Integrand size = 26, antiderivative size = 379 \[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}-\frac {7 A b-5 a B}{5 a^2 e^3 \sqrt {e x} \sqrt {a+b x^2}}+\frac {3 (7 A b-5 a B) \sqrt {a+b x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {3 \sqrt {b} (7 A b-5 a B) \sqrt {e x} \sqrt {a+b x^2}}{5 a^3 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {3 \sqrt [4]{b} (7 A b-5 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 a^{11/4} e^{7/2} \sqrt {a+b x^2}}-\frac {3 \sqrt [4]{b} (7 A b-5 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{10 a^{11/4} e^{7/2} \sqrt {a+b x^2}} \]
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Time = 0.21 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {464, 296, 331, 335, 311, 226, 1210} \[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=-\frac {3 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (7 A b-5 a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{10 a^{11/4} e^{7/2} \sqrt {a+b x^2}}+\frac {3 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (7 A b-5 a B) E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 a^{11/4} e^{7/2} \sqrt {a+b x^2}}-\frac {3 \sqrt {b} \sqrt {e x} \sqrt {a+b x^2} (7 A b-5 a B)}{5 a^3 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {3 \sqrt {a+b x^2} (7 A b-5 a B)}{5 a^3 e^3 \sqrt {e x}}-\frac {7 A b-5 a B}{5 a^2 e^3 \sqrt {e x} \sqrt {a+b x^2}}-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}} \]
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Rule 226
Rule 296
Rule 311
Rule 331
Rule 335
Rule 464
Rule 1210
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}-\frac {(7 A b-5 a B) \int \frac {1}{(e x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx}{5 a e^2} \\ & = -\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}-\frac {7 A b-5 a B}{5 a^2 e^3 \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 (7 A b-5 a B)) \int \frac {1}{(e x)^{3/2} \sqrt {a+b x^2}} \, dx}{10 a^2 e^2} \\ & = -\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}-\frac {7 A b-5 a B}{5 a^2 e^3 \sqrt {e x} \sqrt {a+b x^2}}+\frac {3 (7 A b-5 a B) \sqrt {a+b x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {(3 b (7 A b-5 a B)) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{10 a^3 e^4} \\ & = -\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}-\frac {7 A b-5 a B}{5 a^2 e^3 \sqrt {e x} \sqrt {a+b x^2}}+\frac {3 (7 A b-5 a B) \sqrt {a+b x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {(3 b (7 A b-5 a B)) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 a^3 e^5} \\ & = -\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}-\frac {7 A b-5 a B}{5 a^2 e^3 \sqrt {e x} \sqrt {a+b x^2}}+\frac {3 (7 A b-5 a B) \sqrt {a+b x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {\left (3 \sqrt {b} (7 A b-5 a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 a^{5/2} e^4}+\frac {\left (3 \sqrt {b} (7 A b-5 a B)\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a} e}}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 a^{5/2} e^4} \\ & = -\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}-\frac {7 A b-5 a B}{5 a^2 e^3 \sqrt {e x} \sqrt {a+b x^2}}+\frac {3 (7 A b-5 a B) \sqrt {a+b x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {3 \sqrt {b} (7 A b-5 a B) \sqrt {e x} \sqrt {a+b x^2}}{5 a^3 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {3 \sqrt [4]{b} (7 A b-5 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 {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 a^{11/4} e^{7/2} \sqrt {a+b x^2}}-\frac {3 \sqrt [4]{b} (7 A b-5 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 )}{10 a^{11/4} e^{7/2} \sqrt {a+b x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.21 \[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {x \left (-2 a A+2 (7 A b-5 a B) x^2 \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{2},\frac {3}{4},-\frac {b x^2}{a}\right )\right )}{5 a^2 (e x)^{7/2} \sqrt {a+b x^2}} \]
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Time = 3.77 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.85
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {2 A \sqrt {b e \,x^{3}+a e x}}{5 e^{4} a^{2} x^{3}}+\frac {2 \left (b e \,x^{2}+a e \right ) \left (8 A b -5 B a \right )}{5 e^{4} a^{3} \sqrt {x \left (b e \,x^{2}+a e \right )}}+\frac {b \,x^{2} \left (A b -B a \right )}{e^{3} a^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\left (-\frac {b \left (8 A b -5 B a \right )}{5 a^{3} e^{3}}-\frac {b \left (A b -B a \right )}{2 a^{3} e^{3}}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(324\) |
default | \(-\frac {42 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}-21 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}-30 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}+15 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}-42 A \,b^{2} x^{4}+30 B a b \,x^{4}-28 a A b \,x^{2}+20 a^{2} B \,x^{2}+4 a^{2} A}{10 x^{2} \sqrt {b \,x^{2}+a}\, e^{3} \sqrt {e x}\, a^{3}}\) | \(417\) |
risch | \(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-8 A b \,x^{2}+5 B a \,x^{2}+A a \right )}{5 a^{3} x^{2} e^{3} \sqrt {e x}}-\frac {b^{2} \left (\frac {\left (8 A b -5 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b^{2} \sqrt {b e \,x^{3}+a e x}}-\frac {5 \left (A b -B a \right ) a \left (\frac {x^{2}}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}-\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a b \sqrt {b e \,x^{3}+a e x}}\right )}{b}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{5 a^{3} e^{3} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(451\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.37 \[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=-\frac {3 \, {\left ({\left (5 \, B a b - 7 \, A b^{2}\right )} x^{5} + {\left (5 \, B a^{2} - 7 \, A a b\right )} x^{3}\right )} \sqrt {b e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (3 \, {\left (5 \, B a b - 7 \, A b^{2}\right )} x^{4} + 2 \, A a^{2} + 2 \, {\left (5 \, B a^{2} - 7 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{5 \, {\left (a^{3} b e^{4} x^{5} + a^{4} e^{4} x^{3}\right )}} \]
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Result contains complex when optimal does not.
Time = 54.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.27 \[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {A \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {B \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {7}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \]
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\[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {B\,x^2+A}{{\left (e\,x\right )}^{7/2}\,{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
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