Integrand size = 24, antiderivative size = 152 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{9/2}} \, dx=\frac {2 (A b-7 a B) \sqrt {a+b x^2}}{21 a x^{3/2}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}}-\frac {2 b^{3/4} (A b-7 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{21 a^{5/4} \sqrt {a+b x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {464, 283, 335, 226} \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{9/2}} \, dx=-\frac {2 b^{3/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (A b-7 a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{21 a^{5/4} \sqrt {a+b x^2}}+\frac {2 \sqrt {a+b x^2} (A b-7 a B)}{21 a x^{3/2}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}} \]
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Rule 226
Rule 283
Rule 335
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}}-\frac {\left (2 \left (\frac {A b}{2}-\frac {7 a B}{2}\right )\right ) \int \frac {\sqrt {a+b x^2}}{x^{5/2}} \, dx}{7 a} \\ & = \frac {2 (A b-7 a B) \sqrt {a+b x^2}}{21 a x^{3/2}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}}-\frac {(2 b (A b-7 a B)) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{21 a} \\ & = \frac {2 (A b-7 a B) \sqrt {a+b x^2}}{21 a x^{3/2}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}}-\frac {(4 b (A b-7 a B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{21 a} \\ & = \frac {2 (A b-7 a B) \sqrt {a+b x^2}}{21 a x^{3/2}}-\frac {2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}}-\frac {2 b^{3/4} (A b-7 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 )}{21 a^{5/4} \sqrt {a+b x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{9/2}} \, dx=\frac {2 \sqrt {a+b x^2} \left (-3 A \left (a+b x^2\right )+\frac {(A b-7 a B) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {1}{2},\frac {1}{4},-\frac {b x^2}{a}\right )}{\sqrt {1+\frac {b x^2}{a}}}\right )}{21 a x^{7/2}} \]
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Time = 3.06 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.16
method | result | size |
risch | \(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (2 A b \,x^{2}+7 B a \,x^{2}+3 A a \right )}{21 x^{\frac {7}{2}} a}-\frac {2 \left (A b -7 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x \left (b \,x^{2}+a \right )}}{21 a \sqrt {b \,x^{3}+a x}\, \sqrt {x}\, \sqrt {b \,x^{2}+a}}\) | \(177\) |
elliptic | \(\frac {\sqrt {x \left (b \,x^{2}+a \right )}\, \left (-\frac {2 A \sqrt {b \,x^{3}+a x}}{7 x^{4}}-\frac {2 \left (2 A b +7 B a \right ) \sqrt {b \,x^{3}+a x}}{21 a \,x^{2}}+\frac {\left (B b -\frac {b \left (2 A b +7 B a \right )}{21 a}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b \,x^{3}+a x}}\right )}{\sqrt {x}\, \sqrt {b \,x^{2}+a}}\) | \(197\) |
default | \(-\frac {2 \left (A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, b \,x^{3}-7 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a \,x^{3}+2 A \,b^{2} x^{4}+7 B a b \,x^{4}+5 a A b \,x^{2}+7 a^{2} B \,x^{2}+3 a^{2} A \right )}{21 \sqrt {b \,x^{2}+a}\, x^{\frac {7}{2}} a}\) | \(242\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{9/2}} \, dx=\frac {2 \, {\left (2 \, {\left (7 \, B a - A b\right )} \sqrt {b} x^{4} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left ({\left (7 \, B a + 2 \, A b\right )} x^{2} + 3 \, A a\right )} \sqrt {b x^{2} + a} \sqrt {x}\right )}}{21 \, a x^{4}} \]
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Result contains complex when optimal does not.
Time = 10.60 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{9/2}} \, dx=\frac {A \sqrt {a} \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 x^{\frac {7}{2}} \Gamma \left (- \frac {3}{4}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \]
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\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{9/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a}}{x^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{9/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \sqrt {b x^{2} + a}}{x^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{9/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {b\,x^2+a}}{x^{9/2}} \,d x \]
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