Integrand size = 15, antiderivative size = 122 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{11}} \, dx=-\frac {b \sqrt [4]{a+b x^4}}{12 x^6}-\frac {b^2 \sqrt [4]{a+b x^4}}{24 a x^2}-\frac {\left (a+b x^4\right )^{5/4}}{10 x^{10}}-\frac {b^{5/2} \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{24 \sqrt {a} \left (a+b x^4\right )^{3/4}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {281, 283, 331, 239, 237} \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{11}} \, dx=-\frac {b^{5/2} \left (\frac {b x^4}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{24 \sqrt {a} \left (a+b x^4\right )^{3/4}}-\frac {b^2 \sqrt [4]{a+b x^4}}{24 a x^2}-\frac {\left (a+b x^4\right )^{5/4}}{10 x^{10}}-\frac {b \sqrt [4]{a+b x^4}}{12 x^6} \]
[In]
[Out]
Rule 237
Rule 239
Rule 281
Rule 283
Rule 331
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\left (a+b x^2\right )^{5/4}}{x^6} \, dx,x,x^2\right ) \\ & = -\frac {\left (a+b x^4\right )^{5/4}}{10 x^{10}}+\frac {1}{4} b \text {Subst}\left (\int \frac {\sqrt [4]{a+b x^2}}{x^4} \, dx,x,x^2\right ) \\ & = -\frac {b \sqrt [4]{a+b x^4}}{12 x^6}-\frac {\left (a+b x^4\right )^{5/4}}{10 x^{10}}+\frac {1}{24} b^2 \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right ) \\ & = -\frac {b \sqrt [4]{a+b x^4}}{12 x^6}-\frac {b^2 \sqrt [4]{a+b x^4}}{24 a x^2}-\frac {\left (a+b x^4\right )^{5/4}}{10 x^{10}}-\frac {b^3 \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )}{48 a} \\ & = -\frac {b \sqrt [4]{a+b x^4}}{12 x^6}-\frac {b^2 \sqrt [4]{a+b x^4}}{24 a x^2}-\frac {\left (a+b x^4\right )^{5/4}}{10 x^{10}}-\frac {\left (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 )}{48 a \left (a+b x^4\right )^{3/4}} \\ & = -\frac {b \sqrt [4]{a+b x^4}}{12 x^6}-\frac {b^2 \sqrt [4]{a+b x^4}}{24 a x^2}-\frac {\left (a+b x^4\right )^{5/4}}{10 x^{10}}-\frac {b^{5/2} \left (1+\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{24 \sqrt {a} \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 = 0.43 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{11}} \, dx=-\frac {a \sqrt [4]{a+b x^4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {5}{4},-\frac {3}{2},-\frac {b x^4}{a}\right )}{10 x^{10} \sqrt [4]{1+\frac {b x^4}{a}}} \]
[In]
[Out]
\[\int \frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}}}{x^{11}}d x\]
[In]
[Out]
\[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{11}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{x^{11}} \,d x } \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.28 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{11}} \, dx=- \frac {a^{\frac {5}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, - \frac {5}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{10 x^{10}} \]
[In]
[Out]
\[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{11}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{x^{11}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{11}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{x^{11}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{11}} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{5/4}}{x^{11}} \,d x \]
[In]
[Out]