Integrand size = 26, antiderivative size = 349 \[ \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {(A b-a B) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac {(A b-7 a B) e (e x)^{3/2}}{6 a b^2 \sqrt {a+b x^2}}-\frac {(A b-7 a B) e^2 \sqrt {e x} \sqrt {a+b x^2}}{2 a b^{5/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {(A b-7 a B) e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 a^{3/4} b^{11/4} \sqrt {a+b x^2}}-\frac {(A b-7 a B) e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 a^{3/4} b^{11/4} \sqrt {a+b x^2}} \]
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Time = 0.20 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {468, 294, 335, 311, 226, 1210} \[ \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (A b-7 a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 a^{3/4} b^{11/4} \sqrt {a+b x^2}}+\frac {e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (A b-7 a B) E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 a^{3/4} b^{11/4} \sqrt {a+b x^2}}-\frac {e^2 \sqrt {e x} \sqrt {a+b x^2} (A b-7 a B)}{2 a b^{5/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {e (e x)^{3/2} (A b-7 a B)}{6 a b^2 \sqrt {a+b x^2}}+\frac {(e x)^{7/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}} \]
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Rule 226
Rule 294
Rule 311
Rule 335
Rule 468
Rule 1210
Rubi steps \begin{align*} \text {integral}& = \frac {(A b-a B) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac {\left (-\frac {A b}{2}+\frac {7 a B}{2}\right ) \int \frac {(e x)^{5/2}}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a b} \\ & = \frac {(A b-a B) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac {(A b-7 a B) e (e x)^{3/2}}{6 a b^2 \sqrt {a+b x^2}}-\frac {\left ((A b-7 a B) e^2\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{4 a b^2} \\ & = \frac {(A b-a B) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac {(A b-7 a B) e (e x)^{3/2}}{6 a b^2 \sqrt {a+b x^2}}-\frac {((A b-7 a B) e) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a b^2} \\ & = \frac {(A b-a B) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac {(A b-7 a B) e (e x)^{3/2}}{6 a b^2 \sqrt {a+b x^2}}-\frac {\left ((A b-7 a B) e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 \sqrt {a} b^{5/2}}+\frac {\left ((A b-7 a B) e^2\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a} e}}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 \sqrt {a} b^{5/2}} \\ & = \frac {(A b-a B) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac {(A b-7 a B) e (e x)^{3/2}}{6 a b^2 \sqrt {a+b x^2}}-\frac {(A b-7 a B) e^2 \sqrt {e x} \sqrt {a+b x^2}}{2 a b^{5/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {(A b-7 a B) e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 a^{3/4} b^{11/4} \sqrt {a+b x^2}}-\frac {(A b-7 a B) e^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{4 a^{3/4} b^{11/4} \sqrt {a+b x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.28 \[ \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {2 e (e x)^{3/2} \left (a \left (A b-7 a B-3 b B x^2\right )+(-A b+7 a B) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{3 a b^2 \left (a+b x^2\right )^{3/2}} \]
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Time = 3.02 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.86
method | result | size |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {e^{2} x \left (A b -B a \right ) \sqrt {b e \,x^{3}+a e x}}{3 b^{4} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {e^{3} x^{2} \left (A b -3 B a \right )}{2 b^{2} a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\left (\frac {B \,e^{3}}{b^{2}}-\frac {e^{3} \left (A b -3 B a \right )}{4 b^{2} a}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(300\) |
default | \(-\frac {\left (6 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{2}-3 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{2}-42 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,x^{2}+21 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,x^{2}+6 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b -3 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b -42 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}+21 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-6 A \,b^{3} x^{4}+18 B a \,b^{2} x^{4}-2 a A \,b^{2} x^{2}+14 B \,a^{2} b \,x^{2}\right ) e^{2} \sqrt {e x}}{12 x \,b^{3} a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(767\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.51 \[ \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {3 \, {\left ({\left (7 \, B a b^{2} - A b^{3}\right )} e^{2} x^{4} + 2 \, {\left (7 \, B a^{2} b - A a b^{2}\right )} e^{2} x^{2} + {\left (7 \, B a^{3} - A a^{2} b\right )} e^{2}\right )} \sqrt {b e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (3 \, {\left (3 \, B a b^{2} - A b^{3}\right )} e^{2} x^{3} + {\left (7 \, B a^{2} b - A a b^{2}\right )} e^{2} x\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{6 \, {\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \]
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Timed out. \[ \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^{5/2}}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \]
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