Integrand size = 15, antiderivative size = 153 \[ \int \frac {1}{x^{14} \sqrt [4]{a+b x^4}} \, dx=\frac {8 b^3}{39 a^3 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}+\frac {10 b \left (a+b x^4\right )^{3/4}}{117 a^2 x^9}-\frac {4 b^2 \left (a+b x^4\right )^{3/4}}{39 a^3 x^5}-\frac {8 b^{7/2} \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{39 a^{7/2} \sqrt [4]{a+b x^4}} \]
8/39*b^3/a^3/x/(b*x^4+a)^(1/4)-1/13*(b*x^4+a)^(3/4)/a/x^13+10/117*b*(b*x^4 +a)^(3/4)/a^2/x^9-4/39*b^2*(b*x^4+a)^(3/4)/a^3/x^5-8/39*b^(7/2)*(1+a/b/x^4 )^(1/4)*x*(cos(1/2*arccot(x^2*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arccot(x^ 2*b^(1/2)/a^(1/2)))*EllipticE(sin(1/2*arccot(x^2*b^(1/2)/a^(1/2))),2^(1/2) )/a^(7/2)/(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 = 51, normalized size of antiderivative = 0.33 \[ \int \frac {1}{x^{14} \sqrt [4]{a+b x^4}} \, dx=-\frac {\sqrt [4]{1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {13}{4},\frac {1}{4},-\frac {9}{4},-\frac {b x^4}{a}\right )}{13 x^{13} \sqrt [4]{a+b x^4}} \]
-1/13*((1 + (b*x^4)/a)^(1/4)*Hypergeometric2F1[-13/4, 1/4, -9/4, -((b*x^4) /a)])/(x^13*(a + b*x^4)^(1/4))
Time = 0.33 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {847, 847, 847, 841, 813, 858, 807, 212}
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.
\(\displaystyle \int \frac {1}{x^{14} \sqrt [4]{a+b x^4}} \, dx\) |
\(\Big \downarrow \) 847 |
\(\displaystyle -\frac {10 b \int \frac {1}{x^{10} \sqrt [4]{b x^4+a}}dx}{13 a}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle -\frac {10 b \left (-\frac {2 b \int \frac {1}{x^6 \sqrt [4]{b x^4+a}}dx}{3 a}-\frac {\left (a+b x^4\right )^{3/4}}{9 a x^9}\right )}{13 a}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle -\frac {10 b \left (-\frac {2 b \left (-\frac {2 b \int \frac {1}{x^2 \sqrt [4]{b x^4+a}}dx}{5 a}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^5}\right )}{3 a}-\frac {\left (a+b x^4\right )^{3/4}}{9 a x^9}\right )}{13 a}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}\) |
\(\Big \downarrow \) 841 |
\(\displaystyle -\frac {10 b \left (-\frac {2 b \left (-\frac {2 b \left (-b \int \frac {x^2}{\left (b x^4+a\right )^{5/4}}dx-\frac {1}{x \sqrt [4]{a+b x^4}}\right )}{5 a}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^5}\right )}{3 a}-\frac {\left (a+b x^4\right )^{3/4}}{9 a x^9}\right )}{13 a}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}\) |
\(\Big \downarrow \) 813 |
\(\displaystyle -\frac {10 b \left (-\frac {2 b \left (-\frac {2 b \left (-\frac {x \sqrt [4]{\frac {a}{b x^4}+1} \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{5/4} x^3}dx}{\sqrt [4]{a+b x^4}}-\frac {1}{x \sqrt [4]{a+b x^4}}\right )}{5 a}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^5}\right )}{3 a}-\frac {\left (a+b x^4\right )^{3/4}}{9 a x^9}\right )}{13 a}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\frac {10 b \left (-\frac {2 b \left (-\frac {2 b \left (\frac {x \sqrt [4]{\frac {a}{b x^4}+1} \int \frac {1}{\left (\frac {a}{b x^4}+1\right )^{5/4} x}d\frac {1}{x}}{\sqrt [4]{a+b x^4}}-\frac {1}{x \sqrt [4]{a+b x^4}}\right )}{5 a}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^5}\right )}{3 a}-\frac {\left (a+b x^4\right )^{3/4}}{9 a x^9}\right )}{13 a}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle -\frac {10 b \left (-\frac {2 b \left (-\frac {2 b \left (\frac {x \sqrt [4]{\frac {a}{b x^4}+1} \int \frac {1}{\left (\frac {a}{b x^2}+1\right )^{5/4}}d\frac {1}{x^2}}{2 \sqrt [4]{a+b x^4}}-\frac {1}{x \sqrt [4]{a+b x^4}}\right )}{5 a}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^5}\right )}{3 a}-\frac {\left (a+b x^4\right )^{3/4}}{9 a x^9}\right )}{13 a}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle -\frac {10 b \left (-\frac {2 b \left (-\frac {2 b \left (\frac {\sqrt {b} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{a+b x^4}}-\frac {1}{x \sqrt [4]{a+b x^4}}\right )}{5 a}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^5}\right )}{3 a}-\frac {\left (a+b x^4\right )^{3/4}}{9 a x^9}\right )}{13 a}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}\) |
-1/13*(a + b*x^4)^(3/4)/(a*x^13) - (10*b*(-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*(-(1/(x*(a + b*x^4)^(1/4))) + (Sqrt[b]*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcTan[Sqrt[a]/(Sqrt[b]*x^2)]/ 2, 2])/(Sqrt[a]*(a + b*x^4)^(1/4))))/(5*a)))/(3*a)))/(13*a)
3.12.9.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) )*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[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[(x_)^2/((a_) + (b_.)*(x_)^4)^(5/4), x_Symbol] :> Simp[x*((1 + a/(b*x^4) )^(1/4)/(b*(a + b*x^4)^(1/4))) Int[1/(x^3*(1 + a/(b*x^4))^(5/4)), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[1/((x_)^2*((a_) + (b_.)*(x_)^4)^(1/4)), x_Symbol] :> -Simp[(x*(a + b*x^ 4)^(1/4))^(-1), x] - Simp[b Int[x^2/(a + b*x^4)^(5/4), x], x] /; FreeQ[{a , b}, x] && PosQ[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]
\[\int \frac {1}{x^{14} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}d x\]
\[ \int \frac {1}{x^{14} \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{14}} \,d x } \]
Result contains complex when optimal does not.
Time = 0.90 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.29 \[ \int \frac {1}{x^{14} \sqrt [4]{a+b x^4}} \, dx=\frac {\Gamma \left (- \frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {13}{4}, \frac {1}{4} \\ - \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt [4]{a} x^{13} \Gamma \left (- \frac {9}{4}\right )} \]
gamma(-13/4)*hyper((-13/4, 1/4), (-9/4,), b*x**4*exp_polar(I*pi)/a)/(4*a** (1/4)*x**13*gamma(-9/4))
\[ \int \frac {1}{x^{14} \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{14}} \,d x } \]
\[ \int \frac {1}{x^{14} \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{14}} \,d x } \]
Timed out. \[ \int \frac {1}{x^{14} \sqrt [4]{a+b x^4}} \, dx=\int \frac {1}{x^{14}\,{\left (b\,x^4+a\right )}^{1/4}} \,d x \]