Integrand size = 24, antiderivative size = 56 \[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {\sqrt {-x+x^4} \left (2 b-2 b x^3+3 a x^6\right )}{9 x^5}-\frac {1}{3} a \text {arctanh}\left (\frac {x^2}{\sqrt {-x+x^4}}\right ) \] Output:
1/9*(x^4-x)^(1/2)*(3*a*x^6-2*b*x^3+2*b)/x^5-1/3*a*arctanh(x^2/(x^4-x)^(1/2 ))
Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {\sqrt {x \left (-1+x^3\right )} \left (\sqrt {-1+x^3} \left (3 a x^6-2 b \left (-1+x^3\right )\right )-3 a x^{9/2} \log \left (x^{3/2}+\sqrt {-1+x^3}\right )\right )}{9 x^5 \sqrt {-1+x^3}} \] Input:
Integrate[(Sqrt[-x + x^4]*(-b + a*x^6))/x^6,x]
Output:
(Sqrt[x*(-1 + x^3)]*(Sqrt[-1 + x^3]*(3*a*x^6 - 2*b*(-1 + x^3)) - 3*a*x^(9/ 2)*Log[x^(3/2) + Sqrt[-1 + x^3]]))/(9*x^5*Sqrt[-1 + x^3])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.48 (sec) , antiderivative size = 178, normalized size of antiderivative = 3.18, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2445, 2449, 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.
\(\displaystyle \int \frac {\sqrt {x^4-x} \left (a x^6-b\right )}{x^6} \, dx\) |
\(\Big \downarrow \) 2445 |
\(\displaystyle \int \frac {\left (-a x^2-b\right ) \sqrt {x^4-x}}{x^6}dx+\frac {a \left (x^4-x\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 2449 |
\(\displaystyle \int \left (-\frac {\sqrt {x^4-x} a}{x^4}-\frac {b \sqrt {x^4-x}}{x^6}\right )dx+\frac {a \left (x^4-x\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3^{3/4} a (1-x) x \sqrt {\frac {x^2+x+1}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5 \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x^4-x}}+\frac {a \left (x^4-x\right )^{3/2}}{3 x^3}+\frac {2 a \sqrt {x^4-x}}{5 x^3}-\frac {2 b \left (x^4-x\right )^{3/2}}{9 x^6}\) |
Input:
Int[(Sqrt[-x + x^4]*(-b + a*x^6))/x^6,x]
Output:
(2*a*Sqrt[-x + x^4])/(5*x^3) - (2*b*(-x + x^4)^(3/2))/(9*x^6) + (a*(-x + x ^4)^(3/2))/(3*x^3) - (3^(3/4)*a*(1 - x)*x*Sqrt[(1 + x + x^2)/(1 - (1 + Sqr t[3])*x)^2]*EllipticF[ArcCos[(1 - (1 - Sqrt[3])*x)/(1 - (1 + Sqrt[3])*x)], (2 + Sqrt[3])/4])/(5*Sqrt[-(((1 - x)*x)/(1 - (1 + Sqrt[3])*x)^2)]*Sqrt[-x + x^4])
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_S ymbol] :> With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Int[(c*x)^ m*ExpandToSum[Pq - Pqq*x^q - a*Pqq*(m + q - n + 1)*(x^(q - n)/(b*(m + q + n *p + 1))), x]*(a*x^j + b*x^n)^p, x] + Simp[Pqq*(c*x)^(m + q - n + 1)*((a*x^ j + b*x^n)^(p + 1)/(b*c^(q - n + 1)*(m + q + n*p + 1))), x]] /; GtQ[q, n - 1] && NeQ[m + q + n*p + 1, 0] && (IntegerQ[2*p] || IntegerQ[p + (q + 1)/(2* n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && !IntegerQ[p] && IGtQ [j, 0] && IGtQ[n, 0] && LtQ[j, n]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_S ymbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ [{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !Integer Q[p] && NeQ[n, j]
Time = 0.84 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.98
method | result | size |
trager | \(\frac {\sqrt {x^{4}-x}\, \left (3 a \,x^{6}-2 b \,x^{3}+2 b \right )}{9 x^{5}}-\frac {a \ln \left (-2 x^{3}-2 x \sqrt {x^{4}-x}+1\right )}{6}\) | \(55\) |
risch | \(\frac {\left (x^{3}-1\right ) \left (3 a \,x^{6}-2 b \,x^{3}+2 b \right )}{9 x^{4} \sqrt {x \left (x^{3}-1\right )}}-\frac {a \ln \left (-2 x^{3}-2 x \sqrt {x^{4}-x}+1\right )}{6}\) | \(60\) |
pseudoelliptic | \(\frac {\left (6 a \,x^{6}-4 b \,x^{3}+4 b \right ) \sqrt {x^{4}-x}+3 \left (\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )\right ) a \,x^{5}}{18 x^{5}}\) | \(80\) |
meijerg | \(\frac {i a \sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (-2 i \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {-x^{3}+1}-2 i \sqrt {\pi }\, \arcsin \left (x^{\frac {3}{2}}\right )\right )}{6 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}}+\frac {2 \sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, b \left (-x^{3}+1\right )^{\frac {3}{2}}}{9 \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, x^{\frac {9}{2}}}\) | \(89\) |
default | \(a \left (\frac {x \sqrt {x^{4}-x}}{3}-\frac {\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}+\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}\right )-b \left (-\frac {2 \sqrt {x^{4}-x}}{9 x^{5}}+\frac {2 \sqrt {x^{4}-x}}{9 x^{2}}\right )\) | \(91\) |
elliptic | \(\frac {2 b \sqrt {x^{4}-x}}{9 x^{5}}-\frac {2 b \sqrt {x^{4}-x}}{9 x^{2}}+\frac {a x \sqrt {x^{4}-x}}{3}-\frac {a \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\operatorname {EllipticPi}\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(333\) |
Input:
int((x^4-x)^(1/2)*(a*x^6-b)/x^6,x,method=_RETURNVERBOSE)
Output:
1/9*(x^4-x)^(1/2)*(3*a*x^6-2*b*x^3+2*b)/x^5-1/6*a*ln(-2*x^3-2*x*(x^4-x)^(1 /2)+1)
Time = 0.17 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {3 \, a x^{5} \log \left (2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x - 1\right ) + 2 \, {\left (3 \, a x^{6} - 2 \, b x^{3} + 2 \, b\right )} \sqrt {x^{4} - x}}{18 \, x^{5}} \] Input:
integrate((x^4-x)^(1/2)*(a*x^6-b)/x^6,x, algorithm="fricas")
Output:
1/18*(3*a*x^5*log(2*x^3 - 2*sqrt(x^4 - x)*x - 1) + 2*(3*a*x^6 - 2*b*x^3 + 2*b)*sqrt(x^4 - x))/x^5
\[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\int \frac {\sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (a x^{6} - b\right )}{x^{6}}\, dx \] Input:
integrate((x**4-x)**(1/2)*(a*x**6-b)/x**6,x)
Output:
Integral(sqrt(x*(x - 1)*(x**2 + x + 1))*(a*x**6 - b)/x**6, x)
\[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\int { \frac {{\left (a x^{6} - b\right )} \sqrt {x^{4} - x}}{x^{6}} \,d x } \] Input:
integrate((x^4-x)^(1/2)*(a*x^6-b)/x^6,x, algorithm="maxima")
Output:
integrate((a*x^6 - b)*sqrt(x^4 - x)/x^6, x)
Time = 0.46 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {1}{3} \, \sqrt {x^{4} - x} a x - \frac {2}{9} \, b {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {3}{2}} - \frac {1}{6} \, a \log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right ) + \frac {1}{6} \, a \log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right ) \] Input:
integrate((x^4-x)^(1/2)*(a*x^6-b)/x^6,x, algorithm="giac")
Output:
1/3*sqrt(x^4 - x)*a*x - 2/9*b*(-1/x^3 + 1)^(3/2) - 1/6*a*log(sqrt(-1/x^3 + 1) + 1) + 1/6*a*log(abs(sqrt(-1/x^3 + 1) - 1))
Time = 8.46 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {2\,a\,x\,\sqrt {x^4-x}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {1}{2};\ \frac {3}{2};\ x^3\right )}{3\,\sqrt {1-x^3}}-\frac {2\,b\,\sqrt {x^4-x}\,\left (x^3-1\right )}{9\,x^5} \] Input:
int(-((x^4 - x)^(1/2)*(b - a*x^6))/x^6,x)
Output:
(2*a*x*(x^4 - x)^(1/2)*hypergeom([-1/2, 1/2], 3/2, x^3))/(3*(1 - x^3)^(1/2 )) - (2*b*(x^4 - x)^(1/2)*(x^3 - 1))/(9*x^5)
Time = 0.14 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {6 \sqrt {x^{3}-1}\, a \,x^{6}-4 \sqrt {x^{3}-1}\, b \,x^{3}+4 \sqrt {x^{3}-1}\, b +3 \sqrt {x}\, \mathrm {log}\left (\sqrt {x^{3}-1}-\sqrt {x}\, x \right ) a \,x^{4}-3 \sqrt {x}\, \mathrm {log}\left (\sqrt {x^{3}-1}+\sqrt {x}\, x \right ) a \,x^{4}}{18 \sqrt {x}\, x^{4}} \] Input:
int((x^4-x)^(1/2)*(a*x^6-b)/x^6,x)
Output:
(6*sqrt(x**3 - 1)*a*x**6 - 4*sqrt(x**3 - 1)*b*x**3 + 4*sqrt(x**3 - 1)*b + 3*sqrt(x)*log(sqrt(x**3 - 1) - sqrt(x)*x)*a*x**4 - 3*sqrt(x)*log(sqrt(x**3 - 1) + sqrt(x)*x)*a*x**4)/(18*sqrt(x)*x**4)