Integrand size = 26, antiderivative size = 208 \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {(A b-3 a B) e (e x)^{5/2}}{3 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}}-\frac {5 (A b-3 a B) e^3 \sqrt {e x}}{6 b^3 \sqrt {a+b x^2}}+\frac {5 (A b-3 a B) e^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{12 \sqrt [4]{a} b^{13/4} \sqrt {a+b x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {470, 294, 335, 226} \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {5 e^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (A b-3 a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{12 \sqrt [4]{a} b^{13/4} \sqrt {a+b x^2}}-\frac {5 e^3 \sqrt {e x} (A b-3 a B)}{6 b^3 \sqrt {a+b x^2}}-\frac {e (e x)^{5/2} (A b-3 a B)}{3 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}} \]
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Rule 226
Rule 294
Rule 335
Rule 470
Rubi steps \begin{align*} \text {integral}& = \frac {2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}}-\frac {\left (2 \left (-\frac {3 A b}{2}+\frac {9 a B}{2}\right )\right ) \int \frac {(e x)^{7/2}}{\left (a+b x^2\right )^{5/2}} \, dx}{3 b} \\ & = -\frac {(A b-3 a B) e (e x)^{5/2}}{3 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}}+\frac {\left (5 (A b-3 a B) e^2\right ) \int \frac {(e x)^{3/2}}{\left (a+b x^2\right )^{3/2}} \, dx}{6 b^2} \\ & = -\frac {(A b-3 a B) e (e x)^{5/2}}{3 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}}-\frac {5 (A b-3 a B) e^3 \sqrt {e x}}{6 b^3 \sqrt {a+b x^2}}+\frac {\left (5 (A b-3 a B) e^4\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{12 b^3} \\ & = -\frac {(A b-3 a B) e (e x)^{5/2}}{3 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}}-\frac {5 (A b-3 a B) e^3 \sqrt {e x}}{6 b^3 \sqrt {a+b x^2}}+\frac {\left (5 (A b-3 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 )}{6 b^3} \\ & = -\frac {(A b-3 a B) e (e x)^{5/2}}{3 b^2 \left (a+b x^2\right )^{3/2}}+\frac {2 B (e x)^{9/2}}{3 b e \left (a+b x^2\right )^{3/2}}-\frac {5 (A b-3 a B) e^3 \sqrt {e x}}{6 b^3 \sqrt {a+b x^2}}+\frac {5 (A b-3 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 )}{12 \sqrt [4]{a} 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 = 116, normalized size of antiderivative = 0.56 \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {e^3 \sqrt {e x} \left (15 a^2 B+b^2 x^2 \left (-7 A+4 B x^2\right )+a \left (-5 A b+21 b B x^2\right )+5 (A b-3 a B) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{6 b^3 \left (a+b x^2\right )^{3/2}} \]
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Time = 4.22 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.36
method | result | size |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {a \,e^{3} \left (A b -B a \right ) \sqrt {b e \,x^{3}+a e x}}{3 b^{5} \left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {e^{4} x \left (7 A b -13 B a \right )}{6 b^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {2 B \,e^{3} \sqrt {b e \,x^{3}+a e x}}{3 b^{3}}+\frac {\left (\frac {\left (A b -2 B a \right ) e^{4}}{b^{3}}-\frac {e^{4} \left (7 A b -13 B a \right )}{12 b^{3}}-\frac {B \,e^{4} a}{3 b^{3}}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(283\) |
default | \(\frac {\left (5 A \sqrt {-a b}\, \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b^{2} x^{2}-15 B \sqrt {-a b}\, \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}+5 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a b -15 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{2}+8 b^{3} B \,x^{5}-14 A \,b^{3} x^{3}+42 B a \,b^{2} x^{3}-10 a \,b^{2} A x +30 a^{2} b B x \right ) e^{3} \sqrt {e x}}{12 x \,b^{4} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(439\) |
risch | \(\frac {2 B x \sqrt {b \,x^{2}+a}\, e^{4}}{3 b^{3} \sqrt {e x}}+\frac {\left (\frac {3 A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {b e \,x^{3}+a e x}}-\frac {7 B a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}+3 a^{2} \left (A b -B a \right ) \left (\frac {\sqrt {b e \,x^{3}+a e x}}{3 a e \,b^{2} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {5 x}{6 a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {5 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{12 a^{2} b \sqrt {b e \,x^{3}+a e x}}\right )-3 a \left (2 A b -3 B a \right ) \left (\frac {x}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 a b \sqrt {b e \,x^{3}+a e x}}\right )\right ) e^{4} \sqrt {\left (b \,x^{2}+a \right ) e x}}{3 b^{3} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(610\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.86 \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {5 \, {\left ({\left (3 \, B a b^{2} - A b^{3}\right )} e^{3} x^{4} + 2 \, {\left (3 \, B a^{2} b - A a b^{2}\right )} e^{3} x^{2} + {\left (3 \, B a^{3} - A a^{2} b\right )} e^{3}\right )} \sqrt {b e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left (4 \, B b^{3} e^{3} x^{4} + 7 \, {\left (3 \, B a b^{2} - A b^{3}\right )} e^{3} x^{2} + 5 \, {\left (3 \, B a^{2} b - A a b^{2}\right )} e^{3}\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{6 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} \]
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Timed out. \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^{7/2}}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \]
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