Integrand size = 26, antiderivative size = 174 \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {2 (7 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{21 b^2}+\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e}-\frac {a^{3/4} (7 A b-5 a B) e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{21 b^{9/4} \sqrt {a+b x^2}} \]
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Time = 0.07 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {470, 327, 335, 226} \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=-\frac {a^{3/4} e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (7 A b-5 a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{21 b^{9/4} \sqrt {a+b x^2}}+\frac {2 e \sqrt {e x} \sqrt {a+b x^2} (7 A b-5 a B)}{21 b^2}+\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e} \]
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Rule 226
Rule 327
Rule 335
Rule 470
Rubi steps \begin{align*} \text {integral}& = \frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e}-\frac {\left (2 \left (-\frac {7 A b}{2}+\frac {5 a B}{2}\right )\right ) \int \frac {(e x)^{3/2}}{\sqrt {a+b x^2}} \, dx}{7 b} \\ & = \frac {2 (7 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{21 b^2}+\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e}-\frac {\left (a (7 A b-5 a B) e^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{21 b^2} \\ & = \frac {2 (7 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{21 b^2}+\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e}-\frac {(2 a (7 A b-5 a B) e) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 b^2} \\ & = \frac {2 (7 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{21 b^2}+\frac {2 B (e x)^{5/2} \sqrt {a+b x^2}}{7 b e}-\frac {a^{3/4} (7 A b-5 a B) e^{3/2} \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 )}{21 b^{9/4} \sqrt {a+b x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.55 \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {2 e \sqrt {e x} \left (-\left (\left (a+b x^2\right ) \left (-7 A b+5 a B-3 b B x^2\right )\right )+a (-7 A b+5 a B) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{21 b^2 \sqrt {a+b x^2}} \]
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Time = 3.03 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.09
method | result | size |
risch | \(\frac {2 \left (3 b B \,x^{2}+7 A b -5 B a \right ) x \sqrt {b \,x^{2}+a}\, e^{2}}{21 b^{2} \sqrt {e x}}-\frac {a \left (7 A b -5 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) e^{2} \sqrt {\left (b \,x^{2}+a \right ) e x}}{21 b^{3} \sqrt {b e \,x^{3}+a e x}\, \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(190\) |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {2 B e \,x^{2} \sqrt {b e \,x^{3}+a e x}}{7 b}+\frac {2 \left (A \,e^{2}-\frac {5 B \,e^{2} a}{7 b}\right ) \sqrt {b e \,x^{3}+a e x}}{3 b e}-\frac {\left (A \,e^{2}-\frac {5 B \,e^{2} a}{7 b}\right ) a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 b^{2} \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(222\) |
default | \(-\frac {e \sqrt {e x}\, \left (7 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a b -5 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{2}-6 b^{3} B \,x^{5}-14 A \,b^{3} x^{3}+4 B a \,b^{2} x^{3}-14 a \,b^{2} A x +10 a^{2} b B x \right )}{21 x \sqrt {b \,x^{2}+a}\, b^{3}}\) | \(250\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.43 \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {2 \, {\left ({\left (5 \, B a^{2} - 7 \, A a b\right )} \sqrt {b e} e {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (3 \, B b^{2} e x^{2} - {\left (5 \, B a b - 7 \, A b^{2}\right )} e\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{21 \, b^{3}} \]
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Result contains complex when optimal does not.
Time = 5.01 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.54 \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {A e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {B e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \]
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\[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {3}{2}}}{\sqrt {b x^{2} + a}} \,d x } \]
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\[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {3}{2}}}{\sqrt {b x^{2} + a}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^{3/2}}{\sqrt {b\,x^2+a}} \,d x \]
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