Integrand size = 16, antiderivative size = 109 \[ \int \frac {1}{x^{10} \sqrt [4]{a-b x^4}} \, dx=-\frac {\left (a-b x^4\right )^{3/4}}{9 a x^9}-\frac {2 b \left (a-b x^4\right )^{3/4}}{15 a^2 x^5}-\frac {4 b^{5/2} \sqrt [4]{1-\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{15 a^{5/2} \sqrt [4]{a-b x^4}} \] Output:
-1/9*(-b*x^4+a)^(3/4)/a/x^9-2/15*b*(-b*x^4+a)^(3/4)/a^2/x^5-4/15*b^(5/2)*( 1-a/b/x^4)^(1/4)*x*EllipticE(sin(1/2*arccsc(b^(1/2)*x^2/a^(1/2))),2^(1/2)) /a^(5/2)/(-b*x^4+a)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.48 \[ \int \frac {1}{x^{10} \sqrt [4]{a-b x^4}} \, dx=-\frac {\sqrt [4]{1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {1}{4},-\frac {5}{4},\frac {b x^4}{a}\right )}{9 x^9 \sqrt [4]{a-b x^4}} \] Input:
Integrate[1/(x^10*(a - b*x^4)^(1/4)),x]
Output:
-1/9*((1 - (b*x^4)/a)^(1/4)*Hypergeometric2F1[-9/4, 1/4, -5/4, (b*x^4)/a]) /(x^9*(a - b*x^4)^(1/4))
Time = 0.27 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {847, 847, 842, 858, 807, 226}
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.
\(\displaystyle \int \frac {1}{x^{10} \sqrt [4]{a-b x^4}} \, dx\) |
\(\Big \downarrow \) 847 |
\(\displaystyle \frac {2 b \int \frac {1}{x^6 \sqrt [4]{a-b x^4}}dx}{3 a}-\frac {\left (a-b x^4\right )^{3/4}}{9 a x^9}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle \frac {2 b \left (\frac {2 b \int \frac {1}{x^2 \sqrt [4]{a-b x^4}}dx}{5 a}-\frac {\left (a-b x^4\right )^{3/4}}{5 a x^5}\right )}{3 a}-\frac {\left (a-b x^4\right )^{3/4}}{9 a x^9}\) |
\(\Big \downarrow \) 842 |
\(\displaystyle \frac {2 b \left (\frac {2 b x \sqrt [4]{1-\frac {a}{b x^4}} \int \frac {1}{\sqrt [4]{1-\frac {a}{b x^4}} x^3}dx}{5 a \sqrt [4]{a-b x^4}}-\frac {\left (a-b x^4\right )^{3/4}}{5 a x^5}\right )}{3 a}-\frac {\left (a-b x^4\right )^{3/4}}{9 a x^9}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {2 b \left (-\frac {2 b x \sqrt [4]{1-\frac {a}{b x^4}} \int \frac {1}{\sqrt [4]{1-\frac {a}{b x^4}} x}d\frac {1}{x}}{5 a \sqrt [4]{a-b x^4}}-\frac {\left (a-b x^4\right )^{3/4}}{5 a x^5}\right )}{3 a}-\frac {\left (a-b x^4\right )^{3/4}}{9 a x^9}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {2 b \left (-\frac {b x \sqrt [4]{1-\frac {a}{b x^4}} \int \frac {1}{\sqrt [4]{1-\frac {a}{b x^2}}}d\frac {1}{x^2}}{5 a \sqrt [4]{a-b x^4}}-\frac {\left (a-b x^4\right )^{3/4}}{5 a x^5}\right )}{3 a}-\frac {\left (a-b x^4\right )^{3/4}}{9 a x^9}\) |
\(\Big \downarrow \) 226 |
\(\displaystyle \frac {2 b \left (-\frac {2 b^{3/2} x \sqrt [4]{1-\frac {a}{b x^4}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a-b x^4}}-\frac {\left (a-b x^4\right )^{3/4}}{5 a x^5}\right )}{3 a}-\frac {\left (a-b x^4\right )^{3/4}}{9 a x^9}\) |
Input:
Int[1/(x^10*(a - b*x^4)^(1/4)),x]
Output:
-1/9*(a - b*x^4)^(3/4)/(a*x^9) + (2*b*(-1/5*(a - b*x^4)^(3/4)/(a*x^5) - (2 *b^(3/2)*(1 - a/(b*x^4))^(1/4)*x*EllipticE[ArcSin[Sqrt[a]/(Sqrt[b]*x^2)]/2 , 2])/(5*a^(3/2)*(a - b*x^4)^(1/4))))/(3*a)
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2/(a^(1/4)*Rt[-b/a, 2] ))*EllipticE[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ [a, 0] && NegQ[b/a]
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[1/((x_)^2*((a_) + (b_.)*(x_)^4)^(1/4)), x_Symbol] :> Simp[x*((1 + a/(b* x^4))^(1/4)/(a + b*x^4)^(1/4)) Int[1/(x^3*(1 + a/(b*x^4))^(1/4)), x], x] /; FreeQ[{a, b}, x] && NegQ[b/a]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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]
\[\int \frac {1}{x^{10} \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}d x\]
Input:
int(1/x^10/(-b*x^4+a)^(1/4),x)
Output:
int(1/x^10/(-b*x^4+a)^(1/4),x)
\[ \int \frac {1}{x^{10} \sqrt [4]{a-b x^4}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{10}} \,d x } \] Input:
integrate(1/x^10/(-b*x^4+a)^(1/4),x, algorithm="fricas")
Output:
integral(-(-b*x^4 + a)^(3/4)/(b*x^14 - a*x^10), x)
Result contains complex when optimal does not.
Time = 0.68 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.28 \[ \int \frac {1}{x^{10} \sqrt [4]{a-b x^4}} \, dx=\frac {i e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{10 \sqrt [4]{b} x^{10}} \] Input:
integrate(1/x**10/(-b*x**4+a)**(1/4),x)
Output:
I*exp(I*pi/4)*hyper((1/4, 5/2), (7/2,), a/(b*x**4))/(10*b**(1/4)*x**10)
\[ \int \frac {1}{x^{10} \sqrt [4]{a-b x^4}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{10}} \,d x } \] Input:
integrate(1/x^10/(-b*x^4+a)^(1/4),x, algorithm="maxima")
Output:
integrate(1/((-b*x^4 + a)^(1/4)*x^10), x)
\[ \int \frac {1}{x^{10} \sqrt [4]{a-b x^4}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{10}} \,d x } \] Input:
integrate(1/x^10/(-b*x^4+a)^(1/4),x, algorithm="giac")
Output:
integrate(1/((-b*x^4 + a)^(1/4)*x^10), x)
Timed out. \[ \int \frac {1}{x^{10} \sqrt [4]{a-b x^4}} \, dx=\int \frac {1}{x^{10}\,{\left (a-b\,x^4\right )}^{1/4}} \,d x \] Input:
int(1/(x^10*(a - b*x^4)^(1/4)),x)
Output:
int(1/(x^10*(a - b*x^4)^(1/4)), x)
\[ \int \frac {1}{x^{10} \sqrt [4]{a-b x^4}} \, dx=\int \frac {1}{\left (-b \,x^{4}+a \right )^{\frac {1}{4}} x^{10}}d x \] Input:
int(1/x^10/(-b*x^4+a)^(1/4),x)
Output:
int(1/((a - b*x**4)**(1/4)*x**10),x)