Integrand size = 16, antiderivative size = 131 \[ \int x^8 \sqrt [4]{a-b x^4} \, dx=-\frac {a^2 x \sqrt [4]{a-b x^4}}{24 b^2}-\frac {a x^5 \sqrt [4]{a-b x^4}}{60 b}+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}-\frac {a^{5/2} \left (1-\frac {a}{b x^4}\right )^{3/4} x^3 \operatorname {EllipticF}\left (\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{24 b^{3/2} \left (a-b x^4\right )^{3/4}} \]
-1/24*a^2*x*(-b*x^4+a)^(1/4)/b^2-1/60*a*x^5*(-b*x^4+a)^(1/4)/b+1/10*x^9*(- b*x^4+a)^(1/4)-1/24*a^(5/2)*(1-a/b/x^4)^(3/4)*x^3*(cos(1/2*arccsc(x^2*b^(1 /2)/a^(1/2)))^2)^(1/2)/cos(1/2*arccsc(x^2*b^(1/2)/a^(1/2)))*EllipticF(sin( 1/2*arccsc(x^2*b^(1/2)/a^(1/2))),2^(1/2))/b^(3/2)/(-b*x^4+a)^(3/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 7.48 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.73 \[ \int x^8 \sqrt [4]{a-b x^4} \, dx=\frac {x \sqrt [4]{a-b x^4} \left (-\sqrt [4]{1-\frac {b x^4}{a}} \left (5 a^2+a b x^4-6 b^2 x^8\right )+5 a^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{4},\frac {5}{4},\frac {b x^4}{a}\right )\right )}{60 b^2 \sqrt [4]{1-\frac {b x^4}{a}}} \]
(x*(a - b*x^4)^(1/4)*(-((1 - (b*x^4)/a)^(1/4)*(5*a^2 + a*b*x^4 - 6*b^2*x^8 )) + 5*a^2*Hypergeometric2F1[-1/4, 1/4, 5/4, (b*x^4)/a]))/(60*b^2*(1 - (b* x^4)/a)^(1/4))
Time = 0.30 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {811, 843, 843, 768, 858, 807, 230}
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.
\(\displaystyle \int x^8 \sqrt [4]{a-b x^4} \, dx\) |
\(\Big \downarrow \) 811 |
\(\displaystyle \frac {1}{10} a \int \frac {x^8}{\left (a-b x^4\right )^{3/4}}dx+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {1}{10} a \left (\frac {5 a \int \frac {x^4}{\left (a-b x^4\right )^{3/4}}dx}{6 b}-\frac {x^5 \sqrt [4]{a-b x^4}}{6 b}\right )+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {1}{10} a \left (\frac {5 a \left (\frac {a \int \frac {1}{\left (a-b x^4\right )^{3/4}}dx}{2 b}-\frac {x \sqrt [4]{a-b x^4}}{2 b}\right )}{6 b}-\frac {x^5 \sqrt [4]{a-b x^4}}{6 b}\right )+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}\) |
\(\Big \downarrow \) 768 |
\(\displaystyle \frac {1}{10} a \left (\frac {5 a \left (\frac {a x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} \int \frac {1}{\left (1-\frac {a}{b x^4}\right )^{3/4} x^3}dx}{2 b \left (a-b x^4\right )^{3/4}}-\frac {x \sqrt [4]{a-b x^4}}{2 b}\right )}{6 b}-\frac {x^5 \sqrt [4]{a-b x^4}}{6 b}\right )+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {1}{10} a \left (\frac {5 a \left (-\frac {a x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} \int \frac {1}{\left (1-\frac {a}{b x^4}\right )^{3/4} x}d\frac {1}{x}}{2 b \left (a-b x^4\right )^{3/4}}-\frac {x \sqrt [4]{a-b x^4}}{2 b}\right )}{6 b}-\frac {x^5 \sqrt [4]{a-b x^4}}{6 b}\right )+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{10} a \left (\frac {5 a \left (-\frac {a x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} \int \frac {1}{\left (1-\frac {a}{b x^2}\right )^{3/4}}d\frac {1}{x^2}}{4 b \left (a-b x^4\right )^{3/4}}-\frac {x \sqrt [4]{a-b x^4}}{2 b}\right )}{6 b}-\frac {x^5 \sqrt [4]{a-b x^4}}{6 b}\right )+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}\) |
\(\Big \downarrow \) 230 |
\(\displaystyle \frac {1}{10} a \left (\frac {5 a \left (-\frac {\sqrt {a} x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right ),2\right )}{2 \sqrt {b} \left (a-b x^4\right )^{3/4}}-\frac {x \sqrt [4]{a-b x^4}}{2 b}\right )}{6 b}-\frac {x^5 \sqrt [4]{a-b x^4}}{6 b}\right )+\frac {1}{10} x^9 \sqrt [4]{a-b x^4}\) |
(x^9*(a - b*x^4)^(1/4))/10 + (a*(-1/6*(x^5*(a - b*x^4)^(1/4))/b + (5*a*(-1 /2*(x*(a - b*x^4)^(1/4))/b - (Sqrt[a]*(1 - a/(b*x^4))^(3/4)*x^3*EllipticF[ ArcSin[Sqrt[a]/(Sqrt[b]*x^2)]/2, 2])/(2*Sqrt[b]*(a - b*x^4)^(3/4))))/(6*b) ))/10
3.12.97.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[-b/a, 2] ))*EllipticF[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ [a, 0] && NegQ[b/a]
Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Simp[x^3*((1 + a/(b*x^4))^(3 /4)/(a + b*x^4)^(3/4)) Int[1/(x^3*(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ [{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
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[((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] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
\[\int x^{8} \left (-b \,x^{4}+a \right )^{\frac {1}{4}}d x\]
\[ \int x^8 \sqrt [4]{a-b x^4} \, dx=\int { {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{8} \,d x } \]
Result contains complex when optimal does not.
Time = 0.62 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.31 \[ \int x^8 \sqrt [4]{a-b x^4} \, dx=\frac {\sqrt [4]{a} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} \]
a**(1/4)*x**9*gamma(9/4)*hyper((-1/4, 9/4), (13/4,), b*x**4*exp_polar(2*I* pi)/a)/(4*gamma(13/4))
\[ \int x^8 \sqrt [4]{a-b x^4} \, dx=\int { {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{8} \,d x } \]
\[ \int x^8 \sqrt [4]{a-b x^4} \, dx=\int { {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{8} \,d x } \]
Timed out. \[ \int x^8 \sqrt [4]{a-b x^4} \, dx=\int x^8\,{\left (a-b\,x^4\right )}^{1/4} \,d x \]