Integrand size = 19, antiderivative size = 264 \[ \int \frac {x^{3/2}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {165}{112 b^3 x^{7/2}}+\frac {55 c}{16 b^4 x^{3/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}-\frac {165 c^{7/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{19/4}}+\frac {165 c^{7/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{19/4}}-\frac {165 c^{7/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}}+\frac {165 c^{7/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}} \]
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Time = 0.16 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {1598, 296, 331, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{3/2}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {165 c^{7/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{19/4}}+\frac {165 c^{7/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{19/4}}-\frac {165 c^{7/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}}+\frac {165 c^{7/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}}+\frac {55 c}{16 b^4 x^{3/2}}-\frac {165}{112 b^3 x^{7/2}}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2} \]
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Rule 210
Rule 217
Rule 296
Rule 331
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^{9/2} \left (b+c x^2\right )^3} \, dx \\ & = \frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15 \int \frac {1}{x^{9/2} \left (b+c x^2\right )^2} \, dx}{8 b} \\ & = \frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac {165 \int \frac {1}{x^{9/2} \left (b+c x^2\right )} \, dx}{32 b^2} \\ & = -\frac {165}{112 b^3 x^{7/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}-\frac {(165 c) \int \frac {1}{x^{5/2} \left (b+c x^2\right )} \, dx}{32 b^3} \\ & = -\frac {165}{112 b^3 x^{7/2}}+\frac {55 c}{16 b^4 x^{3/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac {\left (165 c^2\right ) \int \frac {1}{\sqrt {x} \left (b+c x^2\right )} \, dx}{32 b^4} \\ & = -\frac {165}{112 b^3 x^{7/2}}+\frac {55 c}{16 b^4 x^{3/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac {\left (165 c^2\right ) \text {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,\sqrt {x}\right )}{16 b^4} \\ & = -\frac {165}{112 b^3 x^{7/2}}+\frac {55 c}{16 b^4 x^{3/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac {\left (165 c^2\right ) \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^{9/2}}+\frac {\left (165 c^2\right ) \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^{9/2}} \\ & = -\frac {165}{112 b^3 x^{7/2}}+\frac {55 c}{16 b^4 x^{3/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}+\frac {\left (165 c^{3/2}\right ) \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^{9/2}}+\frac {\left (165 c^{3/2}\right ) \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^{9/2}}-\frac {\left (165 c^{7/4}\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^{19/4}}-\frac {\left (165 c^{7/4}\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^{19/4}} \\ & = -\frac {165}{112 b^3 x^{7/2}}+\frac {55 c}{16 b^4 x^{3/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}-\frac {165 c^{7/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}}+\frac {165 c^{7/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}}+\frac {\left (165 c^{7/4}\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^{19/4}}-\frac {\left (165 c^{7/4}\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^{19/4}} \\ & = -\frac {165}{112 b^3 x^{7/2}}+\frac {55 c}{16 b^4 x^{3/2}}+\frac {1}{4 b x^{7/2} \left (b+c x^2\right )^2}+\frac {15}{16 b^2 x^{7/2} \left (b+c x^2\right )}-\frac {165 c^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{19/4}}+\frac {165 c^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{19/4}}-\frac {165 c^{7/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}}+\frac {165 c^{7/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{19/4}} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.61 \[ \int \frac {x^{3/2}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\frac {4 b^{3/4} \left (-32 b^3+160 b^2 c x^2+605 b c^2 x^4+385 c^3 x^6\right )}{x^{7/2} \left (b+c x^2\right )^2}-1155 \sqrt {2} c^{7/4} \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )+1155 \sqrt {2} c^{7/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{448 b^{19/4}} \]
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Time = 0.17 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.59
method | result | size |
risch | \(-\frac {2 \left (-7 c \,x^{2}+b \right )}{7 b^{4} x^{\frac {7}{2}}}+\frac {c^{2} \left (\frac {\frac {23 c \,x^{\frac {5}{2}}}{16}+\frac {27 b \sqrt {x}}{16}}{\left (c \,x^{2}+b \right )^{2}}+\frac {165 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b}\right )}{b^{4}}\) | \(155\) |
derivativedivides | \(-\frac {2}{7 b^{3} x^{\frac {7}{2}}}+\frac {2 c}{b^{4} x^{\frac {3}{2}}}+\frac {2 c^{2} \left (\frac {\frac {23 c \,x^{\frac {5}{2}}}{32}+\frac {27 b \sqrt {x}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {165 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 b}\right )}{b^{4}}\) | \(156\) |
default | \(-\frac {2}{7 b^{3} x^{\frac {7}{2}}}+\frac {2 c}{b^{4} x^{\frac {3}{2}}}+\frac {2 c^{2} \left (\frac {\frac {23 c \,x^{\frac {5}{2}}}{32}+\frac {27 b \sqrt {x}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {165 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 b}\right )}{b^{4}}\) | \(156\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.25 \[ \int \frac {x^{3/2}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {1155 \, {\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\right )} \left (-\frac {c^{7}}{b^{19}}\right )^{\frac {1}{4}} \log \left (165 \, b^{5} \left (-\frac {c^{7}}{b^{19}}\right )^{\frac {1}{4}} + 165 \, c^{2} \sqrt {x}\right ) - 1155 \, {\left (-i \, b^{4} c^{2} x^{8} - 2 i \, b^{5} c x^{6} - i \, b^{6} x^{4}\right )} \left (-\frac {c^{7}}{b^{19}}\right )^{\frac {1}{4}} \log \left (165 i \, b^{5} \left (-\frac {c^{7}}{b^{19}}\right )^{\frac {1}{4}} + 165 \, c^{2} \sqrt {x}\right ) - 1155 \, {\left (i \, b^{4} c^{2} x^{8} + 2 i \, b^{5} c x^{6} + i \, b^{6} x^{4}\right )} \left (-\frac {c^{7}}{b^{19}}\right )^{\frac {1}{4}} \log \left (-165 i \, b^{5} \left (-\frac {c^{7}}{b^{19}}\right )^{\frac {1}{4}} + 165 \, c^{2} \sqrt {x}\right ) - 1155 \, {\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\right )} \left (-\frac {c^{7}}{b^{19}}\right )^{\frac {1}{4}} \log \left (-165 \, b^{5} \left (-\frac {c^{7}}{b^{19}}\right )^{\frac {1}{4}} + 165 \, c^{2} \sqrt {x}\right ) + 4 \, {\left (385 \, c^{3} x^{6} + 605 \, b c^{2} x^{4} + 160 \, b^{2} c x^{2} - 32 \, b^{3}\right )} \sqrt {x}}{448 \, {\left (b^{4} c^{2} x^{8} + 2 \, b^{5} c x^{6} + b^{6} x^{4}\right )}} \]
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Timed out. \[ \int \frac {x^{3/2}}{\left (b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.93 \[ \int \frac {x^{3/2}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {385 \, c^{3} x^{6} + 605 \, b c^{2} x^{4} + 160 \, b^{2} c x^{2} - 32 \, b^{3}}{112 \, {\left (b^{4} c^{2} x^{\frac {15}{2}} + 2 \, b^{5} c x^{\frac {11}{2}} + b^{6} x^{\frac {7}{2}}\right )}} + \frac {165 \, {\left (\frac {2 \, \sqrt {2} c^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} c^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} c^{\frac {7}{4}} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}}} - \frac {\sqrt {2} c^{\frac {7}{4}} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}}}\right )}}{128 \, b^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.85 \[ \int \frac {x^{3/2}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {165 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} c \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{5}} + \frac {165 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} c \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{5}} + \frac {165 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} c \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{5}} - \frac {165 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} c \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{5}} + \frac {23 \, c^{3} x^{\frac {5}{2}} + 27 \, b c^{2} \sqrt {x}}{16 \, {\left (c x^{2} + b\right )}^{2} b^{4}} + \frac {2 \, {\left (7 \, c x^{2} - b\right )}}{7 \, b^{4} x^{\frac {7}{2}}} \]
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Time = 13.39 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.41 \[ \int \frac {x^{3/2}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\frac {10\,c\,x^2}{7\,b^2}-\frac {2}{7\,b}+\frac {605\,c^2\,x^4}{112\,b^3}+\frac {55\,c^3\,x^6}{16\,b^4}}{b^2\,x^{7/2}+c^2\,x^{15/2}+2\,b\,c\,x^{11/2}}+\frac {165\,{\left (-c\right )}^{7/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{32\,b^{19/4}}+\frac {165\,{\left (-c\right )}^{7/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{32\,b^{19/4}} \]
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