Integrand size = 15, antiderivative size = 152 \[ \int \frac {1}{x^{11} \sqrt [4]{a+b x^4}} \, dx=\frac {7 b^3 x^2}{40 a^3 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{10 a x^{10}}+\frac {7 b \left (a+b x^4\right )^{3/4}}{60 a^2 x^6}-\frac {7 b^2 \left (a+b x^4\right )^{3/4}}{40 a^3 x^2}-\frac {7 b^{5/2} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{40 a^{5/2} \sqrt [4]{a+b x^4}} \]
7/40*b^3*x^2/a^3/(b*x^4+a)^(1/4)-1/10*(b*x^4+a)^(3/4)/a/x^10+7/60*b*(b*x^4 +a)^(3/4)/a^2/x^6-7/40*b^2*(b*x^4+a)^(3/4)/a^3/x^2-7/40*b^(5/2)*(1+b*x^4/a )^(1/4)*(cos(1/2*arctan(x^2*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arctan(x^2* b^(1/2)/a^(1/2)))*EllipticE(sin(1/2*arctan(x^2*b^(1/2)/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 = 51, normalized size of antiderivative = 0.34 \[ \int \frac {1}{x^{11} \sqrt [4]{a+b x^4}} \, dx=-\frac {\sqrt [4]{1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4},-\frac {3}{2},-\frac {b x^4}{a}\right )}{10 x^{10} \sqrt [4]{a+b x^4}} \]
-1/10*((1 + (b*x^4)/a)^(1/4)*Hypergeometric2F1[-5/2, 1/4, -3/2, -((b*x^4)/ a)])/(x^10*(a + b*x^4)^(1/4))
Time = 0.26 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {807, 264, 264, 264, 227, 225, 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.
\(\displaystyle \int \frac {1}{x^{11} \sqrt [4]{a+b x^4}} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^{12} \sqrt [4]{b x^4+a}}dx^2\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {1}{2} \left (-\frac {7 b \int \frac {1}{x^8 \sqrt [4]{b x^4+a}}dx^2}{10 a}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^{10}}\right )\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {1}{2} \left (-\frac {7 b \left (-\frac {b \int \frac {1}{x^4 \sqrt [4]{b x^4+a}}dx^2}{2 a}-\frac {\left (a+b x^4\right )^{3/4}}{3 a x^6}\right )}{10 a}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^{10}}\right )\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {1}{2} \left (-\frac {7 b \left (-\frac {b \left (\frac {b \int \frac {1}{\sqrt [4]{b x^4+a}}dx^2}{2 a}-\frac {\left (a+b x^4\right )^{3/4}}{a x^2}\right )}{2 a}-\frac {\left (a+b x^4\right )^{3/4}}{3 a x^6}\right )}{10 a}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^{10}}\right )\) |
\(\Big \downarrow \) 227 |
\(\displaystyle \frac {1}{2} \left (-\frac {7 b \left (-\frac {b \left (\frac {b \sqrt [4]{\frac {b x^4}{a}+1} \int \frac {1}{\sqrt [4]{\frac {b x^4}{a}+1}}dx^2}{2 a \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{a x^2}\right )}{2 a}-\frac {\left (a+b x^4\right )^{3/4}}{3 a x^6}\right )}{10 a}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^{10}}\right )\) |
\(\Big \downarrow \) 225 |
\(\displaystyle \frac {1}{2} \left (-\frac {7 b \left (-\frac {b \left (\frac {b \sqrt [4]{\frac {b x^4}{a}+1} \left (\frac {2 x^2}{\sqrt [4]{\frac {b x^4}{a}+1}}-\int \frac {1}{\left (\frac {b x^4}{a}+1\right )^{5/4}}dx^2\right )}{2 a \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{a x^2}\right )}{2 a}-\frac {\left (a+b x^4\right )^{3/4}}{3 a x^6}\right )}{10 a}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^{10}}\right )\) |
\(\Big \downarrow \) 212 |
\(\displaystyle \frac {1}{2} \left (-\frac {7 b \left (-\frac {b \left (\frac {b \sqrt [4]{\frac {b x^4}{a}+1} \left (\frac {2 x^2}{\sqrt [4]{\frac {b x^4}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{2 a \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{a x^2}\right )}{2 a}-\frac {\left (a+b x^4\right )^{3/4}}{3 a x^6}\right )}{10 a}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^{10}}\right )\) |
(-1/5*(a + b*x^4)^(3/4)/(a*x^10) - (7*b*(-1/3*(a + b*x^4)^(3/4)/(a*x^6) - (b*(-((a + b*x^4)^(3/4)/(a*x^2)) + (b*(1 + (b*x^4)/a)^(1/4)*((2*x^2)/(1 + (b*x^4)/a)^(1/4) - (2*Sqrt[a]*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2 ])/Sqrt[b]))/(2*a*(a + b*x^4)^(1/4))))/(2*a)))/(10*a))/2
3.11.94.3.1 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[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[2*(x/(a + b*x^2)^(1/4)) , x] - Simp[a Int[1/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b}, x] && GtQ[ a, 0] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( a + b*x^2)^(1/4) Int[1/(1 + b*(x^2/a))^(1/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, 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]
\[\int \frac {1}{x^{11} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}d x\]
\[ \int \frac {1}{x^{11} \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{11}} \,d x } \]
Result contains complex when optimal does not.
Time = 0.76 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.21 \[ \int \frac {1}{x^{11} \sqrt [4]{a+b x^4}} \, dx=- \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {1}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{10 \sqrt [4]{a} x^{10}} \]
\[ \int \frac {1}{x^{11} \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{11}} \,d x } \]
\[ \int \frac {1}{x^{11} \sqrt [4]{a+b x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{11}} \,d x } \]
Timed out. \[ \int \frac {1}{x^{11} \sqrt [4]{a+b x^4}} \, dx=\int \frac {1}{x^{11}\,{\left (b\,x^4+a\right )}^{1/4}} \,d x \]