Integrand size = 30, antiderivative size = 338 \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=-\frac {c e^3 \sqrt {e x}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-2 a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b d^{5/4} (b c-a d) \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}} \]
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Time = 0.38 (sec) , antiderivative size = 338, 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} \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\frac {\sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (b c-2 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b d^{5/4} \sqrt {c-d x^2} (b c-a d)}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}+\frac {a \sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}-\frac {c e^3 \sqrt {e x}}{d \sqrt {c-d x^2} (b c-a d)} \]
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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 ) \left (c-\frac {d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt {e x}\right )}{e} \\ & = -\frac {c e^3 \sqrt {e x}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {e^3 \text {Subst}\left (\int \frac {a c-\frac {(b c-2 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 )}{d (b c-a d)} \\ & = -\frac {c e^3 \sqrt {e x}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {\left (2 a^2 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 )}{b (b c-a d)}+\frac {\left ((b c-2 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 )}{b d (b c-a d)} \\ & = -\frac {c e^3 \sqrt {e x}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {\left (a 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 )}{b (b c-a d)}+\frac {\left (a 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 )}{b (b c-a d)}+\frac {\left ((b c-2 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 )}{b d (b c-a d) \sqrt {c-d x^2}} \\ & = -\frac {c e^3 \sqrt {e x}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-2 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 )}{b d^{5/4} (b c-a d) \sqrt {c-d x^2}}+\frac {\left (a 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 )}{b (b c-a d) \sqrt {c-d x^2}}+\frac {\left (a 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 )}{b (b c-a d) \sqrt {c-d x^2}} \\ & = -\frac {c e^3 \sqrt {e x}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-2 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 )}{b d^{5/4} (b c-a d) \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} 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 )}{b \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {a \sqrt [4]{c} 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 )}{b \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 = 11.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.44 \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=-\frac {e^3 \sqrt {e x} \left (-5 a c+5 a c \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+(-b c+2 a d) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{5 a d (-b c+a d) \sqrt {c-d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(593\) vs. \(2(264)=528\).
Time = 3.09 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.76
method | result | size |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (\frac {e^{4} x c}{d \left (a d -b c \right ) \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d e x}}+\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) e^{4}}{d^{2} \sqrt {-d e \,x^{3}+c e x}\, b}+\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) c \,e^{4}}{2 d^{2} \sqrt {-d e \,x^{3}+c e x}\, \left (a d -b c \right )}+\frac {e^{4} a^{2} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right ) b \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {e^{4} a^{2} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right ) b \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-d \,x^{2}+c}}\) | \(594\) |
default | \(-\frac {\left (2 F\left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} d^{2} \sqrt {c d}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}-3 F\left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b c d \sqrt {c d}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}+F\left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, b^{2} c^{2} \sqrt {c d}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}+\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} b c \,d^{2}-\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}\, \Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {c d}\, a^{2} d^{2}-\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} b c \,d^{2}-\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}\, \Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {c d}\, a^{2} d^{2}+2 a b c \,d^{2} x \sqrt {a b}-2 b^{2} c^{2} d x \sqrt {a b}\right ) e^{3} \sqrt {e x}}{2 d x \sqrt {-d \,x^{2}+c}\, \left (\sqrt {c d}\, b -\sqrt {a b}\, d \right ) \left (\sqrt {c d}\, b +\sqrt {a b}\, d \right ) \sqrt {a b}\, \left (a d -b c \right )}\) | \(815\) |
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Timed out. \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=- \int \frac {\left (e x\right )^{\frac {7}{2}}}{- a c \sqrt {c - d x^{2}} + a d x^{2} \sqrt {c - d x^{2}} + b c x^{2} \sqrt {c - d x^{2}} - b d x^{4} \sqrt {c - d x^{2}}}\, dx \]
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\[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int { -\frac {\left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int { -\frac {\left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}}{\left (a-b\,x^2\right )\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \]
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