Integrand size = 16, antiderivative size = 136 \[ \int \frac {1}{x^{12} \left (a-b x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}}-\frac {10 b \sqrt [4]{a-b x^4}}{77 a^2 x^7}-\frac {20 b^2 \sqrt [4]{a-b x^4}}{77 a^3 x^3}-\frac {40 b^{7/2} \left (1-\frac {a}{b x^4}\right )^{3/4} x^3 \operatorname {EllipticF}\left (\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{77 a^{7/2} \left (a-b x^4\right )^{3/4}} \]
-1/11*(-b*x^4+a)^(1/4)/a/x^11-10/77*b*(-b*x^4+a)^(1/4)/a^2/x^7-20/77*b^2*( -b*x^4+a)^(1/4)/a^3/x^3-40/77*b^(7/2)*(1-a/b/x^4)^(3/4)*x^3*(cos(1/2*arccs c(x^2*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arccsc(x^2*b^(1/2)/a^(1/2)))*Elli pticF(sin(1/2*arccsc(x^2*b^(1/2)/a^(1/2))),2^(1/2))/a^(7/2)/(-b*x^4+a)^(3/ 4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.38 \[ \int \frac {1}{x^{12} \left (a-b x^4\right )^{3/4}} \, dx=-\frac {\left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},\frac {3}{4},-\frac {7}{4},\frac {b x^4}{a}\right )}{11 x^{11} \left (a-b x^4\right )^{3/4}} \]
-1/11*((1 - (b*x^4)/a)^(3/4)*Hypergeometric2F1[-11/4, 3/4, -7/4, (b*x^4)/a ])/(x^11*(a - b*x^4)^(3/4))
Time = 0.32 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {847, 847, 847, 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.
\(\displaystyle \int \frac {1}{x^{12} \left (a-b x^4\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 847 |
\(\displaystyle \frac {10 b \int \frac {1}{x^8 \left (a-b x^4\right )^{3/4}}dx}{11 a}-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle \frac {10 b \left (\frac {6 b \int \frac {1}{x^4 \left (a-b x^4\right )^{3/4}}dx}{7 a}-\frac {\sqrt [4]{a-b x^4}}{7 a x^7}\right )}{11 a}-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle \frac {10 b \left (\frac {6 b \left (\frac {2 b \int \frac {1}{\left (a-b x^4\right )^{3/4}}dx}{3 a}-\frac {\sqrt [4]{a-b x^4}}{3 a x^3}\right )}{7 a}-\frac {\sqrt [4]{a-b x^4}}{7 a x^7}\right )}{11 a}-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}}\) |
\(\Big \downarrow \) 768 |
\(\displaystyle \frac {10 b \left (\frac {6 b \left (\frac {2 b x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} \int \frac {1}{\left (1-\frac {a}{b x^4}\right )^{3/4} x^3}dx}{3 a \left (a-b x^4\right )^{3/4}}-\frac {\sqrt [4]{a-b x^4}}{3 a x^3}\right )}{7 a}-\frac {\sqrt [4]{a-b x^4}}{7 a x^7}\right )}{11 a}-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {10 b \left (\frac {6 b \left (-\frac {2 b x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} \int \frac {1}{\left (1-\frac {a}{b x^4}\right )^{3/4} x}d\frac {1}{x}}{3 a \left (a-b x^4\right )^{3/4}}-\frac {\sqrt [4]{a-b x^4}}{3 a x^3}\right )}{7 a}-\frac {\sqrt [4]{a-b x^4}}{7 a x^7}\right )}{11 a}-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {10 b \left (\frac {6 b \left (-\frac {b x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} \int \frac {1}{\left (1-\frac {a}{b x^2}\right )^{3/4}}d\frac {1}{x^2}}{3 a \left (a-b x^4\right )^{3/4}}-\frac {\sqrt [4]{a-b x^4}}{3 a x^3}\right )}{7 a}-\frac {\sqrt [4]{a-b x^4}}{7 a x^7}\right )}{11 a}-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}}\) |
\(\Big \downarrow \) 230 |
\(\displaystyle \frac {10 b \left (\frac {6 b \left (-\frac {2 b^{3/2} x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {b} x^2}\right ),2\right )}{3 a^{3/2} \left (a-b x^4\right )^{3/4}}-\frac {\sqrt [4]{a-b x^4}}{3 a x^3}\right )}{7 a}-\frac {\sqrt [4]{a-b x^4}}{7 a x^7}\right )}{11 a}-\frac {\sqrt [4]{a-b x^4}}{11 a x^{11}}\) |
-1/11*(a - b*x^4)^(1/4)/(a*x^11) + (10*b*(-1/7*(a - b*x^4)^(1/4)/(a*x^7) + (6*b*(-1/3*(a - b*x^4)^(1/4)/(a*x^3) - (2*b^(3/2)*(1 - a/(b*x^4))^(3/4)*x ^3*EllipticF[ArcSin[Sqrt[a]/(Sqrt[b]*x^2)]/2, 2])/(3*a^(3/2)*(a - b*x^4)^( 3/4))))/(7*a)))/(11*a)
3.13.62.3.1 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 + 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]
\[\int \frac {1}{x^{12} \left (-b \,x^{4}+a \right )^{\frac {3}{4}}}d x\]
\[ \int \frac {1}{x^{12} \left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}} x^{12}} \,d x } \]
Result contains complex when optimal does not.
Time = 0.92 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x^{12} \left (a-b x^4\right )^{3/4}} \, dx=- \frac {i e^{\frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{14 b^{\frac {3}{4}} x^{14}} \]
\[ \int \frac {1}{x^{12} \left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}} x^{12}} \,d x } \]
\[ \int \frac {1}{x^{12} \left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}} x^{12}} \,d x } \]
Timed out. \[ \int \frac {1}{x^{12} \left (a-b x^4\right )^{3/4}} \, dx=\int \frac {1}{x^{12}\,{\left (a-b\,x^4\right )}^{3/4}} \,d x \]