Integrand size = 19, antiderivative size = 251 \[ \int \frac {x^{23/2}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {45 \sqrt {x}}{16 c^3}-\frac {x^{9/2}}{4 c \left (b+c x^2\right )^2}-\frac {9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}+\frac {45 \sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{13/4}}-\frac {45 \sqrt [4]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{13/4}}+\frac {45 \sqrt [4]{b} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{13/4}}-\frac {45 \sqrt [4]{b} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{13/4}} \]
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Time = 0.14 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {1598, 294, 327, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^{23/2}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {45 \sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{13/4}}-\frac {45 \sqrt [4]{b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} c^{13/4}}+\frac {45 \sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} c^{13/4}}-\frac {45 \sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} c^{13/4}}-\frac {9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}-\frac {x^{9/2}}{4 c \left (b+c x^2\right )^2}+\frac {45 \sqrt {x}}{16 c^3} \]
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Rule 210
Rule 217
Rule 294
Rule 327
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{11/2}}{\left (b+c x^2\right )^3} \, dx \\ & = -\frac {x^{9/2}}{4 c \left (b+c x^2\right )^2}+\frac {9 \int \frac {x^{7/2}}{\left (b+c x^2\right )^2} \, dx}{8 c} \\ & = -\frac {x^{9/2}}{4 c \left (b+c x^2\right )^2}-\frac {9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}+\frac {45 \int \frac {x^{3/2}}{b+c x^2} \, dx}{32 c^2} \\ & = \frac {45 \sqrt {x}}{16 c^3}-\frac {x^{9/2}}{4 c \left (b+c x^2\right )^2}-\frac {9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}-\frac {(45 b) \int \frac {1}{\sqrt {x} \left (b+c x^2\right )} \, dx}{32 c^3} \\ & = \frac {45 \sqrt {x}}{16 c^3}-\frac {x^{9/2}}{4 c \left (b+c x^2\right )^2}-\frac {9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}-\frac {(45 b) \text {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,\sqrt {x}\right )}{16 c^3} \\ & = \frac {45 \sqrt {x}}{16 c^3}-\frac {x^{9/2}}{4 c \left (b+c x^2\right )^2}-\frac {9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}-\frac {\left (45 \sqrt {b}\right ) \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 c^3}-\frac {\left (45 \sqrt {b}\right ) \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 c^3} \\ & = \frac {45 \sqrt {x}}{16 c^3}-\frac {x^{9/2}}{4 c \left (b+c x^2\right )^2}-\frac {9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}-\frac {\left (45 \sqrt {b}\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 c^{7/2}}-\frac {\left (45 \sqrt {b}\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 c^{7/2}}+\frac {\left (45 \sqrt [4]{b}\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} c^{13/4}}+\frac {\left (45 \sqrt [4]{b}\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} c^{13/4}} \\ & = \frac {45 \sqrt {x}}{16 c^3}-\frac {x^{9/2}}{4 c \left (b+c x^2\right )^2}-\frac {9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}+\frac {45 \sqrt [4]{b} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{13/4}}-\frac {45 \sqrt [4]{b} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{13/4}}-\frac {\left (45 \sqrt [4]{b}\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} c^{13/4}}+\frac {\left (45 \sqrt [4]{b}\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} c^{13/4}} \\ & = \frac {45 \sqrt {x}}{16 c^3}-\frac {x^{9/2}}{4 c \left (b+c x^2\right )^2}-\frac {9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}+\frac {45 \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{13/4}}-\frac {45 \sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{13/4}}+\frac {45 \sqrt [4]{b} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{13/4}}-\frac {45 \sqrt [4]{b} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{13/4}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.59 \[ \int \frac {x^{23/2}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\frac {4 \sqrt [4]{c} \sqrt {x} \left (45 b^2+81 b c x^2+32 c^2 x^4\right )}{\left (b+c x^2\right )^2}+45 \sqrt {2} \sqrt [4]{b} \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )-45 \sqrt {2} \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{64 c^{13/4}} \]
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Time = 0.23 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.58
method | result | size |
derivativedivides | \(\frac {2 \sqrt {x}}{c^{3}}-\frac {2 b \left (\frac {-\frac {17 c \,x^{\frac {5}{2}}}{32}-\frac {13 b \sqrt {x}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {45 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 b}\right )}{c^{3}}\) | \(145\) |
default | \(\frac {2 \sqrt {x}}{c^{3}}-\frac {2 b \left (\frac {-\frac {17 c \,x^{\frac {5}{2}}}{32}-\frac {13 b \sqrt {x}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {45 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 b}\right )}{c^{3}}\) | \(145\) |
risch | \(\frac {2 \sqrt {x}}{c^{3}}-\frac {b \left (\frac {-\frac {17 c \,x^{\frac {5}{2}}}{16}-\frac {13 b \sqrt {x}}{16}}{\left (c \,x^{2}+b \right )^{2}}+\frac {45 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b}\right )}{c^{3}}\) | \(146\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.10 \[ \int \frac {x^{23/2}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {45 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )} \left (-\frac {b}{c^{13}}\right )^{\frac {1}{4}} \log \left (45 \, c^{3} \left (-\frac {b}{c^{13}}\right )^{\frac {1}{4}} + 45 \, \sqrt {x}\right ) + 45 \, {\left (i \, c^{5} x^{4} + 2 i \, b c^{4} x^{2} + i \, b^{2} c^{3}\right )} \left (-\frac {b}{c^{13}}\right )^{\frac {1}{4}} \log \left (45 i \, c^{3} \left (-\frac {b}{c^{13}}\right )^{\frac {1}{4}} + 45 \, \sqrt {x}\right ) + 45 \, {\left (-i \, c^{5} x^{4} - 2 i \, b c^{4} x^{2} - i \, b^{2} c^{3}\right )} \left (-\frac {b}{c^{13}}\right )^{\frac {1}{4}} \log \left (-45 i \, c^{3} \left (-\frac {b}{c^{13}}\right )^{\frac {1}{4}} + 45 \, \sqrt {x}\right ) - 45 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )} \left (-\frac {b}{c^{13}}\right )^{\frac {1}{4}} \log \left (-45 \, c^{3} \left (-\frac {b}{c^{13}}\right )^{\frac {1}{4}} + 45 \, \sqrt {x}\right ) - 4 \, {\left (32 \, c^{2} x^{4} + 81 \, b c x^{2} + 45 \, b^{2}\right )} \sqrt {x}}{64 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} \]
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Timed out. \[ \int \frac {x^{23/2}}{\left (b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
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none
Time = 0.28 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.91 \[ \int \frac {x^{23/2}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {17 \, b c x^{\frac {5}{2}} + 13 \, b^{2} \sqrt {x}}{16 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} - \frac {45 \, {\left (\frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} b^{\frac {1}{4}} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{c^{\frac {1}{4}}} - \frac {\sqrt {2} b^{\frac {1}{4}} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{c^{\frac {1}{4}}}\right )}}{128 \, c^{3}} + \frac {2 \, \sqrt {x}}{c^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.83 \[ \int \frac {x^{23/2}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {45 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, c^{4}} - \frac {45 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, c^{4}} - \frac {45 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, c^{4}} + \frac {45 \, \sqrt {2} \left (b c^{3}\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, c^{4}} + \frac {2 \, \sqrt {x}}{c^{3}} + \frac {17 \, b c x^{\frac {5}{2}} + 13 \, b^{2} \sqrt {x}}{16 \, {\left (c x^{2} + b\right )}^{2} c^{3}} \]
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Time = 12.91 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.40 \[ \int \frac {x^{23/2}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\frac {13\,b^2\,\sqrt {x}}{16}+\frac {17\,b\,c\,x^{5/2}}{16}}{b^2\,c^3+2\,b\,c^4\,x^2+c^5\,x^4}+\frac {2\,\sqrt {x}}{c^3}-\frac {45\,{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{32\,c^{13/4}}+\frac {{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-b\right )}^{1/4}}\right )\,45{}\mathrm {i}}{32\,c^{13/4}} \]
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