Integrand size = 28, antiderivative size = 405 \[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )^{3/2}} \, dx=-\frac {2 A}{9 b x^{7/2} \sqrt {b x^2+c x^4}}+\frac {9 b B-11 A c}{9 b^2 x^{3/2} \sqrt {b x^2+c x^4}}-\frac {7 c^{3/2} (9 b B-11 A c) x^{3/2} \left (b+c x^2\right )}{15 b^4 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {7 (9 b B-11 A c) \sqrt {b x^2+c x^4}}{45 b^3 x^{7/2}}+\frac {7 c (9 b B-11 A c) \sqrt {b x^2+c x^4}}{15 b^4 x^{3/2}}+\frac {7 c^{5/4} (9 b B-11 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{15/4} \sqrt {b x^2+c x^4}}-\frac {7 c^{5/4} (9 b B-11 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{30 b^{15/4} \sqrt {b x^2+c x^4}} \]
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Time = 0.35 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2063, 2048, 2050, 2057, 335, 311, 226, 1210} \[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )^{3/2}} \, dx=-\frac {7 c^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (9 b B-11 A c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{30 b^{15/4} \sqrt {b x^2+c x^4}}+\frac {7 c^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (9 b B-11 A c) E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{15/4} \sqrt {b x^2+c x^4}}-\frac {7 c^{3/2} x^{3/2} \left (b+c x^2\right ) (9 b B-11 A c)}{15 b^4 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {7 c \sqrt {b x^2+c x^4} (9 b B-11 A c)}{15 b^4 x^{3/2}}-\frac {7 \sqrt {b x^2+c x^4} (9 b B-11 A c)}{45 b^3 x^{7/2}}+\frac {9 b B-11 A c}{9 b^2 x^{3/2} \sqrt {b x^2+c x^4}}-\frac {2 A}{9 b x^{7/2} \sqrt {b x^2+c x^4}} \]
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2048
Rule 2050
Rule 2057
Rule 2063
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A}{9 b x^{7/2} \sqrt {b x^2+c x^4}}-\frac {\left (2 \left (-\frac {9 b B}{2}+\frac {11 A c}{2}\right )\right ) \int \frac {1}{\sqrt {x} \left (b x^2+c x^4\right )^{3/2}} \, dx}{9 b} \\ & = -\frac {2 A}{9 b x^{7/2} \sqrt {b x^2+c x^4}}+\frac {9 b B-11 A c}{9 b^2 x^{3/2} \sqrt {b x^2+c x^4}}+\frac {(7 (9 b B-11 A c)) \int \frac {1}{x^{5/2} \sqrt {b x^2+c x^4}} \, dx}{18 b^2} \\ & = -\frac {2 A}{9 b x^{7/2} \sqrt {b x^2+c x^4}}+\frac {9 b B-11 A c}{9 b^2 x^{3/2} \sqrt {b x^2+c x^4}}-\frac {7 (9 b B-11 A c) \sqrt {b x^2+c x^4}}{45 b^3 x^{7/2}}-\frac {(7 c (9 b B-11 A c)) \int \frac {1}{\sqrt {x} \sqrt {b x^2+c x^4}} \, dx}{30 b^3} \\ & = -\frac {2 A}{9 b x^{7/2} \sqrt {b x^2+c x^4}}+\frac {9 b B-11 A c}{9 b^2 x^{3/2} \sqrt {b x^2+c x^4}}-\frac {7 (9 b B-11 A c) \sqrt {b x^2+c x^4}}{45 b^3 x^{7/2}}+\frac {7 c (9 b B-11 A c) \sqrt {b x^2+c x^4}}{15 b^4 x^{3/2}}-\frac {\left (7 c^2 (9 b B-11 A c)\right ) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx}{30 b^4} \\ & = -\frac {2 A}{9 b x^{7/2} \sqrt {b x^2+c x^4}}+\frac {9 b B-11 A c}{9 b^2 x^{3/2} \sqrt {b x^2+c x^4}}-\frac {7 (9 b B-11 A c) \sqrt {b x^2+c x^4}}{45 b^3 x^{7/2}}+\frac {7 c (9 b B-11 A c) \sqrt {b x^2+c x^4}}{15 b^4 x^{3/2}}-\frac {\left (7 c^2 (9 b B-11 A c) x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{30 b^4 \sqrt {b x^2+c x^4}} \\ & = -\frac {2 A}{9 b x^{7/2} \sqrt {b x^2+c x^4}}+\frac {9 b B-11 A c}{9 b^2 x^{3/2} \sqrt {b x^2+c x^4}}-\frac {7 (9 b B-11 A c) \sqrt {b x^2+c x^4}}{45 b^3 x^{7/2}}+\frac {7 c (9 b B-11 A c) \sqrt {b x^2+c x^4}}{15 b^4 x^{3/2}}-\frac {\left (7 c^2 (9 b B-11 A c) x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{15 b^4 \sqrt {b x^2+c x^4}} \\ & = -\frac {2 A}{9 b x^{7/2} \sqrt {b x^2+c x^4}}+\frac {9 b B-11 A c}{9 b^2 x^{3/2} \sqrt {b x^2+c x^4}}-\frac {7 (9 b B-11 A c) \sqrt {b x^2+c x^4}}{45 b^3 x^{7/2}}+\frac {7 c (9 b B-11 A c) \sqrt {b x^2+c x^4}}{15 b^4 x^{3/2}}-\frac {\left (7 c^{3/2} (9 b B-11 A c) x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{15 b^{7/2} \sqrt {b x^2+c x^4}}+\frac {\left (7 c^{3/2} (9 b B-11 A c) x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {b}}}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{15 b^{7/2} \sqrt {b x^2+c x^4}} \\ & = -\frac {2 A}{9 b x^{7/2} \sqrt {b x^2+c x^4}}+\frac {9 b B-11 A c}{9 b^2 x^{3/2} \sqrt {b x^2+c x^4}}-\frac {7 c^{3/2} (9 b B-11 A c) x^{3/2} \left (b+c x^2\right )}{15 b^4 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {7 (9 b B-11 A c) \sqrt {b x^2+c x^4}}{45 b^3 x^{7/2}}+\frac {7 c (9 b B-11 A c) \sqrt {b x^2+c x^4}}{15 b^4 x^{3/2}}+\frac {7 c^{5/4} (9 b B-11 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{15/4} \sqrt {b x^2+c x^4}}-\frac {7 c^{5/4} (9 b B-11 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{30 b^{15/4} \sqrt {b x^2+c x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.20 \[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )^{3/2}} \, dx=\frac {-10 A b+2 (-9 b B+11 A c) x^2 \sqrt {1+\frac {c x^2}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{2},-\frac {1}{4},-\frac {c x^2}{b}\right )}{45 b^2 x^{7/2} \sqrt {x^2 \left (b+c x^2\right )}} \]
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Time = 2.64 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.11
method | result | size |
default | \(\frac {\left (c \,x^{2}+b \right ) \left (462 A \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{2} x^{4}-231 A \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{2} x^{4}-378 B \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c \,x^{4}+189 B \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c \,x^{4}-462 A \,c^{3} x^{6}+378 x^{6} B b \,c^{2}-308 A b \,c^{2} x^{4}+252 x^{4} B \,b^{2} c +44 A \,b^{2} c \,x^{2}-36 b^{3} B \,x^{2}-20 b^{3} A \right )}{90 \left (x^{4} c +b \,x^{2}\right )^{\frac {3}{2}} x^{\frac {3}{2}} b^{4}}\) | \(450\) |
risch | \(-\frac {2 \left (c \,x^{2}+b \right ) \left (93 A \,c^{2} x^{4}-72 x^{4} B b c -16 A b c \,x^{2}+9 b^{2} B \,x^{2}+5 b^{2} A \right )}{45 b^{4} x^{\frac {7}{2}} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}+\frac {c^{2} \left (\frac {\left (31 A c -24 B b \right ) \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c \,x^{3}+b x}}-15 b \left (A c -B b \right ) \left (\frac {x^{2}}{b \sqrt {\left (x^{2}+\frac {b}{c}\right ) c x}}-\frac {\sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{2 b c \sqrt {c \,x^{3}+b x}}\right )\right ) \sqrt {x}\, \sqrt {x \left (c \,x^{2}+b \right )}}{15 b^{4} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\) | \(467\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.40 \[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )^{3/2}} \, dx=\frac {21 \, {\left ({\left (9 \, B b c^{2} - 11 \, A c^{3}\right )} x^{8} + {\left (9 \, B b^{2} c - 11 \, A b c^{2}\right )} x^{6}\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) + {\left (21 \, {\left (9 \, B b c^{2} - 11 \, A c^{3}\right )} x^{6} + 14 \, {\left (9 \, B b^{2} c - 11 \, A b c^{2}\right )} x^{4} - 10 \, A b^{3} - 2 \, {\left (9 \, B b^{3} - 11 \, A b^{2} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}}{45 \, {\left (b^{4} c x^{8} + b^{5} x^{6}\right )}} \]
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Timed out. \[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {B\,x^2+A}{x^{5/2}\,{\left (c\,x^4+b\,x^2\right )}^{3/2}} \,d x \]
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