Integrand size = 30, antiderivative size = 417 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=-\frac {2 c \sqrt {c-d x^2}}{a e \sqrt {e x}}-\frac {2 c^{3/4} \sqrt [4]{d} (b c+a d) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a b e^{3/2} \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} (b c+a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a b e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}} \]
[Out]
Time = 0.55 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {477, 485, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=-\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a b e^{3/2} \sqrt {c-d x^2}}-\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (a d+b c) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a b e^{3/2} \sqrt {c-d x^2}}-\frac {2 c \sqrt {c-d x^2}}{a e \sqrt {e x}} \]
[In]
[Out]
Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 485
Rule 504
Rule 598
Rule 1213
Rule 1214
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {\left (c-\frac {d x^4}{e^2}\right )^{3/2}}{x^2 \left (a-\frac {b x^4}{e^2}\right )} \, dx,x,\sqrt {e x}\right )}{e} \\ & = -\frac {2 c \sqrt {c-d x^2}}{a e \sqrt {e x}}+\frac {2 \text {Subst}\left (\int \frac {x^2 \left (\frac {c (b c-3 a d)}{e^2}+\frac {d (b c+a 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 )}{a e} \\ & = -\frac {2 c \sqrt {c-d x^2}}{a e \sqrt {e x}}+\frac {2 \text {Subst}\left (\int \left (-\frac {d (b c+a d) x^2}{b e^2 \sqrt {c-\frac {d x^4}{e^2}}}+\frac {\left (b^2 c^2-2 a b c d+a^2 d^2\right ) x^2}{b 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 )}{a e} \\ & = -\frac {2 c \sqrt {c-d x^2}}{a e \sqrt {e x}}+\frac {\left (2 (b c-a d)^2\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 )}{a b e^3}-\frac {(2 d (b c+a d)) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a b e^3} \\ & = -\frac {2 c \sqrt {c-d x^2}}{a e \sqrt {e x}}+\frac {\left (2 \sqrt {c} \sqrt {d} (b c+a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a b e^2}-\frac {\left (2 \sqrt {c} \sqrt {d} (b c+a d)\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 )}{a b e^2}+\frac {(b c-a d)^2 \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 )}{a b^{3/2} e}-\frac {(b c-a d)^2 \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 )}{a b^{3/2} e} \\ & = -\frac {2 c \sqrt {c-d x^2}}{a e \sqrt {e x}}+\frac {\left (2 \sqrt {c} \sqrt {d} (b c+a d) \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 )}{a b e^2 \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {c} \sqrt {d} (b c+a d) \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 )}{a b e^2 \sqrt {c-d x^2}}+\frac {\left ((b c-a d)^2 \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 )}{a b^{3/2} e \sqrt {c-d x^2}}-\frac {\left ((b c-a d)^2 \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 )}{a b^{3/2} e \sqrt {c-d x^2}} \\ & = -\frac {2 c \sqrt {c-d x^2}}{a e \sqrt {e x}}+\frac {2 c^{3/4} \sqrt [4]{d} (b c+a d) \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 )}{a b e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d)^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 )}{a^{3/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^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 )}{a^{3/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {c} \sqrt {d} (b c+a d) \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 )}{a b e^2 \sqrt {c-d x^2}} \\ & = -\frac {2 c \sqrt {c-d x^2}}{a e \sqrt {e x}}-\frac {2 c^{3/4} \sqrt [4]{d} (b c+a d) \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 )}{a b e^{3/2} \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} (b c+a d) \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 )}{a b e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d)^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 )}{a^{3/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^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 )}{a^{3/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.36 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\frac {x \left (-42 a c \left (c-d x^2\right )+14 c (b c-3 a d) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+6 d (b c+a d) x^4 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{21 a^2 (e x)^{3/2} \sqrt {c-d x^2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1291\) vs. \(2(317)=634\).
Time = 3.15 (sec) , antiderivative size = 1292, normalized size of antiderivative = 3.10
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1292\) |
default | \(\text {Expression too large to display}\) | \(1747\) |
[In]
[Out]
Timed out. \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=- \int \frac {c \sqrt {c - d x^{2}}}{- a \left (e x\right )^{\frac {3}{2}} + b x^{2} \left (e x\right )^{\frac {3}{2}}}\, dx - \int \left (- \frac {d x^{2} \sqrt {c - d x^{2}}}{- a \left (e x\right )^{\frac {3}{2}} + b x^{2} \left (e x\right )^{\frac {3}{2}}}\right )\, dx \]
[In]
[Out]
\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\int { -\frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\int { -\frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\int \frac {{\left (c-d\,x^2\right )}^{3/2}}{{\left (e\,x\right )}^{3/2}\,\left (a-b\,x^2\right )} \,d x \]
[In]
[Out]