Integrand size = 21, antiderivative size = 371 \[ \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\frac {b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac {\sqrt {b} \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{10 a^{3/2} c (b c-a d)^3 \sqrt [4]{a+b x^2}}-\frac {\sqrt [4]{a} d^{3/2} (11 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (-b c+a d)^{7/2} x}+\frac {\sqrt [4]{a} d^{3/2} (11 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (-b c+a d)^{7/2} x} \]
[Out]
Time = 0.45 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {425, 541, 544, 235, 233, 202, 408, 504, 1232} \[ \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\frac {\sqrt {b} \sqrt [4]{\frac {b x^2}{a}+1} \left (-5 a^2 d^2-52 a b c d+12 b^2 c^2\right ) E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{10 a^{3/2} c \sqrt [4]{a+b x^2} (b c-a d)^3}-\frac {\sqrt [4]{a} d^{3/2} \sqrt {-\frac {b x^2}{a}} (11 b c-2 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{4 c x (a d-b c)^{7/2}}+\frac {\sqrt [4]{a} d^{3/2} \sqrt {-\frac {b x^2}{a}} (11 b c-2 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{4 c x (a d-b c)^{7/2}}-\frac {d x}{2 c \left (a+b x^2\right )^{5/4} \left (c+d x^2\right ) (b c-a d)}+\frac {b x (5 a d+4 b c)}{10 a c \left (a+b x^2\right )^{5/4} (b c-a d)^2} \]
[In]
[Out]
Rule 202
Rule 233
Rule 235
Rule 408
Rule 425
Rule 504
Rule 541
Rule 544
Rule 1232
Rubi steps \begin{align*} \text {integral}& = -\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac {\int \frac {2 b c-a d-\frac {7}{2} b d x^2}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)} \\ & = \frac {b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (-6 b^2 c^2+20 a b c d-5 a^2 d^2\right )-\frac {3}{4} b d (4 b c+5 a d) x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx}{5 a c (b c-a d)^2} \\ & = \frac {b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}+\frac {b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) x}{10 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^2}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac {2 \int \frac {\frac {1}{4} \left (-6 b^3 c^3+26 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right )-\frac {1}{8} b d \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) x^2}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{5 a^2 c (b c-a d)^3} \\ & = \frac {b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}+\frac {b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) x}{10 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^2}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac {\left (d^2 (11 b c-2 a d)\right ) \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{4 c (b c-a d)^3}-\frac {\left (b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right )\right ) \int \frac {1}{\sqrt [4]{a+b x^2}} \, dx}{20 a^2 c (b c-a d)^3} \\ & = \frac {b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}+\frac {b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) x}{10 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^2}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac {\left (d^2 (11 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{a}} \left (b c-a d+d x^4\right )} \, dx,x,\sqrt [4]{a+b x^2}\right )}{2 c (b c-a d)^3 x}-\frac {\left (b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^2}} \\ & = \frac {b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}-\frac {\left (d^{3/2} (11 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-b c+a d}-\sqrt {d} x^2\right ) \sqrt {1-\frac {x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c (b c-a d)^3 x}+\frac {\left (d^{3/2} (11 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-b c+a d}+\sqrt {d} x^2\right ) \sqrt {1-\frac {x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c (b c-a d)^3 x}+\frac {\left (b \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx}{20 a^2 c (b c-a d)^3 \sqrt [4]{a+b x^2}} \\ & = \frac {b (4 b c+5 a d) x}{10 a c (b c-a d)^2 \left (a+b x^2\right )^{5/4}}-\frac {d x}{2 c (b c-a d) \left (a+b x^2\right )^{5/4} \left (c+d x^2\right )}+\frac {\sqrt {b} \left (12 b^2 c^2-52 a b c d-5 a^2 d^2\right ) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{10 a^{3/2} c (b c-a d)^3 \sqrt [4]{a+b x^2}}-\frac {\sqrt [4]{a} d^{3/2} (11 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c (-b c+a d)^{7/2} x}+\frac {\sqrt [4]{a} d^{3/2} (11 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c (-b c+a d)^{7/2} x} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.70 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\frac {b d \left (-12 b^2 c^2+52 a b c d+5 a^2 d^2\right ) x^3 \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+\frac {6 c \left (-6 a c x \left (10 a^4 d^3+15 a^3 b d^2 \left (-2 c+d x^2\right )-6 b^4 c^2 x^2 \left (c+2 d x^2\right )+a^2 b^2 d \left (30 c^2+26 c d x^2+5 d^2 x^4\right )+2 a b^3 c \left (-5 c^2+5 c d x^2+26 d^2 x^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^3 \left (5 a^4 d^3+10 a^3 b d^3 x^2-12 b^4 c^2 x^2 \left (c+d x^2\right )+a^2 b^2 d \left (56 c^2+56 c d x^2+5 d^2 x^4\right )+4 a b^3 c \left (-4 c^2+9 c d x^2+13 d^2 x^4\right )\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (a+b x^2\right ) \left (c+d x^2\right ) \left (6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}}{60 a^2 c^2 (b c-a d)^3 \sqrt [4]{a+b x^2}} \]
[In]
[Out]
\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {9}{4}} \left (d \,x^{2}+c \right )^{2}}d x\]
[In]
[Out]
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {9}{4}} \left (c + d x^{2}\right )^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{9/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{9/4}\,{\left (d\,x^2+c\right )}^2} \,d x \]
[In]
[Out]