Integrand size = 12, antiderivative size = 90 \[ \int \left (-1-x+x^2\right )^{5/2} \, dx=-\frac {125}{512} (1-2 x) \sqrt {-1-x+x^2}+\frac {25}{192} (1-2 x) \left (-1-x+x^2\right )^{3/2}-\frac {1}{12} (1-2 x) \left (-1-x+x^2\right )^{5/2}+\frac {625 \text {arctanh}\left (\frac {1-2 x}{2 \sqrt {-1-x+x^2}}\right )}{1024} \] Output:
-125/512*(1-2*x)*(x^2-x-1)^(1/2)+25/192*(1-2*x)*(x^2-x-1)^(3/2)-1/12*(1-2* x)*(x^2-x-1)^(5/2)+625/1024*arctanh(1/2*(1-2*x)/(x^2-x-1)^(1/2))
Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \left (-1-x+x^2\right )^{5/2} \, dx=\frac {\sqrt {-1-x+x^2} \left (-703+950 x+1240 x^2-400 x^3-640 x^4+256 x^5\right )}{1536}+\frac {625 \log \left (1-2 x+2 \sqrt {-1-x+x^2}\right )}{1024} \] Input:
Integrate[(-1 - x + x^2)^(5/2),x]
Output:
(Sqrt[-1 - x + x^2]*(-703 + 950*x + 1240*x^2 - 400*x^3 - 640*x^4 + 256*x^5 ))/1536 + (625*Log[1 - 2*x + 2*Sqrt[-1 - x + x^2]])/1024
Time = 0.37 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1087, 1087, 1087, 1092, 219}
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.
\(\displaystyle \int \left (x^2-x-1\right )^{5/2} \, dx\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle -\frac {25}{24} \int \left (x^2-x-1\right )^{3/2}dx-\frac {1}{12} (1-2 x) \left (x^2-x-1\right )^{5/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle -\frac {25}{24} \left (-\frac {15}{16} \int \sqrt {x^2-x-1}dx-\frac {1}{8} (1-2 x) \left (x^2-x-1\right )^{3/2}\right )-\frac {1}{12} (1-2 x) \left (x^2-x-1\right )^{5/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle -\frac {25}{24} \left (-\frac {15}{16} \left (-\frac {5}{8} \int \frac {1}{\sqrt {x^2-x-1}}dx-\frac {1}{4} \sqrt {x^2-x-1} (1-2 x)\right )-\frac {1}{8} (1-2 x) \left (x^2-x-1\right )^{3/2}\right )-\frac {1}{12} (1-2 x) \left (x^2-x-1\right )^{5/2}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle -\frac {25}{24} \left (-\frac {15}{16} \left (-\frac {5}{4} \int \frac {1}{4-\frac {(1-2 x)^2}{x^2-x-1}}d\left (-\frac {1-2 x}{\sqrt {x^2-x-1}}\right )-\frac {1}{4} \sqrt {x^2-x-1} (1-2 x)\right )-\frac {1}{8} (1-2 x) \left (x^2-x-1\right )^{3/2}\right )-\frac {1}{12} (1-2 x) \left (x^2-x-1\right )^{5/2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {25}{24} \left (-\frac {15}{16} \left (\frac {5}{8} \text {arctanh}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )-\frac {1}{4} (1-2 x) \sqrt {x^2-x-1}\right )-\frac {1}{8} (1-2 x) \left (x^2-x-1\right )^{3/2}\right )-\frac {1}{12} (1-2 x) \left (x^2-x-1\right )^{5/2}\) |
Input:
Int[(-1 - x + x^2)^(5/2),x]
Output:
-1/12*((1 - 2*x)*(-1 - x + x^2)^(5/2)) - (25*(-1/8*((1 - 2*x)*(-1 - x + x^ 2)^(3/2)) - (15*(-1/4*((1 - 2*x)*Sqrt[-1 - x + x^2]) + (5*ArcTanh[(1 - 2*x )/(2*Sqrt[-1 - x + x^2])])/8))/16))/24
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Time = 0.64 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.61
method | result | size |
risch | \(\frac {\left (256 x^{5}-640 x^{4}-400 x^{3}+1240 x^{2}+950 x -703\right ) \sqrt {x^{2}-x -1}}{1536}-\frac {625 \ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x -1}\right )}{1024}\) | \(55\) |
trager | \(\left (\frac {1}{6} x^{5}-\frac {5}{12} x^{4}-\frac {25}{96} x^{3}+\frac {155}{192} x^{2}+\frac {475}{768} x -\frac {703}{1536}\right ) \sqrt {x^{2}-x -1}-\frac {625 \ln \left (2 \sqrt {x^{2}-x -1}-1+2 x \right )}{1024}\) | \(58\) |
default | \(\frac {\left (2 x -1\right ) \left (x^{2}-x -1\right )^{\frac {5}{2}}}{12}-\frac {25 \left (x^{2}-x -1\right )^{\frac {3}{2}} \left (2 x -1\right )}{192}+\frac {125 \left (2 x -1\right ) \sqrt {x^{2}-x -1}}{512}-\frac {625 \ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x -1}\right )}{1024}\) | \(69\) |
Input:
int((x^2-x-1)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/1536*(256*x^5-640*x^4-400*x^3+1240*x^2+950*x-703)*(x^2-x-1)^(1/2)-625/10 24*ln(x-1/2+(x^2-x-1)^(1/2))
Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.64 \[ \int \left (-1-x+x^2\right )^{5/2} \, dx=\frac {1}{1536} \, {\left (256 \, x^{5} - 640 \, x^{4} - 400 \, x^{3} + 1240 \, x^{2} + 950 \, x - 703\right )} \sqrt {x^{2} - x - 1} + \frac {625}{1024} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1\right ) \] Input:
integrate((x^2-x-1)^(5/2),x, algorithm="fricas")
Output:
1/1536*(256*x^5 - 640*x^4 - 400*x^3 + 1240*x^2 + 950*x - 703)*sqrt(x^2 - x - 1) + 625/1024*log(-2*x + 2*sqrt(x^2 - x - 1) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (73) = 146\).
Time = 0.51 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.89 \[ \int \left (-1-x+x^2\right )^{5/2} \, dx=\left (\frac {x}{2} - \frac {1}{4}\right ) \sqrt {x^{2} - x - 1} + 2 \left (\frac {x^{2}}{3} - \frac {x}{12} - \frac {11}{24}\right ) \sqrt {x^{2} - x - 1} - \sqrt {x^{2} - x - 1} \left (\frac {x^{3}}{4} - \frac {x^{2}}{24} - \frac {17 x}{96} - \frac {67}{192}\right ) - 2 \sqrt {x^{2} - x - 1} \left (\frac {x^{4}}{5} - \frac {x^{3}}{40} - \frac {23 x^{2}}{240} - \frac {151 x}{960} - \frac {821}{1920}\right ) + \sqrt {x^{2} - x - 1} \left (\frac {x^{5}}{6} - \frac {x^{4}}{60} - \frac {29 x^{3}}{480} - \frac {89 x^{2}}{960} - \frac {793 x}{3840} - \frac {3803}{7680}\right ) - \frac {625 \log {\left (2 x + 2 \sqrt {x^{2} - x - 1} - 1 \right )}}{1024} \] Input:
integrate((x**2-x-1)**(5/2),x)
Output:
(x/2 - 1/4)*sqrt(x**2 - x - 1) + 2*(x**2/3 - x/12 - 11/24)*sqrt(x**2 - x - 1) - sqrt(x**2 - x - 1)*(x**3/4 - x**2/24 - 17*x/96 - 67/192) - 2*sqrt(x* *2 - x - 1)*(x**4/5 - x**3/40 - 23*x**2/240 - 151*x/960 - 821/1920) + sqrt (x**2 - x - 1)*(x**5/6 - x**4/60 - 29*x**3/480 - 89*x**2/960 - 793*x/3840 - 3803/7680) - 625*log(2*x + 2*sqrt(x**2 - x - 1) - 1)/1024
Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.07 \[ \int \left (-1-x+x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (x^{2} - x - 1\right )}^{\frac {5}{2}} x - \frac {1}{12} \, {\left (x^{2} - x - 1\right )}^{\frac {5}{2}} - \frac {25}{96} \, {\left (x^{2} - x - 1\right )}^{\frac {3}{2}} x + \frac {25}{192} \, {\left (x^{2} - x - 1\right )}^{\frac {3}{2}} + \frac {125}{256} \, \sqrt {x^{2} - x - 1} x - \frac {125}{512} \, \sqrt {x^{2} - x - 1} - \frac {625}{1024} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - x - 1} - 1\right ) \] Input:
integrate((x^2-x-1)^(5/2),x, algorithm="maxima")
Output:
1/6*(x^2 - x - 1)^(5/2)*x - 1/12*(x^2 - x - 1)^(5/2) - 25/96*(x^2 - x - 1) ^(3/2)*x + 25/192*(x^2 - x - 1)^(3/2) + 125/256*sqrt(x^2 - x - 1)*x - 125/ 512*sqrt(x^2 - x - 1) - 625/1024*log(2*x + 2*sqrt(x^2 - x - 1) - 1)
Time = 0.17 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.66 \[ \int \left (-1-x+x^2\right )^{5/2} \, dx=\frac {1}{1536} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, x - 5\right )} x - 25\right )} x + 155\right )} x + 475\right )} x - 703\right )} \sqrt {x^{2} - x - 1} + \frac {625}{1024} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1 \right |}\right ) \] Input:
integrate((x^2-x-1)^(5/2),x, algorithm="giac")
Output:
1/1536*(2*(4*(2*(8*(2*x - 5)*x - 25)*x + 155)*x + 475)*x - 703)*sqrt(x^2 - x - 1) + 625/1024*log(abs(-2*x + 2*sqrt(x^2 - x - 1) + 1))
Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.71 \[ \int \left (-1-x+x^2\right )^{5/2} \, dx=\frac {125\,\left (\frac {x}{2}-\frac {1}{4}\right )\,\sqrt {x^2-x-1}}{128}-\frac {625\,\ln \left (x+\sqrt {x^2-x-1}-\frac {1}{2}\right )}{1024}-\frac {25\,\left (x-\frac {1}{2}\right )\,{\left (x^2-x-1\right )}^{3/2}}{96}+\frac {\left (x-\frac {1}{2}\right )\,{\left (x^2-x-1\right )}^{5/2}}{6} \] Input:
int((x^2 - x - 1)^(5/2),x)
Output:
(125*(x/2 - 1/4)*(x^2 - x - 1)^(1/2))/128 - (625*log(x + (x^2 - x - 1)^(1/ 2) - 1/2))/1024 - (25*(x - 1/2)*(x^2 - x - 1)^(3/2))/96 + ((x - 1/2)*(x^2 - x - 1)^(5/2))/6
Time = 0.24 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.16 \[ \int \left (-1-x+x^2\right )^{5/2} \, dx=\frac {\sqrt {x^{2}-x -1}\, x^{5}}{6}-\frac {5 \sqrt {x^{2}-x -1}\, x^{4}}{12}-\frac {25 \sqrt {x^{2}-x -1}\, x^{3}}{96}+\frac {155 \sqrt {x^{2}-x -1}\, x^{2}}{192}+\frac {475 \sqrt {x^{2}-x -1}\, x}{768}-\frac {703 \sqrt {x^{2}-x -1}}{1536}-\frac {625 \,\mathrm {log}\left (\frac {2 \sqrt {x^{2}-x -1}+2 x -1}{\sqrt {5}}\right )}{1024} \] Input:
int((x^2-x-1)^(5/2),x)
Output:
(512*sqrt(x**2 - x - 1)*x**5 - 1280*sqrt(x**2 - x - 1)*x**4 - 800*sqrt(x** 2 - x - 1)*x**3 + 2480*sqrt(x**2 - x - 1)*x**2 + 1900*sqrt(x**2 - x - 1)*x - 1406*sqrt(x**2 - x - 1) - 1875*log((2*sqrt(x**2 - x - 1) + 2*x - 1)/sqr t(5)))/3072