3.98.67 \(\int \frac {2040 x+1008 x^2+144 x^3+16 x^4+(7225+2040 x+504 x^2+48 x^3+4 x^4) \log (\frac {1}{4} (7225+2040 x+504 x^2+48 x^3+4 x^4))}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx\)

Optimal. Leaf size=20 \[ x \log \left (5 x^2+\left (\frac {67}{2}+(3+x)^2\right )^2\right ) \]

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Rubi [B]  time = 0.78, antiderivative size = 70, normalized size of antiderivative = 3.50, number of steps used = 13, number of rules used = 4, integrand size = 88, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6688, 6742, 2100, 2523} \begin {gather*} x \log \left (x^4+12 x^3+126 x^2+510 x+\frac {7225}{4}\right )+3 \log \left (x^4+12 x^3+126 x^2+510 x+\frac {7225}{4}\right )-3 \log \left (4 x^4+48 x^3+504 x^2+2040 x+7225\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2040*x + 1008*x^2 + 144*x^3 + 16*x^4 + (7225 + 2040*x + 504*x^2 + 48*x^3 + 4*x^4)*Log[(7225 + 2040*x + 50
4*x^2 + 48*x^3 + 4*x^4)/4])/(7225 + 2040*x + 504*x^2 + 48*x^3 + 4*x^4),x]

[Out]

3*Log[7225/4 + 510*x + 126*x^2 + 12*x^3 + x^4] + x*Log[7225/4 + 510*x + 126*x^2 + 12*x^3 + x^4] - 3*Log[7225 +
2040*x + 504*x^2 + 48*x^3 + 4*x^4]

Rule 2100

Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Log[Qn])/(n*Coe
ff[Qn, x, n]), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x
], x]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, x]

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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {8 x \left (255+126 x+18 x^2+2 x^3\right )}{7225+2040 x+504 x^2+48 x^3+4 x^4}+\log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )\right ) \, dx\\ &=8 \int \frac {x \left (255+126 x+18 x^2+2 x^3\right )}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx+\int \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right ) \, dx\\ &=x \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )+8 \int \left (\frac {1}{2}-\frac {7225+1530 x+252 x^2+12 x^3}{2 \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )}\right ) \, dx-\int \frac {x \left (510+252 x+36 x^2+4 x^3\right )}{\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4} \, dx\\ &=4 x+x \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )-4 \int \frac {7225+1530 x+252 x^2+12 x^3}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx-\int \left (4-\frac {7225+1530 x+252 x^2+12 x^3}{\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4}\right ) \, dx\\ &=x \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )-3 \log \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )-\frac {1}{4} \int \frac {91120+12384 x+2304 x^2}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx+\int \frac {7225+1530 x+252 x^2+12 x^3}{\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4} \, dx\\ &=3 \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )+x \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )-3 \log \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )+\frac {1}{4} \int \frac {22780+3096 x+576 x^2}{\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4} \, dx-\frac {1}{4} \int \left (\frac {91120}{7225+2040 x+504 x^2+48 x^3+4 x^4}+\frac {12384 x}{7225+2040 x+504 x^2+48 x^3+4 x^4}+\frac {2304 x^2}{7225+2040 x+504 x^2+48 x^3+4 x^4}\right ) \, dx\\ &=3 \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )+x \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )-3 \log \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )+\frac {1}{4} \int \left (\frac {91120}{7225+2040 x+504 x^2+48 x^3+4 x^4}+\frac {12384 x}{7225+2040 x+504 x^2+48 x^3+4 x^4}+\frac {2304 x^2}{7225+2040 x+504 x^2+48 x^3+4 x^4}\right ) \, dx-576 \int \frac {x^2}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx-3096 \int \frac {x}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx-22780 \int \frac {1}{7225+2040 x+504 x^2+48 x^3+4 x^4} \, dx\\ &=3 \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )+x \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right )-3 \log \left (7225+2040 x+504 x^2+48 x^3+4 x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 23, normalized size = 1.15 \begin {gather*} x \log \left (\frac {7225}{4}+510 x+126 x^2+12 x^3+x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2040*x + 1008*x^2 + 144*x^3 + 16*x^4 + (7225 + 2040*x + 504*x^2 + 48*x^3 + 4*x^4)*Log[(7225 + 2040*
x + 504*x^2 + 48*x^3 + 4*x^4)/4])/(7225 + 2040*x + 504*x^2 + 48*x^3 + 4*x^4),x]

[Out]

x*Log[7225/4 + 510*x + 126*x^2 + 12*x^3 + x^4]

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fricas [A]  time = 0.89, size = 21, normalized size = 1.05 \begin {gather*} x \log \left (x^{4} + 12 \, x^{3} + 126 \, x^{2} + 510 \, x + \frac {7225}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+48*x^3+504*x^2+2040*x+7225)*log(x^4+12*x^3+126*x^2+510*x+7225/4)+16*x^4+144*x^3+1008*x^2+204
0*x)/(4*x^4+48*x^3+504*x^2+2040*x+7225),x, algorithm="fricas")

[Out]

x*log(x^4 + 12*x^3 + 126*x^2 + 510*x + 7225/4)

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giac [A]  time = 0.25, size = 21, normalized size = 1.05 \begin {gather*} x \log \left (x^{4} + 12 \, x^{3} + 126 \, x^{2} + 510 \, x + \frac {7225}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+48*x^3+504*x^2+2040*x+7225)*log(x^4+12*x^3+126*x^2+510*x+7225/4)+16*x^4+144*x^3+1008*x^2+204
0*x)/(4*x^4+48*x^3+504*x^2+2040*x+7225),x, algorithm="giac")

[Out]

x*log(x^4 + 12*x^3 + 126*x^2 + 510*x + 7225/4)

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maple [A]  time = 0.04, size = 22, normalized size = 1.10

method	result	size
norman	\(\ln \left (x^{4}+12 x^{3}+126 x^{2}+510 x +\frac {7225}{4}\right ) x\)	\(22\)
risch	\(\ln \left (x^{4}+12 x^{3}+126 x^{2}+510 x +\frac {7225}{4}\right ) x\)	\(22\)
default	\(-2 x \ln \relax (2)+x \ln \left (4 x^{4}+48 x^{3}+504 x^{2}+2040 x +7225\right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^4+48*x^3+504*x^2+2040*x+7225)*ln(x^4+12*x^3+126*x^2+510*x+7225/4)+16*x^4+144*x^3+1008*x^2+2040*x)/(4
*x^4+48*x^3+504*x^2+2040*x+7225),x,method=_RETURNVERBOSE)

[Out]

ln(x^4+12*x^3+126*x^2+510*x+7225/4)*x

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maxima [A]  time = 0.43, size = 29, normalized size = 1.45 \begin {gather*} -2 \, x \log \relax (2) + x \log \left (4 \, x^{4} + 48 \, x^{3} + 504 \, x^{2} + 2040 \, x + 7225\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+48*x^3+504*x^2+2040*x+7225)*log(x^4+12*x^3+126*x^2+510*x+7225/4)+16*x^4+144*x^3+1008*x^2+204
0*x)/(4*x^4+48*x^3+504*x^2+2040*x+7225),x, algorithm="maxima")

[Out]

-2*x*log(2) + x*log(4*x^4 + 48*x^3 + 504*x^2 + 2040*x + 7225)

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mupad [B]  time = 5.88, size = 21, normalized size = 1.05 \begin {gather*} x\,\ln \left (x^4+12\,x^3+126\,x^2+510\,x+\frac {7225}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2040*x + log(510*x + 126*x^2 + 12*x^3 + x^4 + 7225/4)*(2040*x + 504*x^2 + 48*x^3 + 4*x^4 + 7225) + 1008*x
^2 + 144*x^3 + 16*x^4)/(2040*x + 504*x^2 + 48*x^3 + 4*x^4 + 7225),x)

[Out]

x*log(510*x + 126*x^2 + 12*x^3 + x^4 + 7225/4)

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sympy [A]  time = 0.20, size = 22, normalized size = 1.10 \begin {gather*} x \log {\left (x^{4} + 12 x^{3} + 126 x^{2} + 510 x + \frac {7225}{4} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**4+48*x**3+504*x**2+2040*x+7225)*ln(x**4+12*x**3+126*x**2+510*x+7225/4)+16*x**4+144*x**3+1008*
x**2+2040*x)/(4*x**4+48*x**3+504*x**2+2040*x+7225),x)

[Out]

x*log(x**4 + 12*x**3 + 126*x**2 + 510*x + 7225/4)

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