3.70.66 \(\int \frac {6 x+3 x^2+e^4 (6 x+3 x^2)+(6 x+3 x^2) \log (3)+(-170-414 x-241 x^2-53 x^3-4 x^4+e^4 (-170-244 x-82 x^2-8 x^3)+(-170-244 x-82 x^2-8 x^3) \log (3)) \log (\frac {17+4 x}{5+x})}{(170 x+159 x^2+45 x^3+4 x^4) \log (\frac {17+4 x}{5+x})} \, dx\)

Optimal. Leaf size=33 \[ -x+\left (1+e^4+\log (3)\right ) \log \left (\frac {5 \log \left (4-\frac {3}{5+x}\right )}{x (2+x)}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 1.17, antiderivative size = 57, normalized size of antiderivative = 1.73, number of steps used = 7, number of rules used = 5, integrand size = 140, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {6, 6688, 6742, 893, 2504} \begin {gather*} -x-\frac {1}{2} \left (2+2 e^4+\log (9)\right ) \log (x)-\frac {1}{2} \left (2+2 e^4+\log (9)\right ) \log (x+2)+\left (1+e^4+\log (3)\right ) \log \left (\log \left (\frac {4 x+17}{x+5}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6*x + 3*x^2 + E^4*(6*x + 3*x^2) + (6*x + 3*x^2)*Log[3] + (-170 - 414*x - 241*x^2 - 53*x^3 - 4*x^4 + E^4*(
-170 - 244*x - 82*x^2 - 8*x^3) + (-170 - 244*x - 82*x^2 - 8*x^3)*Log[3])*Log[(17 + 4*x)/(5 + x)])/((170*x + 15
9*x^2 + 45*x^3 + 4*x^4)*Log[(17 + 4*x)/(5 + x)]),x]

[Out]

-x - ((2 + 2*E^4 + Log[9])*Log[x])/2 - ((2 + 2*E^4 + Log[9])*Log[2 + x])/2 + (1 + E^4 + Log[3])*Log[Log[(17 +
4*x)/(5 + x)]]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 2504

Int[(u_)/Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)], x_Symbol] :> With[{h
= Simplify[u*(a + b*x)*(c + d*x)]}, Simp[(h*Log[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]])/(p*r*(b*c - a*d)), x] /
; FreeQ[h, x]] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 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 \frac {6 x+3 x^2+\left (6 x+3 x^2\right ) \left (e^4+\log (3)\right )+\left (-170-414 x-241 x^2-53 x^3-4 x^4+e^4 \left (-170-244 x-82 x^2-8 x^3\right )+\left (-170-244 x-82 x^2-8 x^3\right ) \log (3)\right ) \log \left (\frac {17+4 x}{5+x}\right )}{\left (170 x+159 x^2+45 x^3+4 x^4\right ) \log \left (\frac {17+4 x}{5+x}\right )} \, dx\\ &=\int \frac {6 x+3 x^2+3 x (2+x) \left (e^4+\log (3)\right )-\left (85+37 x+4 x^2\right ) \left (2+x^2+2 e^4 (1+x)+\log (9)+x (4+\log (9))\right ) \log \left (\frac {17+4 x}{5+x}\right )}{x \left (170+159 x+45 x^2+4 x^3\right ) \log \left (\frac {17+4 x}{5+x}\right )} \, dx\\ &=\int \left (\frac {-2-2 e^4-x^2-\log (9)-x \left (4+2 e^4+\log (9)\right )}{x (2+x)}+\frac {3 \left (1+e^4+\log (3)\right )}{(5+x) (17+4 x) \log \left (\frac {17+4 x}{5+x}\right )}\right ) \, dx\\ &=\left (3 \left (1+e^4+\log (3)\right )\right ) \int \frac {1}{(5+x) (17+4 x) \log \left (\frac {17+4 x}{5+x}\right )} \, dx+\int \frac {-2-2 e^4-x^2-\log (9)-x \left (4+2 e^4+\log (9)\right )}{x (2+x)} \, dx\\ &=\left (1+e^4+\log (3)\right ) \log \left (\log \left (\frac {17+4 x}{5+x}\right )\right )+\int \left (-1+\frac {-2-2 e^4-\log (9)}{2 x}+\frac {-2-2 e^4-\log (9)}{2 (2+x)}\right ) \, dx\\ &=-x-\frac {1}{2} \left (2+2 e^4+\log (9)\right ) \log (x)-\frac {1}{2} \left (2+2 e^4+\log (9)\right ) \log (2+x)+\left (1+e^4+\log (3)\right ) \log \left (\log \left (\frac {17+4 x}{5+x}\right )\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 57, normalized size = 1.73 \begin {gather*} -x-\frac {1}{2} \left (2+2 e^4+\log (9)\right ) \log (x)-\frac {1}{2} \left (2+2 e^4+\log (9)\right ) \log (2+x)+\left (1+e^4+\log (3)\right ) \log \left (\log \left (\frac {17+4 x}{5+x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6*x + 3*x^2 + E^4*(6*x + 3*x^2) + (6*x + 3*x^2)*Log[3] + (-170 - 414*x - 241*x^2 - 53*x^3 - 4*x^4 +
E^4*(-170 - 244*x - 82*x^2 - 8*x^3) + (-170 - 244*x - 82*x^2 - 8*x^3)*Log[3])*Log[(17 + 4*x)/(5 + x)])/((170*
x + 159*x^2 + 45*x^3 + 4*x^4)*Log[(17 + 4*x)/(5 + x)]),x]

[Out]

-x - ((2 + 2*E^4 + Log[9])*Log[x])/2 - ((2 + 2*E^4 + Log[9])*Log[2 + x])/2 + (1 + E^4 + Log[3])*Log[Log[(17 +
4*x)/(5 + x)]]

________________________________________________________________________________________

fricas [A]  time = 0.78, size = 40, normalized size = 1.21 \begin {gather*} -{\left (e^{4} + \log \relax (3) + 1\right )} \log \left (x^{2} + 2 \, x\right ) + {\left (e^{4} + \log \relax (3) + 1\right )} \log \left (\log \left (\frac {4 \, x + 17}{x + 5}\right )\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*x^3-82*x^2-244*x-170)*log(3)+(-8*x^3-82*x^2-244*x-170)*exp(4)-4*x^4-53*x^3-241*x^2-414*x-170)*
log((4*x+17)/(5+x))+(3*x^2+6*x)*log(3)+(3*x^2+6*x)*exp(4)+3*x^2+6*x)/(4*x^4+45*x^3+159*x^2+170*x)/log((4*x+17)
/(5+x)),x, algorithm="fricas")

[Out]

-(e^4 + log(3) + 1)*log(x^2 + 2*x) + (e^4 + log(3) + 1)*log(log((4*x + 17)/(x + 5))) - x

________________________________________________________________________________________

giac [B]  time = 0.48, size = 502, normalized size = 15.21 \begin {gather*} -\frac {\frac {{\left (4 \, x + 17\right )} e^{4} \log \left (\frac {5 \, {\left (4 \, x + 17\right )}^{2}}{{\left (x + 5\right )}^{2}} - \frac {32 \, {\left (4 \, x + 17\right )}}{x + 5} + 51\right )}{x + 5} - 4 \, e^{4} \log \left (\frac {5 \, {\left (4 \, x + 17\right )}^{2}}{{\left (x + 5\right )}^{2}} - \frac {32 \, {\left (4 \, x + 17\right )}}{x + 5} + 51\right ) + \frac {{\left (4 \, x + 17\right )} \log \relax (3) \log \left (\frac {5 \, {\left (4 \, x + 17\right )}^{2}}{{\left (x + 5\right )}^{2}} - \frac {32 \, {\left (4 \, x + 17\right )}}{x + 5} + 51\right )}{x + 5} - 4 \, \log \relax (3) \log \left (\frac {5 \, {\left (4 \, x + 17\right )}^{2}}{{\left (x + 5\right )}^{2}} - \frac {32 \, {\left (4 \, x + 17\right )}}{x + 5} + 51\right ) - \frac {2 \, {\left (4 \, x + 17\right )} e^{4} \log \left (\frac {4 \, x + 17}{x + 5} - 4\right )}{x + 5} + 8 \, e^{4} \log \left (\frac {4 \, x + 17}{x + 5} - 4\right ) - \frac {2 \, {\left (4 \, x + 17\right )} \log \relax (3) \log \left (\frac {4 \, x + 17}{x + 5} - 4\right )}{x + 5} + 8 \, \log \relax (3) \log \left (\frac {4 \, x + 17}{x + 5} - 4\right ) - \frac {{\left (4 \, x + 17\right )} e^{4} \log \left (\log \left (\frac {4 \, x + 17}{x + 5}\right )\right )}{x + 5} + 4 \, e^{4} \log \left (\log \left (\frac {4 \, x + 17}{x + 5}\right )\right ) - \frac {{\left (4 \, x + 17\right )} \log \relax (3) \log \left (\log \left (\frac {4 \, x + 17}{x + 5}\right )\right )}{x + 5} + 4 \, \log \relax (3) \log \left (\log \left (\frac {4 \, x + 17}{x + 5}\right )\right ) + \frac {{\left (4 \, x + 17\right )} \log \left (\frac {5 \, {\left (4 \, x + 17\right )}^{2}}{{\left (x + 5\right )}^{2}} - \frac {32 \, {\left (4 \, x + 17\right )}}{x + 5} + 51\right )}{x + 5} - \frac {2 \, {\left (4 \, x + 17\right )} \log \left (\frac {4 \, x + 17}{x + 5} - 4\right )}{x + 5} - \frac {{\left (4 \, x + 17\right )} \log \left (\log \left (\frac {4 \, x + 17}{x + 5}\right )\right )}{x + 5} - 4 \, \log \left (\frac {5 \, {\left (4 \, x + 17\right )}^{2}}{{\left (x + 5\right )}^{2}} - \frac {32 \, {\left (4 \, x + 17\right )}}{x + 5} + 51\right ) + 8 \, \log \left (\frac {4 \, x + 17}{x + 5} - 4\right ) + 4 \, \log \left (\log \left (\frac {4 \, x + 17}{x + 5}\right )\right ) - 3}{\frac {4 \, x + 17}{x + 5} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*x^3-82*x^2-244*x-170)*log(3)+(-8*x^3-82*x^2-244*x-170)*exp(4)-4*x^4-53*x^3-241*x^2-414*x-170)*
log((4*x+17)/(5+x))+(3*x^2+6*x)*log(3)+(3*x^2+6*x)*exp(4)+3*x^2+6*x)/(4*x^4+45*x^3+159*x^2+170*x)/log((4*x+17)
/(5+x)),x, algorithm="giac")

[Out]

-((4*x + 17)*e^4*log(5*(4*x + 17)^2/(x + 5)^2 - 32*(4*x + 17)/(x + 5) + 51)/(x + 5) - 4*e^4*log(5*(4*x + 17)^2
/(x + 5)^2 - 32*(4*x + 17)/(x + 5) + 51) + (4*x + 17)*log(3)*log(5*(4*x + 17)^2/(x + 5)^2 - 32*(4*x + 17)/(x +
5) + 51)/(x + 5) - 4*log(3)*log(5*(4*x + 17)^2/(x + 5)^2 - 32*(4*x + 17)/(x + 5) + 51) - 2*(4*x + 17)*e^4*log
((4*x + 17)/(x + 5) - 4)/(x + 5) + 8*e^4*log((4*x + 17)/(x + 5) - 4) - 2*(4*x + 17)*log(3)*log((4*x + 17)/(x +
5) - 4)/(x + 5) + 8*log(3)*log((4*x + 17)/(x + 5) - 4) - (4*x + 17)*e^4*log(log((4*x + 17)/(x + 5)))/(x + 5)
+ 4*e^4*log(log((4*x + 17)/(x + 5))) - (4*x + 17)*log(3)*log(log((4*x + 17)/(x + 5)))/(x + 5) + 4*log(3)*log(l
og((4*x + 17)/(x + 5))) + (4*x + 17)*log(5*(4*x + 17)^2/(x + 5)^2 - 32*(4*x + 17)/(x + 5) + 51)/(x + 5) - 2*(4
*x + 17)*log((4*x + 17)/(x + 5) - 4)/(x + 5) - (4*x + 17)*log(log((4*x + 17)/(x + 5)))/(x + 5) - 4*log(5*(4*x
+ 17)^2/(x + 5)^2 - 32*(4*x + 17)/(x + 5) + 51) + 8*log((4*x + 17)/(x + 5) - 4) + 4*log(log((4*x + 17)/(x + 5)
)) - 3)/((4*x + 17)/(x + 5) - 4)

________________________________________________________________________________________

maple [A]  time = 0.13, size = 53, normalized size = 1.61

method	result	size
norman	\(-x +\left (-1-\ln \relax (3)-{\mathrm e}^{4}\right ) \ln \relax (x )+\left (-1-\ln \relax (3)-{\mathrm e}^{4}\right ) \ln \left (2+x \right )+\left (1+{\mathrm e}^{4}+\ln \relax (3)\right ) \ln \left (\ln \left (\frac {4 x +17}{5+x}\right )\right )\)	\(53\)
risch	\(-\ln \left (x^{2}+2 x \right ) \ln \relax (3)-\ln \left (x^{2}+2 x \right ) {\mathrm e}^{4}-\ln \left (x^{2}+2 x \right )-x +\ln \left (\ln \left (\frac {4 x +17}{5+x}\right )\right ) \ln \relax (3)+\ln \left (\ln \left (\frac {4 x +17}{5+x}\right )\right ) {\mathrm e}^{4}+\ln \left (\ln \left (\frac {4 x +17}{5+x}\right )\right )\)	\(84\)
derivativedivides	\(-\ln \relax (3) \ln \left (3-\frac {15}{5+x}\right )+\ln \relax (3) \ln \left (\ln \left (4-\frac {3}{5+x}\right )\right )-\ln \relax (3) \ln \left (1-\frac {3}{5+x}\right )+2 \ln \relax (3) \ln \left (-\frac {3}{5+x}\right )+\ln \left (\ln \left (4-\frac {3}{5+x}\right )\right ) {\mathrm e}^{4}+\ln \left (\ln \left (4-\frac {3}{5+x}\right )\right )-\ln \left (1-\frac {3}{5+x}\right ) {\mathrm e}^{4}-\ln \left (1-\frac {3}{5+x}\right )+2 \ln \left (-\frac {3}{5+x}\right ) {\mathrm e}^{4}+2 \ln \left (-\frac {3}{5+x}\right )-5-x -\ln \left (3-\frac {15}{5+x}\right ) {\mathrm e}^{4}-\ln \left (3-\frac {15}{5+x}\right )\)	\(159\)
default	\(-\ln \relax (3) \ln \left (3-\frac {15}{5+x}\right )+\ln \relax (3) \ln \left (\ln \left (4-\frac {3}{5+x}\right )\right )-\ln \relax (3) \ln \left (1-\frac {3}{5+x}\right )+2 \ln \relax (3) \ln \left (-\frac {3}{5+x}\right )+\ln \left (\ln \left (4-\frac {3}{5+x}\right )\right ) {\mathrm e}^{4}+\ln \left (\ln \left (4-\frac {3}{5+x}\right )\right )-\ln \left (1-\frac {3}{5+x}\right ) {\mathrm e}^{4}-\ln \left (1-\frac {3}{5+x}\right )+2 \ln \left (-\frac {3}{5+x}\right ) {\mathrm e}^{4}+2 \ln \left (-\frac {3}{5+x}\right )-5-x -\ln \left (3-\frac {15}{5+x}\right ) {\mathrm e}^{4}-\ln \left (3-\frac {15}{5+x}\right )\)	\(159\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-8*x^3-82*x^2-244*x-170)*ln(3)+(-8*x^3-82*x^2-244*x-170)*exp(4)-4*x^4-53*x^3-241*x^2-414*x-170)*ln((4*x
+17)/(5+x))+(3*x^2+6*x)*ln(3)+(3*x^2+6*x)*exp(4)+3*x^2+6*x)/(4*x^4+45*x^3+159*x^2+170*x)/ln((4*x+17)/(5+x)),x,
method=_RETURNVERBOSE)

[Out]

-x+(-1-ln(3)-exp(4))*ln(x)+(-1-ln(3)-exp(4))*ln(2+x)+(1+exp(4)+ln(3))*ln(ln((4*x+17)/(5+x)))

________________________________________________________________________________________

maxima [A]  time = 0.51, size = 47, normalized size = 1.42 \begin {gather*} -{\left (e^{4} + \log \relax (3) + 1\right )} \log \left (x + 2\right ) - {\left (e^{4} + \log \relax (3) + 1\right )} \log \relax (x) + {\left (e^{4} + \log \relax (3) + 1\right )} \log \left (\log \left (4 \, x + 17\right ) - \log \left (x + 5\right )\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*x^3-82*x^2-244*x-170)*log(3)+(-8*x^3-82*x^2-244*x-170)*exp(4)-4*x^4-53*x^3-241*x^2-414*x-170)*
log((4*x+17)/(5+x))+(3*x^2+6*x)*log(3)+(3*x^2+6*x)*exp(4)+3*x^2+6*x)/(4*x^4+45*x^3+159*x^2+170*x)/log((4*x+17)
/(5+x)),x, algorithm="maxima")

[Out]

-(e^4 + log(3) + 1)*log(x + 2) - (e^4 + log(3) + 1)*log(x) + (e^4 + log(3) + 1)*log(log(4*x + 17) - log(x + 5)
) - x

________________________________________________________________________________________

mupad [B]  time = 0.65, size = 38, normalized size = 1.15 \begin {gather*} \ln \left (\ln \left (\frac {4\,x+17}{x+5}\right )\right )\,\left ({\mathrm {e}}^4+\ln \relax (3)+1\right )-x-\ln \left (x\,\left (x+2\right )\right )\,\left ({\mathrm {e}}^4+\ln \relax (3)+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x + exp(4)*(6*x + 3*x^2) + log(3)*(6*x + 3*x^2) - log((4*x + 17)/(x + 5))*(414*x + exp(4)*(244*x + 82*x
^2 + 8*x^3 + 170) + log(3)*(244*x + 82*x^2 + 8*x^3 + 170) + 241*x^2 + 53*x^3 + 4*x^4 + 170) + 3*x^2)/(log((4*x
+ 17)/(x + 5))*(170*x + 159*x^2 + 45*x^3 + 4*x^4)),x)

[Out]

log(log((4*x + 17)/(x + 5)))*(exp(4) + log(3) + 1) - x - log(x*(x + 2))*(exp(4) + log(3) + 1)

________________________________________________________________________________________

sympy [A]  time = 0.42, size = 37, normalized size = 1.12 \begin {gather*} - x - \left (1 + \log {\relax (3 )} + e^{4}\right ) \log {\left (x^{2} + 2 x \right )} + \left (1 + \log {\relax (3 )} + e^{4}\right ) \log {\left (\log {\left (\frac {4 x + 17}{x + 5} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*x**3-82*x**2-244*x-170)*ln(3)+(-8*x**3-82*x**2-244*x-170)*exp(4)-4*x**4-53*x**3-241*x**2-414*x
-170)*ln((4*x+17)/(5+x))+(3*x**2+6*x)*ln(3)+(3*x**2+6*x)*exp(4)+3*x**2+6*x)/(4*x**4+45*x**3+159*x**2+170*x)/ln
((4*x+17)/(5+x)),x)

[Out]

-x - (1 + log(3) + exp(4))*log(x**2 + 2*x) + (1 + log(3) + exp(4))*log(log((4*x + 17)/(x + 5)))

________________________________________________________________________________________