Optimal. Leaf size=173 \[ \frac {\sqrt {\sqrt {b} \left (\sqrt {a}-\sqrt {b}\right )} \tan ^{-1}\left (\frac {x \sqrt {x^4-x} \sqrt {\sqrt {a} \sqrt {b}-b}}{\sqrt {b} (x-1) \left (x^2+x+1\right )}\right )}{3 \sqrt {a} b}-\frac {\sqrt {-\left (\sqrt {b} \left (\sqrt {a}+\sqrt {b}\right )\right )} \tan ^{-1}\left (\frac {x \sqrt {x^4-x} \sqrt {-\sqrt {a} \sqrt {b}-b}}{\sqrt {b} (x-1) \left (x^2+x+1\right )}\right )}{3 \sqrt {a} b} \]
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Rubi [A] time = 0.28, antiderivative size = 185, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2056, 1493, 1491, 1175, 402, 217, 206, 377, 205, 208} \begin {gather*} \frac {\sqrt {x^4-x} \sqrt {\sqrt {a}-\sqrt {b}} \tan ^{-1}\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}} \tanh ^{-1}\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 206
Rule 208
Rule 217
Rule 377
Rule 402
Rule 1175
Rule 1491
Rule 1493
Rule 2056
Rubi steps
\begin {align*} \int \frac {\sqrt {-x+x^4}}{-b+a x^6} \, dx &=\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \tan ^{-1}\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} \tanh ^{-1}\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*}
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Mathematica [C] time = 0.26, size = 280, normalized size = 1.62 \begin {gather*} \frac {\sqrt {x} \sqrt {x^3-1} \left (\sqrt {\sqrt {a}-\sqrt {b}} \left (\tan ^{-1}\left (\frac {\sqrt [4]{a}-\sqrt [4]{b} x^{3/2}}{\sqrt {x^3-1} \sqrt {\sqrt {a}-\sqrt {b}}}\right )-\tan ^{-1}\left (\frac {\sqrt [4]{a}+\sqrt [4]{b} x^{3/2}}{\sqrt {x^3-1} \sqrt {\sqrt {a}-\sqrt {b}}}\right )\right )-\sqrt {\sqrt {a}+\sqrt {b}} \tanh ^{-1}\left (\frac {-\sqrt [4]{b} x^{3/2}+i \sqrt [4]{a}}{\sqrt {x^3-1} \sqrt {\sqrt {a}+\sqrt {b}}}\right )+\sqrt {\sqrt {a}+\sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{b} x^{3/2}+i \sqrt [4]{a}}{\sqrt {x^3-1} \sqrt {\sqrt {a}+\sqrt {b}}}\right )\right )}{6 \sqrt {a} b^{3/4} \sqrt {x \left (x^3-1\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.79, size = 173, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {-\left (\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b}\right )} \tan ^{-1}\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}} \tan ^{-1}\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} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.07, size = 1083, normalized size = 6.26
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 207, normalized size = 1.20 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.54, size = 365, normalized size = 2.11
method | result | size |
default | \(-\frac {\sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{6}-b \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+1\right ) \left (-1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{5}+\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (1-i \sqrt {3}\right ) \sqrt {\frac {x \left (-3+i \sqrt {3}\right )}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1-i \sqrt {3}\right )}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\frac {\underline {\hspace {1.25 ex}}\alpha ^{5} a \EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {i \underline {\hspace {1.25 ex}}\alpha ^{5} \sqrt {3}\, a -3 \underline {\hspace {1.25 ex}}\alpha ^{5} a -i \sqrt {3}\, b +3 b}{6 b}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{b}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{4} \left (a -b \right ) \left (-3+i \sqrt {3}\right ) \sqrt {x \left (-1+x \right ) \left (i \sqrt {3}+2 x +1\right ) \left (1+2 x -i \sqrt {3}\right )}}\right )}{3}\) | \(365\) |
elliptic | \(-\frac {\sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{6}-b \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+1\right ) \left (-1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{5}+\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (1-i \sqrt {3}\right ) \sqrt {\frac {x \left (-3+i \sqrt {3}\right )}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1-i \sqrt {3}\right )}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\frac {\underline {\hspace {1.25 ex}}\alpha ^{5} a \EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {i \underline {\hspace {1.25 ex}}\alpha ^{5} \sqrt {3}\, a -3 \underline {\hspace {1.25 ex}}\alpha ^{5} a -i \sqrt {3}\, b +3 b}{6 b}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{b}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{4} \left (a -b \right ) \left (-3+i \sqrt {3}\right ) \sqrt {x \left (-1+x \right ) \left (i \sqrt {3}+2 x +1\right ) \left (1+2 x -i \sqrt {3}\right )}}\right )}{3}\) | \(365\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} - x}}{a x^{6} - b}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\sqrt {x^4-x}}{b-a\,x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{a x^{6} - b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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