Optimal. Leaf size=25 \[ -\frac {2 \sqrt {a x^2+b x^3}}{a x^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2014} \[ -\frac {2 \sqrt {a x^2+b x^3}}{a x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^3}} \, dx &=-\frac {2 \sqrt {a x^2+b x^3}}{a x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.92 \[ -\frac {2 \sqrt {x^2 (a+b x)}}{a x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 21, normalized size = 0.84 \[ -\frac {2 \, \sqrt {b x^{3} + a x^{2}}}{a x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 30, normalized size = 1.20 \[ \frac {4 \, \sqrt {b}}{{\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2} - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 27, normalized size = 1.08 \[ -\frac {2 \left (b x +a \right ) \sqrt {x}}{\sqrt {b \,x^{3}+a \,x^{2}}\, a} \]
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 \[ \int \frac {1}{\sqrt {b x^{3} + a x^{2}} \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\sqrt {x}\,\sqrt {b\,x^3+a\,x^2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {x} \sqrt {x^{2} \left (a + b x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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