Integrand size = 26, antiderivative size = 174 \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {(3 A b-5 a B) e \sqrt {e x}}{3 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{5/2}}{3 b e \sqrt {a+b x^2}}+\frac {(3 A b-5 a B) e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{6 \sqrt [4]{a} b^{9/4} \sqrt {a+b x^2}} \]
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Time = 0.08 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {470, 294, 335, 226} \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-5 a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{6 \sqrt [4]{a} b^{9/4} \sqrt {a+b x^2}}-\frac {e \sqrt {e x} (3 A b-5 a B)}{3 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{5/2}}{3 b e \sqrt {a+b x^2}} \]
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Rule 226
Rule 294
Rule 335
Rule 470
Rubi steps \begin{align*} \text {integral}& = \frac {2 B (e x)^{5/2}}{3 b e \sqrt {a+b x^2}}-\frac {\left (2 \left (-\frac {3 A b}{2}+\frac {5 a B}{2}\right )\right ) \int \frac {(e x)^{3/2}}{\left (a+b x^2\right )^{3/2}} \, dx}{3 b} \\ & = -\frac {(3 A b-5 a B) e \sqrt {e x}}{3 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{5/2}}{3 b e \sqrt {a+b x^2}}+\frac {\left ((3 A b-5 a B) e^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{6 b^2} \\ & = -\frac {(3 A b-5 a B) e \sqrt {e x}}{3 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{5/2}}{3 b e \sqrt {a+b x^2}}+\frac {((3 A b-5 a B) e) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{3 b^2} \\ & = -\frac {(3 A b-5 a B) e \sqrt {e x}}{3 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{5/2}}{3 b e \sqrt {a+b x^2}}+\frac {(3 A b-5 a B) e^{3/2} \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 )}{6 \sqrt [4]{a} b^{9/4} \sqrt {a+b x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.49 \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {e \sqrt {e x} \left (-3 A b+5 a B+2 b B x^2+(3 A b-5 a B) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{3 b^2 \sqrt {a+b x^2}} \]
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Time = 3.54 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.28
method | result | size |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {e^{2} x \left (A b -B a \right )}{b^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {2 B e \sqrt {b e \,x^{3}+a e x}}{3 b^{2}}+\frac {\left (\frac {\left (A b -B a \right ) e^{2}}{2 b^{2}}-\frac {B \,e^{2} a}{3 b^{2}}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(223\) |
default | \(\frac {e \sqrt {e x}\, \left (3 A \sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b -5 B \sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a +4 b^{2} B \,x^{3}-6 A \,b^{2} x +10 B a b x \right )}{6 x \sqrt {b \,x^{2}+a}\, b^{3}}\) | \(225\) |
risch | \(\frac {2 B x \sqrt {b \,x^{2}+a}\, e^{2}}{3 b^{2} \sqrt {e x}}+\frac {\left (\frac {3 A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {b e \,x^{3}+a e x}}-\frac {4 B a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}-3 a \left (A b -B a \right ) \left (\frac {x}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 a b \sqrt {b e \,x^{3}+a e x}}\right )\right ) e^{2} \sqrt {\left (b \,x^{2}+a \right ) e x}}{3 b^{2} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(426\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.61 \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {{\left ({\left (5 \, B a b - 3 \, A b^{2}\right )} e x^{2} + {\left (5 \, B a^{2} - 3 \, A a b\right )} e\right )} \sqrt {b e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left (2 \, B b^{2} e x^{2} + {\left (5 \, B a b - 3 \, A b^{2}\right )} e\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{3 \, {\left (b^{4} x^{2} + a b^{3}\right )}} \]
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Result contains complex when optimal does not.
Time = 14.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.54 \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {A e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {B e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {13}{4}\right )} \]
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\[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^{3/2}}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
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