Integrand size = 19, antiderivative size = 239 \[ \int \frac {x^{21/2}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {x^{7/2}}{4 c \left (b+c x^2\right )^2}-\frac {7 x^{3/2}}{16 c^2 \left (b+c x^2\right )}-\frac {21 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} \sqrt [4]{b} c^{11/4}}+\frac {21 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} \sqrt [4]{b} c^{11/4}}+\frac {21 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} \sqrt [4]{b} c^{11/4}}-\frac {21 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} \sqrt [4]{b} c^{11/4}} \]
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Time = 0.12 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {1598, 294, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {x^{21/2}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {21 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} \sqrt [4]{b} c^{11/4}}+\frac {21 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} \sqrt [4]{b} c^{11/4}}+\frac {21 \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} \sqrt [4]{b} c^{11/4}}-\frac {21 \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} \sqrt [4]{b} c^{11/4}}-\frac {7 x^{3/2}}{16 c^2 \left (b+c x^2\right )}-\frac {x^{7/2}}{4 c \left (b+c x^2\right )^2} \]
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Rule 210
Rule 294
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{9/2}}{\left (b+c x^2\right )^3} \, dx \\ & = -\frac {x^{7/2}}{4 c \left (b+c x^2\right )^2}+\frac {7 \int \frac {x^{5/2}}{\left (b+c x^2\right )^2} \, dx}{8 c} \\ & = -\frac {x^{7/2}}{4 c \left (b+c x^2\right )^2}-\frac {7 x^{3/2}}{16 c^2 \left (b+c x^2\right )}+\frac {21 \int \frac {\sqrt {x}}{b+c x^2} \, dx}{32 c^2} \\ & = -\frac {x^{7/2}}{4 c \left (b+c x^2\right )^2}-\frac {7 x^{3/2}}{16 c^2 \left (b+c x^2\right )}+\frac {21 \text {Subst}\left (\int \frac {x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{16 c^2} \\ & = -\frac {x^{7/2}}{4 c \left (b+c x^2\right )^2}-\frac {7 x^{3/2}}{16 c^2 \left (b+c x^2\right )}-\frac {21 \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 c^{5/2}}+\frac {21 \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 c^{5/2}} \\ & = -\frac {x^{7/2}}{4 c \left (b+c x^2\right )^2}-\frac {7 x^{3/2}}{16 c^2 \left (b+c x^2\right )}+\frac {21 \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 c^3}+\frac {21 \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 c^3}+\frac {21 \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} \sqrt [4]{b} c^{11/4}}+\frac {21 \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} \sqrt [4]{b} c^{11/4}} \\ & = -\frac {x^{7/2}}{4 c \left (b+c x^2\right )^2}-\frac {7 x^{3/2}}{16 c^2 \left (b+c x^2\right )}+\frac {21 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} \sqrt [4]{b} c^{11/4}}-\frac {21 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} \sqrt [4]{b} c^{11/4}}+\frac {21 \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} \sqrt [4]{b} c^{11/4}}-\frac {21 \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} \sqrt [4]{b} c^{11/4}} \\ & = -\frac {x^{7/2}}{4 c \left (b+c x^2\right )^2}-\frac {7 x^{3/2}}{16 c^2 \left (b+c x^2\right )}-\frac {21 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} \sqrt [4]{b} c^{11/4}}+\frac {21 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} \sqrt [4]{b} c^{11/4}}+\frac {21 \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} \sqrt [4]{b} c^{11/4}}-\frac {21 \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} \sqrt [4]{b} c^{11/4}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.58 \[ \int \frac {x^{21/2}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {-\frac {4 c^{3/4} x^{3/2} \left (7 b+11 c x^2\right )}{\left (b+c x^2\right )^2}-\frac {21 \sqrt {2} \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt [4]{b}}-\frac {21 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{\sqrt [4]{b}}}{64 c^{11/4}} \]
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Time = 0.16 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.57
method | result | size |
derivativedivides | \(\frac {-\frac {11 x^{\frac {7}{2}}}{16 c}-\frac {7 b \,x^{\frac {3}{2}}}{16 c^{2}}}{\left (c \,x^{2}+b \right )^{2}}+\frac {21 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c^{3} \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(136\) |
default | \(\frac {-\frac {11 x^{\frac {7}{2}}}{16 c}-\frac {7 b \,x^{\frac {3}{2}}}{16 c^{2}}}{\left (c \,x^{2}+b \right )^{2}}+\frac {21 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c^{3} \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(136\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.15 \[ \int \frac {x^{21/2}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {21 \, {\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )} \left (-\frac {1}{b c^{11}}\right )^{\frac {1}{4}} \log \left (b c^{8} \left (-\frac {1}{b c^{11}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) - 21 \, {\left (i \, c^{4} x^{4} + 2 i \, b c^{3} x^{2} + i \, b^{2} c^{2}\right )} \left (-\frac {1}{b c^{11}}\right )^{\frac {1}{4}} \log \left (i \, b c^{8} \left (-\frac {1}{b c^{11}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) - 21 \, {\left (-i \, c^{4} x^{4} - 2 i \, b c^{3} x^{2} - i \, b^{2} c^{2}\right )} \left (-\frac {1}{b c^{11}}\right )^{\frac {1}{4}} \log \left (-i \, b c^{8} \left (-\frac {1}{b c^{11}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) - 21 \, {\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )} \left (-\frac {1}{b c^{11}}\right )^{\frac {1}{4}} \log \left (-b c^{8} \left (-\frac {1}{b c^{11}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) - 4 \, {\left (11 \, c x^{3} + 7 \, b x\right )} \sqrt {x}}{64 \, {\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )}} \]
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Timed out. \[ \int \frac {x^{21/2}}{\left (b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.91 \[ \int \frac {x^{21/2}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {11 \, c x^{\frac {7}{2}} + 7 \, b x^{\frac {3}{2}}}{16 \, {\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )}} + \frac {21 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{128 \, c^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.87 \[ \int \frac {x^{21/2}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {11 \, c x^{\frac {7}{2}} + 7 \, b x^{\frac {3}{2}}}{16 \, {\left (c x^{2} + b\right )}^{2} c^{2}} + \frac {21 \, \sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b c^{5}} + \frac {21 \, \sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b c^{5}} - \frac {21 \, \sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b c^{5}} + \frac {21 \, \sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b c^{5}} \]
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Time = 12.81 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.36 \[ \int \frac {x^{21/2}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {21\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{32\,{\left (-b\right )}^{1/4}\,c^{11/4}}-\frac {\frac {11\,x^{7/2}}{16\,c}+\frac {7\,b\,x^{3/2}}{16\,c^2}}{b^2+2\,b\,c\,x^2+c^2\,x^4}-\frac {21\,\mathrm {atanh}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{32\,{\left (-b\right )}^{1/4}\,c^{11/4}} \]
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