Integrand size = 30, antiderivative size = 376 \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\frac {a e^3 \sqrt {e x} \sqrt {c-d x^2}}{2 b (b c-a d) \left (a-b x^2\right )}+\frac {\sqrt [4]{c} (4 b c-3 a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 b^2 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-3 a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^2 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-3 a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^2 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}} \]
[Out]
Time = 0.39 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {477, 481, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\frac {\sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (4 b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 b^2 \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}-\frac {\sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-3 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^2 \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}-\frac {\sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-3 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^2 \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}+\frac {a e^3 \sqrt {e x} \sqrt {c-d x^2}}{2 b \left (a-b x^2\right ) (b c-a d)} \]
[In]
[Out]
Rule 227
Rule 230
Rule 418
Rule 477
Rule 481
Rule 537
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x^8}{\left (a-\frac {b x^4}{e^2}\right )^2 \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {a e^3 \sqrt {e x} \sqrt {c-d x^2}}{2 b (b c-a d) \left (a-b x^2\right )}-\frac {e^3 \text {Subst}\left (\int \frac {a c+\frac {(4 b c-3 a d) x^4}{e^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 (b c-a d)} \\ & = \frac {a e^3 \sqrt {e x} \sqrt {c-d x^2}}{2 b (b c-a d) \left (a-b x^2\right )}+\frac {\left ((4 b c-3 a d) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 b^2 (b c-a d)}-\frac {\left (a (5 b c-3 a d) e^3\right ) \text {Subst}\left (\int \frac {1}{\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^2 (b c-a d)} \\ & = \frac {a e^3 \sqrt {e x} \sqrt {c-d x^2}}{2 b (b c-a d) \left (a-b x^2\right )}-\frac {\left ((5 b c-3 a d) e^3\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 b^2 (b c-a d)}-\frac {\left ((5 b c-3 a d) e^3\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 b^2 (b c-a d)}+\frac {\left ((4 b c-3 a d) e^3 \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^2 (b c-a d) \sqrt {c-d x^2}} \\ & = \frac {a e^3 \sqrt {e x} \sqrt {c-d x^2}}{2 b (b c-a d) \left (a-b x^2\right )}+\frac {\sqrt [4]{c} (4 b c-3 a d) e^{7/2} \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^2 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}-\frac {\left ((5 b c-3 a d) e^3 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{4 b^2 (b c-a d) \sqrt {c-d x^2}}-\frac {\left ((5 b c-3 a d) e^3 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{4 b^2 (b c-a d) \sqrt {c-d x^2}} \\ & = \frac {a e^3 \sqrt {e x} \sqrt {c-d x^2}}{2 b (b c-a d) \left (a-b x^2\right )}+\frac {\sqrt [4]{c} (4 b c-3 a d) e^{7/2} \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^2 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-3 a d) e^{7/2} \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 b^2 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-3 a d) e^{7/2} \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 b^2 \sqrt [4]{d} (b c-a d) \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 = 184, normalized size of antiderivative = 0.49 \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\frac {e^3 \sqrt {e x} \left (-5 a^2 \left (c-d x^2\right )+5 a c \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )-(-4 b c+3 a d) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{10 a b (-b c+a d) \left (a-b x^2\right ) \sqrt {c-d x^2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(922\) vs. \(2(294)=588\).
Time = 3.08 (sec) , antiderivative size = 923, normalized size of antiderivative = 2.45
method | result | size |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (-\frac {e^{3} a \sqrt {-d e \,x^{3}+c e x}}{2 \left (a d -b c \right ) b \left (-b \,x^{2}+a \right )}+\frac {\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}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) e^{4}}{d \sqrt {-d e \,x^{3}+c e x}\, b^{2}}-\frac {\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}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a \,e^{4}}{4 \sqrt {-d e \,x^{3}+c e x}\, b^{2} \left (a d -b c \right )}+\frac {3 a^{2} e^{4} \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 )}{8 b^{2} \left (a d -b c \right ) \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {5 a \,e^{4} \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 b \left (a d -b c \right ) \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {3 a^{2} e^{4} \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 )}{8 b^{2} \left (a d -b c \right ) \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {5 a \,e^{4} \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 b \left (a d -b c \right ) \sqrt {a b}\, 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}}\) | \(923\) |
default | \(\text {Expression too large to display}\) | \(2508\) |
[In]
[Out]
Timed out. \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int { \frac {\left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )}^{2} \sqrt {-d x^{2} + c}} \,d x } \]
[In]
[Out]
\[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int { \frac {\left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )}^{2} \sqrt {-d x^{2} + c}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}}{{\left (a-b\,x^2\right )}^2\,\sqrt {c-d\,x^2}} \,d x \]
[In]
[Out]