Integrand size = 24, antiderivative size = 331 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{11/2}} \, dx=\frac {2 (A b-3 a B) \sqrt {a+b x^2}}{15 a x^{5/2}}+\frac {4 b (A b-3 a B) \sqrt {a+b x^2}}{15 a^2 \sqrt {x}}-\frac {4 b^{3/2} (A b-3 a B) \sqrt {x} \sqrt {a+b x^2}}{15 a^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}}+\frac {4 b^{5/4} (A b-3 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{7/4} \sqrt {a+b x^2}}-\frac {2 b^{5/4} (A b-3 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15 a^{7/4} \sqrt {a+b x^2}} \]
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Time = 0.17 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {464, 283, 331, 335, 311, 226, 1210} \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{11/2}} \, dx=-\frac {2 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (A b-3 a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15 a^{7/4} \sqrt {a+b x^2}}+\frac {4 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (A b-3 a B) E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{7/4} \sqrt {a+b x^2}}-\frac {4 b^{3/2} \sqrt {x} \sqrt {a+b x^2} (A b-3 a B)}{15 a^2 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {4 b \sqrt {a+b x^2} (A b-3 a B)}{15 a^2 \sqrt {x}}+\frac {2 \sqrt {a+b x^2} (A b-3 a B)}{15 a x^{5/2}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}} \]
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Rule 226
Rule 283
Rule 311
Rule 331
Rule 335
Rule 464
Rule 1210
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}}-\frac {\left (2 \left (\frac {3 A b}{2}-\frac {9 a B}{2}\right )\right ) \int \frac {\sqrt {a+b x^2}}{x^{7/2}} \, dx}{9 a} \\ & = \frac {2 (A b-3 a B) \sqrt {a+b x^2}}{15 a x^{5/2}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}}-\frac {(2 b (A b-3 a B)) \int \frac {1}{x^{3/2} \sqrt {a+b x^2}} \, dx}{15 a} \\ & = \frac {2 (A b-3 a B) \sqrt {a+b x^2}}{15 a x^{5/2}}+\frac {4 b (A b-3 a B) \sqrt {a+b x^2}}{15 a^2 \sqrt {x}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}}-\frac {\left (2 b^2 (A b-3 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x^2}} \, dx}{15 a^2} \\ & = \frac {2 (A b-3 a B) \sqrt {a+b x^2}}{15 a x^{5/2}}+\frac {4 b (A b-3 a B) \sqrt {a+b x^2}}{15 a^2 \sqrt {x}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}}-\frac {\left (4 b^2 (A b-3 a B)\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{15 a^2} \\ & = \frac {2 (A b-3 a B) \sqrt {a+b x^2}}{15 a x^{5/2}}+\frac {4 b (A b-3 a B) \sqrt {a+b x^2}}{15 a^2 \sqrt {x}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}}-\frac {\left (4 b^{3/2} (A b-3 a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{15 a^{3/2}}+\frac {\left (4 b^{3/2} (A b-3 a B)\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{15 a^{3/2}} \\ & = \frac {2 (A b-3 a B) \sqrt {a+b x^2}}{15 a x^{5/2}}+\frac {4 b (A b-3 a B) \sqrt {a+b x^2}}{15 a^2 \sqrt {x}}-\frac {4 b^{3/2} (A b-3 a B) \sqrt {x} \sqrt {a+b x^2}}{15 a^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}}+\frac {4 b^{5/4} (A b-3 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 {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{7/4} \sqrt {a+b x^2}}-\frac {2 b^{5/4} (A b-3 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 {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{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 = 80, normalized size of antiderivative = 0.24 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{11/2}} \, dx=\frac {2 \sqrt {a+b x^2} \left (-5 A \left (a+b x^2\right )+\frac {3 (A b-3 a B) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {1}{2},-\frac {1}{4},-\frac {b x^2}{a}\right )}{\sqrt {1+\frac {b x^2}{a}}}\right )}{45 a x^{9/2}} \]
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Time = 3.10 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-6 A \,b^{2} x^{4}+18 B a b \,x^{4}+2 a A b \,x^{2}+9 a^{2} B \,x^{2}+5 a^{2} A \right )}{45 x^{\frac {9}{2}} a^{2}}-\frac {2 b \left (A b -3 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) \sqrt {x \left (b \,x^{2}+a \right )}}{15 a^{2} \sqrt {b \,x^{3}+a x}\, \sqrt {x}\, \sqrt {b \,x^{2}+a}}\) | \(250\) |
elliptic | \(\frac {\sqrt {x \left (b \,x^{2}+a \right )}\, \left (-\frac {2 A \sqrt {b \,x^{3}+a x}}{9 x^{5}}-\frac {2 \left (2 A b +9 B a \right ) \sqrt {b \,x^{3}+a x}}{45 a \,x^{3}}+\frac {4 \left (b \,x^{2}+a \right ) b \left (A b -3 B a \right )}{15 a^{2} \sqrt {x \left (b \,x^{2}+a \right )}}-\frac {2 b \left (A b -3 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{15 a^{2} \sqrt {b \,x^{3}+a x}}\right )}{\sqrt {x}\, \sqrt {b \,x^{2}+a}}\) | \(270\) |
default | \(-\frac {2 \left (6 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{4}-3 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{4}-18 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,x^{4}+9 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,x^{4}-6 x^{6} b^{3} A +18 x^{6} a \,b^{2} B -4 A a \,b^{2} x^{4}+27 B \,a^{2} b \,x^{4}+7 A \,a^{2} b \,x^{2}+9 B \,a^{3} x^{2}+5 a^{3} A \right )}{45 \sqrt {b \,x^{2}+a}\, x^{\frac {9}{2}} a^{2}}\) | \(439\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.30 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{11/2}} \, dx=-\frac {2 \, {\left (6 \, {\left (3 \, B a b - A b^{2}\right )} \sqrt {b} x^{5} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (6 \, {\left (3 \, B a b - A b^{2}\right )} x^{4} + 5 \, A a^{2} + {\left (9 \, B a^{2} + 2 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {x}\right )}}{45 \, a^{2} x^{5}} \]
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Result contains complex when optimal does not.
Time = 28.95 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.30 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{11/2}} \, dx=\frac {A \sqrt {a} \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {1}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 x^{\frac {9}{2}} \Gamma \left (- \frac {5}{4}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} \]
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\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{11/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a}}{x^{\frac {11}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{11/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a}}{x^{\frac {11}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{11/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {b\,x^2+a}}{x^{11/2}} \,d x \]
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