Integrand size = 30, antiderivative size = 413 \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\frac {e (e x)^{3/2} \sqrt {c-d x^2}}{2 b \left (a-b x^2\right )}-\frac {5 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 b^2 \sqrt {c-d x^2}}+\frac {5 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 b^2 \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (3 b c-5 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (3 b c-5 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}} \]
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Time = 0.51 (sec) , antiderivative size = 413, 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, 478, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\frac {\sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (3 b c-5 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (3 b c-5 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {e (e x)^{3/2} \sqrt {c-d x^2}}{2 b \left (a-b x^2\right )}+\frac {5 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 b^2 \sqrt {c-d x^2}}-\frac {5 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 b^2 \sqrt {c-d x^2}} \]
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Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 478
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 {x^6 \sqrt {c-\frac {d x^4}{e^2}}}{\left (a-\frac {b x^4}{e^2}\right )^2} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {e (e x)^{3/2} \sqrt {c-d x^2}}{2 b \left (a-b x^2\right )}-\frac {e \text {Subst}\left (\int \frac {x^2 \left (3 c-\frac {5 d x^4}{e^2}\right )}{\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} \\ & = \frac {e (e x)^{3/2} \sqrt {c-d x^2}}{2 b \left (a-b x^2\right )}-\frac {e \text {Subst}\left (\int \left (\frac {5 d x^2}{b \sqrt {c-\frac {d x^4}{e^2}}}+\frac {(3 b c-5 a d) x^2}{b \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}\right ) \, dx,x,\sqrt {e x}\right )}{2 b} \\ & = \frac {e (e x)^{3/2} \sqrt {c-d x^2}}{2 b \left (a-b x^2\right )}-\frac {(5 d e) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 b^2}-\frac {((3 b c-5 a d) e) \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 )}{2 b^2} \\ & = \frac {e (e x)^{3/2} \sqrt {c-d x^2}}{2 b \left (a-b x^2\right )}+\frac {\left (5 \sqrt {c} \sqrt {d} e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 b^2}-\frac {\left (5 \sqrt {c} \sqrt {d} e^2\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 )}{2 b^2}-\frac {\left ((3 b c-5 a d) e^3\right ) \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 )}{4 b^{5/2}}+\frac {\left ((3 b c-5 a d) e^3\right ) \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 )}{4 b^{5/2}} \\ & = \frac {e (e x)^{3/2} \sqrt {c-d x^2}}{2 b \left (a-b x^2\right )}+\frac {\left (5 \sqrt {c} \sqrt {d} e^2 \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 \sqrt {c-d x^2}}-\frac {\left (5 \sqrt {c} \sqrt {d} e^2 \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 )}{2 b^2 \sqrt {c-d x^2}}-\frac {\left ((3 b c-5 a d) e^3 \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 )}{4 b^{5/2} \sqrt {c-d x^2}}+\frac {\left ((3 b c-5 a d) e^3 \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 )}{4 b^{5/2} \sqrt {c-d x^2}} \\ & = \frac {e (e x)^{3/2} \sqrt {c-d x^2}}{2 b \left (a-b x^2\right )}+\frac {5 c^{3/4} \sqrt [4]{d} e^{5/2} \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 {c-d x^2}}+\frac {\sqrt [4]{c} (3 b c-5 a d) e^{5/2} \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 \sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (3 b c-5 a d) e^{5/2} \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 \sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\left (5 \sqrt {c} \sqrt {d} e^2 \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 )}{2 b^2 \sqrt {c-d x^2}} \\ & = \frac {e (e x)^{3/2} \sqrt {c-d x^2}}{2 b \left (a-b x^2\right )}-\frac {5 c^{3/4} \sqrt [4]{d} e^{5/2} \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 )}{2 b^2 \sqrt {c-d x^2}}+\frac {5 c^{3/4} \sqrt [4]{d} e^{5/2} \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 {c-d x^2}}+\frac {\sqrt [4]{c} (3 b c-5 a d) e^{5/2} \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 \sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (3 b c-5 a d) e^{5/2} \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 \sqrt {a} b^{5/2} \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 = 11.14 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.39 \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\frac {e (e x)^{3/2} \left (-7 a \left (c-d x^2\right )+7 c \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+5 d x^2 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{14 a b \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(825\) vs. \(2(303)=606\).
Time = 3.16 (sec) , antiderivative size = 826, normalized size of antiderivative = 2.00
method | result | size |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (\frac {e^{2} x \sqrt {-d e \,x^{3}+c e x}}{2 b \left (-b \,x^{2}+a \right )}+\frac {5 e^{3} c \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{2 b^{2} \sqrt {-d e \,x^{3}+c e x}}-\frac {5 e^{3} c \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 b^{2} \sqrt {-d e \,x^{3}+c e x}}-\frac {5 e^{3} \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 b^{3} \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {3 e^{3} \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^{2} d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {5 e^{3} \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 b^{3} \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {3 e^{3} \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^{2} 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}}\) | \(826\) |
default | \(\text {Expression too large to display}\) | \(2530\) |
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Timed out. \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\int \frac {\left (e x\right )^{\frac {5}{2}} \sqrt {c - d x^{2}}}{\left (- a + b x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {\sqrt {-d x^{2} + c} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]
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\[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {\sqrt {-d x^{2} + c} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}\,\sqrt {c-d\,x^2}}{{\left (a-b\,x^2\right )}^2} \,d x \]
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