Integrand size = 26, antiderivative size = 211 \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {(7 A b-9 a B) e (e x)^{5/2}}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}+\frac {5 (7 A b-9 a B) e^3 \sqrt {e x} \sqrt {a+b x^2}}{21 b^3}-\frac {5 a^{3/4} (7 A b-9 a B) e^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{42 b^{13/4} \sqrt {a+b x^2}} \]
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Time = 0.11 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {470, 294, 327, 335, 226} \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {5 a^{3/4} e^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (7 A b-9 a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{42 b^{13/4} \sqrt {a+b x^2}}+\frac {5 e^3 \sqrt {e x} \sqrt {a+b x^2} (7 A b-9 a B)}{21 b^3}-\frac {e (e x)^{5/2} (7 A b-9 a B)}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}} \]
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Rule 226
Rule 294
Rule 327
Rule 335
Rule 470
Rubi steps \begin{align*} \text {integral}& = \frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}-\frac {\left (2 \left (-\frac {7 A b}{2}+\frac {9 a B}{2}\right )\right ) \int \frac {(e x)^{7/2}}{\left (a+b x^2\right )^{3/2}} \, dx}{7 b} \\ & = -\frac {(7 A b-9 a B) e (e x)^{5/2}}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}+\frac {\left (5 (7 A b-9 a B) e^2\right ) \int \frac {(e x)^{3/2}}{\sqrt {a+b x^2}} \, dx}{14 b^2} \\ & = -\frac {(7 A b-9 a B) e (e x)^{5/2}}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}+\frac {5 (7 A b-9 a B) e^3 \sqrt {e x} \sqrt {a+b x^2}}{21 b^3}-\frac {\left (5 a (7 A b-9 a B) e^4\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{42 b^3} \\ & = -\frac {(7 A b-9 a B) e (e x)^{5/2}}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}+\frac {5 (7 A b-9 a B) e^3 \sqrt {e x} \sqrt {a+b x^2}}{21 b^3}-\frac {\left (5 a (7 A b-9 a B) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 b^3} \\ & = -\frac {(7 A b-9 a B) e (e x)^{5/2}}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}+\frac {5 (7 A b-9 a B) e^3 \sqrt {e x} \sqrt {a+b x^2}}{21 b^3}-\frac {5 a^{3/4} (7 A b-9 a B) e^{7/2} \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 )}{42 b^{13/4} \sqrt {a+b x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.53 \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {e^3 \sqrt {e x} \left (-45 a^2 B+a b \left (35 A-18 B x^2\right )+2 b^2 x^2 \left (7 A+3 B x^2\right )+5 a (-7 A b+9 a B) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{21 b^3 \sqrt {a+b x^2}} \]
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Time = 3.65 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.19
method | result | size |
default | \(-\frac {e^{3} \sqrt {e x}\, \left (35 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a b -45 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{2}-12 b^{3} B \,x^{5}-28 A \,b^{3} x^{3}+36 B a \,b^{2} x^{3}-70 a \,b^{2} A x +90 a^{2} b B x \right )}{42 x \sqrt {b \,x^{2}+a}\, b^{4}}\) | \(252\) |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {e^{4} x a \left (A b -B a \right )}{b^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {2 B \,e^{3} x^{2} \sqrt {b e \,x^{3}+a e x}}{7 b^{2}}+\frac {2 \left (\frac {\left (A b -B a \right ) e^{4}}{b^{2}}-\frac {5 B \,e^{4} a}{7 b^{2}}\right ) \sqrt {b e \,x^{3}+a e x}}{3 b e}+\frac {\left (-\frac {a \left (A b -B a \right ) e^{4}}{2 b^{3}}-\frac {\left (\frac {\left (A b -B a \right ) e^{4}}{b^{2}}-\frac {5 B \,e^{4} a}{7 b^{2}}\right ) a}{3 b}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(298\) |
risch | \(\frac {2 \left (3 b B \,x^{2}+7 A b -12 B a \right ) x \sqrt {b \,x^{2}+a}\, e^{4}}{21 b^{3} \sqrt {e x}}-\frac {a \left (\frac {28 A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {b e \,x^{3}+a e x}}-\frac {33 B a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}-21 a \left (A b -B a \right ) \left (\frac {x}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 a b \sqrt {b e \,x^{3}+a e x}}\right )\right ) e^{4} \sqrt {\left (b \,x^{2}+a \right ) e x}}{21 b^{3} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(442\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.69 \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {5 \, {\left ({\left (9 \, B a^{2} b - 7 \, A a b^{2}\right )} e^{3} x^{2} + {\left (9 \, B a^{3} - 7 \, A a^{2} b\right )} e^{3}\right )} \sqrt {b e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (6 \, B b^{3} e^{3} x^{4} - 2 \, {\left (9 \, B a b^{2} - 7 \, A b^{3}\right )} e^{3} x^{2} - 5 \, {\left (9 \, B a^{2} b - 7 \, A a b^{2}\right )} e^{3}\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{21 \, {\left (b^{5} x^{2} + a b^{4}\right )}} \]
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Result contains complex when optimal does not.
Time = 163.87 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.45 \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {A e^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {13}{4}\right )} + \frac {B e^{\frac {7}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {17}{4}\right )} \]
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\[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^{7/2}}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
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