3.10.38 \(\int \frac {8+2 x+(4+18 x+8 x^2) \log (\frac {1+8 x+16 x^2}{x^2})+(3+12 x+400 x^3+1725 x^4+500 x^5) \log ^2(\frac {1+8 x+16 x^2}{x^2})}{(1+4 x) \log ^2(\frac {1+8 x+16 x^2}{x^2})} \, dx\)

Optimal. Leaf size=29 \[ (4+x) \left (3+25 x^4+\frac {x}{\log \left (\frac {\left (x+4 x^2\right )^2}{x^4}\right )}\right ) \]

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Rubi [F]  time = 0.49, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {8+2 x+\left (4+18 x+8 x^2\right ) \log \left (\frac {1+8 x+16 x^2}{x^2}\right )+\left (3+12 x+400 x^3+1725 x^4+500 x^5\right ) \log ^2\left (\frac {1+8 x+16 x^2}{x^2}\right )}{(1+4 x) \log ^2\left (\frac {1+8 x+16 x^2}{x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8 + 2*x + (4 + 18*x + 8*x^2)*Log[(1 + 8*x + 16*x^2)/x^2] + (3 + 12*x + 400*x^3 + 1725*x^4 + 500*x^5)*Log[
(1 + 8*x + 16*x^2)/x^2]^2)/((1 + 4*x)*Log[(1 + 8*x + 16*x^2)/x^2]^2),x]

[Out]

3*x + 100*x^4 + 25*x^5 + 2*Defer[Int][(4 + x)/((1 + 4*x)*Log[(1 + 4*x)^2/x^2]^2), x] + 2*Defer[Int][(2 + x)/Lo
g[(1 + 4*x)^2/x^2], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8+2 x+\left (4+18 x+8 x^2\right ) \log \left (\frac {1+8 x+16 x^2}{x^2}\right )+\left (3+12 x+400 x^3+1725 x^4+500 x^5\right ) \log ^2\left (\frac {1+8 x+16 x^2}{x^2}\right )}{(1+4 x) \log ^2\left (16+\frac {1}{x^2}+\frac {8}{x}\right )} \, dx\\ &=\int \left (3+400 x^3+125 x^4+\frac {2 (4+x)}{(1+4 x) \log ^2\left (\frac {(1+4 x)^2}{x^2}\right )}+\frac {2 (2+x)}{\log \left (\frac {(1+4 x)^2}{x^2}\right )}\right ) \, dx\\ &=3 x+100 x^4+25 x^5+2 \int \frac {4+x}{(1+4 x) \log ^2\left (\frac {(1+4 x)^2}{x^2}\right )} \, dx+2 \int \frac {2+x}{\log \left (\frac {(1+4 x)^2}{x^2}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 33, normalized size = 1.14 \begin {gather*} 3 x+100 x^4+25 x^5+\frac {x (4+x)}{\log \left (\frac {(1+4 x)^2}{x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 + 2*x + (4 + 18*x + 8*x^2)*Log[(1 + 8*x + 16*x^2)/x^2] + (3 + 12*x + 400*x^3 + 1725*x^4 + 500*x^5
)*Log[(1 + 8*x + 16*x^2)/x^2]^2)/((1 + 4*x)*Log[(1 + 8*x + 16*x^2)/x^2]^2),x]

[Out]

3*x + 100*x^4 + 25*x^5 + (x*(4 + x))/Log[(1 + 4*x)^2/x^2]

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fricas [A]  time = 0.67, size = 55, normalized size = 1.90 \begin {gather*} \frac {x^{2} + {\left (25 \, x^{5} + 100 \, x^{4} + 3 \, x\right )} \log \left (\frac {16 \, x^{2} + 8 \, x + 1}{x^{2}}\right ) + 4 \, x}{\log \left (\frac {16 \, x^{2} + 8 \, x + 1}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((500*x^5+1725*x^4+400*x^3+12*x+3)*log((16*x^2+8*x+1)/x^2)^2+(8*x^2+18*x+4)*log((16*x^2+8*x+1)/x^2)+
2*x+8)/(4*x+1)/log((16*x^2+8*x+1)/x^2)^2,x, algorithm="fricas")

[Out]

(x^2 + (25*x^5 + 100*x^4 + 3*x)*log((16*x^2 + 8*x + 1)/x^2) + 4*x)/log((16*x^2 + 8*x + 1)/x^2)

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giac [A]  time = 1.24, size = 39, normalized size = 1.34 \begin {gather*} 25 \, x^{5} + 100 \, x^{4} + 3 \, x + \frac {x^{2} + 4 \, x}{\log \left (\frac {16 \, x^{2} + 8 \, x + 1}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((500*x^5+1725*x^4+400*x^3+12*x+3)*log((16*x^2+8*x+1)/x^2)^2+(8*x^2+18*x+4)*log((16*x^2+8*x+1)/x^2)+
2*x+8)/(4*x+1)/log((16*x^2+8*x+1)/x^2)^2,x, algorithm="giac")

[Out]

25*x^5 + 100*x^4 + 3*x + (x^2 + 4*x)/log((16*x^2 + 8*x + 1)/x^2)

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maple [A]  time = 0.16, size = 37, normalized size = 1.28

method	result	size
risch	\(25 x^{5}+100 x^{4}+3 x +\frac {\left (4+x \right ) x}{\ln \left (\frac {16 x^{2}+8 x +1}{x^{2}}\right )}\)	\(37\)
norman	\(\frac {x^{2}+4 x +100 x^{4} \ln \left (\frac {16 x^{2}+8 x +1}{x^{2}}\right )+25 x^{5} \ln \left (\frac {16 x^{2}+8 x +1}{x^{2}}\right )+3 \ln \left (\frac {16 x^{2}+8 x +1}{x^{2}}\right ) x}{\ln \left (\frac {16 x^{2}+8 x +1}{x^{2}}\right )}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((500*x^5+1725*x^4+400*x^3+12*x+3)*ln((16*x^2+8*x+1)/x^2)^2+(8*x^2+18*x+4)*ln((16*x^2+8*x+1)/x^2)+2*x+8)/(
4*x+1)/ln((16*x^2+8*x+1)/x^2)^2,x,method=_RETURNVERBOSE)

[Out]

25*x^5+100*x^4+3*x+(4+x)*x/ln((16*x^2+8*x+1)/x^2)

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maxima [B]  time = 0.58, size = 62, normalized size = 2.14 \begin {gather*} \frac {x^{2} + 2 \, {\left (25 \, x^{5} + 100 \, x^{4} + 3 \, x\right )} \log \left (4 \, x + 1\right ) - 2 \, {\left (25 \, x^{5} + 100 \, x^{4} + 3 \, x\right )} \log \relax (x) + 4 \, x}{2 \, {\left (\log \left (4 \, x + 1\right ) - \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((500*x^5+1725*x^4+400*x^3+12*x+3)*log((16*x^2+8*x+1)/x^2)^2+(8*x^2+18*x+4)*log((16*x^2+8*x+1)/x^2)+
2*x+8)/(4*x+1)/log((16*x^2+8*x+1)/x^2)^2,x, algorithm="maxima")

[Out]

1/2*(x^2 + 2*(25*x^5 + 100*x^4 + 3*x)*log(4*x + 1) - 2*(25*x^5 + 100*x^4 + 3*x)*log(x) + 4*x)/(log(4*x + 1) -
log(x))

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mupad [B]  time = 1.00, size = 37, normalized size = 1.28 \begin {gather*} x\,\left (25\,x^4+100\,x^3+3\right )+\frac {x\,\left (x+4\right )}{\ln \left (\frac {16\,x^2+8\,x+1}{x^2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + log((8*x + 16*x^2 + 1)/x^2)^2*(12*x + 400*x^3 + 1725*x^4 + 500*x^5 + 3) + log((8*x + 16*x^2 + 1)/x^
2)*(18*x + 8*x^2 + 4) + 8)/(log((8*x + 16*x^2 + 1)/x^2)^2*(4*x + 1)),x)

[Out]

x*(100*x^3 + 25*x^4 + 3) + (x*(x + 4))/log((8*x + 16*x^2 + 1)/x^2)

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sympy [A]  time = 0.14, size = 34, normalized size = 1.17 \begin {gather*} 25 x^{5} + 100 x^{4} + 3 x + \frac {x^{2} + 4 x}{\log {\left (\frac {16 x^{2} + 8 x + 1}{x^{2}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((500*x**5+1725*x**4+400*x**3+12*x+3)*ln((16*x**2+8*x+1)/x**2)**2+(8*x**2+18*x+4)*ln((16*x**2+8*x+1)
/x**2)+2*x+8)/(4*x+1)/ln((16*x**2+8*x+1)/x**2)**2,x)

[Out]

25*x**5 + 100*x**4 + 3*x + (x**2 + 4*x)/log((16*x**2 + 8*x + 1)/x**2)

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