Integrand size = 19, antiderivative size = 217 \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=-\frac {2 b x^{3/2}}{3 c^2}+\frac {2 x^{7/2}}{7 c}-\frac {b^{7/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{11/4}}+\frac {b^{7/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{11/4}}+\frac {b^{7/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{11/4}}-\frac {b^{7/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{11/4}} \]
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Time = 0.15 (sec) , antiderivative size = 217, 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, 327, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=-\frac {b^{7/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{11/4}}+\frac {b^{7/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} c^{11/4}}+\frac {b^{7/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} c^{11/4}}-\frac {b^{7/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} c^{11/4}}-\frac {2 b x^{3/2}}{3 c^2}+\frac {2 x^{7/2}}{7 c} \]
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Rule 210
Rule 303
Rule 327
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{9/2}}{b+c x^2} \, dx \\ & = \frac {2 x^{7/2}}{7 c}-\frac {b \int \frac {x^{5/2}}{b+c x^2} \, dx}{c} \\ & = -\frac {2 b x^{3/2}}{3 c^2}+\frac {2 x^{7/2}}{7 c}+\frac {b^2 \int \frac {\sqrt {x}}{b+c x^2} \, dx}{c^2} \\ & = -\frac {2 b x^{3/2}}{3 c^2}+\frac {2 x^{7/2}}{7 c}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{c^2} \\ & = -\frac {2 b x^{3/2}}{3 c^2}+\frac {2 x^{7/2}}{7 c}-\frac {b^2 \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{c^{5/2}}+\frac {b^2 \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{c^{5/2}} \\ & = -\frac {2 b x^{3/2}}{3 c^2}+\frac {2 x^{7/2}}{7 c}+\frac {b^2 \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 )}{2 c^3}+\frac {b^2 \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 )}{2 c^3}+\frac {b^{7/4} \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 )}{2 \sqrt {2} c^{11/4}}+\frac {b^{7/4} \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 )}{2 \sqrt {2} c^{11/4}} \\ & = -\frac {2 b x^{3/2}}{3 c^2}+\frac {2 x^{7/2}}{7 c}+\frac {b^{7/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{11/4}}-\frac {b^{7/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{11/4}}+\frac {b^{7/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{11/4}}-\frac {b^{7/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{11/4}} \\ & = -\frac {2 b x^{3/2}}{3 c^2}+\frac {2 x^{7/2}}{7 c}-\frac {b^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{11/4}}+\frac {b^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{11/4}}+\frac {b^{7/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{11/4}}-\frac {b^{7/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{11/4}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.59 \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=\frac {4 c^{3/4} x^{3/2} \left (-7 b+3 c x^2\right )-21 \sqrt {2} b^{7/4} \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )-21 \sqrt {2} b^{7/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{42 c^{11/4}} \]
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Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.59
method | result | size |
derivativedivides | \(-\frac {2 \left (-\frac {c \,x^{\frac {7}{2}}}{7}+\frac {b \,x^{\frac {3}{2}}}{3}\right )}{c^{2}}+\frac {b^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{3} \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(128\) |
default | \(-\frac {2 \left (-\frac {c \,x^{\frac {7}{2}}}{7}+\frac {b \,x^{\frac {3}{2}}}{3}\right )}{c^{2}}+\frac {b^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{3} \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(128\) |
risch | \(-\frac {2 x^{\frac {3}{2}} \left (-3 c \,x^{2}+7 b \right )}{21 c^{2}}+\frac {b^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{3} \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(128\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.82 \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=\frac {21 \, c^{2} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {1}{4}} \log \left (c^{8} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {3}{4}} + b^{5} \sqrt {x}\right ) - 21 i \, c^{2} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {1}{4}} \log \left (i \, c^{8} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {3}{4}} + b^{5} \sqrt {x}\right ) + 21 i \, c^{2} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {1}{4}} \log \left (-i \, c^{8} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {3}{4}} + b^{5} \sqrt {x}\right ) - 21 \, c^{2} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {1}{4}} \log \left (-c^{8} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {3}{4}} + b^{5} \sqrt {x}\right ) + 4 \, {\left (3 \, c x^{3} - 7 \, b x\right )} \sqrt {x}}{42 \, c^{2}} \]
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Timed out. \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.91 \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=\frac {b^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{4 \, c^{2}} + \frac {2 \, {\left (3 \, c x^{\frac {7}{2}} - 7 \, b x^{\frac {3}{2}}\right )}}{21 \, c^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.91 \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=\frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{5}} + \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{5}} - \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} b \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, c^{5}} + \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} b \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, c^{5}} + \frac {2 \, {\left (3 \, c^{6} x^{\frac {7}{2}} - 7 \, b c^{5} x^{\frac {3}{2}}\right )}}{21 \, c^{7}} \]
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Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.30 \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=\frac {2\,x^{7/2}}{7\,c}-\frac {2\,b\,x^{3/2}}{3\,c^2}+\frac {{\left (-b\right )}^{7/4}\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{c^{11/4}}+\frac {{\left (-b\right )}^{7/4}\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-b\right )}^{1/4}}\right )\,1{}\mathrm {i}}{c^{11/4}} \]
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