Integrand size = 23, antiderivative size = 173 \[ \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx=-\frac {\sqrt {-\left (\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}\right )} \arctan \left (\frac {\sqrt {-\sqrt {a} \sqrt {b}-b} x \sqrt {-x+x^4}}{\sqrt {b} (-1+x) \left (1+x+x^2\right )}\right )}{3 \sqrt {a} b}+\frac {\sqrt {\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b}} \arctan \left (\frac {\sqrt {\sqrt {a} \sqrt {b}-b} x \sqrt {-x+x^4}}{\sqrt {b} (-1+x) \left (1+x+x^2\right )}\right )}{3 \sqrt {a} b} \]
-1/3*(-(a^(1/2)+b^(1/2))*b^(1/2))^(1/2)*arctan((-a^(1/2)*b^(1/2)-b)^(1/2)* x*(x^4-x)^(1/2)/b^(1/2)/(-1+x)/(x^2+x+1))/a^(1/2)/b+1/3*((a^(1/2)-b^(1/2)) *b^(1/2))^(1/2)*arctan((a^(1/2)*b^(1/2)-b)^(1/2)*x*(x^4-x)^(1/2)/b^(1/2)/( -1+x)/(x^2+x+1))/a^(1/2)/b
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.22 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx=\frac {\sqrt {x \left (-1+x^3\right )} \text {RootSum}\left [16 a-16 b+32 a \text {$\#$1}-32 b \text {$\#$1}+24 a \text {$\#$1}^2-16 b \text {$\#$1}^2+8 a \text {$\#$1}^3+a \text {$\#$1}^4\&,\frac {\log \left (-2+2 x^3+2 x^{3/2} \sqrt {-1+x^3}-\text {$\#$1}\right ) \text {$\#$1}^2}{8 a-8 b+12 a \text {$\#$1}-8 b \text {$\#$1}+6 a \text {$\#$1}^2+a \text {$\#$1}^3}\&\right ]}{3 \sqrt {x} \sqrt {-1+x^3}} \]
(Sqrt[x*(-1 + x^3)]*RootSum[16*a - 16*b + 32*a*#1 - 32*b*#1 + 24*a*#1^2 - 16*b*#1^2 + 8*a*#1^3 + a*#1^4 & , (Log[-2 + 2*x^3 + 2*x^(3/2)*Sqrt[-1 + x^ 3] - #1]*#1^2)/(8*a - 8*b + 12*a*#1 - 8*b*#1 + 6*a*#1^2 + a*#1^3) & ])/(3* Sqrt[x]*Sqrt[-1 + x^3])
Time = 0.45 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.35, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2467, 25, 1817, 1815, 1489, 27, 301, 224, 219, 291, 218, 221}
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.
| \(\displaystyle \int \frac {\sqrt {x^4-x}}{a x^6-b} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x^4-x} \int -\frac {\sqrt {x} \sqrt {x^3-1}}{b-a x^6}dx}{\sqrt {x} \sqrt {x^3-1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x^4-x} \int \frac {\sqrt {x} \sqrt {x^3-1}}{b-a x^6}dx}{\sqrt {x} \sqrt {x^3-1}}\) |
| \(\Big \downarrow \) 1817 |
| \(\displaystyle -\frac {2 \sqrt {x^4-x} \int \frac {x \sqrt {x^3-1}}{b-a x^6}d\sqrt {x}}{\sqrt {x} \sqrt {x^3-1}}\) |
| \(\Big \downarrow \) 1815 |
| \(\displaystyle -\frac {2 \sqrt {x^4-x} \int \frac {\sqrt {x-1}}{b-a x^2}dx^{3/2}}{3 \sqrt {x} \sqrt {x^3-1}}\) |
| \(\Big \downarrow \) 1489 |
| \(\displaystyle -\frac {2 \sqrt {x^4-x} \left (\frac {\sqrt {a} \int \frac {\sqrt {x-1}}{\sqrt {a} \left (\sqrt {b}-\sqrt {a} x\right )}dx^{3/2}}{2 \sqrt {b}}+\frac {\sqrt {a} \int \frac {\sqrt {x-1}}{\sqrt {a} \left (\sqrt {a} x+\sqrt {b}\right )}dx^{3/2}}{2 \sqrt {b}}\right )}{3 \sqrt {x} \sqrt {x^3-1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \sqrt {x^4-x} \left (\frac {\int \frac {\sqrt {x-1}}{\sqrt {b}-\sqrt {a} x}dx^{3/2}}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {x-1}}{\sqrt {a} x+\sqrt {b}}dx^{3/2}}{2 \sqrt {b}}\right )}{3 \sqrt {x} \sqrt {x^3-1}}\) |
| \(\Big \downarrow \) 301 |
| \(\displaystyle -\frac {2 \sqrt {x^4-x} \left (\frac {-\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \int \frac {1}{\sqrt {x-1} \left (\sqrt {b}-\sqrt {a} x\right )}dx^{3/2}-\frac {\int \frac {1}{\sqrt {x-1}}dx^{3/2}}{\sqrt {a}}}{2 \sqrt {b}}+\frac {\frac {\int \frac {1}{\sqrt {x-1}}dx^{3/2}}{\sqrt {a}}-\left (\frac {\sqrt {b}}{\sqrt {a}}+1\right ) \int \frac {1}{\sqrt {x-1} \left (\sqrt {a} x+\sqrt {b}\right )}dx^{3/2}}{2 \sqrt {b}}\right )}{3 \sqrt {x} \sqrt {x^3-1}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle -\frac {2 \sqrt {x^4-x} \left (\frac {-\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \int \frac {1}{\sqrt {x-1} \left (\sqrt {b}-\sqrt {a} x\right )}dx^{3/2}-\frac {\int \frac {1}{1-x}d\frac {x^{3/2}}{\sqrt {x-1}}}{\sqrt {a}}}{2 \sqrt {b}}+\frac {\frac {\int \frac {1}{1-x}d\frac {x^{3/2}}{\sqrt {x-1}}}{\sqrt {a}}-\left (\frac {\sqrt {b}}{\sqrt {a}}+1\right ) \int \frac {1}{\sqrt {x-1} \left (\sqrt {a} x+\sqrt {b}\right )}dx^{3/2}}{2 \sqrt {b}}\right )}{3 \sqrt {x} \sqrt {x^3-1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2 \sqrt {x^4-x} \left (\frac {-\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \int \frac {1}{\sqrt {x-1} \left (\sqrt {b}-\sqrt {a} x\right )}dx^{3/2}-\frac {\text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x-1}}\right )}{\sqrt {a}}}{2 \sqrt {b}}+\frac {\frac {\text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x-1}}\right )}{\sqrt {a}}-\left (\frac {\sqrt {b}}{\sqrt {a}}+1\right ) \int \frac {1}{\sqrt {x-1} \left (\sqrt {a} x+\sqrt {b}\right )}dx^{3/2}}{2 \sqrt {b}}\right )}{3 \sqrt {x} \sqrt {x^3-1}}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle -\frac {2 \sqrt {x^4-x} \left (\frac {-\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \int \frac {1}{\sqrt {b}-\left (\sqrt {b}-\sqrt {a}\right ) x}d\frac {x^{3/2}}{\sqrt {x-1}}-\frac {\text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x-1}}\right )}{\sqrt {a}}}{2 \sqrt {b}}+\frac {\frac {\text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x-1}}\right )}{\sqrt {a}}-\left (\frac {\sqrt {b}}{\sqrt {a}}+1\right ) \int \frac {1}{\sqrt {b}-\left (\sqrt {a}+\sqrt {b}\right ) x}d\frac {x^{3/2}}{\sqrt {x-1}}}{2 \sqrt {b}}\right )}{3 \sqrt {x} \sqrt {x^3-1}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {2 \sqrt {x^4-x} \left (\frac {\frac {\text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x-1}}\right )}{\sqrt {a}}-\left (\frac {\sqrt {b}}{\sqrt {a}}+1\right ) \int \frac {1}{\sqrt {b}-\left (\sqrt {a}+\sqrt {b}\right ) x}d\frac {x^{3/2}}{\sqrt {x-1}}}{2 \sqrt {b}}+\frac {-\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \arctan \left (\frac {x^{3/2} \sqrt {\sqrt {a}-\sqrt {b}}}{\sqrt [4]{b} \sqrt {x-1}}\right )}{\sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x-1}}\right )}{\sqrt {a}}}{2 \sqrt {b}}\right )}{3 \sqrt {x} \sqrt {x^3-1}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {2 \sqrt {x^4-x} \left (\frac {-\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \arctan \left (\frac {x^{3/2} \sqrt {\sqrt {a}-\sqrt {b}}}{\sqrt [4]{b} \sqrt {x-1}}\right )}{\sqrt [4]{b} \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x-1}}\right )}{\sqrt {a}}}{2 \sqrt {b}}+\frac {\frac {\text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x-1}}\right )}{\sqrt {a}}-\frac {\left (\frac {\sqrt {b}}{\sqrt {a}}+1\right ) \text {arctanh}\left (\frac {x^{3/2} \sqrt {\sqrt {a}+\sqrt {b}}}{\sqrt [4]{b} \sqrt {x-1}}\right )}{\sqrt [4]{b} \sqrt {\sqrt {a}+\sqrt {b}}}}{2 \sqrt {b}}\right )}{3 \sqrt {x} \sqrt {x^3-1}}\) |
(-2*Sqrt[-x + x^4]*((-(((1 - Sqrt[b]/Sqrt[a])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[ b]]*x^(3/2))/(b^(1/4)*Sqrt[-1 + x])])/(Sqrt[Sqrt[a] - Sqrt[b]]*b^(1/4))) - ArcTanh[x^(3/2)/Sqrt[-1 + x]]/Sqrt[a])/(2*Sqrt[b]) + (ArcTanh[x^(3/2)/Sqr t[-1 + x]]/Sqrt[a] - ((1 + Sqrt[b]/Sqrt[a])*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b ]]*x^(3/2))/(b^(1/4)*Sqrt[-1 + x])])/(Sqrt[Sqrt[a] + Sqrt[b]]*b^(1/4)))/(2 *Sqrt[b])))/(3*Sqrt[x]*Sqrt[-1 + x^3])
3.23.78.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ d Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d Int[(a + b*x^2)^ (p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E qQ[b*c + 3*a*d, 0]))
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{r = Rt[(-a)*c, 2]}, Simp[-c/(2*r) Int[(d + e*x^2)^q/(r - c*x^2), x], x] - S imp[c/(2*r) Int[(d + e*x^2)^q/(r + c*x^2), x], x]] /; FreeQ[{a, c, d, e, q}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[q]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_ .), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/ k - 1)*(d + e*x^(n/k))^q*(a + c*x^(2*(n/k)))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IntegerQ[m ]
Int[((f_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n _))^(q_.), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/f Subst[Int[x^( k*(m + 1) - 1)*(d + e*(x^(k*n)/f))^q*(a + c*(x^(2*k*n)/f))^p, x], x, (f*x)^ (1/k)], x]] /; FreeQ[{a, c, d, e, f, p, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0 ] && FractionQ[m] && IntegerQ[p]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 1.34 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.61
| method | result | size |
| default | \(-\frac {-\frac {\left (\sqrt {a b}+b \right ) \operatorname {arctanh}\left (\frac {b \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {\left (\sqrt {a b}-b \right ) \arctan \left (\frac {b \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}}{3 \sqrt {a b}}\) | \(106\) |
| pseudoelliptic | \(-\frac {-\frac {\left (\sqrt {a b}+b \right ) \operatorname {arctanh}\left (\frac {b \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {\left (\sqrt {a b}-b \right ) \arctan \left (\frac {b \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}}{3 \sqrt {a b}}\) | \(106\) |
| elliptic | \(-\frac {\sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}-b \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+1\right ) \left (-1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{5}+\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (1-i \sqrt {3}\right ) \sqrt {\frac {x \left (i \sqrt {3}-3\right )}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1-i \sqrt {3}\right )}}\, \sqrt {\frac {-i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1+i \sqrt {3}\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 )-\frac {\underline {\hspace {1.25 ex}}\alpha ^{5} a \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 {i \underline {\hspace {1.25 ex}}\alpha ^{5} \sqrt {3}\, a -3 \underline {\hspace {1.25 ex}}\alpha ^{5} a -i \sqrt {3}\, b +3 b}{6 b}, \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 )}{b}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{4} \left (a -b \right ) \left (i \sqrt {3}-3\right ) \sqrt {x \left (-1+x \right ) \left (i \sqrt {3}+2 x +1\right ) \left (-i \sqrt {3}+2 x +1\right )}}\right )}{3}\) | \(365\) |
-1/3/(a*b)^(1/2)*(-((a*b)^(1/2)+b)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(b*(x* (x^3-1))^(1/2)/x^2/(((a*b)^(1/2)+b)*b)^(1/2))+((a*b)^(1/2)-b)/(((a*b)^(1/2 )-b)*b)^(1/2)*arctan(b*(x*(x^3-1))^(1/2)/x^2/(((a*b)^(1/2)-b)*b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 1083 vs. \(2 (129) = 258\).
Time = 0.61 (sec) , antiderivative size = 1083, normalized size of antiderivative = 6.26 \[ \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx=\text {Too large to display} \]
-1/12*sqrt(-(a*b*sqrt(1/(a*b^3)) - 1)/(a*b))*log((2*((a^2 - 3*a*b + 4*b^2) *x^4 + 2*(a*b - 2*b^2)*x + (2*(a^2*b^2 - 2*a*b^3)*x^4 + (a^2*b^2 - 3*a*b^3 + 4*b^4)*x)*sqrt(1/(a*b^3)))*sqrt(x^4 - x) + ((a^2*b - 4*a*b^2)*x^6 + 2*( a^2*b - 3*a*b^2 + 4*b^3)*x^3 + 3*a*b^2 - 4*b^3 + ((a^3*b^2 - 6*a^2*b^3 + 8 *a*b^4)*x^6 + a^2*b^3 + 4*(a^2*b^3 - 2*a*b^4)*x^3)*sqrt(1/(a*b^3)))*sqrt(- (a*b*sqrt(1/(a*b^3)) - 1)/(a*b)))/(a*x^6 - b)) + 1/12*sqrt(-(a*b*sqrt(1/(a *b^3)) - 1)/(a*b))*log((2*((a^2 - 3*a*b + 4*b^2)*x^4 + 2*(a*b - 2*b^2)*x + (2*(a^2*b^2 - 2*a*b^3)*x^4 + (a^2*b^2 - 3*a*b^3 + 4*b^4)*x)*sqrt(1/(a*b^3 )))*sqrt(x^4 - x) - ((a^2*b - 4*a*b^2)*x^6 + 2*(a^2*b - 3*a*b^2 + 4*b^3)*x ^3 + 3*a*b^2 - 4*b^3 + ((a^3*b^2 - 6*a^2*b^3 + 8*a*b^4)*x^6 + a^2*b^3 + 4* (a^2*b^3 - 2*a*b^4)*x^3)*sqrt(1/(a*b^3)))*sqrt(-(a*b*sqrt(1/(a*b^3)) - 1)/ (a*b)))/(a*x^6 - b)) - 1/12*sqrt((a*b*sqrt(1/(a*b^3)) + 1)/(a*b))*log((2*( (a^2 - 3*a*b + 4*b^2)*x^4 + 2*(a*b - 2*b^2)*x - (2*(a^2*b^2 - 2*a*b^3)*x^4 + (a^2*b^2 - 3*a*b^3 + 4*b^4)*x)*sqrt(1/(a*b^3)))*sqrt(x^4 - x) + ((a^2*b - 4*a*b^2)*x^6 + 2*(a^2*b - 3*a*b^2 + 4*b^3)*x^3 + 3*a*b^2 - 4*b^3 - ((a^ 3*b^2 - 6*a^2*b^3 + 8*a*b^4)*x^6 + a^2*b^3 + 4*(a^2*b^3 - 2*a*b^4)*x^3)*sq rt(1/(a*b^3)))*sqrt((a*b*sqrt(1/(a*b^3)) + 1)/(a*b)))/(a*x^6 - b)) + 1/12* sqrt((a*b*sqrt(1/(a*b^3)) + 1)/(a*b))*log((2*((a^2 - 3*a*b + 4*b^2)*x^4 + 2*(a*b - 2*b^2)*x - (2*(a^2*b^2 - 2*a*b^3)*x^4 + (a^2*b^2 - 3*a*b^3 + 4*b^ 4)*x)*sqrt(1/(a*b^3)))*sqrt(x^4 - x) - ((a^2*b - 4*a*b^2)*x^6 + 2*(a^2*...
\[ \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx=\int \frac {\sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{a x^{6} - b}\, dx \]
\[ \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx=\int { \frac {\sqrt {x^{4} - x}}{a x^{6} - b} \,d x } \]
Time = 0.38 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx=\frac {{\left (4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b\right )} {\left | b \right |} \arctan \left (\frac {\sqrt {-\frac {1}{x^{3}} + 1}}{\sqrt {-\frac {b + \sqrt {{\left (a - b\right )} b + b^{2}}}{b}}}\right )}{3 \, {\left (4 \, a^{2} b^{3} + 5 \, a b^{4}\right )}} - \frac {{\left (4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b\right )} {\left | b \right |} \arctan \left (\frac {\sqrt {-\frac {1}{x^{3}} + 1}}{\sqrt {-\frac {b - \sqrt {{\left (a - b\right )} b + b^{2}}}{b}}}\right )}{3 \, {\left (4 \, a^{2} b^{3} + 5 \, a b^{4}\right )}} \]
1/3*(4*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a + 5*sqrt(a*b)*sqrt(-b^2 - sqrt (a*b)*b)*b)*abs(b)*arctan(sqrt(-1/x^3 + 1)/sqrt(-(b + sqrt((a - b)*b + b^2 ))/b))/(4*a^2*b^3 + 5*a*b^4) - 1/3*(4*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a + 5*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*b)*abs(b)*arctan(sqrt(-1/x^3 + 1)/ sqrt(-(b - sqrt((a - b)*b + b^2))/b))/(4*a^2*b^3 + 5*a*b^4)
Timed out. \[ \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx=-\int \frac {\sqrt {x^4-x}}{b-a\,x^6} \,d x \]