Integrand size = 15, antiderivative size = 50 \[ \int x^6 \sqrt {-x+x^4} \, dx=\frac {1}{72} \sqrt {-x+x^4} \left (-3 x-2 x^4+8 x^7\right )-\frac {1}{24} \text {arctanh}\left (\frac {x^2}{\sqrt {-x+x^4}}\right ) \]
Time = 0.14 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.32 \[ \int x^6 \sqrt {-x+x^4} \, dx=\frac {\sqrt {x \left (-1+x^3\right )} \left (x^{3/2} \left (-3-2 x^3+8 x^6\right )-\frac {3 \log \left (x^{3/2}+\sqrt {-1+x^3}\right )}{\sqrt {-1+x^3}}\right )}{72 \sqrt {x}} \]
(Sqrt[x*(-1 + x^3)]*(x^(3/2)*(-3 - 2*x^3 + 8*x^6) - (3*Log[x^(3/2) + Sqrt[ -1 + x^3]])/Sqrt[-1 + x^3]))/(72*Sqrt[x])
Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.66, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1927, 1930, 1930, 1935, 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.
\(\displaystyle \int x^6 \sqrt {x^4-x} \, dx\) |
\(\Big \downarrow \) 1927 |
\(\displaystyle \frac {1}{9} x^7 \sqrt {x^4-x}-\frac {1}{6} \int \frac {x^7}{\sqrt {x^4-x}}dx\) |
\(\Big \downarrow \) 1930 |
\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \int \frac {x^4}{\sqrt {x^4-x}}dx-\frac {1}{6} \sqrt {x^4-x} x^4\right )+\frac {1}{9} \sqrt {x^4-x} x^7\) |
\(\Big \downarrow \) 1930 |
\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \int \frac {x}{\sqrt {x^4-x}}dx+\frac {1}{3} \sqrt {x^4-x} x\right )-\frac {1}{6} \sqrt {x^4-x} x^4\right )+\frac {1}{9} \sqrt {x^4-x} x^7\) |
\(\Big \downarrow \) 1935 |
\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{3} \int \frac {1}{1-\frac {x^4}{x^4-x}}d\frac {x^2}{\sqrt {x^4-x}}+\frac {1}{3} \sqrt {x^4-x} x\right )-\frac {1}{6} \sqrt {x^4-x} x^4\right )+\frac {1}{9} \sqrt {x^4-x} x^7\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{3} \text {arctanh}\left (\frac {x^2}{\sqrt {x^4-x}}\right )+\frac {1}{3} \sqrt {x^4-x} x\right )-\frac {1}{6} \sqrt {x^4-x} x^4\right )+\frac {1}{9} \sqrt {x^4-x} x^7\) |
(x^7*Sqrt[-x + x^4])/9 + (-1/6*(x^4*Sqrt[-x + x^4]) - (3*((x*Sqrt[-x + x^4 ])/3 + ArcTanh[x^2/Sqrt[-x + x^4]]/3))/4)/6
3.7.35.3.1 Defintions of rubi rules used
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[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(c*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a* (n - j)*(p/(c^j*(m + n*p + 1))) Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1) , x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[p] && LtQ[0, j, n] && (Int egersQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && NeQ[m + n*p + 1, 0]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a*x^j + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*p + 1))) I nt[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && Gt Q[m + j*p - n + j + 1, 0] && NeQ[m + n*p + 1, 0]
Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp [-2/(n - j) Subst[Int[1/(1 - a*x^2), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]
Time = 3.39 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96
method | result | size |
trager | \(\frac {x \left (8 x^{6}-2 x^{3}-3\right ) \sqrt {x^{4}-x}}{72}-\frac {\ln \left (-2 x^{3}-2 x \sqrt {x^{4}-x}+1\right )}{48}\) | \(48\) |
risch | \(\frac {x^{2} \left (8 x^{6}-2 x^{3}-3\right ) \left (x^{3}-1\right )}{72 \sqrt {x \left (x^{3}-1\right )}}-\frac {\ln \left (-2 x^{3}-2 x \sqrt {x^{4}-x}+1\right )}{48}\) | \(55\) |
meijerg | \(\frac {i \sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (\frac {i \sqrt {\pi }\, x^{\frac {3}{2}} \left (-40 x^{6}+10 x^{3}+15\right ) \sqrt {-x^{3}+1}}{60}-\frac {i \sqrt {\pi }\, \arcsin \left (x^{\frac {3}{2}}\right )}{4}\right )}{6 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}}\) | \(66\) |
default | \(\frac {x^{3} \left (\left (16 x^{7}-4 x^{4}-6 x \right ) \sqrt {x^{4}-x}+3 \ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )-3 \ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )\right )}{144 \left (x^{2}-\sqrt {x^{4}-x}\right )^{3} \left (x^{2}+\sqrt {x^{4}-x}\right )^{3}}\) | \(105\) |
pseudoelliptic | \(\frac {x^{3} \left (\left (16 x^{7}-4 x^{4}-6 x \right ) \sqrt {x^{4}-x}+3 \ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )-3 \ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )\right )}{144 \left (x^{2}-\sqrt {x^{4}-x}\right )^{3} \left (x^{2}+\sqrt {x^{4}-x}\right )^{3}}\) | \(105\) |
elliptic | \(\frac {x^{7} \sqrt {x^{4}-x}}{9}-\frac {x^{4} \sqrt {x^{4}-x}}{36}-\frac {x \sqrt {x^{4}-x}}{24}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \left (x -1\right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\operatorname {EllipticPi}\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{8 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (x -1\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(329\) |
Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96 \[ \int x^6 \sqrt {-x+x^4} \, dx=\frac {1}{72} \, {\left (8 \, x^{7} - 2 \, x^{4} - 3 \, x\right )} \sqrt {x^{4} - x} + \frac {1}{48} \, \log \left (2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x - 1\right ) \]
\[ \int x^6 \sqrt {-x+x^4} \, dx=\int x^{6} \sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
\[ \int x^6 \sqrt {-x+x^4} \, dx=\int { \sqrt {x^{4} - x} x^{6} \,d x } \]
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.12 \[ \int x^6 \sqrt {-x+x^4} \, dx=\frac {1}{72} \, {\left (2 \, {\left (4 \, x^{3} - 1\right )} x^{3} - 3\right )} \sqrt {x^{4} - x} x - \frac {1}{48} \, \log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right ) + \frac {1}{48} \, \log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right ) \]
1/72*(2*(4*x^3 - 1)*x^3 - 3)*sqrt(x^4 - x)*x - 1/48*log(sqrt(-1/x^3 + 1) + 1) + 1/48*log(abs(sqrt(-1/x^3 + 1) - 1))
Timed out. \[ \int x^6 \sqrt {-x+x^4} \, dx=\int x^6\,\sqrt {x^4-x} \,d x \]