Integrand size = 16, antiderivative size = 133 \[ \int \frac {1}{x^{11} \left (a-b x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{a-b x^4}}{10 a x^{10}}-\frac {3 b \sqrt [4]{a-b x^4}}{20 a^2 x^6}-\frac {3 b^2 \sqrt [4]{a-b x^4}}{8 a^3 x^2}+\frac {3 b^{5/2} \left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{8 a^{5/2} \left (a-b x^4\right )^{3/4}} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {281, 331, 239, 238} \[ \int \frac {1}{x^{11} \left (a-b x^4\right )^{3/4}} \, dx=\frac {3 b^{5/2} \left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{8 a^{5/2} \left (a-b x^4\right )^{3/4}}-\frac {3 b^2 \sqrt [4]{a-b x^4}}{8 a^3 x^2}-\frac {3 b \sqrt [4]{a-b x^4}}{20 a^2 x^6}-\frac {\sqrt [4]{a-b x^4}}{10 a x^{10}} \]
[In]
[Out]
Rule 238
Rule 239
Rule 281
Rule 331
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^6 \left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt [4]{a-b x^4}}{10 a x^{10}}+\frac {(9 b) \text {Subst}\left (\int \frac {1}{x^4 \left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{20 a} \\ & = -\frac {\sqrt [4]{a-b x^4}}{10 a x^{10}}-\frac {3 b \sqrt [4]{a-b x^4}}{20 a^2 x^6}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{8 a^2} \\ & = -\frac {\sqrt [4]{a-b x^4}}{10 a x^{10}}-\frac {3 b \sqrt [4]{a-b x^4}}{20 a^2 x^6}-\frac {3 b^2 \sqrt [4]{a-b x^4}}{8 a^3 x^2}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {1}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{16 a^3} \\ & = -\frac {\sqrt [4]{a-b x^4}}{10 a x^{10}}-\frac {3 b \sqrt [4]{a-b x^4}}{20 a^2 x^6}-\frac {3 b^2 \sqrt [4]{a-b x^4}}{8 a^3 x^2}+\frac {\left (3 b^3 \left (1-\frac {b x^4}{a}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{16 a^3 \left (a-b x^4\right )^{3/4}} \\ & = -\frac {\sqrt [4]{a-b x^4}}{10 a x^{10}}-\frac {3 b \sqrt [4]{a-b x^4}}{20 a^2 x^6}-\frac {3 b^2 \sqrt [4]{a-b x^4}}{8 a^3 x^2}+\frac {3 b^{5/2} \left (1-\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{8 a^{5/2} \left (a-b x^4\right )^{3/4}} \\ \end{align*}
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 = 0.39 \[ \int \frac {1}{x^{11} \left (a-b x^4\right )^{3/4}} \, dx=-\frac {\left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{4},-\frac {3}{2},\frac {b x^4}{a}\right )}{10 x^{10} \left (a-b x^4\right )^{3/4}} \]
[In]
[Out]
\[\int \frac {1}{x^{11} \left (-b \,x^{4}+a \right )^{\frac {3}{4}}}d x\]
[In]
[Out]
\[ \int \frac {1}{x^{11} \left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}} x^{11}} \,d x } \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.86 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^{11} \left (a-b x^4\right )^{3/4}} \, dx=- \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {3}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{10 a^{\frac {3}{4}} x^{10}} \]
[In]
[Out]
\[ \int \frac {1}{x^{11} \left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}} x^{11}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{x^{11} \left (a-b x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}} x^{11}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{x^{11} \left (a-b x^4\right )^{3/4}} \, dx=\int \frac {1}{x^{11}\,{\left (a-b\,x^4\right )}^{3/4}} \,d x \]
[In]
[Out]