Integrand size = 13, antiderivative size = 54 \[ \int \frac {5+\sqrt [4]{x}}{-6+x} \, dx=4 \sqrt [4]{x}-2 \sqrt [4]{6} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{6}}\right )-2 \sqrt [4]{6} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{6}}\right )+5 \log (6-x) \] Output:
4*x^(1/4)-2*6^(1/4)*arctan(1/6*x^(1/4)*6^(3/4))-2*6^(1/4)*arctanh(1/6*x^(1 /4)*6^(3/4))+5*ln(6-x)
Time = 0.14 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.96 \[ \int \frac {5+\sqrt [4]{x}}{-6+x} \, dx=4 \sqrt [4]{x}-2 \sqrt [4]{6} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{6}}\right )-2 \sqrt [4]{6} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{6}}\right )+5 \log (-6+x) \] Input:
Integrate[(5 + x^(1/4))/(-6 + x),x]
Output:
4*x^(1/4) - 2*6^(1/4)*ArcTan[x^(1/4)/6^(1/4)] - 2*6^(1/4)*ArcTanh[x^(1/4)/ 6^(1/4)] + 5*Log[-6 + x]
Time = 0.50 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {7267, 25, 2370, 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.
\(\displaystyle \int \frac {\sqrt [4]{x}+5}{x-6} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 4 \int -\frac {\left (\sqrt [4]{x}+5\right ) x^{3/4}}{6-x}d\sqrt [4]{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -4 \int \frac {\left (\sqrt [4]{x}+5\right ) x^{3/4}}{6-x}d\sqrt [4]{x}\) |
\(\Big \downarrow \) 2370 |
\(\displaystyle -4 \int \left (\frac {x}{6-x}+\frac {5 x^{3/4}}{6-x}\right )d\sqrt [4]{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \left (-\frac {\sqrt [4]{3} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{6}}\right )}{2^{3/4}}-\frac {\sqrt [4]{3} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{6}}\right )}{2^{3/4}}+\sqrt [4]{x}+\frac {5}{4} \log (6-x)\right )\) |
Input:
Int[(5 + x^(1/4))/(-6 + x),x]
Output:
4*(x^(1/4) - (3^(1/4)*ArcTan[x^(1/4)/6^(1/4)])/2^(3/4) - (3^(1/4)*ArcTanh[ x^(1/4)/6^(1/4)])/2^(3/4) + (5*Log[6 - x])/4)
Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[ {v = Sum[(c*x)^(m + ii)*((Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2) )/(c^ii*(a + b*x^n))), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{ a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 3.67 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(4 x^{\frac {1}{4}}-6^{\frac {1}{4}} \left (\ln \left (\frac {x^{\frac {1}{4}}+6^{\frac {1}{4}}}{x^{\frac {1}{4}}-6^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x^{\frac {1}{4}} 6^{\frac {3}{4}}}{6}\right )\right )+5 \ln \left (-6+x \right )\) | \(50\) |
default | \(4 x^{\frac {1}{4}}-6^{\frac {1}{4}} \left (\ln \left (\frac {x^{\frac {1}{4}}+6^{\frac {1}{4}}}{x^{\frac {1}{4}}-6^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x^{\frac {1}{4}} 6^{\frac {3}{4}}}{6}\right )\right )+5 \ln \left (-6+x \right )\) | \(50\) |
meijerg | \(5 \ln \left (1-\frac {x}{6}\right )-6^{\frac {1}{4}} \left (-1\right )^{\frac {3}{4}} \left (\frac {2 \,8^{\frac {1}{4}} 3^{\frac {3}{4}} x^{\frac {1}{4}} \left (-1\right )^{\frac {1}{4}}}{3}+\left (-1\right )^{\frac {1}{4}} \left (\ln \left (1-\frac {x^{\frac {1}{4}} 6^{\frac {3}{4}}}{6}\right )-\ln \left (1+\frac {x^{\frac {1}{4}} 6^{\frac {3}{4}}}{6}\right )-2 \arctan \left (\frac {x^{\frac {1}{4}} 6^{\frac {3}{4}}}{6}\right )\right )\right )\) | \(73\) |
trager | \(\text {Expression too large to display}\) | \(2948\) |
Input:
int((5+x^(1/4))/(-6+x),x,method=_RETURNVERBOSE)
Output:
4*x^(1/4)-6^(1/4)*(ln((x^(1/4)+6^(1/4))/(x^(1/4)-6^(1/4)))+2*arctan(1/6*x^ (1/4)*6^(3/4)))+5*ln(-6+x)
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13 \[ \int \frac {5+\sqrt [4]{x}}{-6+x} \, dx=-{\left (6^{\frac {1}{4}} - 5\right )} \log \left (6^{\frac {1}{4}} + x^{\frac {1}{4}}\right ) + {\left (6^{\frac {1}{4}} + 5\right )} \log \left (-6^{\frac {1}{4}} + x^{\frac {1}{4}}\right ) - 2 \cdot 6^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 6^{\frac {3}{4}} x^{\frac {1}{4}}\right ) + 4 \, x^{\frac {1}{4}} + 5 \, \log \left (\sqrt {6} + \sqrt {x}\right ) \] Input:
integrate((5+x^(1/4))/(x-6),x, algorithm="fricas")
Output:
-(6^(1/4) - 5)*log(6^(1/4) + x^(1/4)) + (6^(1/4) + 5)*log(-6^(1/4) + x^(1/ 4)) - 2*6^(1/4)*arctan(1/6*6^(3/4)*x^(1/4)) + 4*x^(1/4) + 5*log(sqrt(6) + sqrt(x))
Time = 0.58 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.85 \[ \int \frac {5+\sqrt [4]{x}}{-6+x} \, dx=4 \sqrt [4]{x} + \sqrt [4]{6} \log {\left (\sqrt [4]{x} - \sqrt [4]{6} \right )} + 5 \log {\left (\sqrt [4]{x} - \sqrt [4]{6} \right )} - \sqrt [4]{6} \log {\left (\sqrt [4]{x} + \sqrt [4]{6} \right )} + 5 \log {\left (\sqrt [4]{x} + \sqrt [4]{6} \right )} + 5 \log {\left (\sqrt {x} + \sqrt {6} \right )} - 2 \cdot \sqrt [4]{6} \operatorname {atan}{\left (\frac {6^{\frac {3}{4}} \sqrt [4]{x}}{6} \right )} \] Input:
integrate((5+x**(1/4))/(x-6),x)
Output:
4*x**(1/4) + 6**(1/4)*log(x**(1/4) - 6**(1/4)) + 5*log(x**(1/4) - 6**(1/4) ) - 6**(1/4)*log(x**(1/4) + 6**(1/4)) + 5*log(x**(1/4) + 6**(1/4)) + 5*log (sqrt(x) + sqrt(6)) - 2*6**(1/4)*atan(6**(3/4)*x**(1/4)/6)
Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.24 \[ \int \frac {5+\sqrt [4]{x}}{-6+x} \, dx=-2 \cdot 6^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 6^{\frac {3}{4}} x^{\frac {1}{4}}\right ) + 6^{\frac {1}{4}} \log \left (-\frac {6^{\frac {1}{4}} - x^{\frac {1}{4}}}{6^{\frac {1}{4}} + x^{\frac {1}{4}}}\right ) + 4 \, x^{\frac {1}{4}} + 5 \, \log \left (\sqrt {6} + \sqrt {x}\right ) + 5 \, \log \left (-\sqrt {6} + \sqrt {x}\right ) \] Input:
integrate((5+x^(1/4))/(x-6),x, algorithm="maxima")
Output:
-2*6^(1/4)*arctan(1/6*6^(3/4)*x^(1/4)) + 6^(1/4)*log(-(6^(1/4) - x^(1/4))/ (6^(1/4) + x^(1/4))) + 4*x^(1/4) + 5*log(sqrt(6) + sqrt(x)) + 5*log(-sqrt( 6) + sqrt(x))
Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.02 \[ \int \frac {5+\sqrt [4]{x}}{-6+x} \, dx=-2 \cdot 6^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 6^{\frac {3}{4}} x^{\frac {1}{4}}\right ) - 6^{\frac {1}{4}} \log \left (6^{\frac {1}{4}} + x^{\frac {1}{4}}\right ) + 6^{\frac {1}{4}} \log \left ({\left | -6^{\frac {1}{4}} + x^{\frac {1}{4}} \right |}\right ) + 4 \, x^{\frac {1}{4}} + 5 \, \log \left ({\left | x - 6 \right |}\right ) \] Input:
integrate((5+x^(1/4))/(x-6),x, algorithm="giac")
Output:
-2*6^(1/4)*arctan(1/6*6^(3/4)*x^(1/4)) - 6^(1/4)*log(6^(1/4) + x^(1/4)) + 6^(1/4)*log(abs(-6^(1/4) + x^(1/4))) + 4*x^(1/4) + 5*log(abs(x - 6))
Time = 0.11 (sec) , antiderivative size = 162, normalized size of antiderivative = 3.00 \[ \int \frac {5+\sqrt [4]{x}}{-6+x} \, dx=\ln \left (11520\,x^{1/4}-\left (6^{1/4}+5\right )\,\left (2304\,x^{1/4}-2304\,6^{1/4}+11520\right )+57600\right )\,\left (6^{1/4}+5\right )-\ln \left (\left (6^{1/4}-5\right )\,\left (2304\,6^{1/4}+2304\,x^{1/4}+11520\right )+11520\,x^{1/4}+57600\right )\,\left (6^{1/4}-5\right )-\ln \left (11520\,x^{1/4}+\left (\sqrt {-\sqrt {6}}-5\right )\,\left (2304\,\sqrt {-\sqrt {6}}+2304\,x^{1/4}+11520\right )+57600\right )\,\left (\sqrt {-\sqrt {6}}-5\right )+\ln \left (11520\,x^{1/4}-\left (\sqrt {-\sqrt {6}}+5\right )\,\left (2304\,x^{1/4}-2304\,\sqrt {-\sqrt {6}}+11520\right )+57600\right )\,\left (\sqrt {-\sqrt {6}}+5\right )+4\,x^{1/4} \] Input:
int((x^(1/4) + 5)/(x - 6),x)
Output:
log(11520*x^(1/4) - (6^(1/4) + 5)*(2304*x^(1/4) - 2304*6^(1/4) + 11520) + 57600)*(6^(1/4) + 5) - log((6^(1/4) - 5)*(2304*6^(1/4) + 2304*x^(1/4) + 11 520) + 11520*x^(1/4) + 57600)*(6^(1/4) - 5) - log(11520*x^(1/4) + ((-6^(1/ 2))^(1/2) - 5)*(2304*(-6^(1/2))^(1/2) + 2304*x^(1/4) + 11520) + 57600)*((- 6^(1/2))^(1/2) - 5) + log(11520*x^(1/4) - ((-6^(1/2))^(1/2) + 5)*(2304*x^( 1/4) - 2304*(-6^(1/2))^(1/2) + 11520) + 57600)*((-6^(1/2))^(1/2) + 5) + 4* x^(1/4)
Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {5+\sqrt [4]{x}}{-6+x} \, dx=-2 \,6^{\frac {1}{4}} \mathit {atan} \left (\frac {x^{\frac {1}{4}} 6^{\frac {3}{4}}}{6}\right )+4 x^{\frac {1}{4}}-6^{\frac {1}{4}} \mathrm {log}\left (x^{\frac {1}{4}}+6^{\frac {1}{4}}\right )+6^{\frac {1}{4}} \mathrm {log}\left (x^{\frac {1}{4}}-6^{\frac {1}{4}}\right )+5 \,\mathrm {log}\left (x -6\right ) \] Input:
int((5+x^(1/4))/(x-6),x)
Output:
- 2*6**(1/4)*atan(x**(1/4)/6**(1/4)) + 4*x**(1/4) - 6**(1/4)*log(x**(1/4) + 6**(1/4)) + 6**(1/4)*log(x**(1/4) - 6**(1/4)) + 5*log(x - 6)