Integrand size = 30, antiderivative size = 391 \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {3 d e \sqrt {e x}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} d^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c+5 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c+5 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}} \]
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Time = 0.45 (sec) , antiderivative size = 391, 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, 482, 541, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {3 \sqrt [4]{c} d^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {c-d x^2} (b c-a d)^2}-\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (5 a d+b c) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^2}-\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (5 a d+b c) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^2}+\frac {3 d e \sqrt {e x}}{2 \sqrt {c-d x^2} (b c-a d)^2}+\frac {e \sqrt {e x}}{2 \left (a-b x^2\right ) \sqrt {c-d x^2} (b c-a d)} \]
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Rule 227
Rule 230
Rule 418
Rule 477
Rule 482
Rule 537
Rule 541
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x^4}{\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 \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {e \text {Subst}\left (\int \frac {c+\frac {5 d x^4}{e^2}}{\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 \sqrt {e x}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {e^3 \text {Subst}\left (\int \frac {-\frac {2 c (b c+2 a d)}{e^2}-\frac {6 b c 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 )}{4 c (b c-a d)^2} \\ & = \frac {3 d e \sqrt {e x}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e \sqrt {e x}}{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 {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 (b c-a d)^2}-\frac {((b c+5 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 c-a d)^2} \\ & = \frac {3 d e \sqrt {e x}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {((b c+5 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 c-a d)^2}-\frac {((b c+5 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 c-a d)^2}+\frac {\left (3 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 )}{2 (b c-a d)^2 \sqrt {c-d x^2}} \\ & = \frac {3 d e \sqrt {e x}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} d^{3/4} 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 )}{2 (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\left ((b c+5 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 c-a d)^2 \sqrt {c-d x^2}}-\frac {\left ((b c+5 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 c-a d)^2 \sqrt {c-d x^2}} \\ & = \frac {3 d e \sqrt {e x}}{2 (b c-a d)^2 \sqrt {c-d x^2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} d^{3/4} 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 )}{2 (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c+5 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 \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c+5 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 \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 = 11.23 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.48 \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=-\frac {e \sqrt {e x} \left (5 a \left (2 a d+b \left (c-3 d x^2\right )\right )+5 (b c+2 a d) \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b d x^2 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{10 a (b c-a d)^2 \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(839\) vs. \(2(303)=606\).
Time = 3.17 (sec) , antiderivative size = 840, normalized size of antiderivative = 2.15
method | result | size |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (\frac {b e \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^{2} x}{\left (a d -b c \right )^{2} \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d e x}}+\frac {3 e^{2} \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 )}{4 \left (a d -b c \right )^{2} \sqrt {-d e \,x^{3}+c e x}}+\frac {5 e^{2} \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} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {e^{2} \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 ) b c}{8 \left (a d -b c \right )^{2} \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 {5 e^{2} \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} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}-\frac {e^{2} \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 ) b c}{8 \left (a d -b c \right )^{2} \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}}\) | \(840\) |
default | \(\text {Expression too large to display}\) | \(2265\) |
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Timed out. \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \]
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