Optimal. Leaf size=19 \[ \frac {2 \sqrt {a x^3-b x}}{x} \]
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Rubi [A] time = 0.10, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2036} \begin {gather*} \frac {2 \sqrt {a x^3-b x}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2036
Rubi steps
\begin {align*} \int \frac {b+a x^2}{x \sqrt {-b x+a x^3}} \, dx &=\frac {2 \sqrt {-b x+a x^3}}{x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a x^3-b x}}{x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {-b x+a x^3}}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 17, normalized size = 0.89 \begin {gather*} \frac {2 \, \sqrt {a x^{3} - b x}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{2} + b}{\sqrt {a x^{3} - b x} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 18, normalized size = 0.95
method | result | size |
trager | \(\frac {2 \sqrt {a \,x^{3}-b x}}{x}\) | \(18\) |
gosper | \(\frac {2 a \,x^{2}-2 b}{\sqrt {a \,x^{3}-b x}}\) | \(24\) |
risch | \(\frac {2 a \,x^{2}-2 b}{\sqrt {x \left (a \,x^{2}-b \right )}}\) | \(25\) |
elliptic | \(\frac {2 a \,x^{2}-2 b}{\sqrt {x \left (a \,x^{2}-b \right )}}\) | \(25\) |
default | \(\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{\sqrt {a \,x^{3}-b x}}+b \left (\frac {2 a \,x^{2}-2 b}{b \sqrt {x \left (a \,x^{2}-b \right )}}-\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{b \sqrt {a \,x^{3}-b x}}\right )\) | \(322\) |
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 {a x^{2} + b}{\sqrt {a x^{3} - b x} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 17, normalized size = 0.89 \begin {gather*} \frac {2\,\sqrt {a\,x^3-b\,x}}{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 {a x^{2} + b}{x \sqrt {x \left (a x^{2} - b\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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