Integrand size = 30, antiderivative size = 485 \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} (b c+a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} \sqrt {b} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {3 \sqrt [4]{c} (b c+a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} \sqrt {b} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}} \]
[Out]
Time = 0.72 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {477, 482, 593, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {c-d x^2} (b c-a d)^2}-\frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 \sqrt {c-d x^2} (b c-a d)^2}+\frac {3 \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (a d+b c) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} \sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^2}-\frac {3 \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (a d+b c) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} \sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^2}+\frac {3 d e (e x)^{3/2}}{2 \sqrt {c-d x^2} (b c-a d)^2}+\frac {e (e x)^{3/2}}{2 \left (a-b x^2\right ) \sqrt {c-d x^2} (b c-a d)} \]
[In]
[Out]
Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 482
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^6}{\left (a-\frac {b x^4}{e^2}\right )^2 \left (c-\frac {d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {e \text {Subst}\left (\int \frac {x^2 \left (3 c+\frac {3 d x^4}{e^2}\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 )}{2 (b c-a d)} \\ & = \frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {e^3 \text {Subst}\left (\int \frac {x^2 \left (-\frac {6 c (b c+2 a d)}{e^2}+\frac {6 b c d x^4}{e^4}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 c (b c-a d)^2} \\ & = \frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {e^3 \text {Subst}\left (\int \left (-\frac {6 c d x^2}{e^2 \sqrt {c-\frac {d x^4}{e^2}}}-\frac {6 \left (b c^2+a c d\right ) x^2}{e^2 \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}\right ) \, dx,x,\sqrt {e x}\right )}{4 c (b c-a d)^2} \\ & = \frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {(3 d e) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 (b c-a d)^2}-\frac {(3 (b c+a d) e) \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 (b c-a d)^2} \\ & = \frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {\left (3 \sqrt {c} \sqrt {d} e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 (b c-a d)^2}-\frac {\left (3 \sqrt {c} \sqrt {d} e^2\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 (b c-a d)^2}-\frac {\left (3 (b c+a d) e^3\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 \sqrt {b} (b c-a d)^2}+\frac {\left (3 (b c+a d) e^3\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 \sqrt {b} (b c-a d)^2} \\ & = \frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {\left (3 \sqrt {c} \sqrt {d} e^2 \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 (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\left (3 \sqrt {c} \sqrt {d} e^2 \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 (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\left (3 (b c+a d) e^3 \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 \sqrt {b} (b c-a d)^2 \sqrt {c-d x^2}}+\frac {\left (3 (b c+a d) e^3 \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 \sqrt {b} (b c-a d)^2 \sqrt {c-d x^2}} \\ & = \frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \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 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} (b c+a d) e^{5/2} \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 \sqrt {a} \sqrt {b} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {3 \sqrt [4]{c} (b c+a d) e^{5/2} \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 \sqrt {a} \sqrt {b} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\left (3 \sqrt {c} \sqrt {d} e^2 \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 (b c-a d)^2 \sqrt {c-d x^2}} \\ & = \frac {3 d e (e x)^{3/2}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \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 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \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 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} (b c+a d) e^{5/2} \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 \sqrt {a} \sqrt {b} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {3 \sqrt [4]{c} (b c+a d) e^{5/2} \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 \sqrt {a} \sqrt {b} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.17 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.38 \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {e (e x)^{3/2} \left (-7 a \left (2 a d+b \left (c-3 d x^2\right )\right )+7 (b c+2 a d) \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b d x^2 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{14 a (b c-a d)^2 \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(916\) vs. \(2(369)=738\).
Time = 3.10 (sec) , antiderivative size = 917, normalized size of antiderivative = 1.89
method | result | size |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (\frac {b \,e^{2} x \sqrt {-d e \,x^{3}+c e x}}{2 \left (a d -b c \right )^{2} \left (-b \,x^{2}+a \right )}+\frac {d \,e^{3} x^{2}}{\left (a d -b c \right )^{2} \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d e x}}+\frac {3 e^{3} c \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right )^{2} \sqrt {-d e \,x^{3}+c e x}}-\frac {3 e^{3} c \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{4 \left (a d -b c \right )^{2} \sqrt {-d e \,x^{3}+c e x}}+\frac {3 e^{3} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) a}{8 \left (a d -b c \right )^{2} b \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {3 e^{3} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{8 \left (a d -b c \right )^{2} d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {3 e^{3} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) a}{8 \left (a d -b c \right )^{2} b \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {3 e^{3} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{8 \left (a d -b c \right )^{2} d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-d \,x^{2}+c}}\) | \(917\) |
default | \(\text {Expression too large to display}\) | \(2549\) |
[In]
[Out]
Timed out. \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \]
[In]
[Out]