Integrand size = 26, antiderivative size = 176 \[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/2}} \, dx=-\frac {2 A}{3 a e (e x)^{3/2} \sqrt {a+b x^2}}-\frac {(5 A b-3 a B) \sqrt {e x}}{3 a^2 e^3 \sqrt {a+b x^2}}-\frac {(5 A b-3 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{6 a^{9/4} \sqrt [4]{b} e^{5/2} \sqrt {a+b x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 176, 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 = {464, 296, 335, 226} \[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/2}} \, dx=-\frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (5 A b-3 a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{6 a^{9/4} \sqrt [4]{b} e^{5/2} \sqrt {a+b x^2}}-\frac {\sqrt {e x} (5 A b-3 a B)}{3 a^2 e^3 \sqrt {a+b x^2}}-\frac {2 A}{3 a e (e x)^{3/2} \sqrt {a+b x^2}} \]
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Rule 226
Rule 296
Rule 335
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A}{3 a e (e x)^{3/2} \sqrt {a+b x^2}}-\frac {(5 A b-3 a B) \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^{3/2}} \, dx}{3 a e^2} \\ & = -\frac {2 A}{3 a e (e x)^{3/2} \sqrt {a+b x^2}}-\frac {(5 A b-3 a B) \sqrt {e x}}{3 a^2 e^3 \sqrt {a+b x^2}}-\frac {(5 A b-3 a B) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{6 a^2 e^2} \\ & = -\frac {2 A}{3 a e (e x)^{3/2} \sqrt {a+b x^2}}-\frac {(5 A b-3 a B) \sqrt {e x}}{3 a^2 e^3 \sqrt {a+b x^2}}-\frac {(5 A b-3 a B) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{3 a^2 e^3} \\ & = -\frac {2 A}{3 a e (e x)^{3/2} \sqrt {a+b x^2}}-\frac {(5 A b-3 a B) \sqrt {e x}}{3 a^2 e^3 \sqrt {a+b x^2}}-\frac {(5 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 {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{6 a^{9/4} \sqrt [4]{b} e^{5/2} \sqrt {a+b x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.52 \[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {x \left (-2 a A-5 A b x^2+3 a B x^2+(-5 A b+3 a B) x^2 \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{3 a^2 (e x)^{5/2} \sqrt {a+b x^2}} \]
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Time = 3.47 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.26
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {x \left (A b -B a \right )}{e^{2} a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}-\frac {2 A \sqrt {b e \,x^{3}+a e x}}{3 a^{2} e^{3} x^{2}}+\frac {\left (-\frac {A b -B a}{2 a^{2} e^{2}}-\frac {b A}{3 a^{2} e^{2}}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(222\) |
default | \(-\frac {5 A \sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b x -3 B \sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a x +10 A \,b^{2} x^{2}-6 B a b \,x^{2}+4 a b A}{6 x \sqrt {b \,x^{2}+a}\, b \,a^{2} e^{2} \sqrt {e x}}\) | \(232\) |
risch | \(-\frac {2 A \sqrt {b \,x^{2}+a}}{3 a^{2} x \,e^{2} \sqrt {e x}}-\frac {\left (\frac {A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {b e \,x^{3}+a e x}}+3 a \left (A b -B a \right ) \left (\frac {x}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 a b \sqrt {b e \,x^{3}+a e x}}\right )\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{3 a^{2} e^{2} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(315\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.66 \[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{4} + {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt {b e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left (2 \, A a b - {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{3 \, {\left (a^{2} b^{2} e^{3} x^{4} + a^{3} b e^{3} x^{2}\right )}} \]
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Result contains complex when optimal does not.
Time = 23.14 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.55 \[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {A \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {B \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} \]
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\[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {B\,x^2+A}{{\left (e\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
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