Integrand size = 69, antiderivative size = 26 \[ \int \frac {-3 x^2+12 x^4+16 x^5+5 x^6+\left (-6 x+16 x^3+20 x^4+6 x^5\right ) \log (5)-3 \log ^2(5)}{3 x^2+6 x \log (5)+3 \log ^2(5)} \, dx=-x+\frac {x^2 \left (2 x+x^2\right )^2}{3 (x+\log (5))} \] Output:
1/3*(x^2+2*x)^2*x^2/(ln(5)+x)-x
Leaf count is larger than twice the leaf count of optimal. \(111\) vs. \(2(26)=52\).
Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 4.27 \[ \int \frac {-3 x^2+12 x^4+16 x^5+5 x^6+\left (-6 x+16 x^3+20 x^4+6 x^5\right ) \log (5)-3 \log ^2(5)}{3 x^2+6 x \log (5)+3 \log ^2(5)} \, dx=\frac {24 x^5+6 x^6+x^4 \left (24+15 \log ^2(5)-5 \log (5) \log (125)\right )+6 x^2 \left (-3+15 \log ^4(5)-5 \log ^3(5) \log (125)\right )+6 \log ^4(5) \left (4+25 \log ^2(5)-4 \log (5) (1+\log (15625))\right )+6 x \log (5) \left (-3+4 \log ^2(5)+25 \log ^4(5)-4 \log ^3(5) (1+\log (15625))\right )}{18 (x+\log (5))} \] Input:
Integrate[(-3*x^2 + 12*x^4 + 16*x^5 + 5*x^6 + (-6*x + 16*x^3 + 20*x^4 + 6* x^5)*Log[5] - 3*Log[5]^2)/(3*x^2 + 6*x*Log[5] + 3*Log[5]^2),x]
Output:
(24*x^5 + 6*x^6 + x^4*(24 + 15*Log[5]^2 - 5*Log[5]*Log[125]) + 6*x^2*(-3 + 15*Log[5]^4 - 5*Log[5]^3*Log[125]) + 6*Log[5]^4*(4 + 25*Log[5]^2 - 4*Log[ 5]*(1 + Log[15625])) + 6*x*Log[5]*(-3 + 4*Log[5]^2 + 25*Log[5]^4 - 4*Log[5 ]^3*(1 + Log[15625])))/(18*(x + Log[5]))
Leaf count is larger than twice the leaf count of optimal. \(100\) vs. \(2(26)=52\).
Time = 0.38 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.85, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2007, 2389, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {5 x^6+16 x^5+12 x^4-3 x^2+\left (6 x^5+20 x^4+16 x^3-6 x\right ) \log (5)-3 \log ^2(5)}{3 x^2+6 x \log (5)+3 \log ^2(5)} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {5 x^6+16 x^5+12 x^4-3 x^2+\left (6 x^5+20 x^4+16 x^3-6 x\right ) \log (5)-3 \log ^2(5)}{\left (\sqrt {3} x+\sqrt {3} \log (5)\right )^2}dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (\frac {5 x^4}{3}-\frac {4}{3} x^3 (\log (5)-4)+x^2 (\log (5)-2)^2-\frac {(\log (5)-2)^2 \log ^4(5)}{3 (x+\log (5))^2}-\frac {2}{3} x (\log (5)-2)^2 \log (5)+\frac {1}{3} \left (-3+\log ^4(5)-4 \log ^3(5)+4 \log ^2(5)\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^5}{3}+\frac {1}{3} x^4 (4-\log (5))+\frac {1}{3} x^3 (2-\log (5))^2-\frac {1}{3} x^2 (2-\log (5))^2 \log (5)+\frac {(2-\log (5))^2 \log ^4(5)}{3 (x+\log (5))}-\frac {1}{3} x \left (3-\log ^4(5)+4 \log ^3(5)-4 \log ^2(5)\right )\) |
Input:
Int[(-3*x^2 + 12*x^4 + 16*x^5 + 5*x^6 + (-6*x + 16*x^3 + 20*x^4 + 6*x^5)*L og[5] - 3*Log[5]^2)/(3*x^2 + 6*x*Log[5] + 3*Log[5]^2),x]
Output:
x^5/3 + (x^3*(2 - Log[5])^2)/3 + (x^4*(4 - Log[5]))/3 - (x^2*(2 - Log[5])^ 2*Log[5])/3 + ((2 - Log[5])^2*Log[5]^4)/(3*(x + Log[5])) - (x*(3 - 4*Log[5 ]^2 + 4*Log[5]^3 - Log[5]^4))/3
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Time = 0.36 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27
method | result | size |
norman | \(\frac {-x^{2}+\frac {4 x^{4}}{3}+\frac {4 x^{5}}{3}+\frac {x^{6}}{3}+\ln \left (5\right )^{2}}{\ln \left (5\right )+x}\) | \(33\) |
gosper | \(\frac {x^{6}+4 x^{5}+4 x^{4}+3 \ln \left (5\right )^{2}-3 x^{2}}{3 \ln \left (5\right )+3 x}\) | \(34\) |
parallelrisch | \(\frac {x^{6}+4 x^{5}+4 x^{4}+3 \ln \left (5\right )^{2}-3 x^{2}}{3 \ln \left (5\right )+3 x}\) | \(34\) |
default | \(\frac {\ln \left (5\right )^{4} x}{3}-\frac {\ln \left (5\right )^{3} x^{2}}{3}+\frac {x^{3} \ln \left (5\right )^{2}}{3}-\frac {x^{4} \ln \left (5\right )}{3}+\frac {x^{5}}{3}-\frac {4 \ln \left (5\right )^{3} x}{3}+\frac {4 x^{2} \ln \left (5\right )^{2}}{3}-\frac {4 x^{3} \ln \left (5\right )}{3}+\frac {4 x^{4}}{3}+\frac {4 x \ln \left (5\right )^{2}}{3}-\frac {4 x^{2} \ln \left (5\right )}{3}+\frac {4 x^{3}}{3}-x +\frac {\ln \left (5\right )^{4} \left (\ln \left (5\right )^{2}-4 \ln \left (5\right )+4\right )}{3 \ln \left (5\right )+3 x}\) | \(111\) |
risch | \(\frac {\ln \left (5\right )^{4} x}{3}-\frac {\ln \left (5\right )^{3} x^{2}}{3}+\frac {x^{3} \ln \left (5\right )^{2}}{3}-\frac {x^{4} \ln \left (5\right )}{3}+\frac {x^{5}}{3}-\frac {4 \ln \left (5\right )^{3} x}{3}+\frac {4 x^{2} \ln \left (5\right )^{2}}{3}-\frac {4 x^{3} \ln \left (5\right )}{3}+\frac {4 x^{4}}{3}+\frac {4 x \ln \left (5\right )^{2}}{3}-\frac {4 x^{2} \ln \left (5\right )}{3}+\frac {4 x^{3}}{3}-x +\frac {\ln \left (5\right )^{6}}{3 \ln \left (5\right )+3 x}-\frac {4 \ln \left (5\right )^{5}}{3 \left (\ln \left (5\right )+x \right )}+\frac {4 \ln \left (5\right )^{4}}{3 \left (\ln \left (5\right )+x \right )}\) | \(125\) |
meijerg | \(-\frac {x}{1+\frac {x}{\ln \left (5\right )}}+\frac {5 \ln \left (5\right )^{5} \left (\frac {x \left (\frac {14 x^{5}}{\ln \left (5\right )^{5}}-\frac {21 x^{4}}{\ln \left (5\right )^{4}}+\frac {35 x^{3}}{\ln \left (5\right )^{3}}-\frac {70 x^{2}}{\ln \left (5\right )^{2}}+\frac {210 x}{\ln \left (5\right )}+420\right )}{70 \ln \left (5\right ) \left (1+\frac {x}{\ln \left (5\right )}\right )}-6 \ln \left (1+\frac {x}{\ln \left (5\right )}\right )\right )}{3}-\ln \left (5\right ) \left (\frac {x \left (\frac {3 x}{\ln \left (5\right )}+6\right )}{3 \ln \left (5\right ) \left (1+\frac {x}{\ln \left (5\right )}\right )}-2 \ln \left (1+\frac {x}{\ln \left (5\right )}\right )\right )+\ln \left (5\right )^{4} \left (2 \ln \left (5\right )+\frac {16}{3}\right ) \left (-\frac {x \left (-\frac {3 x^{4}}{\ln \left (5\right )^{4}}+\frac {5 x^{3}}{\ln \left (5\right )^{3}}-\frac {10 x^{2}}{\ln \left (5\right )^{2}}+\frac {30 x}{\ln \left (5\right )}+60\right )}{12 \ln \left (5\right ) \left (1+\frac {x}{\ln \left (5\right )}\right )}+5 \ln \left (1+\frac {x}{\ln \left (5\right )}\right )\right )+\ln \left (5\right )^{3} \left (\frac {20 \ln \left (5\right )}{3}+4\right ) \left (\frac {x \left (\frac {5 x^{3}}{\ln \left (5\right )^{3}}-\frac {10 x^{2}}{\ln \left (5\right )^{2}}+\frac {30 x}{\ln \left (5\right )}+60\right )}{15 \ln \left (5\right ) \left (1+\frac {x}{\ln \left (5\right )}\right )}-4 \ln \left (1+\frac {x}{\ln \left (5\right )}\right )\right )+\frac {16 \ln \left (5\right )^{3} \left (-\frac {x \left (-\frac {2 x^{2}}{\ln \left (5\right )^{2}}+\frac {6 x}{\ln \left (5\right )}+12\right )}{4 \ln \left (5\right ) \left (1+\frac {x}{\ln \left (5\right )}\right )}+3 \ln \left (1+\frac {x}{\ln \left (5\right )}\right )\right )}{3}-2 \ln \left (5\right ) \left (-\frac {x}{\ln \left (5\right ) \left (1+\frac {x}{\ln \left (5\right )}\right )}+\ln \left (1+\frac {x}{\ln \left (5\right )}\right )\right )\) | \(364\) |
Input:
int((-3*ln(5)^2+(6*x^5+20*x^4+16*x^3-6*x)*ln(5)+5*x^6+16*x^5+12*x^4-3*x^2) /(3*ln(5)^2+6*x*ln(5)+3*x^2),x,method=_RETURNVERBOSE)
Output:
(-x^2+4/3*x^4+4/3*x^5+1/3*x^6+ln(5)^2)/(ln(5)+x)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {-3 x^2+12 x^4+16 x^5+5 x^6+\left (-6 x+16 x^3+20 x^4+6 x^5\right ) \log (5)-3 \log ^2(5)}{3 x^2+6 x \log (5)+3 \log ^2(5)} \, dx=\frac {x^{6} + {\left (x - 4\right )} \log \left (5\right )^{5} + \log \left (5\right )^{6} + 4 \, x^{5} - 4 \, {\left (x - 1\right )} \log \left (5\right )^{4} + 4 \, x^{4} + 4 \, x \log \left (5\right )^{3} - 3 \, x^{2} - 3 \, x \log \left (5\right )}{3 \, {\left (x + \log \left (5\right )\right )}} \] Input:
integrate((-3*log(5)^2+(6*x^5+20*x^4+16*x^3-6*x)*log(5)+5*x^6+16*x^5+12*x^ 4-3*x^2)/(3*log(5)^2+6*x*log(5)+3*x^2),x, algorithm="fricas")
Output:
1/3*(x^6 + (x - 4)*log(5)^5 + log(5)^6 + 4*x^5 - 4*(x - 1)*log(5)^4 + 4*x^ 4 + 4*x*log(5)^3 - 3*x^2 - 3*x*log(5))/(x + log(5))
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (20) = 40\).
Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.46 \[ \int \frac {-3 x^2+12 x^4+16 x^5+5 x^6+\left (-6 x+16 x^3+20 x^4+6 x^5\right ) \log (5)-3 \log ^2(5)}{3 x^2+6 x \log (5)+3 \log ^2(5)} \, dx=\frac {x^{5}}{3} + x^{4} \cdot \left (\frac {4}{3} - \frac {\log {\left (5 \right )}}{3}\right ) + x^{3} \left (- \frac {4 \log {\left (5 \right )}}{3} + \frac {\log {\left (5 \right )}^{2}}{3} + \frac {4}{3}\right ) + x^{2} \left (- \frac {4 \log {\left (5 \right )}}{3} - \frac {\log {\left (5 \right )}^{3}}{3} + \frac {4 \log {\left (5 \right )}^{2}}{3}\right ) + x \left (- \frac {4 \log {\left (5 \right )}^{3}}{3} - 1 + \frac {\log {\left (5 \right )}^{4}}{3} + \frac {4 \log {\left (5 \right )}^{2}}{3}\right ) + \frac {- 4 \log {\left (5 \right )}^{5} + \log {\left (5 \right )}^{6} + 4 \log {\left (5 \right )}^{4}}{3 x + 3 \log {\left (5 \right )}} \] Input:
integrate((-3*ln(5)**2+(6*x**5+20*x**4+16*x**3-6*x)*ln(5)+5*x**6+16*x**5+1 2*x**4-3*x**2)/(3*ln(5)**2+6*x*ln(5)+3*x**2),x)
Output:
x**5/3 + x**4*(4/3 - log(5)/3) + x**3*(-4*log(5)/3 + log(5)**2/3 + 4/3) + x**2*(-4*log(5)/3 - log(5)**3/3 + 4*log(5)**2/3) + x*(-4*log(5)**3/3 - 1 + log(5)**4/3 + 4*log(5)**2/3) + (-4*log(5)**5 + log(5)**6 + 4*log(5)**4)/( 3*x + 3*log(5))
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (24) = 48\).
Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.69 \[ \int \frac {-3 x^2+12 x^4+16 x^5+5 x^6+\left (-6 x+16 x^3+20 x^4+6 x^5\right ) \log (5)-3 \log ^2(5)}{3 x^2+6 x \log (5)+3 \log ^2(5)} \, dx=\frac {1}{3} \, x^{5} - \frac {1}{3} \, x^{4} {\left (\log \left (5\right ) - 4\right )} + \frac {1}{3} \, {\left (\log \left (5\right )^{2} - 4 \, \log \left (5\right ) + 4\right )} x^{3} - \frac {1}{3} \, {\left (\log \left (5\right )^{3} - 4 \, \log \left (5\right )^{2} + 4 \, \log \left (5\right )\right )} x^{2} + \frac {1}{3} \, {\left (\log \left (5\right )^{4} - 4 \, \log \left (5\right )^{3} + 4 \, \log \left (5\right )^{2} - 3\right )} x + \frac {\log \left (5\right )^{6} - 4 \, \log \left (5\right )^{5} + 4 \, \log \left (5\right )^{4}}{3 \, {\left (x + \log \left (5\right )\right )}} \] Input:
integrate((-3*log(5)^2+(6*x^5+20*x^4+16*x^3-6*x)*log(5)+5*x^6+16*x^5+12*x^ 4-3*x^2)/(3*log(5)^2+6*x*log(5)+3*x^2),x, algorithm="maxima")
Output:
1/3*x^5 - 1/3*x^4*(log(5) - 4) + 1/3*(log(5)^2 - 4*log(5) + 4)*x^3 - 1/3*( log(5)^3 - 4*log(5)^2 + 4*log(5))*x^2 + 1/3*(log(5)^4 - 4*log(5)^3 + 4*log (5)^2 - 3)*x + 1/3*(log(5)^6 - 4*log(5)^5 + 4*log(5)^4)/(x + log(5))
Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (24) = 48\).
Time = 0.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.35 \[ \int \frac {-3 x^2+12 x^4+16 x^5+5 x^6+\left (-6 x+16 x^3+20 x^4+6 x^5\right ) \log (5)-3 \log ^2(5)}{3 x^2+6 x \log (5)+3 \log ^2(5)} \, dx=\frac {1}{3} \, x^{5} - \frac {1}{3} \, x^{4} \log \left (5\right ) + \frac {1}{3} \, x^{3} \log \left (5\right )^{2} - \frac {1}{3} \, x^{2} \log \left (5\right )^{3} + \frac {1}{3} \, x \log \left (5\right )^{4} + \frac {4}{3} \, x^{4} - \frac {4}{3} \, x^{3} \log \left (5\right ) + \frac {4}{3} \, x^{2} \log \left (5\right )^{2} - \frac {4}{3} \, x \log \left (5\right )^{3} + \frac {4}{3} \, x^{3} - \frac {4}{3} \, x^{2} \log \left (5\right ) + \frac {4}{3} \, x \log \left (5\right )^{2} - x + \frac {\log \left (5\right )^{6} - 4 \, \log \left (5\right )^{5} + 4 \, \log \left (5\right )^{4}}{3 \, {\left (x + \log \left (5\right )\right )}} \] Input:
integrate((-3*log(5)^2+(6*x^5+20*x^4+16*x^3-6*x)*log(5)+5*x^6+16*x^5+12*x^ 4-3*x^2)/(3*log(5)^2+6*x*log(5)+3*x^2),x, algorithm="giac")
Output:
1/3*x^5 - 1/3*x^4*log(5) + 1/3*x^3*log(5)^2 - 1/3*x^2*log(5)^3 + 1/3*x*log (5)^4 + 4/3*x^4 - 4/3*x^3*log(5) + 4/3*x^2*log(5)^2 - 4/3*x*log(5)^3 + 4/3 *x^3 - 4/3*x^2*log(5) + 4/3*x*log(5)^2 - x + 1/3*(log(5)^6 - 4*log(5)^5 + 4*log(5)^4)/(x + log(5))
Time = 0.07 (sec) , antiderivative size = 196, normalized size of antiderivative = 7.54 \[ \int \frac {-3 x^2+12 x^4+16 x^5+5 x^6+\left (-6 x+16 x^3+20 x^4+6 x^5\right ) \log (5)-3 \log ^2(5)}{3 x^2+6 x \log (5)+3 \log ^2(5)} \, dx=\frac {4\,{\ln \left (5\right )}^4-4\,{\ln \left (5\right )}^5+{\ln \left (5\right )}^6}{3\,x+3\,\ln \left (5\right )}-x\,\left (2\,\ln \left (5\right )\,\left (\frac {16\,\ln \left (5\right )}{3}-2\,\ln \left (5\right )\,\left (\frac {20\,\ln \left (5\right )}{3}-\frac {5\,{\ln \left (5\right )}^2}{3}+2\,\ln \left (5\right )\,\left (\frac {4\,\ln \left (5\right )}{3}-\frac {16}{3}\right )+4\right )+{\ln \left (5\right )}^2\,\left (\frac {4\,\ln \left (5\right )}{3}-\frac {16}{3}\right )\right )+{\ln \left (5\right )}^2\,\left (\frac {20\,\ln \left (5\right )}{3}-\frac {5\,{\ln \left (5\right )}^2}{3}+2\,\ln \left (5\right )\,\left (\frac {4\,\ln \left (5\right )}{3}-\frac {16}{3}\right )+4\right )+1\right )-x^4\,\left (\frac {\ln \left (5\right )}{3}-\frac {4}{3}\right )+x^3\,\left (\frac {20\,\ln \left (5\right )}{9}-\frac {5\,{\ln \left (5\right )}^2}{9}+\frac {2\,\ln \left (5\right )\,\left (\frac {4\,\ln \left (5\right )}{3}-\frac {16}{3}\right )}{3}+\frac {4}{3}\right )+x^2\,\left (\frac {8\,\ln \left (5\right )}{3}-\ln \left (5\right )\,\left (\frac {20\,\ln \left (5\right )}{3}-\frac {5\,{\ln \left (5\right )}^2}{3}+2\,\ln \left (5\right )\,\left (\frac {4\,\ln \left (5\right )}{3}-\frac {16}{3}\right )+4\right )+\frac {{\ln \left (5\right )}^2\,\left (\frac {4\,\ln \left (5\right )}{3}-\frac {16}{3}\right )}{2}\right )+\frac {x^5}{3} \] Input:
int((log(5)*(16*x^3 - 6*x + 20*x^4 + 6*x^5) - 3*log(5)^2 - 3*x^2 + 12*x^4 + 16*x^5 + 5*x^6)/(6*x*log(5) + 3*log(5)^2 + 3*x^2),x)
Output:
(4*log(5)^4 - 4*log(5)^5 + log(5)^6)/(3*x + 3*log(5)) - x*(2*log(5)*((16*l og(5))/3 - 2*log(5)*((20*log(5))/3 - (5*log(5)^2)/3 + 2*log(5)*((4*log(5)) /3 - 16/3) + 4) + log(5)^2*((4*log(5))/3 - 16/3)) + log(5)^2*((20*log(5))/ 3 - (5*log(5)^2)/3 + 2*log(5)*((4*log(5))/3 - 16/3) + 4) + 1) - x^4*(log(5 )/3 - 4/3) + x^3*((20*log(5))/9 - (5*log(5)^2)/9 + (2*log(5)*((4*log(5))/3 - 16/3))/3 + 4/3) + x^2*((8*log(5))/3 - log(5)*((20*log(5))/3 - (5*log(5) ^2)/3 + 2*log(5)*((4*log(5))/3 - 16/3) + 4) + (log(5)^2*((4*log(5))/3 - 16 /3))/2) + x^5/3
Time = 0.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {-3 x^2+12 x^4+16 x^5+5 x^6+\left (-6 x+16 x^3+20 x^4+6 x^5\right ) \log (5)-3 \log ^2(5)}{3 x^2+6 x \log (5)+3 \log ^2(5)} \, dx=\frac {x \left (-3 \,\mathrm {log}\left (5\right )+x^{5}+4 x^{4}+4 x^{3}-3 x \right )}{3 \,\mathrm {log}\left (5\right )+3 x} \] Input:
int((-3*log(5)^2+(6*x^5+20*x^4+16*x^3-6*x)*log(5)+5*x^6+16*x^5+12*x^4-3*x^ 2)/(3*log(5)^2+6*x*log(5)+3*x^2),x)
Output:
(x*( - 3*log(5) + x**5 + 4*x**4 + 4*x**3 - 3*x))/(3*(log(5) + x))