Integrand size = 26, antiderivative size = 301 \[ \int \frac {\sqrt {e x} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {(A b-a B) (e x)^{3/2}}{a b e \sqrt {a+b x^2}}-\frac {(A b-3 a B) \sqrt {e x} \sqrt {a+b x^2}}{a b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {(A b-3 a B) \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{a^{3/4} b^{7/4} \sqrt {a+b x^2}}-\frac {(A b-3 a B) \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 a^{3/4} b^{7/4} \sqrt {a+b x^2}} \]
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Time = 0.17 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {468, 335, 311, 226, 1210} \[ \int \frac {\sqrt {e x} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (A b-3 a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 a^{3/4} b^{7/4} \sqrt {a+b x^2}}+\frac {\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (A b-3 a B) E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{a^{3/4} b^{7/4} \sqrt {a+b x^2}}-\frac {\sqrt {e x} \sqrt {a+b x^2} (A b-3 a B)}{a b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {(e x)^{3/2} (A b-a B)}{a b e \sqrt {a+b x^2}} \]
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Rule 226
Rule 311
Rule 335
Rule 468
Rule 1210
Rubi steps \begin{align*} \text {integral}& = \frac {(A b-a B) (e x)^{3/2}}{a b e \sqrt {a+b x^2}}+\frac {\left (-\frac {A b}{2}+\frac {3 a B}{2}\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{a b} \\ & = \frac {(A b-a B) (e x)^{3/2}}{a b e \sqrt {a+b x^2}}-\frac {(A b-3 a B) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a b e} \\ & = \frac {(A b-a B) (e x)^{3/2}}{a b e \sqrt {a+b x^2}}-\frac {(A b-3 a B) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{\sqrt {a} b^{3/2}}+\frac {(A b-3 a B) \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 )}{\sqrt {a} b^{3/2}} \\ & = \frac {(A b-a B) (e x)^{3/2}}{a b e \sqrt {a+b x^2}}-\frac {(A b-3 a B) \sqrt {e x} \sqrt {a+b x^2}}{a b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {(A b-3 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 )}{a^{3/4} b^{7/4} \sqrt {a+b x^2}}-\frac {(A b-3 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 )}{2 a^{3/4} 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 = 76, normalized size of antiderivative = 0.25 \[ \int \frac {\sqrt {e x} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {2 x \sqrt {e x} \left (3 a B+(A b-3 a B) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{3 a b \sqrt {a+b x^2}} \]
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Time = 3.10 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.84
method | result | size |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {e \,x^{2} \left (A b -B a \right )}{b a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\left (\frac {B e}{b}-\frac {\left (A b -B a \right ) e}{2 a b}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(252\) |
default | \(-\frac {\sqrt {e x}\, \left (2 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b -A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b -6 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2}+3 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2}-2 A \,b^{2} x^{2}+2 B a b \,x^{2}\right )}{2 \sqrt {b \,x^{2}+a}\, b^{2} x a}\) | \(382\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt {e x} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {{\left (B a b - A b^{2}\right )} \sqrt {b x^{2} + a} \sqrt {e x} x + {\left (3 \, B a^{2} - A a b + {\left (3 \, B a b - A b^{2}\right )} x^{2}\right )} \sqrt {b e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right )}{a b^{3} x^{2} + a^{2} b^{2}} \]
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Result contains complex when optimal does not.
Time = 6.18 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.31 \[ \int \frac {\sqrt {e x} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {A \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} \]
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\[ \int \frac {\sqrt {e x} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {e x}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {e x} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {e x}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {e x} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {e\,x}}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
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