3.14.96 \(\int \frac {-2-79 x-39 x^2+(2+7 x+6 x^2) \log (x)+(6 x+3 x^2) \log (2 x+x^2)}{8 x^2+4 x^3+(16 x+632 x^2+312 x^3) \log (x)+(8+628 x+12480 x^2+6084 x^3) \log ^2(x)+((-48 x^2-24 x^3) \log (x)+(-48 x-1896 x^2-936 x^3) \log ^2(x)) \log (2 x+x^2)+(72 x^2+36 x^3) \log ^2(x) \log ^2(2 x+x^2)} \, dx\)

Optimal. Leaf size=24 \[ \frac {x}{4 (x+\log (x)-3 x \log (x) (-13+\log (x (2+x))))} \]

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Rubi [A]  time = 0.34, antiderivative size = 30, normalized size of antiderivative = 1.25, number of steps used = 4, number of rules used = 4, integrand size = 161, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {6688, 12, 6711, 32} \begin {gather*} -\frac {1}{4 \left (\frac {x}{\log (x) (39 x-3 x \log (x (x+2))+1)}+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 - 79*x - 39*x^2 + (2 + 7*x + 6*x^2)*Log[x] + (6*x + 3*x^2)*Log[2*x + x^2])/(8*x^2 + 4*x^3 + (16*x + 63
2*x^2 + 312*x^3)*Log[x] + (8 + 628*x + 12480*x^2 + 6084*x^3)*Log[x]^2 + ((-48*x^2 - 24*x^3)*Log[x] + (-48*x -
1896*x^2 - 936*x^3)*Log[x]^2)*Log[2*x + x^2] + (72*x^2 + 36*x^3)*Log[x]^2*Log[2*x + x^2]^2),x]

[Out]

-1/4*1/(1 + x/(Log[x]*(1 + 39*x - 3*x*Log[x*(2 + x)])))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 6688

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

Rule 6711

Int[(u_)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w
, x])]}, Dist[c*p, Subst[Int[(b + a*x^p)^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}
, x] && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (2+7 x+6 x^2\right ) \log (x)+(2+x) (-1-39 x+3 x \log (x (2+x)))}{4 (2+x) (x+\log (x) (1+39 x-3 x \log (x (2+x))))^2} \, dx\\ &=\frac {1}{4} \int \frac {\left (2+7 x+6 x^2\right ) \log (x)+(2+x) (-1-39 x+3 x \log (x (2+x)))}{(2+x) (x+\log (x) (1+39 x-3 x \log (x (2+x))))^2} \, dx\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,\frac {x}{\log (x) (1+39 x-3 x \log (x (2+x)))}\right )\\ &=-\frac {1}{4 \left (1+\frac {x}{\log (x) (1+39 x-3 x \log (x (2+x)))}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.81, size = 26, normalized size = 1.08 \begin {gather*} \frac {x}{4 (x+\log (x) (1+39 x-3 x \log (x (2+x))))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 - 79*x - 39*x^2 + (2 + 7*x + 6*x^2)*Log[x] + (6*x + 3*x^2)*Log[2*x + x^2])/(8*x^2 + 4*x^3 + (16*
x + 632*x^2 + 312*x^3)*Log[x] + (8 + 628*x + 12480*x^2 + 6084*x^3)*Log[x]^2 + ((-48*x^2 - 24*x^3)*Log[x] + (-4
8*x - 1896*x^2 - 936*x^3)*Log[x]^2)*Log[2*x + x^2] + (72*x^2 + 36*x^3)*Log[x]^2*Log[2*x + x^2]^2),x]

[Out]

x/(4*(x + Log[x]*(1 + 39*x - 3*x*Log[x*(2 + x)])))

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fricas [A]  time = 0.70, size = 31, normalized size = 1.29 \begin {gather*} -\frac {x}{4 \, {\left (3 \, x \log \left (x^{2} + 2 \, x\right ) \log \relax (x) - {\left (39 \, x + 1\right )} \log \relax (x) - x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+6*x)*log(x^2+2*x)+(6*x^2+7*x+2)*log(x)-39*x^2-79*x-2)/((36*x^3+72*x^2)*log(x)^2*log(x^2+2*x)
^2+((-936*x^3-1896*x^2-48*x)*log(x)^2+(-24*x^3-48*x^2)*log(x))*log(x^2+2*x)+(6084*x^3+12480*x^2+628*x+8)*log(x
)^2+(312*x^3+632*x^2+16*x)*log(x)+4*x^3+8*x^2),x, algorithm="fricas")

[Out]

-1/4*x/(3*x*log(x^2 + 2*x)*log(x) - (39*x + 1)*log(x) - x)

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giac [A]  time = 0.89, size = 34, normalized size = 1.42 \begin {gather*} -\frac {x}{4 \, {\left (3 \, x \log \left (x + 2\right ) \log \relax (x) + 3 \, x \log \relax (x)^{2} - 39 \, x \log \relax (x) - x - \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+6*x)*log(x^2+2*x)+(6*x^2+7*x+2)*log(x)-39*x^2-79*x-2)/((36*x^3+72*x^2)*log(x)^2*log(x^2+2*x)
^2+((-936*x^3-1896*x^2-48*x)*log(x)^2+(-24*x^3-48*x^2)*log(x))*log(x^2+2*x)+(6084*x^3+12480*x^2+628*x+8)*log(x
)^2+(312*x^3+632*x^2+16*x)*log(x)+4*x^3+8*x^2),x, algorithm="giac")

[Out]

-1/4*x/(3*x*log(x + 2)*log(x) + 3*x*log(x)^2 - 39*x*log(x) - x - log(x))

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maple [C]  time = 0.10, size = 127, normalized size = 5.29

method	result	size
risch	\(-\frac {i x}{2 \left (3 \ln \relax (x ) \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (2+x \right )\right ) \mathrm {csgn}\left (i x \left (2+x \right )\right )-3 \ln \relax (x ) \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (2+x \right )\right )^{2}-3 \ln \relax (x ) \pi x \,\mathrm {csgn}\left (i \left (2+x \right )\right ) \mathrm {csgn}\left (i x \left (2+x \right )\right )^{2}+3 \ln \relax (x ) \pi x \mathrm {csgn}\left (i x \left (2+x \right )\right )^{3}+6 i x \ln \relax (x )^{2}+6 i x \ln \relax (x ) \ln \left (2+x \right )-78 i x \ln \relax (x )-2 i x -2 i \ln \relax (x )\right )}\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^2+6*x)*ln(x^2+2*x)+(6*x^2+7*x+2)*ln(x)-39*x^2-79*x-2)/((36*x^3+72*x^2)*ln(x)^2*ln(x^2+2*x)^2+((-936*
x^3-1896*x^2-48*x)*ln(x)^2+(-24*x^3-48*x^2)*ln(x))*ln(x^2+2*x)+(6084*x^3+12480*x^2+628*x+8)*ln(x)^2+(312*x^3+6
32*x^2+16*x)*ln(x)+4*x^3+8*x^2),x,method=_RETURNVERBOSE)

[Out]

-1/2*I*x/(3*ln(x)*Pi*x*csgn(I*x)*csgn(I*(2+x))*csgn(I*x*(2+x))-3*ln(x)*Pi*x*csgn(I*x)*csgn(I*x*(2+x))^2-3*ln(x
)*Pi*x*csgn(I*(2+x))*csgn(I*x*(2+x))^2+3*ln(x)*Pi*x*csgn(I*x*(2+x))^3+6*I*x*ln(x)^2+6*I*x*ln(x)*ln(2+x)-78*I*x
*ln(x)-2*I*x-2*I*ln(x))

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maxima [A]  time = 0.57, size = 34, normalized size = 1.42 \begin {gather*} -\frac {x}{4 \, {\left (3 \, x \log \left (x + 2\right ) \log \relax (x) + 3 \, x \log \relax (x)^{2} - {\left (39 \, x + 1\right )} \log \relax (x) - x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+6*x)*log(x^2+2*x)+(6*x^2+7*x+2)*log(x)-39*x^2-79*x-2)/((36*x^3+72*x^2)*log(x)^2*log(x^2+2*x)
^2+((-936*x^3-1896*x^2-48*x)*log(x)^2+(-24*x^3-48*x^2)*log(x))*log(x^2+2*x)+(6084*x^3+12480*x^2+628*x+8)*log(x
)^2+(312*x^3+632*x^2+16*x)*log(x)+4*x^3+8*x^2),x, algorithm="maxima")

[Out]

-1/4*x/(3*x*log(x + 2)*log(x) + 3*x*log(x)^2 - (39*x + 1)*log(x) - x)

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mupad [B]  time = 1.58, size = 121, normalized size = 5.04 \begin {gather*} \frac {\frac {x\,{\left (x^2+2\,x\right )}^3}{4}+\frac {x\,{\ln \relax (x)}^2\,{\left (x^2+2\,x\right )}^2\,\left (6\,x^2+7\,x+2\right )}{4}}{\left (x+2\right )\,\left (x+\ln \relax (x)+39\,x\,\ln \relax (x)-3\,x\,\ln \left (x^2+2\,x\right )\,\ln \relax (x)\right )\,\left (6\,x^5\,{\ln \relax (x)}^2+x^5+19\,x^4\,{\ln \relax (x)}^2+4\,x^4+16\,x^3\,{\ln \relax (x)}^2+4\,x^3+4\,x^2\,{\ln \relax (x)}^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(79*x - log(2*x + x^2)*(6*x + 3*x^2) - log(x)*(7*x + 6*x^2 + 2) + 39*x^2 + 2)/(log(x)^2*(628*x + 12480*x^
2 + 6084*x^3 + 8) - log(2*x + x^2)*(log(x)*(48*x^2 + 24*x^3) + log(x)^2*(48*x + 1896*x^2 + 936*x^3)) + 8*x^2 +
4*x^3 + log(x)*(16*x + 632*x^2 + 312*x^3) + log(2*x + x^2)^2*log(x)^2*(72*x^2 + 36*x^3)),x)

[Out]

((x*(2*x + x^2)^3)/4 + (x*log(x)^2*(2*x + x^2)^2*(7*x + 6*x^2 + 2))/4)/((x + 2)*(x + log(x) + 39*x*log(x) - 3*
x*log(2*x + x^2)*log(x))*(4*x^2*log(x)^2 + 16*x^3*log(x)^2 + 19*x^4*log(x)^2 + 6*x^5*log(x)^2 + 4*x^3 + 4*x^4
+ x^5))

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sympy [A]  time = 0.44, size = 32, normalized size = 1.33 \begin {gather*} - \frac {x}{12 x \log {\relax (x )} \log {\left (x^{2} + 2 x \right )} - 156 x \log {\relax (x )} - 4 x - 4 \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**2+6*x)*ln(x**2+2*x)+(6*x**2+7*x+2)*ln(x)-39*x**2-79*x-2)/((36*x**3+72*x**2)*ln(x)**2*ln(x**2+
2*x)**2+((-936*x**3-1896*x**2-48*x)*ln(x)**2+(-24*x**3-48*x**2)*ln(x))*ln(x**2+2*x)+(6084*x**3+12480*x**2+628*
x+8)*ln(x)**2+(312*x**3+632*x**2+16*x)*ln(x)+4*x**3+8*x**2),x)

[Out]

-x/(12*x*log(x)*log(x**2 + 2*x) - 156*x*log(x) - 4*x - 4*log(x))

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