Integrand size = 26, antiderivative size = 342 \[ \int \frac {A+B x^2}{(e x)^{7/2} \sqrt {a+b x^2}} \, dx=-\frac {2 A \sqrt {a+b x^2}}{5 a e (e x)^{5/2}}+\frac {2 (3 A b-5 a B) \sqrt {a+b x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {2 \sqrt {b} (3 A b-5 a B) \sqrt {e x} \sqrt {a+b x^2}}{5 a^2 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 \sqrt [4]{b} (3 A b-5 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 a^{7/4} e^{7/2} \sqrt {a+b x^2}}-\frac {\sqrt [4]{b} (3 A b-5 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{5 a^{7/4} e^{7/2} \sqrt {a+b x^2}} \]
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Time = 0.19 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {464, 331, 335, 311, 226, 1210} \[ \int \frac {A+B x^2}{(e x)^{7/2} \sqrt {a+b x^2}} \, dx=-\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-5 a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{5 a^{7/4} e^{7/2} \sqrt {a+b x^2}}+\frac {2 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-5 a B) E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 a^{7/4} e^{7/2} \sqrt {a+b x^2}}-\frac {2 \sqrt {b} \sqrt {e x} \sqrt {a+b x^2} (3 A b-5 a B)}{5 a^2 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 \sqrt {a+b x^2} (3 A b-5 a B)}{5 a^2 e^3 \sqrt {e x}}-\frac {2 A \sqrt {a+b x^2}}{5 a e (e x)^{5/2}} \]
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Rule 226
Rule 311
Rule 331
Rule 335
Rule 464
Rule 1210
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A \sqrt {a+b x^2}}{5 a e (e x)^{5/2}}-\frac {(3 A b-5 a B) \int \frac {1}{(e x)^{3/2} \sqrt {a+b x^2}} \, dx}{5 a e^2} \\ & = -\frac {2 A \sqrt {a+b x^2}}{5 a e (e x)^{5/2}}+\frac {2 (3 A b-5 a B) \sqrt {a+b x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {(b (3 A b-5 a B)) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{5 a^2 e^4} \\ & = -\frac {2 A \sqrt {a+b x^2}}{5 a e (e x)^{5/2}}+\frac {2 (3 A b-5 a B) \sqrt {a+b x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {(2 b (3 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^2 e^5} \\ & = -\frac {2 A \sqrt {a+b x^2}}{5 a e (e x)^{5/2}}+\frac {2 (3 A b-5 a B) \sqrt {a+b x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {\left (2 \sqrt {b} (3 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^{3/2} e^4}+\frac {\left (2 \sqrt {b} (3 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^{3/2} e^4} \\ & = -\frac {2 A \sqrt {a+b x^2}}{5 a e (e x)^{5/2}}+\frac {2 (3 A b-5 a B) \sqrt {a+b x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {2 \sqrt {b} (3 A b-5 a B) \sqrt {e x} \sqrt {a+b x^2}}{5 a^2 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 \sqrt [4]{b} (3 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^{7/4} e^{7/2} \sqrt {a+b x^2}}-\frac {\sqrt [4]{b} (3 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 )}{5 a^{7/4} 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 = 10.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.24 \[ \int \frac {A+B x^2}{(e x)^{7/2} \sqrt {a+b x^2}} \, dx=-\frac {2 x \left (A \left (a+b x^2\right )+(-3 A b+5 a B) x^2 \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {b x^2}{a}\right )\right )}{5 a (e x)^{7/2} \sqrt {a+b x^2}} \]
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Time = 3.08 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.71
method | result | size |
risch | \(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-3 A b \,x^{2}+5 B a \,x^{2}+A a \right )}{5 a^{2} x^{2} e^{3} \sqrt {e x}}-\frac {\left (3 A b -5 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{5 a^{2} \sqrt {b e \,x^{3}+a e x}\, e^{3} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(243\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {2 A \sqrt {b e \,x^{3}+a e x}}{5 e^{4} a \,x^{3}}+\frac {2 \left (b e \,x^{2}+a e \right ) \left (3 A b -5 B a \right )}{5 e^{4} a^{2} \sqrt {x \left (b e \,x^{2}+a e \right )}}-\frac {\left (3 A b -5 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{5 a^{2} e^{3} \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(267\) |
default | \(-\frac {6 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}-3 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}-10 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}+5 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}-6 A \,b^{2} x^{4}+10 B a b \,x^{4}-4 a A b \,x^{2}+10 a^{2} B \,x^{2}+2 a^{2} A}{5 x^{2} \sqrt {b \,x^{2}+a}\, e^{3} \sqrt {e x}\, a^{2}}\) | \(417\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.23 \[ \int \frac {A+B x^2}{(e x)^{7/2} \sqrt {a+b x^2}} \, dx=-\frac {2 \, {\left ({\left (5 \, B a - 3 \, A b\right )} \sqrt {b e} x^{3} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left ({\left (5 \, B a - 3 \, A b\right )} x^{2} + A a\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{5 \, a^{2} e^{4} x^{3}} \]
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Result contains complex when optimal does not.
Time = 15.98 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.30 \[ \int \frac {A+B x^2}{(e x)^{7/2} \sqrt {a+b x^2}} \, dx=\frac {A \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {B \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {7}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \]
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\[ \int \frac {A+B x^2}{(e x)^{7/2} \sqrt {a+b x^2}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {b x^{2} + a} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {A+B x^2}{(e x)^{7/2} \sqrt {a+b x^2}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {b x^{2} + a} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B x^2}{(e x)^{7/2} \sqrt {a+b x^2}} \, dx=\int \frac {B\,x^2+A}{{\left (e\,x\right )}^{7/2}\,\sqrt {b\,x^2+a}} \,d x \]
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