Integrand size = 30, antiderivative size = 381 \[ \int \frac {(e x)^{3/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=-\frac {7 d e \sqrt {e x} \sqrt {c-d x^2}}{6 b^2}+\frac {e \sqrt {e x} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {\sqrt [4]{c} d^{3/4} (17 b c-21 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 b^3 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-7 a d) (b c-a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a b^3 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-7 a d) (b c-a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a b^3 \sqrt [4]{d} \sqrt {c-d x^2}} \]
[Out]
Time = 0.54 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {477, 478, 542, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {(e x)^{3/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=-\frac {\sqrt [4]{c} d^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (17 b c-21 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 b^3 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-7 a d) (b c-a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a b^3 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-7 a d) (b c-a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a b^3 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {e \sqrt {e x} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {7 d e \sqrt {e x} \sqrt {c-d x^2}}{6 b^2} \]
[In]
[Out]
Rule 227
Rule 230
Rule 418
Rule 477
Rule 478
Rule 537
Rule 542
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x^4 \left (c-\frac {d x^4}{e^2}\right )^{3/2}}{\left (a-\frac {b x^4}{e^2}\right )^2} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {e \sqrt {e x} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {e \text {Subst}\left (\int \frac {\left (c-\frac {7 d x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}{a-\frac {b x^4}{e^2}} \, dx,x,\sqrt {e x}\right )}{2 b} \\ & = -\frac {7 d e \sqrt {e x} \sqrt {c-d x^2}}{6 b^2}+\frac {e \sqrt {e x} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}+\frac {e^3 \text {Subst}\left (\int \frac {-\frac {c (3 b c-7 a d)}{e^2}+\frac {d (17 b c-21 a d) x^4}{e^4}}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{6 b^2} \\ & = -\frac {7 d e \sqrt {e x} \sqrt {c-d x^2}}{6 b^2}+\frac {e \sqrt {e x} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {(d (17 b c-21 a d) e) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{6 b^3}-\frac {((b c-7 a d) (b c-a d) e) \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^3} \\ & = -\frac {7 d e \sqrt {e x} \sqrt {c-d x^2}}{6 b^2}+\frac {e \sqrt {e x} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {((b c-7 a d) (b c-a d) e) \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 a b^3}-\frac {((b c-7 a d) (b c-a d) e) \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 a b^3}-\frac {\left (d (17 b c-21 a d) e \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 )}{6 b^3 \sqrt {c-d x^2}} \\ & = -\frac {7 d e \sqrt {e x} \sqrt {c-d x^2}}{6 b^2}+\frac {e \sqrt {e x} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {\sqrt [4]{c} d^{3/4} (17 b c-21 a d) e^{3/2} \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 )}{6 b^3 \sqrt {c-d x^2}}-\frac {\left ((b c-7 a d) (b c-a d) e \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 a b^3 \sqrt {c-d x^2}}-\frac {\left ((b c-7 a d) (b c-a d) e \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 a b^3 \sqrt {c-d x^2}} \\ & = -\frac {7 d e \sqrt {e x} \sqrt {c-d x^2}}{6 b^2}+\frac {e \sqrt {e x} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {\sqrt [4]{c} d^{3/4} (17 b c-21 a d) e^{3/2} \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 )}{6 b^3 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-7 a d) (b c-a d) e^{3/2} \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 a b^3 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-7 a d) (b c-a d) e^{3/2} \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 a b^3 \sqrt [4]{d} \sqrt {c-d x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.18 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.51 \[ \int \frac {(e x)^{3/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {e \sqrt {e x} \left (5 a \left (c-d x^2\right ) \left (-3 b c+7 a d-4 b d x^2\right )-5 c (-3 b c+7 a d) \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+d (-17 b c+21 a d) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{30 a b^2 \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1193\) vs. \(2(293)=586\).
Time = 4.29 (sec) , antiderivative size = 1194, normalized size of antiderivative = 3.13
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1194\) |
risch | \(\text {Expression too large to display}\) | \(1291\) |
default | \(\text {Expression too large to display}\) | \(3454\) |
[In]
[Out]
Timed out. \[ \int \frac {(e x)^{3/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(e x)^{3/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int \frac {\left (e x\right )^{\frac {3}{2}} \left (c - d x^{2}\right )^{\frac {3}{2}}}{\left (- a + b x^{2}\right )^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {(e x)^{3/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {(e x)^{3/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(e x)^{3/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}\,{\left (c-d\,x^2\right )}^{3/2}}{{\left (a-b\,x^2\right )}^2} \,d x \]
[In]
[Out]