Integrand size = 30, antiderivative size = 625 \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}-\frac {\sqrt [4]{d} \left (b^2 c^2+5 a b c d-a^2 d^2\right ) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a c^{5/4} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} \left (b^2 c^2+5 a b c d-a^2 d^2\right ) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a c^{5/4} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} (b c-11 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} (b c-11 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}} \]
[Out]
Time = 1.00 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {477, 483, 593, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=-\frac {b^{3/2} \sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-11 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^3}+\frac {b^{3/2} \sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-11 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^3}+\frac {\sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-a^2 d^2+5 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a c^{5/4} \sqrt {c-d x^2} (b c-a d)^3}-\frac {\sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-a^2 d^2+5 a b c d+b^2 c^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a c^{5/4} \sqrt {c-d x^2} (b c-a d)^3}+\frac {d (e x)^{3/2} \left (-a^2 d^2+5 a b c d+b^2 c^2\right )}{2 a c^2 e \sqrt {c-d x^2} (b c-a d)^3}+\frac {b (e x)^{3/2}}{2 a e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac {d (e x)^{3/2} (2 a d+3 b c)}{6 a c e \left (c-d x^2\right )^{3/2} (b c-a d)^2} \]
[In]
[Out]
Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 483
Rule 504
Rule 593
Rule 598
Rule 1213
Rule 1214
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x^2}{\left (a-\frac {b x^4}{e^2}\right )^2 \left (c-\frac {d x^4}{e^2}\right )^{5/2}} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {e \text {Subst}\left (\int \frac {x^2 \left (\frac {b c-4 a d}{e^2}-\frac {7 b d x^4}{e^4}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{5/2}} \, dx,x,\sqrt {e x}\right )}{2 a (b c-a d)} \\ & = \frac {d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}-\frac {e^3 \text {Subst}\left (\int \frac {x^2 \left (-\frac {6 \left (b^2 c^2-8 a b c d+2 a^2 d^2\right )}{e^4}+\frac {6 b d (3 b c+2 a d) x^4}{e^6}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt {e x}\right )}{12 a c (b c-a d)^2} \\ & = \frac {d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}+\frac {e^5 \text {Subst}\left (\int \frac {x^2 \left (\frac {12 \left (b^3 c^3-12 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right )}{e^6}+\frac {12 b d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) x^4}{e^8}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{24 a c^2 (b c-a d)^3} \\ & = \frac {d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}+\frac {e^5 \text {Subst}\left (\int \left (-\frac {12 d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) x^2}{e^6 \sqrt {c-\frac {d x^4}{e^2}}}+\frac {12 \left (b^3 c^3-11 a b^2 c^2 d\right ) x^2}{e^6 \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}\right ) \, dx,x,\sqrt {e x}\right )}{24 a c^2 (b c-a d)^3} \\ & = \frac {d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}+\frac {\left (b^2 (b c-11 a d)\right ) \text {Subst}\left (\int \frac {x^2}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a (b c-a d)^3 e}-\frac {\left (d \left (b^2 c^2+5 a b c d-a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a c^2 (b c-a d)^3 e} \\ & = \frac {d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}+\frac {\left (\sqrt {d} \left (b^2 c^2+5 a b c d-a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a c^{3/2} (b c-a d)^3}-\frac {\left (\sqrt {d} \left (b^2 c^2+5 a b c d-a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a c^{3/2} (b c-a d)^3}+\frac {\left (b^{3/2} (b c-11 a d) e\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} x^2\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a (b c-a d)^3}-\frac {\left (b^{3/2} (b c-11 a d) e\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e+\sqrt {b} x^2\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a (b c-a d)^3} \\ & = \frac {d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}+\frac {\left (\sqrt {d} \left (b^2 c^2+5 a b c d-a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a c^{3/2} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\left (\sqrt {d} \left (b^2 c^2+5 a b c d-a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a c^{3/2} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\left (b^{3/2} (b c-11 a d) e \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\left (b^{3/2} (b c-11 a d) e \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e+\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a (b c-a d)^3 \sqrt {c-d x^2}} \\ & = \frac {d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} \left (b^2 c^2+5 a b c d-a^2 d^2\right ) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a c^{5/4} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} (b c-11 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{3/2} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} (b c-11 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{3/2} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\left (\sqrt {d} \left (b^2 c^2+5 a b c d-a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}}{\sqrt {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}} \, dx,x,\sqrt {e x}\right )}{2 a c^{3/2} (b c-a d)^3 \sqrt {c-d x^2}} \\ & = \frac {d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}-\frac {\sqrt [4]{d} \left (b^2 c^2+5 a b c d-a^2 d^2\right ) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a c^{5/4} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} \left (b^2 c^2+5 a b c d-a^2 d^2\right ) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a c^{5/4} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} (b c-11 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{3/2} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} (b c-11 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{3/2} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.39 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {\sqrt {e x} \left (7 a x \left (a b^2 c d^2 x^2 \left (17 c-15 d x^2\right )+a^3 d^3 \left (5 c-3 d x^2\right )-3 b^3 c^2 \left (c-d x^2\right )^2+a^2 b d^2 \left (-17 c^2+10 c d x^2+3 d^2 x^4\right )\right )+7 \left (b^3 c^3-12 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right ) x \left (-a+b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) x^3 \left (-a+b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^2 c^2 (b c-a d)^3 \left (-a+b x^2\right ) \left (c-d x^2\right )^{3/2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1517\) vs. \(2(503)=1006\).
Time = 3.25 (sec) , antiderivative size = 1518, normalized size of antiderivative = 2.43
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1518\) |
default | \(\text {Expression too large to display}\) | \(5677\) |
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {e x}}{\left (- a + b x^{2}\right )^{2} \left (c - d x^{2}\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {e x}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {e x}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {e\,x}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{5/2}} \,d x \]
[In]
[Out]