Integrand size = 26, antiderivative size = 337 \[ \int \sqrt {e x} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {2 (3 A b-a B) (e x)^{3/2} \sqrt {a+b x^2}}{15 b e}+\frac {4 a (3 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e}-\frac {4 a^{5/4} (3 A b-a B) \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^2}}+\frac {2 a^{5/4} (3 A b-a B) \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^2}} \]
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Time = 0.19 (sec) , antiderivative size = 337, 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 = {470, 285, 335, 311, 226, 1210} \[ \int \sqrt {e x} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {2 a^{5/4} \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^2}}-\frac {4 a^{5/4} \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-a B) E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 b^{7/4} \sqrt {a+b x^2}}+\frac {4 a \sqrt {e x} \sqrt {a+b x^2} (3 A b-a B)}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 (e x)^{3/2} \sqrt {a+b x^2} (3 A b-a B)}{15 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e} \]
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Rule 226
Rule 285
Rule 311
Rule 335
Rule 470
Rule 1210
Rubi steps \begin{align*} \text {integral}& = \frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e}-\frac {\left (2 \left (-\frac {9 A b}{2}+\frac {3 a B}{2}\right )\right ) \int \sqrt {e x} \sqrt {a+b x^2} \, dx}{9 b} \\ & = \frac {2 (3 A b-a B) (e x)^{3/2} \sqrt {a+b x^2}}{15 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e}+\frac {(2 a (3 A b-a B)) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{15 b} \\ & = \frac {2 (3 A b-a B) (e x)^{3/2} \sqrt {a+b x^2}}{15 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e}+\frac {(4 a (3 A b-a B)) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 b e} \\ & = \frac {2 (3 A b-a B) (e x)^{3/2} \sqrt {a+b x^2}}{15 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e}+\frac {\left (4 a^{3/2} (3 A b-a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 b^{3/2}}-\frac {\left (4 a^{3/2} (3 A b-a B)\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 )}{15 b^{3/2}} \\ & = \frac {2 (3 A b-a B) (e x)^{3/2} \sqrt {a+b x^2}}{15 b e}+\frac {4 a (3 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{15 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e}-\frac {4 a^{5/4} (3 A b-a B) \sqrt {e} \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 )}{15 b^{7/4} \sqrt {a+b x^2}}+\frac {2 a^{5/4} (3 A b-a B) \sqrt {e} \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 )}{15 b^{7/4} \sqrt {a+b x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.28 \[ \int \sqrt {e x} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {2 x \sqrt {e x} \sqrt {a+b x^2} \left (B \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}}+(3 A b-a B) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{9 b \sqrt {1+\frac {b x^2}{a}}} \]
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Time = 3.07 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.71
method | result | size |
risch | \(\frac {2 x^{2} \left (5 b B \,x^{2}+9 A b +2 B a \right ) \sqrt {b \,x^{2}+a}\, e}{45 b \sqrt {e x}}+\frac {2 a \left (3 A b -B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) e \sqrt {\left (b \,x^{2}+a \right ) e x}}{15 b^{2} \sqrt {b e \,x^{3}+a e x}\, \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(238\) |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {2 B \,x^{3} \sqrt {b e \,x^{3}+a e x}}{9}+\frac {2 \left (\left (A b +B a \right ) e -\frac {7 B a e}{9}\right ) x \sqrt {b e \,x^{3}+a e x}}{5 b e}+\frac {\left (A a e -\frac {3 \left (\left (A b +B a \right ) e -\frac {7 B a e}{9}\right ) a}{5 b}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(276\) |
default | \(\frac {2 \sqrt {e x}\, \left (5 b^{3} B \,x^{6}+18 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b -9 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b -6 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}+3 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}+9 A \,b^{3} x^{4}+7 B a \,b^{2} x^{4}+9 a A \,b^{2} x^{2}+2 B \,a^{2} b \,x^{2}\right )}{45 \sqrt {b \,x^{2}+a}\, b^{2} x}\) | \(414\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.24 \[ \int \sqrt {e x} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {2 \, {\left (6 \, {\left (B a^{2} - 3 \, A a b\right )} \sqrt {b e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (5 \, B b^{2} x^{3} + {\left (2 \, B a b + 9 \, A b^{2}\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{45 \, b^{2}} \]
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Result contains complex when optimal does not.
Time = 2.38 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.29 \[ \int \sqrt {e x} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {A \sqrt {a} \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} + \frac {B \sqrt {a} \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} \]
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\[ \int \sqrt {e x} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\int { {\left (B x^{2} + A\right )} \sqrt {b x^{2} + a} \sqrt {e x} \,d x } \]
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\[ \int \sqrt {e x} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\int { {\left (B x^{2} + A\right )} \sqrt {b x^{2} + a} \sqrt {e x} \,d x } \]
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Timed out. \[ \int \sqrt {e x} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\int \left (B\,x^2+A\right )\,\sqrt {e\,x}\,\sqrt {b\,x^2+a} \,d x \]
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