3.85.49 \(\int \frac {1+(-30 x^2-34 x^3-17 x^4-25 x^5+6 x^6) \log (-25+5 x+(-5+x) \log (2))}{(-5+x) \log (-25+5 x+(-5+x) \log (2))} \, dx\)

Optimal. Leaf size=25 \[ x^2 \left (2+x^2\right ) \left (x+x^2\right )+\log (\log ((-5+x) (5+\log (2)))) \]

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Rubi [A]  time = 0.31, antiderivative size = 30, normalized size of antiderivative = 1.20, number of steps used = 10, number of rules used = 8, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.131, Rules used = {2444, 6688, 6742, 14, 2390, 12, 2302, 29} \begin {gather*} x^6+x^5+2 x^4+2 x^3+\log (\log (-((5-x) (5+\log (2))))) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + (-30*x^2 - 34*x^3 - 17*x^4 - 25*x^5 + 6*x^6)*Log[-25 + 5*x + (-5 + x)*Log[2]])/((-5 + x)*Log[-25 + 5*
x + (-5 + x)*Log[2]]),x]

[Out]

2*x^3 + 2*x^4 + x^5 + x^6 + Log[Log[-((5 - x)*(5 + Log[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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2444

Int[((a_.) + Log[(c_.)*(v_)^(n_.)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Int[u*(a + b*Log[c*ExpandToSum[v, x]^n])^p
, x] /; FreeQ[{a, b, c, n, p}, x] && LinearQ[v, x] &&  !LinearMatchQ[v, x] &&  !(EqQ[n, 1] && MatchQ[c*v, (e_.
)*((f_) + (g_.)*x) /; FreeQ[{e, f, g}, x]])

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 \frac {1+\left (-30 x^2-34 x^3-17 x^4-25 x^5+6 x^6\right ) \log (-25+5 x+(-5+x) \log (2))}{(-5+x) \log (-5 (5+\log (2))+x (5+\log (2)))} \, dx\\ &=\int \frac {-x^2 \left (-30-34 x-17 x^2-25 x^3+6 x^4\right )-\frac {1}{\log ((-5+x) (5+\log (2)))}}{5-x} \, dx\\ &=\int \left (x^2 \left (6+8 x+5 x^2+6 x^3\right )+\frac {1}{(-5+x) \log (-5 (5+\log (2))+x (5+\log (2)))}\right ) \, dx\\ &=\int x^2 \left (6+8 x+5 x^2+6 x^3\right ) \, dx+\int \frac {1}{(-5+x) \log (-5 (5+\log (2))+x (5+\log (2)))} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {5+\log (2)}{x \log (x)} \, dx,x,-5 (5+\log (2))+x (5+\log (2))\right )}{5+\log (2)}+\int \left (6 x^2+8 x^3+5 x^4+6 x^5\right ) \, dx\\ &=2 x^3+2 x^4+x^5+x^6+\operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,-5 (5+\log (2))+x (5+\log (2))\right )\\ &=2 x^3+2 x^4+x^5+x^6+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log ((-5+x) (5+\log (2)))\right )\\ &=2 x^3+2 x^4+x^5+x^6+\log (\log ((-5+x) (5+\log (2))))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 28, normalized size = 1.12 \begin {gather*} -20250+2 x^3+2 x^4+x^5+x^6+\log (\log ((-5+x) (5+\log (2)))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + (-30*x^2 - 34*x^3 - 17*x^4 - 25*x^5 + 6*x^6)*Log[-25 + 5*x + (-5 + x)*Log[2]])/((-5 + x)*Log[-2
5 + 5*x + (-5 + x)*Log[2]]),x]

[Out]

-20250 + 2*x^3 + 2*x^4 + x^5 + x^6 + Log[Log[(-5 + x)*(5 + Log[2])]]

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fricas [A]  time = 1.05, size = 30, normalized size = 1.20 \begin {gather*} x^{6} + x^{5} + 2 \, x^{4} + 2 \, x^{3} + \log \left (\log \left ({\left (x - 5\right )} \log \relax (2) + 5 \, x - 25\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^6-25*x^5-17*x^4-34*x^3-30*x^2)*log((x-5)*log(2)+5*x-25)+1)/(x-5)/log((x-5)*log(2)+5*x-25),x, a
lgorithm="fricas")

[Out]

x^6 + x^5 + 2*x^4 + 2*x^3 + log(log((x - 5)*log(2) + 5*x - 25))

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giac [A]  time = 0.18, size = 32, normalized size = 1.28 \begin {gather*} x^{6} + x^{5} + 2 \, x^{4} + 2 \, x^{3} + \log \left (\log \left (x \log \relax (2) + 5 \, x - 5 \, \log \relax (2) - 25\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^6-25*x^5-17*x^4-34*x^3-30*x^2)*log((x-5)*log(2)+5*x-25)+1)/(x-5)/log((x-5)*log(2)+5*x-25),x, a
lgorithm="giac")

[Out]

x^6 + x^5 + 2*x^4 + 2*x^3 + log(log(x*log(2) + 5*x - 5*log(2) - 25))

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maple [A]  time = 0.38, size = 31, normalized size = 1.24

method	result	size
norman	\(x^{5}+x^{6}+2 x^{3}+2 x^{4}+\ln \left (\ln \left (\left (x -5\right ) \ln \relax (2)+5 x -25\right )\right )\)	\(31\)
risch	\(x^{5}+x^{6}+2 x^{3}+2 x^{4}+\ln \left (\ln \left (\left (x -5\right ) \ln \relax (2)+5 x -25\right )\right )\)	\(31\)
derivativedivides	\(\frac {\frac {219100 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{2} \ln \relax (2)^{3}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {41880 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{3} \ln \relax (2)^{2}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {4020 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{4} \ln \relax (2)}{\left (\ln \relax (2)+5\right )^{5}}+\frac {5756250 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right ) \ln \relax (2)^{3}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {1643250 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{2} \ln \relax (2)^{2}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {209400 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{3} \ln \relax (2)}{\left (\ln \relax (2)+5\right )^{5}}+\frac {28781250 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right ) \ln \relax (2)^{2}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {575625 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right ) \ln \relax (2)^{4}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {18750 \ln \relax (2) \ln \left (\ln \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {\ln \relax (2)^{6} \ln \left (\ln \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {30 \ln \relax (2)^{5} \ln \left (\ln \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {23025 \ln \relax (2)^{5} \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {10955 \ln \relax (2)^{4} \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{2}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {375 \ln \relax (2)^{4} \ln \left (\ln \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {9375 \ln \relax (2)^{2} \ln \left (\ln \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {2500 \ln \relax (2)^{3} \ln \left (\ln \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {2792 \ln \relax (2)^{3} \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{3}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {402 \ln \relax (2)^{2} \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{4}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {31 \ln \relax (2) \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{5}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {5477500 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{2} \ln \relax (2)}{\left (\ln \relax (2)+5\right )^{5}}+\frac {71953125 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right ) \ln \relax (2)}{\left (\ln \relax (2)+5\right )^{5}}+\frac {\left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{6}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {15625 \ln \left (\ln \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {349000 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{3}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {6846875 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{2}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {155 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{5}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {10050 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{4}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {71953125 \left (\ln \relax (2)+5\right ) x -359765625 \ln \relax (2)-1798828125}{\left (\ln \relax (2)+5\right )^{5}}}{\ln \relax (2)+5}\)	\(683\)
default	\(\frac {\frac {219100 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{2} \ln \relax (2)^{3}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {41880 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{3} \ln \relax (2)^{2}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {4020 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{4} \ln \relax (2)}{\left (\ln \relax (2)+5\right )^{5}}+\frac {5756250 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right ) \ln \relax (2)^{3}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {1643250 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{2} \ln \relax (2)^{2}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {209400 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{3} \ln \relax (2)}{\left (\ln \relax (2)+5\right )^{5}}+\frac {28781250 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right ) \ln \relax (2)^{2}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {575625 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right ) \ln \relax (2)^{4}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {18750 \ln \relax (2) \ln \left (\ln \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {\ln \relax (2)^{6} \ln \left (\ln \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {30 \ln \relax (2)^{5} \ln \left (\ln \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {23025 \ln \relax (2)^{5} \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {10955 \ln \relax (2)^{4} \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{2}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {375 \ln \relax (2)^{4} \ln \left (\ln \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {9375 \ln \relax (2)^{2} \ln \left (\ln \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {2500 \ln \relax (2)^{3} \ln \left (\ln \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {2792 \ln \relax (2)^{3} \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{3}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {402 \ln \relax (2)^{2} \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{4}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {31 \ln \relax (2) \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{5}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {5477500 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{2} \ln \relax (2)}{\left (\ln \relax (2)+5\right )^{5}}+\frac {71953125 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right ) \ln \relax (2)}{\left (\ln \relax (2)+5\right )^{5}}+\frac {\left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{6}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {15625 \ln \left (\ln \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )\right )}{\left (\ln \relax (2)+5\right )^{5}}+\frac {349000 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{3}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {6846875 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{2}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {155 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{5}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {10050 \left (\left (\ln \relax (2)+5\right ) x -5 \ln \relax (2)-25\right )^{4}}{\left (\ln \relax (2)+5\right )^{5}}+\frac {71953125 \left (\ln \relax (2)+5\right ) x -359765625 \ln \relax (2)-1798828125}{\left (\ln \relax (2)+5\right )^{5}}}{\ln \relax (2)+5}\)	\(683\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x^6-25*x^5-17*x^4-34*x^3-30*x^2)*ln((x-5)*ln(2)+5*x-25)+1)/(x-5)/ln((x-5)*ln(2)+5*x-25),x,method=_RETU
RNVERBOSE)

[Out]

x^5+x^6+2*x^3+2*x^4+ln(ln((x-5)*ln(2)+5*x-25))

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maxima [A]  time = 0.53, size = 28, normalized size = 1.12 \begin {gather*} x^{6} + x^{5} + 2 \, x^{4} + 2 \, x^{3} + \log \left (\log \left (x - 5\right ) + \log \left (\log \relax (2) + 5\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^6-25*x^5-17*x^4-34*x^3-30*x^2)*log((x-5)*log(2)+5*x-25)+1)/(x-5)/log((x-5)*log(2)+5*x-25),x, a
lgorithm="maxima")

[Out]

x^6 + x^5 + 2*x^4 + 2*x^3 + log(log(x - 5) + log(log(2) + 5))

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mupad [B]  time = 0.47, size = 30, normalized size = 1.20 \begin {gather*} \ln \left (\ln \left (5\,x+\ln \relax (2)\,\left (x-5\right )-25\right )\right )+2\,x^3+2\,x^4+x^5+x^6 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(5*x + log(2)*(x - 5) - 25)*(30*x^2 + 34*x^3 + 17*x^4 + 25*x^5 - 6*x^6) - 1)/(log(5*x + log(2)*(x - 5
) - 25)*(x - 5)),x)

[Out]

log(log(5*x + log(2)*(x - 5) - 25)) + 2*x^3 + 2*x^4 + x^5 + x^6

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sympy [A]  time = 0.14, size = 31, normalized size = 1.24 \begin {gather*} x^{6} + x^{5} + 2 x^{4} + 2 x^{3} + \log {\left (\log {\left (5 x + \left (x - 5\right ) \log {\relax (2 )} - 25 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x**6-25*x**5-17*x**4-34*x**3-30*x**2)*ln((x-5)*ln(2)+5*x-25)+1)/(x-5)/ln((x-5)*ln(2)+5*x-25),x)

[Out]

x**6 + x**5 + 2*x**4 + 2*x**3 + log(log(5*x + (x - 5)*log(2) - 25))

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