Integrand size = 26, antiderivative size = 213 \[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx=-\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/2}}-\frac {(3 A b-a B) \sqrt {e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/2}}-\frac {5 (3 A b-a B) \sqrt {e x}}{6 a^3 e^3 \sqrt {a+b x^2}}-\frac {5 (3 A b-a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{12 a^{13/4} \sqrt [4]{b} e^{5/2} \sqrt {a+b x^2}} \]
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Time = 0.12 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {464, 296, 335, 226} \[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx=-\frac {5 \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{12 a^{13/4} \sqrt [4]{b} e^{5/2} \sqrt {a+b x^2}}-\frac {5 \sqrt {e x} (3 A b-a B)}{6 a^3 e^3 \sqrt {a+b x^2}}-\frac {\sqrt {e x} (3 A b-a B)}{3 a^2 e^3 \left (a+b x^2\right )^{3/2}}-\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/2}} \]
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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} \left (a+b x^2\right )^{3/2}}-\frac {(3 A b-a B) \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^{5/2}} \, dx}{a e^2} \\ & = -\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/2}}-\frac {(3 A b-a B) \sqrt {e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/2}}-\frac {(5 (3 A b-a B)) \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^{3/2}} \, dx}{6 a^2 e^2} \\ & = -\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/2}}-\frac {(3 A b-a B) \sqrt {e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/2}}-\frac {5 (3 A b-a B) \sqrt {e x}}{6 a^3 e^3 \sqrt {a+b x^2}}-\frac {(5 (3 A b-a B)) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{12 a^3 e^2} \\ & = -\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/2}}-\frac {(3 A b-a B) \sqrt {e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/2}}-\frac {5 (3 A b-a B) \sqrt {e x}}{6 a^3 e^3 \sqrt {a+b x^2}}-\frac {(5 (3 A b-a B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{6 a^3 e^3} \\ & = -\frac {2 A}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/2}}-\frac {(3 A b-a B) \sqrt {e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/2}}-\frac {5 (3 A b-a B) \sqrt {e x}}{6 a^3 e^3 \sqrt {a+b x^2}}-\frac {5 (3 A b-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 )}{12 a^{13/4} \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 = 10.08 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.56 \[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx=\frac {x \left (-15 A b^2 x^4+a^2 \left (-4 A+7 B x^2\right )+a \left (-21 A b x^2+5 b B x^4\right )+5 (-3 A b+a B) x^2 \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{6 a^3 (e x)^{5/2} \left (a+b x^2\right )^{3/2}} \]
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Time = 4.19 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.25
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {\left (A b -B a \right ) \sqrt {b e \,x^{3}+a e x}}{3 a^{2} e^{3} b^{2} \left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {x \left (11 A b -5 B a \right )}{6 e^{2} a^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}-\frac {2 A \sqrt {b e \,x^{3}+a e x}}{3 a^{3} e^{3} x^{2}}+\frac {\left (-\frac {11 A b -5 B a}{12 a^{3} e^{2}}-\frac {b A}{3 a^{3} e^{2}}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(267\) |
default | \(-\frac {15 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, b^{2} x^{3}-5 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a b \,x^{3}+15 A \sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b x -5 B \sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x +30 A \,b^{3} x^{4}-10 B a \,b^{2} x^{4}+42 a A \,b^{2} x^{2}-14 B \,a^{2} b \,x^{2}+8 a^{2} b A}{12 x \,e^{2} \sqrt {e x}\, a^{3} b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(446\) |
risch | \(-\frac {2 A \sqrt {b \,x^{2}+a}}{3 a^{3} x \,e^{2} \sqrt {e x}}-\frac {\left (\frac {A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {b e \,x^{3}+a e x}}+3 a b A \left (\frac {x}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 a b \sqrt {b e \,x^{3}+a e x}}\right )+3 a^{2} \left (A b -B a \right ) \left (\frac {\sqrt {b e \,x^{3}+a e x}}{3 a e \,b^{2} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {5 x}{6 a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {5 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{12 a^{2} b \sqrt {b e \,x^{3}+a e x}}\right )\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{3 a^{3} e^{2} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(492\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx=\frac {5 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{6} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{4} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (5 \, {\left (B a b^{2} - 3 \, A b^{3}\right )} x^{4} - 4 \, A a^{2} b + 7 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{6 \, {\left (a^{3} b^{3} e^{3} x^{6} + 2 \, a^{4} b^{2} e^{3} x^{4} + a^{5} b e^{3} x^{2}\right )}} \]
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Result contains complex when optimal does not.
Time = 128.92 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.46 \[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx=\frac {A \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {B \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} \]
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\[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {B\,x^2+A}{{\left (e\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^{5/2}} \,d x \]
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