Integrand size = 31, antiderivative size = 278 \[ \int \frac {(-7+x) \sqrt [3]{1+x-5 x^2+3 x^3}}{(-5+x) (-1+x)^3} \, dx=\frac {(-1+x)^{4/3} (1+3 x)^{2/3} \left (-\frac {21 \sqrt [3]{3} \left (-16 \sqrt [3]{1+3 x}+7 (1+3 x)^{4/3}\right )}{64 (-3+3 x)^{4/3}}+\frac {3 \sqrt [3]{3} \left (-80 \sqrt [3]{1+3 x}+23 (1+3 x)^{4/3}\right )}{64 (-3+3 x)^{4/3}}+\frac {\sqrt {3} \arctan \left (\frac {3^{5/6} \sqrt [3]{1+3 x}}{2\ 2^{2/3} \sqrt [3]{-3+3 x}+\sqrt [3]{3} \sqrt [3]{1+3 x}}\right )}{4 \sqrt [3]{2}}-\frac {\log \left (6^{2/3} \sqrt [3]{-3+3 x}-3 \sqrt [3]{1+3 x}\right )}{4 \sqrt [3]{2}}+\frac {\log \left (2 \sqrt [3]{6} (-3+3 x)^{2/3}+6^{2/3} \sqrt [3]{-3+3 x} \sqrt [3]{1+3 x}+3 (1+3 x)^{2/3}\right )}{8 \sqrt [3]{2}}\right )}{\left ((-1+x)^2 (1+3 x)\right )^{2/3}} \]
(-1+x)^(4/3)*(1+3*x)^(2/3)*(-21/64*3^(1/3)*(-16*(1+3*x)^(1/3)+7*(1+3*x)^(4 /3))/(-3+3*x)^(4/3)+3/64*3^(1/3)*(-80*(1+3*x)^(1/3)+23*(1+3*x)^(4/3))/(-3+ 3*x)^(4/3)+1/8*3^(1/2)*arctan(3^(5/6)*(1+3*x)^(1/3)/(2*2^(2/3)*(-3+3*x)^(1 /3)+3^(1/3)*(1+3*x)^(1/3)))*2^(2/3)-1/8*ln(6^(2/3)*(-3+3*x)^(1/3)-3*(1+3*x )^(1/3))*2^(2/3)+1/16*ln(2*6^(1/3)*(-3+3*x)^(2/3)+6^(2/3)*(-3+3*x)^(1/3)*( 1+3*x)^(1/3)+3*(1+3*x)^(2/3))*2^(2/3))/((-1+x)^2*(1+3*x))^(2/3)
Time = 0.32 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.70 \[ \int \frac {(-7+x) \sqrt [3]{1+x-5 x^2+3 x^3}}{(-5+x) (-1+x)^3} \, dx=\frac {1}{32} \sqrt [3]{(-1+x)^2 (1+3 x)} \left (\frac {3-39 x}{(-1+x)^2}+\frac {4\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt [3]{-1+x}+\sqrt [3]{2+6 x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{(-1+x)^{2/3} \sqrt [3]{1+3 x}}-\frac {4\ 2^{2/3} \log \left (-2+\frac {\sqrt [3]{2+6 x}}{\sqrt [3]{-1+x}}\right )}{(-1+x)^{2/3} \sqrt [3]{1+3 x}}+\frac {2\ 2^{2/3} \log \left (4+\frac {2 \sqrt [3]{2+6 x}}{\sqrt [3]{-1+x}}+\frac {(2+6 x)^{2/3}}{(-1+x)^{2/3}}\right )}{(-1+x)^{2/3} \sqrt [3]{1+3 x}}\right ) \]
(((-1 + x)^2*(1 + 3*x))^(1/3)*((3 - 39*x)/(-1 + x)^2 + (4*2^(2/3)*Sqrt[3]* ArcTan[((-1 + x)^(1/3) + (2 + 6*x)^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/((-1 + x)^(2/3)*(1 + 3*x)^(1/3)) - (4*2^(2/3)*Log[-2 + (2 + 6*x)^(1/3)/(-1 + x) ^(1/3)])/((-1 + x)^(2/3)*(1 + 3*x)^(1/3)) + (2*2^(2/3)*Log[4 + (2*(2 + 6*x )^(1/3))/(-1 + x)^(1/3) + (2 + 6*x)^(2/3)/(-1 + x)^(2/3)])/((-1 + x)^(2/3) *(1 + 3*x)^(1/3))))/32
Time = 0.87 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {7293, 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.
| \(\displaystyle \int \frac {(x-7) \sqrt [3]{3 x^3-5 x^2+x+1}}{(x-5) (x-1)^3} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\sqrt [3]{3 x^3-5 x^2+x+1}}{32 (x-5)}+\frac {\sqrt [3]{3 x^3-5 x^2+x+1}}{32 (x-1)}+\frac {\sqrt [3]{3 x^3-5 x^2+x+1}}{8 (x-1)^2}+\frac {3 \sqrt [3]{3 x^3-5 x^2+x+1}}{2 (x-1)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {3} \sqrt [3]{3 x^3-5 x^2+x+1} \arctan \left (\frac {2\ 2^{2/3} \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{3 x+1}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{2} (x-1)^{2/3} \sqrt [3]{3 x+1}}-\frac {9 \sqrt [3]{3 x^3-5 x^2+x+1} (3 x+1)}{32 (1-x)^2}+\frac {3 \sqrt [3]{3 x^3-5 x^2+x+1}}{8 (1-x)}+\frac {\sqrt [3]{3 x^3-5 x^2+x+1} \log (x-5)}{8 \sqrt [3]{2} (x-1)^{2/3} \sqrt [3]{3 x+1}}-\frac {3 \sqrt [3]{3 x^3-5 x^2+x+1} \log \left (4 \sqrt [3]{x-1}-2 \sqrt [3]{2} \sqrt [3]{3 x+1}\right )}{8 \sqrt [3]{2} (x-1)^{2/3} \sqrt [3]{3 x+1}}\) |
(3*(1 + x - 5*x^2 + 3*x^3)^(1/3))/(8*(1 - x)) - (9*(1 + 3*x)*(1 + x - 5*x^ 2 + 3*x^3)^(1/3))/(32*(1 - x)^2) - (Sqrt[3]*(1 + x - 5*x^2 + 3*x^3)^(1/3)* ArcTan[1/Sqrt[3] + (2*2^(2/3)*(-1 + x)^(1/3))/(Sqrt[3]*(1 + 3*x)^(1/3))])/ (4*2^(1/3)*(-1 + x)^(2/3)*(1 + 3*x)^(1/3)) + ((1 + x - 5*x^2 + 3*x^3)^(1/3 )*Log[-5 + x])/(8*2^(1/3)*(-1 + x)^(2/3)*(1 + 3*x)^(1/3)) - (3*(1 + x - 5* x^2 + 3*x^3)^(1/3)*Log[4*(-1 + x)^(1/3) - 2*2^(1/3)*(1 + 3*x)^(1/3)])/(8*2 ^(1/3)*(-1 + x)^(2/3)*(1 + 3*x)^(1/3))
3.29.15.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.64 (sec) , antiderivative size = 1546, normalized size of antiderivative = 5.56
-3/32*(13*x-1)/(-1+x)^2*(3*x^3-5*x^2+x+1)^(1/3)+1/8*RootOf(_Z^3+4)*ln((-19 2*RootOf(RootOf(_Z^3+4)^2+48*_Z*RootOf(_Z^3+4)+2304*_Z^2)*RootOf(_Z^3+4)^4 *x^2+23040*RootOf(RootOf(_Z^3+4)^2+48*_Z*RootOf(_Z^3+4)+2304*_Z^2)^2*RootO f(_Z^3+4)^3*x^2+192*RootOf(RootOf(_Z^3+4)^2+48*_Z*RootOf(_Z^3+4)+2304*_Z^2 )*RootOf(_Z^3+4)^4*x-23040*RootOf(RootOf(_Z^3+4)^2+48*_Z*RootOf(_Z^3+4)+23 04*_Z^2)^2*RootOf(_Z^3+4)^3*x+576*(3*x^3-5*x^2+x+1)^(2/3)*RootOf(RootOf(_Z ^3+4)^2+48*_Z*RootOf(_Z^3+4)+2304*_Z^2)*RootOf(_Z^3+4)^2-9*RootOf(_Z^3+4)^ 2*x^2+1080*RootOf(RootOf(_Z^3+4)^2+48*_Z*RootOf(_Z^3+4)+2304*_Z^2)*RootOf( _Z^3+4)*x^2-48*(3*x^3-5*x^2+x+1)^(1/3)*RootOf(_Z^3+4)*x+2016*(3*x^3-5*x^2+ x+1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+48*_Z*RootOf(_Z^3+4)+2304*_Z^2)*x+22*Ro otOf(_Z^3+4)^2*x-2640*RootOf(RootOf(_Z^3+4)^2+48*_Z*RootOf(_Z^3+4)+2304*_Z ^2)*RootOf(_Z^3+4)*x-90*(3*x^3-5*x^2+x+1)^(2/3)+48*(3*x^3-5*x^2+x+1)^(1/3) *RootOf(_Z^3+4)-2016*(3*x^3-5*x^2+x+1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+48*_Z *RootOf(_Z^3+4)+2304*_Z^2)-13*RootOf(_Z^3+4)^2+1560*RootOf(RootOf(_Z^3+4)^ 2+48*_Z*RootOf(_Z^3+4)+2304*_Z^2)*RootOf(_Z^3+4))/(-5+x)/(-1+x))-1/8*ln(-( 960*RootOf(RootOf(_Z^3+4)^2+48*_Z*RootOf(_Z^3+4)+2304*_Z^2)*RootOf(_Z^3+4) ^4*x^2-18432*RootOf(RootOf(_Z^3+4)^2+48*_Z*RootOf(_Z^3+4)+2304*_Z^2)^2*Roo tOf(_Z^3+4)^3*x^2-960*RootOf(RootOf(_Z^3+4)^2+48*_Z*RootOf(_Z^3+4)+2304*_Z ^2)*RootOf(_Z^3+4)^4*x+18432*RootOf(RootOf(_Z^3+4)^2+48*_Z*RootOf(_Z^3+4)+ 2304*_Z^2)^2*RootOf(_Z^3+4)^3*x+1152*(3*x^3-5*x^2+x+1)^(2/3)*RootOf(Roo...
Time = 0.26 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.85 \[ \int \frac {(-7+x) \sqrt [3]{1+x-5 x^2+3 x^3}}{(-5+x) (-1+x)^3} \, dx=-\frac {4 \, \sqrt {3} 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (x - 1\right )} + 2 \cdot 2^{\frac {1}{6}} \left (-1\right )^{\frac {2}{3}} {\left (3 \, x^{3} - 5 \, x^{2} + x + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x - 1\right )}}\right ) + 2 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )} \log \left (-\frac {2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (3 \, x^{3} - 5 \, x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 2 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{2} - 2 \, x + 1\right )} - {\left (3 \, x^{3} - 5 \, x^{2} + x + 1\right )}^{\frac {2}{3}}}{x^{2} - 2 \, x + 1}\right ) - 4 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )} \log \left (\frac {2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x - 1\right )} + {\left (3 \, x^{3} - 5 \, x^{2} + x + 1\right )}^{\frac {1}{3}}}{x - 1}\right ) + 3 \, {\left (3 \, x^{3} - 5 \, x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (13 \, x - 1\right )}}{32 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
-1/32*(4*sqrt(3)*2^(2/3)*(-1)^(1/3)*(x^2 - 2*x + 1)*arctan(1/6*sqrt(3)*2^( 1/6)*(2^(5/6)*(x - 1) + 2*2^(1/6)*(-1)^(2/3)*(3*x^3 - 5*x^2 + x + 1)^(1/3) )/(x - 1)) + 2*2^(2/3)*(-1)^(1/3)*(x^2 - 2*x + 1)*log(-(2^(2/3)*(-1)^(1/3) *(3*x^3 - 5*x^2 + x + 1)^(1/3)*(x - 1) - 2*2^(1/3)*(-1)^(2/3)*(x^2 - 2*x + 1) - (3*x^3 - 5*x^2 + x + 1)^(2/3))/(x^2 - 2*x + 1)) - 4*2^(2/3)*(-1)^(1/ 3)*(x^2 - 2*x + 1)*log((2^(2/3)*(-1)^(1/3)*(x - 1) + (3*x^3 - 5*x^2 + x + 1)^(1/3))/(x - 1)) + 3*(3*x^3 - 5*x^2 + x + 1)^(1/3)*(13*x - 1))/(x^2 - 2* x + 1)
\[ \int \frac {(-7+x) \sqrt [3]{1+x-5 x^2+3 x^3}}{(-5+x) (-1+x)^3} \, dx=\int \frac {\sqrt [3]{\left (x - 1\right )^{2} \cdot \left (3 x + 1\right )} \left (x - 7\right )}{\left (x - 5\right ) \left (x - 1\right )^{3}}\, dx \]
\[ \int \frac {(-7+x) \sqrt [3]{1+x-5 x^2+3 x^3}}{(-5+x) (-1+x)^3} \, dx=\int { \frac {{\left (3 \, x^{3} - 5 \, x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x - 7\right )}}{{\left (x - 1\right )}^{3} {\left (x - 5\right )}} \,d x } \]
\[ \int \frac {(-7+x) \sqrt [3]{1+x-5 x^2+3 x^3}}{(-5+x) (-1+x)^3} \, dx=\int { \frac {{\left (3 \, x^{3} - 5 \, x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x - 7\right )}}{{\left (x - 1\right )}^{3} {\left (x - 5\right )}} \,d x } \]
Timed out. \[ \int \frac {(-7+x) \sqrt [3]{1+x-5 x^2+3 x^3}}{(-5+x) (-1+x)^3} \, dx=\int \frac {\left (x-7\right )\,{\left (3\,x^3-5\,x^2+x+1\right )}^{1/3}}{{\left (x-1\right )}^3\,\left (x-5\right )} \,d x \]