \frac{11 \log{\left (x^{2} + 1 \right )}}{2} + \frac{21 \log{\left (x^{2} + 2 \right )}}{8} + \frac{x^{6} + 13 x^{4} + 8 x^{2} - 2}{4 x^{8} + 12 x^{6} + 8 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

23*log(x)/4 - 11*log(x**2 + 1)/2 + 21*log(x**2 + 2)/8 + (x**6 + 13*x**4 + 8*x**2 - 2)/(4*x**8 + 12*x**6 + 8*x*
*4)

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Giac [A]  time = 1.09415, size = 89, normalized size = 1.39 \begin{align*} \frac{23 \, x^{4} + 51 \, x^{2} + 36}{16 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac{69 \, x^{4} - 22 \, x^{2} + 4}{16 \, x^{4}} + \frac{21}{8} \, \log \left (x^{2} + 2\right ) - \frac{11}{2} \, \log \left (x^{2} + 1\right ) + \frac{23}{8} \, \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(23*x^4 + 51*x^2 + 36)/(x^4 + 3*x^2 + 2) - 1/16*(69*x^4 - 22*x^2 + 4)/x^4 + 21/8*log(x^2 + 2) - 11/2*log(
x^2 + 1) + 23/8*log(x^2)