3.28.66 \(\int \frac {20+1000 x+11 x^2+400 x^3+x^4+40 x^5+(2000+800 x^2+80 x^4) \log (2)}{25+10 x^2+x^4} \, dx\)

Optimal. Leaf size=23 \[ x-\frac {x}{5+x^2}-20 x (-x-4 \log (2)) \]

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Rubi [A]  time = 0.07, antiderivative size = 24, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {28, 1814, 1586} \begin {gather*} 20 x^2-\frac {x}{x^2+5}+x (1+80 \log (2)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(20 + 1000*x + 11*x^2 + 400*x^3 + x^4 + 40*x^5 + (2000 + 800*x^2 + 80*x^4)*Log[2])/(25 + 10*x^2 + x^4),x]

[Out]

20*x^2 - x/(5 + x^2) + x*(1 + 80*Log[2])

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 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {20+1000 x+11 x^2+400 x^3+x^4+40 x^5+\left (2000+800 x^2+80 x^4\right ) \log (2)}{\left (5+x^2\right )^2} \, dx\\ &=-\frac {x}{5+x^2}-\frac {1}{10} \int \frac {-2000 x-400 x^3-50 (1+80 \log (2))-10 x^2 (1+80 \log (2))}{5+x^2} \, dx\\ &=-\frac {x}{5+x^2}-\frac {1}{10} \int (-10-400 x-800 \log (2)) \, dx\\ &=20 x^2-\frac {x}{5+x^2}+x (1+80 \log (2))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 0.87 \begin {gather*} x \left (1+20 x-\frac {1}{5+x^2}+80 \log (2)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20 + 1000*x + 11*x^2 + 400*x^3 + x^4 + 40*x^5 + (2000 + 800*x^2 + 80*x^4)*Log[2])/(25 + 10*x^2 + x^
4),x]

[Out]

x*(1 + 20*x - (5 + x^2)^(-1) + 80*Log[2])

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fricas [A]  time = 0.55, size = 36, normalized size = 1.57 \begin {gather*} \frac {20 \, x^{4} + x^{3} + 100 \, x^{2} + 80 \, {\left (x^{3} + 5 \, x\right )} \log \relax (2) + 4 \, x}{x^{2} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((80*x^4+800*x^2+2000)*log(2)+40*x^5+x^4+400*x^3+11*x^2+1000*x+20)/(x^4+10*x^2+25),x, algorithm="fri
cas")

[Out]

(20*x^4 + x^3 + 100*x^2 + 80*(x^3 + 5*x)*log(2) + 4*x)/(x^2 + 5)

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giac [A]  time = 0.24, size = 22, normalized size = 0.96 \begin {gather*} 20 \, x^{2} + 80 \, x \log \relax (2) + x - \frac {x}{x^{2} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((80*x^4+800*x^2+2000)*log(2)+40*x^5+x^4+400*x^3+11*x^2+1000*x+20)/(x^4+10*x^2+25),x, algorithm="gia
c")

[Out]

20*x^2 + 80*x*log(2) + x - x/(x^2 + 5)

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maple [A]  time = 0.06, size = 23, normalized size = 1.00

method	result	size
default	\(20 x^{2}+80 x \ln \relax (2)+x -\frac {x}{x^{2}+5}\)	\(23\)
risch	\(20 x^{2}+80 x \ln \relax (2)+x -\frac {x}{x^{2}+5}\)	\(23\)
gosper	\(\frac {80 x^{3} \ln \relax (2)+20 x^{4}+x^{3}+400 x \ln \relax (2)+4 x -500}{x^{2}+5}\)	\(34\)
norman	\(\frac {20 x^{4}+\left (80 \ln \relax (2)+1\right ) x^{3}-500+\left (400 \ln \relax (2)+4\right ) x}{x^{2}+5}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((80*x^4+800*x^2+2000)*ln(2)+40*x^5+x^4+400*x^3+11*x^2+1000*x+20)/(x^4+10*x^2+25),x,method=_RETURNVERBOSE)

[Out]

20*x^2+80*x*ln(2)+x-x/(x^2+5)

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maxima [A]  time = 0.39, size = 24, normalized size = 1.04 \begin {gather*} 20 \, x^{2} + x {\left (80 \, \log \relax (2) + 1\right )} - \frac {x}{x^{2} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((80*x^4+800*x^2+2000)*log(2)+40*x^5+x^4+400*x^3+11*x^2+1000*x+20)/(x^4+10*x^2+25),x, algorithm="max
ima")

[Out]

20*x^2 + x*(80*log(2) + 1) - x/(x^2 + 5)

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mupad [B]  time = 0.07, size = 24, normalized size = 1.04 \begin {gather*} x\,\left (80\,\ln \relax (2)+1\right )-\frac {x}{x^2+5}+20\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1000*x + log(2)*(800*x^2 + 80*x^4 + 2000) + 11*x^2 + 400*x^3 + x^4 + 40*x^5 + 20)/(10*x^2 + x^4 + 25),x)

[Out]

x*(80*log(2) + 1) - x/(x^2 + 5) + 20*x^2

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sympy [A]  time = 0.13, size = 19, normalized size = 0.83 \begin {gather*} 20 x^{2} + x \left (1 + 80 \log {\relax (2 )}\right ) - \frac {x}{x^{2} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((80*x**4+800*x**2+2000)*ln(2)+40*x**5+x**4+400*x**3+11*x**2+1000*x+20)/(x**4+10*x**2+25),x)

[Out]

20*x**2 + x*(1 + 80*log(2)) - x/(x**2 + 5)

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