Optimal. Leaf size=86 \[ \frac {2 \sqrt {a-b} \text {ArcTan}\left (\frac {\sqrt {a-b} x \sqrt {-x+x^4}}{\sqrt {b} (-1+x) \left (1+x+x^2\right )}\right )}{3 a \sqrt {b}}+\frac {2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {-x+x^4}}\right )}{3 a} \]
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Rubi [A]
time = 0.09, 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 = {2081, 477, 476,
399, 223, 212, 385, 211} \begin {gather*} \frac {2 \sqrt {x^4-x} \sqrt {a-b} \text {ArcTan}\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 211
Rule 212
Rule 223
Rule 385
Rule 399
Rule 476
Rule 477
Rule 2081
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [A]
time = 0.22, size = 111, normalized size = 1.29 \begin {gather*} \frac {2 \sqrt {x} \sqrt {-1+x^3} \left (\sqrt {a-b} \text {ArcTan}\left (\frac {b-a x^{3/2} \left (x^{3/2}+\sqrt {-1+x^3}\right )}{\sqrt {a-b} \sqrt {b}}\right )+\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {-1+x^3}}{x^{3/2}}\right )\right )}{3 a \sqrt {b} \sqrt {x \left (-1+x^3\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.57, size = 636, normalized size = 7.40 Too large to display
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.57, 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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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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Giac [A]
time = 0.43, 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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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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