Integrand size = 26, antiderivative size = 333 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx=\frac {2 (5 A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{5 a e^3}+\frac {4 (5 A b+a B) \sqrt {e x} \sqrt {a+b x^2}}{5 \sqrt {b} e^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 A \left (a+b x^2\right )^{3/2}}{a e \sqrt {e x}}-\frac {4 \sqrt [4]{a} (5 A b+a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 b^{3/4} e^{3/2} \sqrt {a+b x^2}}+\frac {2 \sqrt [4]{a} (5 A b+a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{5 b^{3/4} e^{3/2} \sqrt {a+b x^2}} \]
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Time = 0.18 (sec) , antiderivative size = 333, 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 = {464, 285, 335, 311, 226, 1210} \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx=\frac {2 \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (a B+5 A b) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{5 b^{3/4} e^{3/2} \sqrt {a+b x^2}}-\frac {4 \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (a B+5 A b) E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 b^{3/4} e^{3/2} \sqrt {a+b x^2}}+\frac {2 (e x)^{3/2} \sqrt {a+b x^2} (a B+5 A b)}{5 a e^3}+\frac {4 \sqrt {e x} \sqrt {a+b x^2} (a B+5 A b)}{5 \sqrt {b} e^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 A \left (a+b x^2\right )^{3/2}}{a e \sqrt {e x}} \]
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Rule 226
Rule 285
Rule 311
Rule 335
Rule 464
Rule 1210
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A \left (a+b x^2\right )^{3/2}}{a e \sqrt {e x}}+\frac {(5 A b+a B) \int \sqrt {e x} \sqrt {a+b x^2} \, dx}{a e^2} \\ & = \frac {2 (5 A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{5 a e^3}-\frac {2 A \left (a+b x^2\right )^{3/2}}{a e \sqrt {e x}}+\frac {(2 (5 A b+a B)) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{5 e^2} \\ & = \frac {2 (5 A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{5 a e^3}-\frac {2 A \left (a+b x^2\right )^{3/2}}{a e \sqrt {e x}}+\frac {(4 (5 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 )}{5 e^3} \\ & = \frac {2 (5 A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{5 a e^3}-\frac {2 A \left (a+b x^2\right )^{3/2}}{a e \sqrt {e x}}+\frac {\left (4 \sqrt {a} (5 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 )}{5 \sqrt {b} e^2}-\frac {\left (4 \sqrt {a} (5 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 )}{5 \sqrt {b} e^2} \\ & = \frac {2 (5 A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{5 a e^3}+\frac {4 (5 A b+a B) \sqrt {e x} \sqrt {a+b x^2}}{5 \sqrt {b} e^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 A \left (a+b x^2\right )^{3/2}}{a e \sqrt {e x}}-\frac {4 \sqrt [4]{a} (5 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}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 b^{3/4} e^{3/2} \sqrt {a+b x^2}}+\frac {2 \sqrt [4]{a} (5 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 )}{5 b^{3/4} e^{3/2} \sqrt {a+b x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.30 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx=\frac {2 x \sqrt {a+b x^2} \left (-3 A \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}}+(5 A b+a B) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{3 a (e x)^{3/2} \sqrt {1+\frac {b x^2}{a}}} \]
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Time = 3.06 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.68
method | result | size |
risch | \(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-x^{2} B +5 A \right )}{5 e \sqrt {e x}}+\frac {\left (2 A b +\frac {2 B a}{5}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{b \sqrt {b e \,x^{3}+a e x}\, e \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(228\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {2 \left (b e \,x^{2}+a e \right ) A}{e^{2} \sqrt {x \left (b e \,x^{2}+a e \right )}}+\frac {2 B x \sqrt {b e \,x^{3}+a e x}}{5 e^{2}}+\frac {\left (\frac {A b +B a}{e}+\frac {b A}{e}-\frac {3 B a}{5 e}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(263\) |
default | \(\frac {4 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b -2 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b +\frac {4 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2}}{5}-\frac {2 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2}}{5}+\frac {2 b^{2} B \,x^{4}}{5}-2 A \,b^{2} x^{2}+\frac {2 B a b \,x^{2}}{5}-2 a b A}{\sqrt {b \,x^{2}+a}\, b e \sqrt {e x}}\) | \(391\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.22 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx=-\frac {2 \, {\left (2 \, {\left (B a + 5 \, A b\right )} \sqrt {b e} x {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (B b x^{2} - 5 \, A b\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{5 \, b e^{2} x} \]
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Result contains complex when optimal does not.
Time = 3.33 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.30 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx=\frac {A \sqrt {a} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B \sqrt {a} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \]
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\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {b\,x^2+a}}{{\left (e\,x\right )}^{3/2}} \,d x \]
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