Integrand size = 15, antiderivative size = 308 \[ \int \frac {\sqrt {x}}{\left (a+c x^4\right )^2} \, dx=\frac {x^{3/2}}{4 a \left (a+c x^4\right )}-\frac {5 \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{13/8} c^{3/8}}+\frac {5 \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{13/8} c^{3/8}}-\frac {5 \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{13/8} c^{3/8}}+\frac {5 \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{13/8} c^{3/8}}+\frac {5 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{13/8} c^{3/8}}-\frac {5 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{13/8} c^{3/8}} \]
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Time = 0.22 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {296, 335, 306, 303, 1176, 631, 210, 1179, 642, 304, 211, 214} \[ \int \frac {\sqrt {x}}{\left (a+c x^4\right )^2} \, dx=-\frac {5 \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{13/8} c^{3/8}}+\frac {5 \arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt {2} (-a)^{13/8} c^{3/8}}-\frac {5 \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{13/8} c^{3/8}}+\frac {5 \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{13/8} c^{3/8}}+\frac {5 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{13/8} c^{3/8}}-\frac {5 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{13/8} c^{3/8}}+\frac {x^{3/2}}{4 a \left (a+c x^4\right )} \]
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Rule 210
Rule 211
Rule 214
Rule 296
Rule 303
Rule 304
Rule 306
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {x^{3/2}}{4 a \left (a+c x^4\right )}+\frac {5 \int \frac {\sqrt {x}}{a+c x^4} \, dx}{8 a} \\ & = \frac {x^{3/2}}{4 a \left (a+c x^4\right )}+\frac {5 \text {Subst}\left (\int \frac {x^2}{a+c x^8} \, dx,x,\sqrt {x}\right )}{4 a} \\ & = \frac {x^{3/2}}{4 a \left (a+c x^4\right )}+\frac {5 \text {Subst}\left (\int \frac {x^2}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{8 (-a)^{3/2}}+\frac {5 \text {Subst}\left (\int \frac {x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{8 (-a)^{3/2}} \\ & = \frac {x^{3/2}}{4 a \left (a+c x^4\right )}+\frac {5 \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{16 (-a)^{3/2} \sqrt [4]{c}}-\frac {5 \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{16 (-a)^{3/2} \sqrt [4]{c}}-\frac {5 \text {Subst}\left (\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{16 (-a)^{3/2} \sqrt [4]{c}}+\frac {5 \text {Subst}\left (\int \frac {\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{16 (-a)^{3/2} \sqrt [4]{c}} \\ & = \frac {x^{3/2}}{4 a \left (a+c x^4\right )}-\frac {5 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{13/8} c^{3/8}}+\frac {5 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{13/8} c^{3/8}}+\frac {5 \text {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{32 (-a)^{3/2} \sqrt {c}}+\frac {5 \text {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{32 (-a)^{3/2} \sqrt {c}}+\frac {5 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}+2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{32 \sqrt {2} (-a)^{13/8} c^{3/8}}+\frac {5 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}-2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{32 \sqrt {2} (-a)^{13/8} c^{3/8}} \\ & = \frac {x^{3/2}}{4 a \left (a+c x^4\right )}-\frac {5 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{13/8} c^{3/8}}+\frac {5 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{13/8} c^{3/8}}+\frac {5 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{13/8} c^{3/8}}-\frac {5 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{13/8} c^{3/8}}+\frac {5 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{13/8} c^{3/8}}-\frac {5 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{13/8} c^{3/8}} \\ & = \frac {x^{3/2}}{4 a \left (a+c x^4\right )}-\frac {5 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{13/8} c^{3/8}}+\frac {5 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{13/8} c^{3/8}}-\frac {5 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{13/8} c^{3/8}}+\frac {5 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{13/8} c^{3/8}}+\frac {5 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{13/8} c^{3/8}}-\frac {5 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{13/8} c^{3/8}} \\ \end{align*}
Time = 1.09 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {x}}{\left (a+c x^4\right )^2} \, dx=\frac {\frac {8 a^{5/8} x^{3/2}}{a+c x^4}+\frac {5 \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{c^{3/8}}-\frac {5 \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{c^{3/8}}+\frac {5 \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{c^{3/8}}-\frac {5 \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{c^{3/8}}}{32 a^{13/8}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.92 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.16
method | result | size |
derivativedivides | \(\frac {x^{\frac {3}{2}}}{4 a \left (x^{4} c +a \right )}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{32 a c}\) | \(50\) |
default | \(\frac {x^{\frac {3}{2}}}{4 a \left (x^{4} c +a \right )}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{32 a c}\) | \(50\) |
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {x}}{\left (a+c x^4\right )^2} \, dx=-\frac {5 \, \sqrt {2} {\left (\left (i - 1\right ) \, a c x^{4} + \left (i - 1\right ) \, a^{2}\right )} \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{5} c \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + 5 \, \sqrt {2} {\left (-\left (i + 1\right ) \, a c x^{4} - \left (i + 1\right ) \, a^{2}\right )} \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{5} c \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + 5 \, \sqrt {2} {\left (\left (i + 1\right ) \, a c x^{4} + \left (i + 1\right ) \, a^{2}\right )} \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{5} c \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + 5 \, \sqrt {2} {\left (-\left (i - 1\right ) \, a c x^{4} - \left (i - 1\right ) \, a^{2}\right )} \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{5} c \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + 10 \, {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {1}{8}} \log \left (a^{5} c \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + 10 \, {\left (-i \, a c x^{4} - i \, a^{2}\right )} \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {1}{8}} \log \left (i \, a^{5} c \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + 10 \, {\left (i \, a c x^{4} + i \, a^{2}\right )} \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {1}{8}} \log \left (-i \, a^{5} c \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) - 10 \, {\left (a c x^{4} + a^{2}\right )} \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {1}{8}} \log \left (-a^{5} c \left (-\frac {1}{a^{13} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) - 16 \, x^{\frac {3}{2}}}{64 \, {\left (a c x^{4} + a^{2}\right )}} \]
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Timed out. \[ \int \frac {\sqrt {x}}{\left (a+c x^4\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {x}}{\left (a+c x^4\right )^2} \, dx=\int { \frac {\sqrt {x}}{{\left (c x^{4} + a\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (207) = 414\).
Time = 0.45 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {x}}{\left (a+c x^4\right )^2} \, dx=\frac {x^{\frac {3}{2}}}{4 \, {\left (c x^{4} + a\right )} a} - \frac {5 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {5 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {5 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {5 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{16 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {5 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \log \left (\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {5 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \log \left (-\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {5 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \log \left (\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {5 \, \left (\frac {a}{c}\right )^{\frac {3}{8}} \log \left (-\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{32 \, a^{2} \sqrt {-2 \, \sqrt {2} + 4}} \]
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Time = 0.15 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {x}}{\left (a+c x^4\right )^2} \, dx=\frac {x^{3/2}}{4\,a\,\left (c\,x^4+a\right )}-\frac {5\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{16\,{\left (-a\right )}^{13/8}\,c^{3/8}}-\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,5{}\mathrm {i}}{16\,{\left (-a\right )}^{13/8}\,c^{3/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {5}{32}-\frac {5}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{13/8}\,c^{3/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {5}{32}+\frac {5}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{13/8}\,c^{3/8}} \]
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