Integrand size = 15, antiderivative size = 308 \[ \int \frac {x^{11/2}}{\left (a+c x^4\right )^2} \, dx=-\frac {x^{5/2}}{4 c \left (a+c x^4\right )}-\frac {5 \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{3/8} c^{13/8}}+\frac {5 \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{3/8} c^{13/8}}-\frac {5 \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{3/8} c^{13/8}}-\frac {5 \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{3/8} c^{13/8}}-\frac {5 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{3/8} c^{13/8}}+\frac {5 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{3/8} c^{13/8}} \]
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Time = 0.19 (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 = {294, 335, 307, 217, 1179, 642, 1176, 631, 210, 218, 214, 211} \[ \int \frac {x^{11/2}}{\left (a+c x^4\right )^2} \, dx=-\frac {5 \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{3/8} c^{13/8}}+\frac {5 \arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt {2} (-a)^{3/8} c^{13/8}}-\frac {5 \arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{3/8} c^{13/8}}-\frac {5 \text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{3/8} c^{13/8}}-\frac {5 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{3/8} c^{13/8}}+\frac {5 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt {2} (-a)^{3/8} c^{13/8}}-\frac {x^{5/2}}{4 c \left (a+c x^4\right )} \]
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Rule 210
Rule 211
Rule 214
Rule 217
Rule 218
Rule 294
Rule 307
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{5/2}}{4 c \left (a+c x^4\right )}+\frac {5 \int \frac {x^{3/2}}{a+c x^4} \, dx}{8 c} \\ & = -\frac {x^{5/2}}{4 c \left (a+c x^4\right )}+\frac {5 \text {Subst}\left (\int \frac {x^4}{a+c x^8} \, dx,x,\sqrt {x}\right )}{4 c} \\ & = -\frac {x^{5/2}}{4 c \left (a+c x^4\right )}-\frac {5 \text {Subst}\left (\int \frac {1}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{8 c^{3/2}}+\frac {5 \text {Subst}\left (\int \frac {1}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{8 c^{3/2}} \\ & = -\frac {x^{5/2}}{4 c \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 \sqrt [4]{-a} c^{3/2}}-\frac {5 \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{16 \sqrt [4]{-a} c^{3/2}}+\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 \sqrt [4]{-a} c^{3/2}}+\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 \sqrt [4]{-a} c^{3/2}} \\ & = -\frac {x^{5/2}}{4 c \left (a+c x^4\right )}-\frac {5 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{3/8} c^{13/8}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{3/8} c^{13/8}}+\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 \sqrt [4]{-a} c^{7/4}}+\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 \sqrt [4]{-a} c^{7/4}}-\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)^{3/8} c^{13/8}}-\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)^{3/8} c^{13/8}} \\ & = -\frac {x^{5/2}}{4 c \left (a+c x^4\right )}-\frac {5 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{3/8} c^{13/8}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{3/8} c^{13/8}}-\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)^{3/8} c^{13/8}}+\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)^{3/8} c^{13/8}}+\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)^{3/8} c^{13/8}}-\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)^{3/8} c^{13/8}} \\ & = -\frac {x^{5/2}}{4 c \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)^{3/8} c^{13/8}}+\frac {5 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 \sqrt {2} (-a)^{3/8} c^{13/8}}-\frac {5 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{3/8} c^{13/8}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{3/8} c^{13/8}}-\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)^{3/8} c^{13/8}}+\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)^{3/8} c^{13/8}} \\ \end{align*}
Time = 1.15 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.90 \[ \int \frac {x^{11/2}}{\left (a+c x^4\right )^2} \, dx=\frac {-\frac {8 c^{5/8} x^{5/2}}{a+c x^4}+\frac {5 \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{a^{3/8}}-\frac {5 \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )}{a^{3/8}}-\frac {5 \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{a^{3/8}}+\frac {5 \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{a^{3/8}}}{32 c^{13/8}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.15
method | result | size |
derivativedivides | \(-\frac {x^{\frac {5}{2}}}{4 c \left (x^{4} c +a \right )}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{32 c^{2}}\) | \(47\) |
default | \(-\frac {x^{\frac {5}{2}}}{4 c \left (x^{4} c +a \right )}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{32 c^{2}}\) | \(47\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.38 \[ \int \frac {x^{11/2}}{\left (a+c x^4\right )^2} \, dx=-\frac {5 \, \sqrt {2} {\left (-\left (i + 1\right ) \, c^{2} x^{4} - \left (i + 1\right ) \, a c\right )} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{2} c^{8} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {5}{8}} + \sqrt {x}\right ) + 5 \, \sqrt {2} {\left (\left (i - 1\right ) \, c^{2} x^{4} + \left (i - 1\right ) \, a c\right )} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{2} c^{8} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {5}{8}} + \sqrt {x}\right ) + 5 \, \sqrt {2} {\left (-\left (i - 1\right ) \, c^{2} x^{4} - \left (i - 1\right ) \, a c\right )} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{2} c^{8} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {5}{8}} + \sqrt {x}\right ) + 5 \, \sqrt {2} {\left (\left (i + 1\right ) \, c^{2} x^{4} + \left (i + 1\right ) \, a c\right )} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{2} c^{8} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {5}{8}} + \sqrt {x}\right ) + 16 \, x^{\frac {5}{2}} + 10 \, {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {1}{8}} \log \left (a^{2} c^{8} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {5}{8}} + \sqrt {x}\right ) + 10 \, {\left (i \, c^{2} x^{4} + i \, a c\right )} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {1}{8}} \log \left (i \, a^{2} c^{8} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {5}{8}} + \sqrt {x}\right ) + 10 \, {\left (-i \, c^{2} x^{4} - i \, a c\right )} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {1}{8}} \log \left (-i \, a^{2} c^{8} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {5}{8}} + \sqrt {x}\right ) - 10 \, {\left (c^{2} x^{4} + a c\right )} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {1}{8}} \log \left (-a^{2} c^{8} \left (-\frac {1}{a^{3} c^{13}}\right )^{\frac {5}{8}} + \sqrt {x}\right )}{64 \, {\left (c^{2} x^{4} + a c\right )}} \]
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Timed out. \[ \int \frac {x^{11/2}}{\left (a+c x^4\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {x^{11/2}}{\left (a+c x^4\right )^2} \, dx=\int { \frac {x^{\frac {11}{2}}}{{\left (c x^{4} + a\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (207) = 414\).
Time = 0.46 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.58 \[ \int \frac {x^{11/2}}{\left (a+c x^4\right )^2} \, dx=-\frac {x^{\frac {5}{2}}}{4 \, {\left (c x^{4} + a\right )} c} - \frac {5 \, \left (\frac {a}{c}\right )^{\frac {5}{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 c \sqrt {2 \, \sqrt {2} + 4}} - \frac {5 \, \left (\frac {a}{c}\right )^{\frac {5}{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 c \sqrt {2 \, \sqrt {2} + 4}} + \frac {5 \, \left (\frac {a}{c}\right )^{\frac {5}{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 c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {5 \, \left (\frac {a}{c}\right )^{\frac {5}{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 c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {5 \, \left (\frac {a}{c}\right )^{\frac {5}{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 c \sqrt {2 \, \sqrt {2} + 4}} + \frac {5 \, \left (\frac {a}{c}\right )^{\frac {5}{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 c \sqrt {2 \, \sqrt {2} + 4}} + \frac {5 \, \left (\frac {a}{c}\right )^{\frac {5}{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 c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {5 \, \left (\frac {a}{c}\right )^{\frac {5}{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 c \sqrt {-2 \, \sqrt {2} + 4}} \]
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Time = 0.15 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.44 \[ \int \frac {x^{11/2}}{\left (a+c x^4\right )^2} \, dx=-\frac {x^{5/2}}{4\,c\,\left (c\,x^4+a\right )}-\frac {5\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{16\,{\left (-a\right )}^{3/8}\,c^{13/8}}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,5{}\mathrm {i}}{16\,{\left (-a\right )}^{3/8}\,c^{13/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {5}{32}+\frac {5}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{3/8}\,c^{13/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {5}{32}-\frac {5}{32}{}\mathrm {i}\right )}{{\left (-a\right )}^{3/8}\,c^{13/8}} \]
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