Optimal. Leaf size=55 \[ \frac {\left (2 b+a x^3\right ) \sqrt {-x+x^4}}{3 x^2}+\frac {1}{3} (-a-2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {-x+x^4}}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.20, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2063, 2029,
2054, 212} \begin {gather*} \frac {1}{3} x \sqrt {x^4-x} (a+2 b)-\frac {1}{3} (a+2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-x}}\right )-\frac {2 b \left (x^4-x\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2029
Rule 2054
Rule 2063
Rubi steps
\begin {align*} \int \frac {\left (-b+a x^3\right ) \sqrt {-x+x^4}}{x^3} \, dx &=-\frac {2 b \left (-x+x^4\right )^{3/2}}{3 x^3}+(a+2 b) \int \sqrt {-x+x^4} \, dx\\ &=\frac {1}{3} (a+2 b) x \sqrt {-x+x^4}-\frac {2 b \left (-x+x^4\right )^{3/2}}{3 x^3}+\frac {1}{2} (-a-2 b) \int \frac {x}{\sqrt {-x+x^4}} \, dx\\ &=\frac {1}{3} (a+2 b) x \sqrt {-x+x^4}-\frac {2 b \left (-x+x^4\right )^{3/2}}{3 x^3}+\frac {1}{3} (-a-2 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {-x+x^4}}\right )\\ &=\frac {1}{3} (a+2 b) x \sqrt {-x+x^4}-\frac {2 b \left (-x+x^4\right )^{3/2}}{3 x^3}-\frac {1}{3} (a+2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {-x+x^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 75, normalized size = 1.36 \begin {gather*} \frac {\sqrt {x \left (-1+x^3\right )} \left (\sqrt {-1+x^3} \left (2 b+a x^3\right )-(a+2 b) x^{3/2} \tanh ^{-1}\left (\frac {\sqrt {-1+x^3}}{x^{3/2}}\right )\right )}{3 x^2 \sqrt {-1+x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.60, size = 609, normalized size = 11.07 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 = 0.36, size = 55, normalized size = 1.00 \begin {gather*} \frac {{\left (a + 2 \, b\right )} x^{2} \log \left (2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x - 1\right ) + 2 \, {\left (a x^{3} + 2 \, b\right )} \sqrt {x^{4} - x}}{6 \, x^{2}} \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 )} \left (a x^{3} - b\right )}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 65, normalized size = 1.18 \begin {gather*} \frac {1}{3} \, \sqrt {x^{4} - x} a x - \frac {1}{6} \, {\left (a + 2 \, b\right )} \log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right ) + \frac {1}{6} \, {\left (a + 2 \, b\right )} \log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right ) + \frac {2}{3} \, b \sqrt {-\frac {1}{x^{3}} + 1} \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 = -0.02 \begin {gather*} -\int \frac {\sqrt {x^4-x}\,\left (b-a\,x^3\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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