Integrand size = 26, antiderivative size = 365 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=\frac {12 b (A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{5 a e^5}+\frac {24 \sqrt {b} (A b+a B) \sqrt {e x} \sqrt {a+b x^2}}{5 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}-\frac {24 \sqrt [4]{a} \sqrt [4]{b} (A b+a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 e^{7/2} \sqrt {a+b x^2}}+\frac {12 \sqrt [4]{a} \sqrt [4]{b} (A b+a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{5 e^{7/2} \sqrt {a+b x^2}} \]
[Out]
Time = 0.21 (sec) , antiderivative size = 365, 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, 283, 285, 335, 311, 226, 1210} \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=\frac {12 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (a B+A b) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{5 e^{7/2} \sqrt {a+b x^2}}-\frac {24 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (a B+A b) E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 e^{7/2} \sqrt {a+b x^2}}+\frac {12 b (e x)^{3/2} \sqrt {a+b x^2} (a B+A b)}{5 a e^5}+\frac {24 \sqrt {b} \sqrt {e x} \sqrt {a+b x^2} (a B+A b)}{5 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 \left (a+b x^2\right )^{3/2} (a B+A b)}{a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}} \]
[In]
[Out]
Rule 226
Rule 283
Rule 285
Rule 311
Rule 335
Rule 464
Rule 1210
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}+\frac {(A b+a B) \int \frac {\left (a+b x^2\right )^{3/2}}{(e x)^{3/2}} \, dx}{a e^2} \\ & = -\frac {2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}+\frac {(6 b (A b+a B)) \int \sqrt {e x} \sqrt {a+b x^2} \, dx}{a e^4} \\ & = \frac {12 b (A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{5 a e^5}-\frac {2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}+\frac {(12 b (A b+a B)) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{5 e^4} \\ & = \frac {12 b (A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{5 a e^5}-\frac {2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}+\frac {(24 b (A b+a B)) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 e^5} \\ & = \frac {12 b (A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{5 a e^5}-\frac {2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}+\frac {\left (24 \sqrt {a} \sqrt {b} (A b+a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 e^4}-\frac {\left (24 \sqrt {a} \sqrt {b} (A b+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 e^4} \\ & = \frac {12 b (A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{5 a e^5}+\frac {24 \sqrt {b} (A b+a B) \sqrt {e x} \sqrt {a+b x^2}}{5 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}-\frac {24 \sqrt [4]{a} \sqrt [4]{b} (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}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 e^{7/2} \sqrt {a+b x^2}}+\frac {12 \sqrt [4]{a} \sqrt [4]{b} (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 )}{5 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 = 10.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=\frac {2 x \sqrt {a+b x^2} \left (-\frac {A \left (a+b x^2\right )^2}{a}-\frac {5 (A b+a B) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{4},\frac {3}{4},-\frac {b x^2}{a}\right )}{\sqrt {1+\frac {b x^2}{a}}}\right )}{5 (e x)^{7/2}} \]
[In]
[Out]
Time = 3.07 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.66
method | result | size |
risch | \(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-b B \,x^{4}+7 A b \,x^{2}+5 B a \,x^{2}+A a \right )}{5 x^{2} e^{3} \sqrt {e x}}+\frac {12 \left (A b +B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{5 \sqrt {b e \,x^{3}+a e x}\, e^{3} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(242\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {2 a A \sqrt {b e \,x^{3}+a e x}}{5 e^{4} x^{3}}-\frac {2 \left (b e \,x^{2}+a e \right ) \left (7 A b +5 B a \right )}{5 e^{4} \sqrt {x \left (b e \,x^{2}+a e \right )}}+\frac {2 b B x \sqrt {b e \,x^{3}+a e x}}{5 e^{4}}+\frac {\left (\frac {b \left (A b +2 B a \right )}{e^{3}}+\frac {b \left (7 A b +5 B a \right )}{5 e^{3}}-\frac {3 b B a}{5 e^{3}}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(307\) |
default | \(\frac {\frac {24 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}}{5}-\frac {12 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}}{5}+\frac {24 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}}{5}-\frac {12 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}}{5}+\frac {2 b^{2} B \,x^{6}}{5}-\frac {14 A \,b^{2} x^{4}}{5}-\frac {8 B a b \,x^{4}}{5}-\frac {16 a A b \,x^{2}}{5}-2 a^{2} B \,x^{2}-\frac {2 a^{2} A}{5}}{x^{2} \sqrt {b \,x^{2}+a}\, e^{3} \sqrt {e x}}\) | \(422\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=-\frac {2 \, {\left (12 \, {\left (B a + A b\right )} \sqrt {b e} x^{3} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (B b x^{4} - {\left (5 \, B a + 7 \, A b\right )} x^{2} - A a\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{5 \, e^{4} x^{3}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 23.28 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.58 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=\frac {A a^{\frac {3}{2}} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {A \sqrt {a} b \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B a^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B \sqrt {a} b x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right )} \]
[In]
[Out]
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {7}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {7}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (b\,x^2+a\right )}^{3/2}}{{\left (e\,x\right )}^{7/2}} \,d x \]
[In]
[Out]