Optimal. Leaf size=86 \[ \frac {2 \sqrt {a-b} \tan ^{-1}\left (\frac {x \sqrt {x^4-x} \sqrt {a-b}}{\sqrt {b} (x-1) \left (x^2+x+1\right )}\right )}{3 a \sqrt {b}}+\frac {2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-x}}\right )}{3 a} \]
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Rubi [A] time = 0.15, antiderivative size = 125, normalized size of antiderivative = 1.45, number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2056, 466, 465, 402, 217, 206, 377, 205} \begin {gather*} \frac {2 \sqrt {x^4-x} \sqrt {a-b} \tan ^{-1}\left (\frac {x^{3/2} \sqrt {a-b}}{\sqrt {b} \sqrt {x^3-1}}\right )}{3 a \sqrt {b} \sqrt {x} \sqrt {x^3-1}}+\frac {2 \sqrt {x^4-x} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {x^3-1}}\right )}{3 a \sqrt {x} \sqrt {x^3-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 206
Rule 217
Rule 377
Rule 402
Rule 465
Rule 466
Rule 2056
Rubi steps
\begin {align*} \int \frac {\sqrt {-x+x^4}}{-b+a x^3} \, dx &=\frac {\sqrt {-x+x^4} \int \frac {\sqrt {x} \sqrt {-1+x^3}}{-b+a x^3} \, 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^6} \, 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^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {\left (2 \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (2 (a-b) \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (-b+a x^2\right )} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {\left (2 \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (2 (a-b) \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{-b-(a-b) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {2 \sqrt {a-b} \sqrt {-x+x^4} \tan ^{-1}\left (\frac {\sqrt {a-b} x^{3/2}}{\sqrt {b} \sqrt {-1+x^3}}\right )}{3 a \sqrt {b} \sqrt {x} \sqrt {-1+x^3}}+\frac {2 \sqrt {-x+x^4} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 52, normalized size = 0.60 \begin {gather*} -\frac {2 x \sqrt {x \left (x^3-1\right )} F_1\left (\frac {1}{2};-\frac {1}{2},1;\frac {3}{2};x^3,\frac {a x^3}{b}\right )}{3 b \sqrt {1-x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.53, size = 86, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a-b} x \sqrt {-x+x^4}}{\sqrt {b} (-1+x) \left (1+x+x^2\right )}\right )}{3 a \sqrt {b}}+\frac {2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {-x+x^4}}\right )}{3 a} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.38, size = 219, normalized size = 2.55 \begin {gather*} \left [\frac {\sqrt {-\frac {a - b}{b}} \log \left (-\frac {{\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} x^{6} + 2 \, {\left (3 \, a b - 4 \, b^{2}\right )} x^{3} + b^{2} + 4 \, {\left ({\left (a b - 2 \, b^{2}\right )} x^{4} + b^{2} x\right )} \sqrt {x^{4} - x} \sqrt {-\frac {a - b}{b}}}{a^{2} x^{6} - 2 \, a b x^{3} + b^{2}}\right ) + 2 \, \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x + 1\right )}{6 \, a}, \frac {\sqrt {\frac {a - b}{b}} \arctan \left (-\frac {2 \, \sqrt {x^{4} - x} b x \sqrt {\frac {a - b}{b}}}{{\left (a - 2 \, b\right )} x^{3} + b}\right ) + \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x + 1\right )}{3 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 80, normalized size = 0.93 \begin {gather*} -\frac {2 \, {\left (a - b\right )} \arctan \left (\frac {b \sqrt {-\frac {1}{x^{3}} + 1}}{\sqrt {a b - b^{2}}}\right )}{3 \, \sqrt {a b - b^{2}} a} + \frac {\log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right )}{3 \, a} - \frac {\log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right )}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 636, normalized size = 7.40
method | result | size |
default | \(\frac {2 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \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 )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \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 )-\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 {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \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 )\right )}{a \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}-\frac {2 \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\left (-1+x \right )^{2} \left (\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 ^{2} 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 ^{2} \sqrt {3}\, a -3 \underline {\hspace {1.25 ex}}\alpha ^{2} 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 \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 a}\) | \(636\) |
elliptic | \(\frac {2 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \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 )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \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 )-\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 {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \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 )\right )}{a \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}-\frac {2 \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\left (-1+x \right )^{2} \left (\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 ^{2} 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 ^{2} \sqrt {3}\, a -3 \underline {\hspace {1.25 ex}}\alpha ^{2} 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 \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 a}\) | \(636\) |
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^{3} - 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^3} \,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^{3} - b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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