3.55.29 \(\int \frac {84+16 x-76 x^2+(-80-320 x-80 x^2) \log (\frac {5+20 x+5 x^2}{x})}{1+4 x+x^2} \, dx\)

Optimal. Leaf size=24 \[ \log (2)+x \left (4-80 \log \left (\frac {5 \left (x+x^2 (4+x)\right )}{x^2}\right )\right ) \]

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Rubi [A]  time = 0.15, antiderivative size = 16, normalized size of antiderivative = 0.67, number of steps used = 13, number of rules used = 5, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {6728, 1657, 632, 31, 2523} \begin {gather*} 4 x-80 x \log \left (5 \left (x+\frac {1}{x}+4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(84 + 16*x - 76*x^2 + (-80 - 320*x - 80*x^2)*Log[(5 + 20*x + 5*x^2)/x])/(1 + 4*x + x^2),x]

[Out]

4*x - 80*x*Log[5*(4 + x^(-1) + x)]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 632

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

Rule 1657

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

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4 \left (-21-4 x+19 x^2\right )}{1+4 x+x^2}-80 \log \left (5 \left (4+\frac {1}{x}+x\right )\right )\right ) \, dx\\ &=-\left (4 \int \frac {-21-4 x+19 x^2}{1+4 x+x^2} \, dx\right )-80 \int \log \left (5 \left (4+\frac {1}{x}+x\right )\right ) \, dx\\ &=-80 x \log \left (5 \left (4+\frac {1}{x}+x\right )\right )-4 \int \left (19-\frac {40 (1+2 x)}{1+4 x+x^2}\right ) \, dx+80 \int \frac {-1+x^2}{1+4 x+x^2} \, dx\\ &=-76 x-80 x \log \left (5 \left (4+\frac {1}{x}+x\right )\right )+80 \int \left (1-\frac {2 (1+2 x)}{1+4 x+x^2}\right ) \, dx+160 \int \frac {1+2 x}{1+4 x+x^2} \, dx\\ &=4 x-80 x \log \left (5 \left (4+\frac {1}{x}+x\right )\right )-160 \int \frac {1+2 x}{1+4 x+x^2} \, dx+\left (80 \left (2-\sqrt {3}\right )\right ) \int \frac {1}{2-\sqrt {3}+x} \, dx+\left (80 \left (2+\sqrt {3}\right )\right ) \int \frac {1}{2+\sqrt {3}+x} \, dx\\ &=4 x+80 \left (2-\sqrt {3}\right ) \log \left (2-\sqrt {3}+x\right )+80 \left (2+\sqrt {3}\right ) \log \left (2+\sqrt {3}+x\right )-80 x \log \left (5 \left (4+\frac {1}{x}+x\right )\right )-\left (80 \left (2-\sqrt {3}\right )\right ) \int \frac {1}{2-\sqrt {3}+x} \, dx-\left (80 \left (2+\sqrt {3}\right )\right ) \int \frac {1}{2+\sqrt {3}+x} \, dx\\ &=4 x-80 x \log \left (5 \left (4+\frac {1}{x}+x\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 16, normalized size = 0.67 \begin {gather*} 4 x-80 x \log \left (5 \left (4+\frac {1}{x}+x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(84 + 16*x - 76*x^2 + (-80 - 320*x - 80*x^2)*Log[(5 + 20*x + 5*x^2)/x])/(1 + 4*x + x^2),x]

[Out]

4*x - 80*x*Log[5*(4 + x^(-1) + x)]

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fricas [A]  time = 0.46, size = 21, normalized size = 0.88 \begin {gather*} -80 \, x \log \left (\frac {5 \, {\left (x^{2} + 4 \, x + 1\right )}}{x}\right ) + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-80*x^2-320*x-80)*log((5*x^2+20*x+5)/x)-76*x^2+16*x+84)/(x^2+4*x+1),x, algorithm="fricas")

[Out]

-80*x*log(5*(x^2 + 4*x + 1)/x) + 4*x

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giac [A]  time = 0.33, size = 21, normalized size = 0.88 \begin {gather*} -80 \, x \log \left (\frac {5 \, {\left (x^{2} + 4 \, x + 1\right )}}{x}\right ) + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-80*x^2-320*x-80)*log((5*x^2+20*x+5)/x)-76*x^2+16*x+84)/(x^2+4*x+1),x, algorithm="giac")

[Out]

-80*x*log(5*(x^2 + 4*x + 1)/x) + 4*x

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maple [A]  time = 0.30, size = 23, normalized size = 0.96

method	result	size
norman	\(4 x -80 x \ln \left (\frac {5 x^{2}+20 x +5}{x}\right )\)	\(23\)
risch	\(4 x -80 x \ln \left (\frac {5 x^{2}+20 x +5}{x}\right )\)	\(23\)
default	\(4 x -80 x \ln \relax (5)-80 x \ln \left (\frac {x^{2}+4 x +1}{x}\right )\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-80*x^2-320*x-80)*ln((5*x^2+20*x+5)/x)-76*x^2+16*x+84)/(x^2+4*x+1),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.57, size = 41, normalized size = 1.71 \begin {gather*} -80 \, x {\left (\log \relax (5) - 1\right )} - 80 \, {\left (x + 2\right )} \log \left (x^{2} + 4 \, x + 1\right ) + 80 \, x \log \relax (x) - 76 \, x + 160 \, \log \left (x^{2} + 4 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-80*x^2-320*x-80)*log((5*x^2+20*x+5)/x)-76*x^2+16*x+84)/(x^2+4*x+1),x, algorithm="maxima")

[Out]

-80*x*(log(5) - 1) - 80*(x + 2)*log(x^2 + 4*x + 1) + 80*x*log(x) - 76*x + 160*log(x^2 + 4*x + 1)

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mupad [B]  time = 3.50, size = 22, normalized size = 0.92 \begin {gather*} 4\,x-80\,x\,\ln \left (\frac {5\,x^2+20\,x+5}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x - 76*x^2 - log((20*x + 5*x^2 + 5)/x)*(320*x + 80*x^2 + 80) + 84)/(4*x + x^2 + 1),x)

[Out]

4*x - 80*x*log((20*x + 5*x^2 + 5)/x)

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sympy [A]  time = 0.15, size = 19, normalized size = 0.79 \begin {gather*} - 80 x \log {\left (\frac {5 x^{2} + 20 x + 5}{x} \right )} + 4 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-80*x**2-320*x-80)*ln((5*x**2+20*x+5)/x)-76*x**2+16*x+84)/(x**2+4*x+1),x)

[Out]

-80*x*log((5*x**2 + 20*x + 5)/x) + 4*x

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