Integrand size = 16, antiderivative size = 263 \[ \int x^6 \sqrt [4]{a-b x^4} \, dx=-\frac {a x^3 \sqrt [4]{a-b x^4}}{32 b}+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}-\frac {3 a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt {2} b^{7/4}}+\frac {3 a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt {2} b^{7/4}}+\frac {3 a^2 \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt {2} b^{7/4}}-\frac {3 a^2 \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt {2} b^{7/4}} \]
-1/32*a*x^3*(-b*x^4+a)^(1/4)/b+1/8*x^7*(-b*x^4+a)^(1/4)+3/128*a^2*arctan(- 1+b^(1/4)*x*2^(1/2)/(-b*x^4+a)^(1/4))/b^(7/4)*2^(1/2)+3/128*a^2*arctan(1+b ^(1/4)*x*2^(1/2)/(-b*x^4+a)^(1/4))/b^(7/4)*2^(1/2)+3/256*a^2*ln(1-b^(1/4)* x*2^(1/2)/(-b*x^4+a)^(1/4)+x^2*b^(1/2)/(-b*x^4+a)^(1/2))/b^(7/4)*2^(1/2)-3 /256*a^2*ln(1+b^(1/4)*x*2^(1/2)/(-b*x^4+a)^(1/4)+x^2*b^(1/2)/(-b*x^4+a)^(1 /2))/b^(7/4)*2^(1/2)
Time = 0.84 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.61 \[ \int x^6 \sqrt [4]{a-b x^4} \, dx=\frac {4 b^{3/4} x^3 \sqrt [4]{a-b x^4} \left (-a+4 b x^4\right )+3 \sqrt {2} a^2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x \sqrt [4]{a-b x^4}}{-\sqrt {b} x^2+\sqrt {a-b x^4}}\right )-3 \sqrt {2} a^2 \text {arctanh}\left (\frac {\sqrt {b} x^2+\sqrt {a-b x^4}}{\sqrt {2} \sqrt [4]{b} x \sqrt [4]{a-b x^4}}\right )}{128 b^{7/4}} \]
(4*b^(3/4)*x^3*(a - b*x^4)^(1/4)*(-a + 4*b*x^4) + 3*Sqrt[2]*a^2*ArcTan[(Sq rt[2]*b^(1/4)*x*(a - b*x^4)^(1/4))/(-(Sqrt[b]*x^2) + Sqrt[a - b*x^4])] - 3 *Sqrt[2]*a^2*ArcTanh[(Sqrt[b]*x^2 + Sqrt[a - b*x^4])/(Sqrt[2]*b^(1/4)*x*(a - b*x^4)^(1/4))])/(128*b^(7/4))
Time = 0.44 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {811, 843, 854, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int x^6 \sqrt [4]{a-b x^4} \, dx\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {1}{8} a \int \frac {x^6}{\left (a-b x^4\right )^{3/4}}dx+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {1}{8} a \left (\frac {3 a \int \frac {x^2}{\left (a-b x^4\right )^{3/4}}dx}{4 b}-\frac {x^3 \sqrt [4]{a-b x^4}}{4 b}\right )+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {1}{8} a \left (\frac {3 a \int \frac {x^2}{\sqrt {a-b x^4} \left (\frac {b x^4}{a-b x^4}+1\right )}d\frac {x}{\sqrt [4]{a-b x^4}}}{4 b}-\frac {x^3 \sqrt [4]{a-b x^4}}{4 b}\right )+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {1}{8} a \left (\frac {3 a \left (\frac {\int \frac {\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}-\frac {\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}\right )}{4 b}-\frac {x^3 \sqrt [4]{a-b x^4}}{4 b}\right )+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{8} a \left (\frac {3 a \left (\frac {\frac {\int \frac {1}{\frac {x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}+\frac {\int \frac {1}{\frac {x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}\right )}{4 b}-\frac {x^3 \sqrt [4]{a-b x^4}}{4 b}\right )+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{8} a \left (\frac {3 a \left (\frac {\frac {\int \frac {1}{-\frac {x^2}{\sqrt {a-b x^4}}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {1}{-\frac {x^2}{\sqrt {a-b x^4}}-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}\right )}{4 b}-\frac {x^3 \sqrt [4]{a-b x^4}}{4 b}\right )+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{8} a \left (\frac {3 a \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}}{\frac {b x^4}{a-b x^4}+1}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}\right )}{4 b}-\frac {x^3 \sqrt [4]{a-b x^4}}{4 b}\right )+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{8} a \left (\frac {3 a \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}}{\sqrt [4]{b} \left (\frac {x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}\right )}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt [4]{b} \left (\frac {x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}\right )}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{4 b}-\frac {x^3 \sqrt [4]{a-b x^4}}{4 b}\right )+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{8} a \left (\frac {3 a \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}}{\sqrt [4]{b} \left (\frac {x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}\right )}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {2} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt [4]{b} \left (\frac {x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}\right )}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{4 b}-\frac {x^3 \sqrt [4]{a-b x^4}}{4 b}\right )+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} a \left (\frac {3 a \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}}{\frac {x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {2} \sqrt {b}}+\frac {\int \frac {\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1}{\frac {x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{b} \sqrt [4]{a-b x^4}}+\frac {1}{\sqrt {b}}}d\frac {x}{\sqrt [4]{a-b x^4}}}{2 \sqrt {b}}}{2 \sqrt {b}}\right )}{4 b}-\frac {x^3 \sqrt [4]{a-b x^4}}{4 b}\right )+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{8} a \left (\frac {3 a \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{\sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1\right )}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1\right )}{2 \sqrt {2} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{4 b}-\frac {x^3 \sqrt [4]{a-b x^4}}{4 b}\right )+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}\) |
(x^7*(a - b*x^4)^(1/4))/8 + (a*(-1/4*(x^3*(a - b*x^4)^(1/4))/b + (3*a*((-( ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/(Sqrt[2]*b^(1/4))) + Arc Tan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/(Sqrt[2]*b^(1/4)))/(2*Sqrt[ b]) - (-1/2*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/(Sqrt[2]*b^(1/4)) + Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/(2*Sqrt[2]*b^(1/4)))/(2*Sqrt[b]) ))/(4*b)))/8
3.12.89.3.1 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, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 6.55 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.77
| method | result | size |
| pseudoelliptic | \(\frac {32 b^{\frac {7}{4}} \left (-b \,x^{4}+a \right )^{\frac {1}{4}} x^{7}-8 a \,x^{3} b^{\frac {3}{4}} \left (-b \,x^{4}+a \right )^{\frac {1}{4}}-3 \ln \left (\frac {b^{\frac {1}{4}} \left (-b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2} \sqrt {b}+\sqrt {-b \,x^{4}+a}}{-b^{\frac {1}{4}} \left (-b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2} \sqrt {b}+\sqrt {-b \,x^{4}+a}}\right ) \sqrt {2}\, a^{2}-6 \arctan \left (\frac {b^{\frac {1}{4}} x +\sqrt {2}\, \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right ) \sqrt {2}\, a^{2}+6 \arctan \left (\frac {b^{\frac {1}{4}} x -\sqrt {2}\, \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right ) \sqrt {2}\, a^{2}}{256 b^{\frac {7}{4}}}\) | \(202\) |
1/256*(32*b^(7/4)*(-b*x^4+a)^(1/4)*x^7-8*a*x^3*b^(3/4)*(-b*x^4+a)^(1/4)-3* ln((b^(1/4)*(-b*x^4+a)^(1/4)*2^(1/2)*x+x^2*b^(1/2)+(-b*x^4+a)^(1/2))/(-b^( 1/4)*(-b*x^4+a)^(1/4)*2^(1/2)*x+x^2*b^(1/2)+(-b*x^4+a)^(1/2)))*2^(1/2)*a^2 -6*arctan((b^(1/4)*x+2^(1/2)*(-b*x^4+a)^(1/4))/b^(1/4)/x)*2^(1/2)*a^2+6*ar ctan((b^(1/4)*x-2^(1/2)*(-b*x^4+a)^(1/4))/b^(1/4)/x)*2^(1/2)*a^2)/b^(7/4)
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.88 \[ \int x^6 \sqrt [4]{a-b x^4} \, dx=-\frac {3 \, \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b \log \left (\frac {3 \, {\left (\left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b^{2} x + {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{2}\right )}}{x}\right ) - 3 \, \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b \log \left (-\frac {3 \, {\left (\left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b^{2} x - {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{2}\right )}}{x}\right ) - 3 i \, \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b \log \left (-\frac {3 \, {\left (i \, \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b^{2} x - {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{2}\right )}}{x}\right ) + 3 i \, \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b \log \left (-\frac {3 \, {\left (-i \, \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b^{2} x - {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{2}\right )}}{x}\right ) - 4 \, {\left (4 \, b x^{7} - a x^{3}\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{128 \, b} \]
-1/128*(3*(-a^8/b^7)^(1/4)*b*log(3*((-a^8/b^7)^(1/4)*b^2*x + (-b*x^4 + a)^ (1/4)*a^2)/x) - 3*(-a^8/b^7)^(1/4)*b*log(-3*((-a^8/b^7)^(1/4)*b^2*x - (-b* x^4 + a)^(1/4)*a^2)/x) - 3*I*(-a^8/b^7)^(1/4)*b*log(-3*(I*(-a^8/b^7)^(1/4) *b^2*x - (-b*x^4 + a)^(1/4)*a^2)/x) + 3*I*(-a^8/b^7)^(1/4)*b*log(-3*(-I*(- a^8/b^7)^(1/4)*b^2*x - (-b*x^4 + a)^(1/4)*a^2)/x) - 4*(4*b*x^7 - a*x^3)*(- b*x^4 + a)^(1/4))/b
Result contains complex when optimal does not.
Time = 1.16 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.16 \[ \int x^6 \sqrt [4]{a-b x^4} \, dx=\frac {\sqrt [4]{a} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} \]
a**(1/4)*x**7*gamma(7/4)*hyper((-1/4, 7/4), (11/4,), b*x**4*exp_polar(2*I* pi)/a)/(4*gamma(11/4))
Time = 0.29 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.03 \[ \int x^6 \sqrt [4]{a-b x^4} \, dx=\frac {\frac {3 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{2} b}{x} - \frac {{\left (-b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{x^{5}}}{32 \, {\left (b^{3} - \frac {2 \, {\left (b x^{4} - a\right )} b^{2}}{x^{4}} + \frac {{\left (b x^{4} - a\right )}^{2} b}{x^{8}}\right )}} - \frac {3 \, {\left (\frac {2 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} a^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} + \frac {\sqrt {2} a^{2} \log \left (\sqrt {b} + \frac {\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{\frac {1}{4}}}{x} + \frac {\sqrt {-b x^{4} + a}}{x^{2}}\right )}{b^{\frac {3}{4}}} - \frac {\sqrt {2} a^{2} \log \left (\sqrt {b} - \frac {\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{\frac {1}{4}}}{x} + \frac {\sqrt {-b x^{4} + a}}{x^{2}}\right )}{b^{\frac {3}{4}}}\right )}}{256 \, b} \]
1/32*(3*(-b*x^4 + a)^(1/4)*a^2*b/x - (-b*x^4 + a)^(5/4)*a^2/x^5)/(b^3 - 2* (b*x^4 - a)*b^2/x^4 + (b*x^4 - a)^2*b/x^8) - 3/256*(2*sqrt(2)*a^2*arctan(1 /2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(-b*x^4 + a)^(1/4)/x)/b^(1/4))/b^(3/4) + 2 *sqrt(2)*a^2*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(-b*x^4 + a)^(1/4)/x )/b^(1/4))/b^(3/4) + sqrt(2)*a^2*log(sqrt(b) + sqrt(2)*(-b*x^4 + a)^(1/4)* b^(1/4)/x + sqrt(-b*x^4 + a)/x^2)/b^(3/4) - sqrt(2)*a^2*log(sqrt(b) - sqrt (2)*(-b*x^4 + a)^(1/4)*b^(1/4)/x + sqrt(-b*x^4 + a)/x^2)/b^(3/4))/b
\[ \int x^6 \sqrt [4]{a-b x^4} \, dx=\int { {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{6} \,d x } \]
Timed out. \[ \int x^6 \sqrt [4]{a-b x^4} \, dx=\int x^6\,{\left (a-b\,x^4\right )}^{1/4} \,d x \]