Integrand size = 26, antiderivative size = 185 \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {(A b-a B) (e x)^{5/2}}{3 a b e \left (a+b x^2\right )^{3/2}}-\frac {(A b+5 a B) e \sqrt {e x}}{6 a b^2 \sqrt {a+b x^2}}+\frac {(A b+5 a B) e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{12 a^{5/4} b^{9/4} \sqrt {a+b x^2}} \]
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Time = 0.09 (sec) , antiderivative size = 185, 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 = {468, 294, 335, 226} \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (5 a B+A b) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{12 a^{5/4} b^{9/4} \sqrt {a+b x^2}}-\frac {e \sqrt {e x} (5 a B+A b)}{6 a b^2 \sqrt {a+b x^2}}+\frac {(e x)^{5/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}} \]
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Rule 226
Rule 294
Rule 335
Rule 468
Rubi steps \begin{align*} \text {integral}& = \frac {(A b-a B) (e x)^{5/2}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac {\left (\frac {A b}{2}+\frac {5 a B}{2}\right ) \int \frac {(e x)^{3/2}}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a b} \\ & = \frac {(A b-a B) (e x)^{5/2}}{3 a b e \left (a+b x^2\right )^{3/2}}-\frac {(A b+5 a B) e \sqrt {e x}}{6 a b^2 \sqrt {a+b x^2}}+\frac {\left ((A b+5 a B) e^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{12 a b^2} \\ & = \frac {(A b-a B) (e x)^{5/2}}{3 a b e \left (a+b x^2\right )^{3/2}}-\frac {(A b+5 a B) e \sqrt {e x}}{6 a b^2 \sqrt {a+b x^2}}+\frac {((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 )}{6 a b^2} \\ & = \frac {(A b-a B) (e x)^{5/2}}{3 a b e \left (a+b x^2\right )^{3/2}}-\frac {(A b+5 a B) e \sqrt {e x}}{6 a b^2 \sqrt {a+b x^2}}+\frac {(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 )}{12 a^{5/4} 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 = 10.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.57 \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {e \sqrt {e x} \left (-5 a^2 B+A b^2 x^2-a b \left (A+7 B x^2\right )+(A b+5 a B) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{6 a b^2 \left (a+b x^2\right )^{3/2}} \]
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Time = 3.01 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.32
method | result | size |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {e \left (A b -B a \right ) \sqrt {b e \,x^{3}+a e x}}{3 b^{4} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {e^{2} x \left (A b -7 B a \right )}{6 b^{2} a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\left (\frac {B \,e^{2}}{b^{2}}+\frac {e^{2} \left (A b -7 B a \right )}{12 b^{2} a}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(245\) |
default | \(\frac {\left (A \sqrt {-a b}\, \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b^{2} x^{2}+5 B \sqrt {-a b}\, \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}+A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a b +5 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{2}+2 A \,b^{3} x^{3}-14 B a \,b^{2} x^{3}-2 a \,b^{2} A x -10 a^{2} b B x \right ) e \sqrt {e x}}{12 x a \,b^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(429\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.83 \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (5 \, B a b^{2} + A b^{3}\right )} e x^{4} + 2 \, {\left (5 \, B a^{2} b + A a b^{2}\right )} e x^{2} + {\left (5 \, B a^{3} + A a^{2} b\right )} e\right )} \sqrt {b e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left ({\left (7 \, B a b^{2} - A b^{3}\right )} e x^{2} + {\left (5 \, B a^{2} b + A a b^{2}\right )} e\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{6 \, {\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \]
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Result contains complex when optimal does not.
Time = 64.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.51 \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {A e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {B e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {9}{4}, \frac {5}{2} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {13}{4}\right )} \]
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\[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^{3/2}}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \]
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