Integrand size = 15, antiderivative size = 128 \[ \int \frac {x^{13}}{\left (a+b x^4\right )^{3/4}} \, dx=\frac {20 a^2 x^2 \sqrt [4]{a+b x^4}}{77 b^3}-\frac {10 a x^6 \sqrt [4]{a+b x^4}}{77 b^2}+\frac {x^{10} \sqrt [4]{a+b x^4}}{11 b}-\frac {40 a^{7/2} \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{77 b^{7/2} \left (a+b x^4\right )^{3/4}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {281, 327, 239, 237} \[ \int \frac {x^{13}}{\left (a+b x^4\right )^{3/4}} \, dx=-\frac {40 a^{7/2} \left (\frac {b x^4}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{77 b^{7/2} \left (a+b x^4\right )^{3/4}}+\frac {20 a^2 x^2 \sqrt [4]{a+b x^4}}{77 b^3}-\frac {10 a x^6 \sqrt [4]{a+b x^4}}{77 b^2}+\frac {x^{10} \sqrt [4]{a+b x^4}}{11 b} \]
[In]
[Out]
Rule 237
Rule 239
Rule 281
Rule 327
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^6}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right ) \\ & = \frac {x^{10} \sqrt [4]{a+b x^4}}{11 b}-\frac {(5 a) \text {Subst}\left (\int \frac {x^4}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )}{11 b} \\ & = -\frac {10 a x^6 \sqrt [4]{a+b x^4}}{77 b^2}+\frac {x^{10} \sqrt [4]{a+b x^4}}{11 b}+\frac {\left (30 a^2\right ) \text {Subst}\left (\int \frac {x^2}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )}{77 b^2} \\ & = \frac {20 a^2 x^2 \sqrt [4]{a+b x^4}}{77 b^3}-\frac {10 a x^6 \sqrt [4]{a+b x^4}}{77 b^2}+\frac {x^{10} \sqrt [4]{a+b x^4}}{11 b}-\frac {\left (20 a^3\right ) \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )}{77 b^3} \\ & = \frac {20 a^2 x^2 \sqrt [4]{a+b x^4}}{77 b^3}-\frac {10 a x^6 \sqrt [4]{a+b x^4}}{77 b^2}+\frac {x^{10} \sqrt [4]{a+b x^4}}{11 b}-\frac {\left (20 a^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 )}{77 b^3 \left (a+b x^4\right )^{3/4}} \\ & = \frac {20 a^2 x^2 \sqrt [4]{a+b x^4}}{77 b^3}-\frac {10 a x^6 \sqrt [4]{a+b x^4}}{77 b^2}+\frac {x^{10} \sqrt [4]{a+b x^4}}{11 b}-\frac {40 a^{7/2} \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 )}{77 b^{7/2} \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 = 8.86 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.71 \[ \int \frac {x^{13}}{\left (a+b x^4\right )^{3/4}} \, dx=\frac {x^2 \left (20 a^3+10 a^2 b x^4-3 a b^2 x^8+7 b^3 x^{12}-20 a^3 \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\frac {b x^4}{a}\right )\right )}{77 b^3 \left (a+b x^4\right )^{3/4}} \]
[In]
[Out]
\[\int \frac {x^{13}}{\left (b \,x^{4}+a \right )^{\frac {3}{4}}}d x\]
[In]
[Out]
\[ \int \frac {x^{13}}{\left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {x^{13}}{{\left (b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.75 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.21 \[ \int \frac {x^{13}}{\left (a+b x^4\right )^{3/4}} \, dx=\frac {x^{14} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{14 a^{\frac {3}{4}}} \]
[In]
[Out]
\[ \int \frac {x^{13}}{\left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {x^{13}}{{\left (b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^{13}}{\left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {x^{13}}{{\left (b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^{13}}{\left (a+b x^4\right )^{3/4}} \, dx=\int \frac {x^{13}}{{\left (b\,x^4+a\right )}^{3/4}} \,d x \]
[In]
[Out]