3.25.44 \(\int \frac {(512 x+1024 x^2+512 x^3) \log (6+x)+(1536+6400 x+5632 x^2+768 x^3) \log ^2(6+x)}{6+x+(-3072 x-6656 x^2-4096 x^3-512 x^4) \log ^2(6+x)+(393216 x^2+1638400 x^3+2621440 x^4+1966080 x^5+655360 x^6+65536 x^7) \log ^4(6+x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {x}{x-256 x^2 (1+x)^2 \log ^2(6+x)} \]

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Rubi [A]  time = 8.40, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 112, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6688, 12, 6742, 6711, 32} \begin {gather*} -\frac {256}{256-\frac {1}{x (x+1)^2 \log ^2(x+6)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((512*x + 1024*x^2 + 512*x^3)*Log[6 + x] + (1536 + 6400*x + 5632*x^2 + 768*x^3)*Log[6 + x]^2)/(6 + x + (-3
072*x - 6656*x^2 - 4096*x^3 - 512*x^4)*Log[6 + x]^2 + (393216*x^2 + 1638400*x^3 + 2621440*x^4 + 1966080*x^5 +
655360*x^6 + 65536*x^7)*Log[6 + x]^4),x]

[Out]

-256/(256 - 1/(x*(1 + x)^2*Log[6 + x]^2))

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]

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 \frac {256 (1+x) \log (6+x) \left (2 x (1+x)+\left (6+19 x+3 x^2\right ) \log (6+x)\right )}{(6+x) \left (1-256 x (1+x)^2 \log ^2(6+x)\right )^2} \, dx\\ &=256 \int \frac {(1+x) \log (6+x) \left (2 x (1+x)+\left (6+19 x+3 x^2\right ) \log (6+x)\right )}{(6+x) \left (1-256 x (1+x)^2 \log ^2(6+x)\right )^2} \, dx\\ &=\operatorname {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,-\frac {1}{256 x (1+x)^2 \log ^2(6+x)}\right )\\ &=-\frac {1}{1-\frac {1}{256 x (1+x)^2 \log ^2(6+x)}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.78, size = 20, normalized size = 0.91 \begin {gather*} -\frac {1}{-1+256 x (1+x)^2 \log ^2(6+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((512*x + 1024*x^2 + 512*x^3)*Log[6 + x] + (1536 + 6400*x + 5632*x^2 + 768*x^3)*Log[6 + x]^2)/(6 + x
+ (-3072*x - 6656*x^2 - 4096*x^3 - 512*x^4)*Log[6 + x]^2 + (393216*x^2 + 1638400*x^3 + 2621440*x^4 + 1966080*
x^5 + 655360*x^6 + 65536*x^7)*Log[6 + x]^4),x]

[Out]

-(-1 + 256*x*(1 + x)^2*Log[6 + x]^2)^(-1)

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fricas [A]  time = 0.85, size = 24, normalized size = 1.09 \begin {gather*} -\frac {1}{256 \, {\left (x^{3} + 2 \, x^{2} + x\right )} \log \left (x + 6\right )^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((768*x^3+5632*x^2+6400*x+1536)*log(x+6)^2+(512*x^3+1024*x^2+512*x)*log(x+6))/((65536*x^7+655360*x^6
+1966080*x^5+2621440*x^4+1638400*x^3+393216*x^2)*log(x+6)^4+(-512*x^4-4096*x^3-6656*x^2-3072*x)*log(x+6)^2+x+6
),x, algorithm="fricas")

[Out]

-1/(256*(x^3 + 2*x^2 + x)*log(x + 6)^2 - 1)

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giac [A]  time = 1.75, size = 37, normalized size = 1.68 \begin {gather*} -\frac {1}{256 \, x^{3} \log \left (x + 6\right )^{2} + 512 \, x^{2} \log \left (x + 6\right )^{2} + 256 \, x \log \left (x + 6\right )^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((768*x^3+5632*x^2+6400*x+1536)*log(x+6)^2+(512*x^3+1024*x^2+512*x)*log(x+6))/((65536*x^7+655360*x^6
+1966080*x^5+2621440*x^4+1638400*x^3+393216*x^2)*log(x+6)^4+(-512*x^4-4096*x^3-6656*x^2-3072*x)*log(x+6)^2+x+6
),x, algorithm="giac")

[Out]

-1/(256*x^3*log(x + 6)^2 + 512*x^2*log(x + 6)^2 + 256*x*log(x + 6)^2 - 1)

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maple [A]  time = 0.04, size = 38, normalized size = 1.73

method	result	size
risch	\(-\frac {1}{256 \ln \left (x +6\right )^{2} x^{3}+512 x^{2} \ln \left (x +6\right )^{2}+256 \ln \left (x +6\right )^{2} x -1}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((768*x^3+5632*x^2+6400*x+1536)*ln(x+6)^2+(512*x^3+1024*x^2+512*x)*ln(x+6))/((65536*x^7+655360*x^6+1966080
*x^5+2621440*x^4+1638400*x^3+393216*x^2)*ln(x+6)^4+(-512*x^4-4096*x^3-6656*x^2-3072*x)*ln(x+6)^2+x+6),x,method
=_RETURNVERBOSE)

[Out]

-1/(256*ln(x+6)^2*x^3+512*x^2*ln(x+6)^2+256*ln(x+6)^2*x-1)

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maxima [A]  time = 0.61, size = 24, normalized size = 1.09 \begin {gather*} -\frac {1}{256 \, {\left (x^{3} + 2 \, x^{2} + x\right )} \log \left (x + 6\right )^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((768*x^3+5632*x^2+6400*x+1536)*log(x+6)^2+(512*x^3+1024*x^2+512*x)*log(x+6))/((65536*x^7+655360*x^6
+1966080*x^5+2621440*x^4+1638400*x^3+393216*x^2)*log(x+6)^4+(-512*x^4-4096*x^3-6656*x^2-3072*x)*log(x+6)^2+x+6
),x, algorithm="maxima")

[Out]

-1/(256*(x^3 + 2*x^2 + x)*log(x + 6)^2 - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\left (768\,x^3+5632\,x^2+6400\,x+1536\right )\,{\ln \left (x+6\right )}^2+\left (512\,x^3+1024\,x^2+512\,x\right )\,\ln \left (x+6\right )}{\left (65536\,x^7+655360\,x^6+1966080\,x^5+2621440\,x^4+1638400\,x^3+393216\,x^2\right )\,{\ln \left (x+6\right )}^4+\left (-512\,x^4-4096\,x^3-6656\,x^2-3072\,x\right )\,{\ln \left (x+6\right )}^2+x+6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + 6)*(512*x + 1024*x^2 + 512*x^3) + log(x + 6)^2*(6400*x + 5632*x^2 + 768*x^3 + 1536))/(x + log(x +
6)^4*(393216*x^2 + 1638400*x^3 + 2621440*x^4 + 1966080*x^5 + 655360*x^6 + 65536*x^7) - log(x + 6)^2*(3072*x +
6656*x^2 + 4096*x^3 + 512*x^4) + 6),x)

[Out]

int((log(x + 6)*(512*x + 1024*x^2 + 512*x^3) + log(x + 6)^2*(6400*x + 5632*x^2 + 768*x^3 + 1536))/(x + log(x +
6)^4*(393216*x^2 + 1638400*x^3 + 2621440*x^4 + 1966080*x^5 + 655360*x^6 + 65536*x^7) - log(x + 6)^2*(3072*x +
6656*x^2 + 4096*x^3 + 512*x^4) + 6), x)

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sympy [A]  time = 0.25, size = 24, normalized size = 1.09 \begin {gather*} - \frac {1}{\left (256 x^{3} + 512 x^{2} + 256 x\right ) \log {\left (x + 6 \right )}^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((768*x**3+5632*x**2+6400*x+1536)*ln(x+6)**2+(512*x**3+1024*x**2+512*x)*ln(x+6))/((65536*x**7+655360
*x**6+1966080*x**5+2621440*x**4+1638400*x**3+393216*x**2)*ln(x+6)**4+(-512*x**4-4096*x**3-6656*x**2-3072*x)*ln
(x+6)**2+x+6),x)

[Out]

-1/((256*x**3 + 512*x**2 + 256*x)*log(x + 6)**2 - 1)

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