Integrand size = 26, antiderivative size = 343 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}+\frac {9 \sqrt [4]{c} (5 b B-13 A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{17/4}}-\frac {9 \sqrt [4]{c} (5 b B-13 A c) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{17/4}}-\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}+\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}} \]
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Time = 0.19 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {1598, 468, 296, 331, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {9 \sqrt [4]{c} (5 b B-13 A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{17/4}}-\frac {9 \sqrt [4]{c} (5 b B-13 A c) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{17/4}}-\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}+\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}+\frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2} \]
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Rule 210
Rule 296
Rule 303
Rule 331
Rule 335
Rule 468
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {A+B x^2}{x^{7/2} \left (b+c x^2\right )^3} \, dx \\ & = -\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}+\frac {\left (-\frac {5 b B}{2}+\frac {13 A c}{2}\right ) \int \frac {1}{x^{7/2} \left (b+c x^2\right )^2} \, dx}{4 b c} \\ & = -\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}-\frac {(9 (5 b B-13 A c)) \int \frac {1}{x^{7/2} \left (b+c x^2\right )} \, dx}{32 b^2 c} \\ & = \frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}+\frac {(9 (5 b B-13 A c)) \int \frac {1}{x^{3/2} \left (b+c x^2\right )} \, dx}{32 b^3} \\ & = \frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}-\frac {(9 c (5 b B-13 A c)) \int \frac {\sqrt {x}}{b+c x^2} \, dx}{32 b^4} \\ & = \frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}-\frac {(9 c (5 b B-13 A c)) \text {Subst}\left (\int \frac {x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{16 b^4} \\ & = \frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}+\frac {\left (9 \sqrt {c} (5 b B-13 A c)\right ) \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^4}-\frac {\left (9 \sqrt {c} (5 b B-13 A c)\right ) \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^4} \\ & = \frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}-\frac {(9 (5 b B-13 A c)) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{64 b^4}-\frac {(9 (5 b B-13 A c)) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{64 b^4}-\frac {\left (9 \sqrt [4]{c} (5 b B-13 A c)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} b^{17/4}}-\frac {\left (9 \sqrt [4]{c} (5 b B-13 A c)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} b^{17/4}} \\ & = \frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}-\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}+\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}-\frac {\left (9 \sqrt [4]{c} (5 b B-13 A c)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{17/4}}+\frac {\left (9 \sqrt [4]{c} (5 b B-13 A c)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{17/4}} \\ & = \frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}+\frac {9 \sqrt [4]{c} (5 b B-13 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{17/4}}-\frac {9 \sqrt [4]{c} (5 b B-13 A c) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{17/4}}-\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}+\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}} \\ \end{align*}
Time = 0.90 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.61 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{b} \left (5 b B x^2 \left (32 b^2+81 b c x^2+45 c^2 x^4\right )+A \left (32 b^3-416 b^2 c x^2-1053 b c^2 x^4-585 c^3 x^6\right )\right )}{x^{5/2} \left (b+c x^2\right )^2}+45 \sqrt {2} \sqrt [4]{c} (5 b B-13 A c) \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )+45 \sqrt {2} \sqrt [4]{c} (5 b B-13 A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{320 b^{17/4}} \]
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Time = 1.89 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.55
method | result | size |
derivativedivides | \(\frac {2 c \left (\frac {\left (\frac {21}{32} A \,c^{2}-\frac {13}{32} B b c \right ) x^{\frac {7}{2}}+\frac {b \left (25 A c -17 B b \right ) x^{\frac {3}{2}}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {\left (\frac {117 A c}{32}-\frac {45 B b}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{4}}-\frac {2 A}{5 b^{3} x^{\frac {5}{2}}}-\frac {2 \left (-3 A c +B b \right )}{b^{4} \sqrt {x}}\) | \(190\) |
default | \(\frac {2 c \left (\frac {\left (\frac {21}{32} A \,c^{2}-\frac {13}{32} B b c \right ) x^{\frac {7}{2}}+\frac {b \left (25 A c -17 B b \right ) x^{\frac {3}{2}}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {\left (\frac {117 A c}{32}-\frac {45 B b}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{4}}-\frac {2 A}{5 b^{3} x^{\frac {5}{2}}}-\frac {2 \left (-3 A c +B b \right )}{b^{4} \sqrt {x}}\) | \(190\) |
risch | \(-\frac {2 \left (-15 A c \,x^{2}+5 b B \,x^{2}+A b \right )}{5 b^{4} x^{\frac {5}{2}}}+\frac {c \left (\frac {2 \left (\frac {21}{32} A \,c^{2}-\frac {13}{32} B b c \right ) x^{\frac {7}{2}}+\frac {b \left (25 A c -17 B b \right ) x^{\frac {3}{2}}}{16}}{\left (c \,x^{2}+b \right )^{2}}+\frac {\left (\frac {117 A c}{32}-\frac {45 B b}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{4}}\) | \(191\) |
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Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 918, normalized size of antiderivative = 2.68 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {45 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )} \left (-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}\right )^{\frac {1}{4}} \log \left (729 \, b^{13} \left (-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}\right )^{\frac {3}{4}} - 729 \, {\left (125 \, B^{3} b^{3} c - 975 \, A B^{2} b^{2} c^{2} + 2535 \, A^{2} B b c^{3} - 2197 \, A^{3} c^{4}\right )} \sqrt {x}\right ) - 45 \, {\left (i \, b^{4} c^{2} x^{7} + 2 i \, b^{5} c x^{5} + i \, b^{6} x^{3}\right )} \left (-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}\right )^{\frac {1}{4}} \log \left (729 i \, b^{13} \left (-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}\right )^{\frac {3}{4}} - 729 \, {\left (125 \, B^{3} b^{3} c - 975 \, A B^{2} b^{2} c^{2} + 2535 \, A^{2} B b c^{3} - 2197 \, A^{3} c^{4}\right )} \sqrt {x}\right ) - 45 \, {\left (-i \, b^{4} c^{2} x^{7} - 2 i \, b^{5} c x^{5} - i \, b^{6} x^{3}\right )} \left (-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}\right )^{\frac {1}{4}} \log \left (-729 i \, b^{13} \left (-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}\right )^{\frac {3}{4}} - 729 \, {\left (125 \, B^{3} b^{3} c - 975 \, A B^{2} b^{2} c^{2} + 2535 \, A^{2} B b c^{3} - 2197 \, A^{3} c^{4}\right )} \sqrt {x}\right ) - 45 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )} \left (-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}\right )^{\frac {1}{4}} \log \left (-729 \, b^{13} \left (-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}\right )^{\frac {3}{4}} - 729 \, {\left (125 \, B^{3} b^{3} c - 975 \, A B^{2} b^{2} c^{2} + 2535 \, A^{2} B b c^{3} - 2197 \, A^{3} c^{4}\right )} \sqrt {x}\right ) - 4 \, {\left (45 \, {\left (5 \, B b c^{2} - 13 \, A c^{3}\right )} x^{6} + 81 \, {\left (5 \, B b^{2} c - 13 \, A b c^{2}\right )} x^{4} + 32 \, A b^{3} + 32 \, {\left (5 \, B b^{3} - 13 \, A b^{2} c\right )} x^{2}\right )} \sqrt {x}}{320 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )}} \]
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Timed out. \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.83 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {45 \, {\left (5 \, B b c^{2} - 13 \, A c^{3}\right )} x^{6} + 81 \, {\left (5 \, B b^{2} c - 13 \, A b c^{2}\right )} x^{4} + 32 \, A b^{3} + 32 \, {\left (5 \, B b^{3} - 13 \, A b^{2} c\right )} x^{2}}{80 \, {\left (b^{4} c^{2} x^{\frac {13}{2}} + 2 \, b^{5} c x^{\frac {9}{2}} + b^{6} x^{\frac {5}{2}}\right )}} - \frac {9 \, {\left (5 \, B b c - 13 \, A c^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{128 \, b^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.95 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {9 \, \sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{5} c^{2}} - \frac {9 \, \sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{5} c^{2}} + \frac {9 \, \sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{5} c^{2}} - \frac {9 \, \sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{5} c^{2}} - \frac {13 \, B b c^{2} x^{\frac {7}{2}} - 21 \, A c^{3} x^{\frac {7}{2}} + 17 \, B b^{2} c x^{\frac {3}{2}} - 25 \, A b c^{2} x^{\frac {3}{2}}}{16 \, {\left (c x^{2} + b\right )}^{2} b^{4}} - \frac {2 \, {\left (5 \, B b x^{2} - 15 \, A c x^{2} + A b\right )}}{5 \, b^{4} x^{\frac {5}{2}}} \]
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Time = 9.13 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.44 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\frac {2\,x^2\,\left (13\,A\,c-5\,B\,b\right )}{5\,b^2}-\frac {2\,A}{5\,b}+\frac {9\,c^2\,x^6\,\left (13\,A\,c-5\,B\,b\right )}{16\,b^4}+\frac {81\,c\,x^4\,\left (13\,A\,c-5\,B\,b\right )}{80\,b^3}}{b^2\,x^{5/2}+c^2\,x^{13/2}+2\,b\,c\,x^{9/2}}+\frac {9\,{\left (-c\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )\,\left (13\,A\,c-5\,B\,b\right )}{32\,b^{17/4}}-\frac {9\,{\left (-c\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )\,\left (13\,A\,c-5\,B\,b\right )}{32\,b^{17/4}} \]
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