Integrand size = 23, antiderivative size = 173 \[ \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx=-\frac {\sqrt {-\left (\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}\right )} \arctan \left (\frac {\sqrt {-\sqrt {a} \sqrt {b}-b} x \sqrt {-x+x^4}}{\sqrt {b} (-1+x) \left (1+x+x^2\right )}\right )}{3 \sqrt {a} b}+\frac {\sqrt {\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b}} \arctan \left (\frac {\sqrt {\sqrt {a} \sqrt {b}-b} x \sqrt {-x+x^4}}{\sqrt {b} (-1+x) \left (1+x+x^2\right )}\right )}{3 \sqrt {a} b} \]
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Time = 0.17 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2081, 1507, 1505, 1189, 399, 223, 212, 385, 211, 214} \[ \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx=\frac {\sqrt {x^4-x} \sqrt {\sqrt {a}-\sqrt {b}} \arctan \left (\frac {x^{3/2} \sqrt {\sqrt {a}-\sqrt {b}}}{\sqrt [4]{b} \sqrt {x^3-1}}\right )}{3 \sqrt {a} b^{3/4} \sqrt {x} \sqrt {x^3-1}}+\frac {\sqrt {x^4-x} \sqrt {\sqrt {a}+\sqrt {b}} \text {arctanh}\left (\frac {x^{3/2} \sqrt {\sqrt {a}+\sqrt {b}}}{\sqrt [4]{b} \sqrt {x^3-1}}\right )}{3 \sqrt {a} b^{3/4} \sqrt {x} \sqrt {x^3-1}} \]
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Rule 211
Rule 212
Rule 214
Rule 223
Rule 385
Rule 399
Rule 1189
Rule 1505
Rule 1507
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-x+x^4} \int \frac {\sqrt {x} \sqrt {-1+x^3}}{-b+a x^6} \, dx}{\sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {\left (2 \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {-1+x^6}}{-b+a x^{12}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {\left (2 \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{-b+a x^4} \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {-1+x^3}} \\ & = -\frac {\left (\sqrt {a} \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\sqrt {a} \sqrt {b}-a x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {b} \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (\sqrt {a} \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\sqrt {a} \sqrt {b}+a x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {b} \sqrt {x} \sqrt {-1+x^3}} \\ & = -\frac {\left (\sqrt {a} \left (-1+\frac {\sqrt {b}}{\sqrt {a}}\right ) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {a} \sqrt {b}-a x^2\right )} \, dx,x,x^{3/2}\right )}{3 \sqrt {b} \sqrt {x} \sqrt {-1+x^3}}+\frac {\left (\sqrt {a} \left (1+\frac {\sqrt {b}}{\sqrt {a}}\right ) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {a} \sqrt {b}+a x^2\right )} \, dx,x,x^{3/2}\right )}{3 \sqrt {b} \sqrt {x} \sqrt {-1+x^3}} \\ & = -\frac {\left (\sqrt {a} \left (-1+\frac {\sqrt {b}}{\sqrt {a}}\right ) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}-\left (-a+\sqrt {a} \sqrt {b}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 \sqrt {b} \sqrt {x} \sqrt {-1+x^3}}+\frac {\left (\sqrt {a} \left (1+\frac {\sqrt {b}}{\sqrt {a}}\right ) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}-\left (a+\sqrt {a} \sqrt {b}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 \sqrt {b} \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {\sqrt {\sqrt {a}-\sqrt {b}} \sqrt {-x+x^4} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} x^{3/2}}{\sqrt [4]{b} \sqrt {-1+x^3}}\right )}{3 \sqrt {a} b^{3/4} \sqrt {x} \sqrt {-1+x^3}}+\frac {\sqrt {\sqrt {a}+\sqrt {b}} \sqrt {-x+x^4} \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} x^{3/2}}{\sqrt [4]{b} \sqrt {-1+x^3}}\right )}{3 \sqrt {a} b^{3/4} \sqrt {x} \sqrt {-1+x^3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.22 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx=\frac {\sqrt {x \left (-1+x^3\right )} \text {RootSum}\left [16 a-16 b+32 a \text {$\#$1}-32 b \text {$\#$1}+24 a \text {$\#$1}^2-16 b \text {$\#$1}^2+8 a \text {$\#$1}^3+a \text {$\#$1}^4\&,\frac {\log \left (-2+2 x^3+2 x^{3/2} \sqrt {-1+x^3}-\text {$\#$1}\right ) \text {$\#$1}^2}{8 a-8 b+12 a \text {$\#$1}-8 b \text {$\#$1}+6 a \text {$\#$1}^2+a \text {$\#$1}^3}\&\right ]}{3 \sqrt {x} \sqrt {-1+x^3}} \]
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Time = 1.34 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.61
method | result | size |
default | \(-\frac {-\frac {\left (\sqrt {a b}+b \right ) \operatorname {arctanh}\left (\frac {b \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {\left (\sqrt {a b}-b \right ) \arctan \left (\frac {b \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}}{3 \sqrt {a b}}\) | \(106\) |
pseudoelliptic | \(-\frac {-\frac {\left (\sqrt {a b}+b \right ) \operatorname {arctanh}\left (\frac {b \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}+\frac {\left (\sqrt {a b}-b \right ) \arctan \left (\frac {b \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}}{3 \sqrt {a b}}\) | \(106\) |
elliptic | \(-\frac {\sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}-b \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+1\right ) \left (-1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{5}+\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (1-i \sqrt {3}\right ) \sqrt {\frac {x \left (i \sqrt {3}-3\right )}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1-i \sqrt {3}\right )}}\, \sqrt {\frac {-i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\frac {\underline {\hspace {1.25 ex}}\alpha ^{5} a \operatorname {EllipticPi}\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {i \underline {\hspace {1.25 ex}}\alpha ^{5} \sqrt {3}\, a -3 \underline {\hspace {1.25 ex}}\alpha ^{5} a -i \sqrt {3}\, b +3 b}{6 b}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{b}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{4} \left (a -b \right ) \left (i \sqrt {3}-3\right ) \sqrt {x \left (-1+x \right ) \left (i \sqrt {3}+2 x +1\right ) \left (-i \sqrt {3}+2 x +1\right )}}\right )}{3}\) | \(365\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1083 vs. \(2 (129) = 258\).
Time = 0.61 (sec) , antiderivative size = 1083, normalized size of antiderivative = 6.26 \[ \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx=\int \frac {\sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{a x^{6} - b}\, dx \]
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\[ \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx=\int { \frac {\sqrt {x^{4} - x}}{a x^{6} - b} \,d x } \]
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Time = 0.38 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx=\frac {{\left (4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b\right )} {\left | b \right |} \arctan \left (\frac {\sqrt {-\frac {1}{x^{3}} + 1}}{\sqrt {-\frac {b + \sqrt {{\left (a - b\right )} b + b^{2}}}{b}}}\right )}{3 \, {\left (4 \, a^{2} b^{3} + 5 \, a b^{4}\right )}} - \frac {{\left (4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b\right )} {\left | b \right |} \arctan \left (\frac {\sqrt {-\frac {1}{x^{3}} + 1}}{\sqrt {-\frac {b - \sqrt {{\left (a - b\right )} b + b^{2}}}{b}}}\right )}{3 \, {\left (4 \, a^{2} b^{3} + 5 \, a b^{4}\right )}} \]
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Timed out. \[ \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx=-\int \frac {\sqrt {x^4-x}}{b-a\,x^6} \,d x \]
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