Integrand size = 26, antiderivative size = 139 \[ \int \frac {A+B x^2}{\sqrt {e x} \sqrt {a+b x^2}} \, dx=\frac {2 B \sqrt {e x} \sqrt {a+b x^2}}{3 b e}+\frac {(3 A b-a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{a} b^{5/4} \sqrt {e} \sqrt {a+b x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {470, 335, 226} \[ \int \frac {A+B x^2}{\sqrt {e x} \sqrt {a+b x^2}} \, dx=\frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{a} b^{5/4} \sqrt {e} \sqrt {a+b x^2}}+\frac {2 B \sqrt {e x} \sqrt {a+b x^2}}{3 b e} \]
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Rule 226
Rule 335
Rule 470
Rubi steps \begin{align*} \text {integral}& = \frac {2 B \sqrt {e x} \sqrt {a+b x^2}}{3 b e}-\frac {\left (2 \left (-\frac {3 A b}{2}+\frac {a B}{2}\right )\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{3 b} \\ & = \frac {2 B \sqrt {e x} \sqrt {a+b x^2}}{3 b e}+\frac {(2 (3 A b-a B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{3 b e} \\ & = \frac {2 B \sqrt {e x} \sqrt {a+b x^2}}{3 b e}+\frac {(3 A b-a B) \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 )}{3 \sqrt [4]{a} b^{5/4} \sqrt {e} \sqrt {a+b x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.57 \[ \int \frac {A+B x^2}{\sqrt {e x} \sqrt {a+b x^2}} \, dx=\frac {2 x \left (B \left (a+b x^2\right )+(3 A b-a B) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{3 b \sqrt {e x} \sqrt {a+b x^2}} \]
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Time = 3.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.21
method | result | size |
risch | \(\frac {2 B x \sqrt {b \,x^{2}+a}}{3 b \sqrt {e x}}+\frac {\left (3 A b -B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{3 b^{2} \sqrt {b e \,x^{3}+a e x}\, \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(168\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {2 B \sqrt {b e \,x^{3}+a e x}}{3 b e}+\frac {\left (A -\frac {a B}{3 b}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(169\) |
default | \(\frac {3 A \sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b -B \sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a +2 b^{2} B \,x^{3}+2 B a b x}{3 \sqrt {b \,x^{2}+a}\, \sqrt {e x}\, b^{2}}\) | \(214\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.36 \[ \int \frac {A+B x^2}{\sqrt {e x} \sqrt {a+b x^2}} \, dx=\frac {2 \, {\left (\sqrt {b x^{2} + a} \sqrt {e x} B b - {\left (B a - 3 \, A b\right )} \sqrt {b e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right )}}{3 \, b^{2} e} \]
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Result contains complex when optimal does not.
Time = 1.76 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.68 \[ \int \frac {A+B x^2}{\sqrt {e x} \sqrt {a+b x^2}} \, dx=\frac {A \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \sqrt {e} \Gamma \left (\frac {5}{4}\right )} + \frac {B x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \sqrt {e} \Gamma \left (\frac {9}{4}\right )} \]
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\[ \int \frac {A+B x^2}{\sqrt {e x} \sqrt {a+b x^2}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {b x^{2} + a} \sqrt {e x}} \,d x } \]
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\[ \int \frac {A+B x^2}{\sqrt {e x} \sqrt {a+b x^2}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {b x^{2} + a} \sqrt {e x}} \,d x } \]
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Timed out. \[ \int \frac {A+B x^2}{\sqrt {e x} \sqrt {a+b x^2}} \, dx=\int \frac {B\,x^2+A}{\sqrt {e\,x}\,\sqrt {b\,x^2+a}} \,d x \]
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