Integrand size = 26, antiderivative size = 212 \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {4 a (11 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{231 b^2}+\frac {2 (11 A b-5 a B) (e x)^{5/2} \sqrt {a+b x^2}}{77 b e}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}-\frac {2 a^{7/4} (11 A b-5 a B) e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a+b x^2}} \]
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Time = 0.12 (sec) , antiderivative size = 212, 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 = {470, 285, 327, 335, 226} \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=-\frac {2 a^{7/4} e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (11 A b-5 a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a+b x^2}}+\frac {4 a e \sqrt {e x} \sqrt {a+b x^2} (11 A b-5 a B)}{231 b^2}+\frac {2 (e x)^{5/2} \sqrt {a+b x^2} (11 A b-5 a B)}{77 b e}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e} \]
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Rule 226
Rule 285
Rule 327
Rule 335
Rule 470
Rubi steps \begin{align*} \text {integral}& = \frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}-\frac {\left (2 \left (-\frac {11 A b}{2}+\frac {5 a B}{2}\right )\right ) \int (e x)^{3/2} \sqrt {a+b x^2} \, dx}{11 b} \\ & = \frac {2 (11 A b-5 a B) (e x)^{5/2} \sqrt {a+b x^2}}{77 b e}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}+\frac {(2 a (11 A b-5 a B)) \int \frac {(e x)^{3/2}}{\sqrt {a+b x^2}} \, dx}{77 b} \\ & = \frac {4 a (11 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{231 b^2}+\frac {2 (11 A b-5 a B) (e x)^{5/2} \sqrt {a+b x^2}}{77 b e}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}-\frac {\left (2 a^2 (11 A b-5 a B) e^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{231 b^2} \\ & = \frac {4 a (11 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{231 b^2}+\frac {2 (11 A b-5 a B) (e x)^{5/2} \sqrt {a+b x^2}}{77 b e}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}-\frac {\left (4 a^2 (11 A b-5 a B) e\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{231 b^2} \\ & = \frac {4 a (11 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{231 b^2}+\frac {2 (11 A b-5 a B) (e x)^{5/2} \sqrt {a+b x^2}}{77 b e}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}-\frac {2 a^{7/4} (11 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 )}{231 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 = 10.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.52 \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {2 e \sqrt {e x} \sqrt {a+b x^2} \left (-\left (\left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \left (-11 A b+5 a B-7 b B x^2\right )\right )+a (-11 A b+5 a B) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{77 b^2 \sqrt {1+\frac {b x^2}{a}}} \]
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Time = 3.19 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.01
method | result | size |
risch | \(\frac {2 \left (21 b^{2} B \,x^{4}+33 A \,b^{2} x^{2}+6 B a b \,x^{2}+22 a b A -10 a^{2} B \right ) x \sqrt {b \,x^{2}+a}\, e^{2}}{231 b^{2} \sqrt {e x}}-\frac {2 a^{2} \left (11 A b -5 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) e^{2} \sqrt {\left (b \,x^{2}+a \right ) e x}}{231 b^{3} \sqrt {b e \,x^{3}+a e x}\, \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(214\) |
default | \(-\frac {2 e \sqrt {e x}\, \left (-21 x^{7} B \,b^{4}+11 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{2} b -33 A \,b^{4} x^{5}-5 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{3}-27 x^{5} B a \,b^{3}-55 A a \,b^{3} x^{3}+4 B \,a^{2} b^{2} x^{3}-22 A x \,a^{2} b^{2}+10 B x \,a^{3} b \right )}{231 x \sqrt {b \,x^{2}+a}\, b^{3}}\) | \(276\) |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {2 B e \,x^{4} \sqrt {b e \,x^{3}+a e x}}{11}+\frac {2 \left (\left (A b +B a \right ) e^{2}-\frac {9 B a \,e^{2}}{11}\right ) x^{2} \sqrt {b e \,x^{3}+a e x}}{7 b e}+\frac {2 \left (A a \,e^{2}-\frac {5 \left (\left (A b +B a \right ) e^{2}-\frac {9 B a \,e^{2}}{11}\right ) a}{7 b}\right ) \sqrt {b e \,x^{3}+a e x}}{3 b e}-\frac {\left (A a \,e^{2}-\frac {5 \left (\left (A b +B a \right ) e^{2}-\frac {9 B a \,e^{2}}{11}\right ) a}{7 b}\right ) a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 b^{2} \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(294\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.48 \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {2 \, {\left (2 \, {\left (5 \, B a^{3} - 11 \, A a^{2} b\right )} \sqrt {b e} e {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (21 \, B b^{3} e x^{4} + 3 \, {\left (2 \, B a b^{2} + 11 \, A b^{3}\right )} e x^{2} - 2 \, {\left (5 \, B a^{2} b - 11 \, A a b^{2}\right )} e\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{231 \, b^{3}} \]
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Result contains complex when optimal does not.
Time = 5.56 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.46 \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {A \sqrt {a} e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {B \sqrt {a} e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{4}\right )} \]
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\[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\int { {\left (B x^{2} + A\right )} \sqrt {b x^{2} + a} \left (e x\right )^{\frac {3}{2}} \,d x } \]
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\[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\int { {\left (B x^{2} + A\right )} \sqrt {b x^{2} + a} \left (e x\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\int \left (B\,x^2+A\right )\,{\left (e\,x\right )}^{3/2}\,\sqrt {b\,x^2+a} \,d x \]
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