Optimal. Leaf size=25 \[ -\frac {2 \sqrt {a x^2+b x^3}}{a x^{3/2}} \]
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Rubi [A]
time = 0.02, 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 = {2039}
\begin {gather*} -\frac {2 \sqrt {a x^2+b x^3}}{a x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
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 = 0.02, size = 23, normalized size = 0.92 \begin {gather*} -\frac {2 \sqrt {x^2 (a+b x)}}{a x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 27, normalized size = 1.08
method | result | size |
risch | \(-\frac {2 \sqrt {x}\, \left (b x +a \right )}{\sqrt {x^{2} \left (b x +a \right )}\, a}\) | \(25\) |
gosper | \(-\frac {2 \sqrt {x}\, \left (b x +a \right )}{a \sqrt {b \,x^{3}+a \,x^{2}}}\) | \(27\) |
default | \(-\frac {2 \sqrt {x}\, \left (b x +a \right )}{a \sqrt {b \,x^{3}+a \,x^{2}}}\) | \(27\) |
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 = 2.86, size = 21, normalized size = 0.84 \begin {gather*} -\frac {2 \, \sqrt {b x^{3} + a x^{2}}}{a x^{\frac {3}{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 {1}{\sqrt {x} \sqrt {x^{2} \left (a + b x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.81, size = 34, normalized size = 1.36 \begin {gather*} \frac {4 \, \sqrt {b}}{{\left ({\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2} - a\right )} \mathrm {sgn}\left (x\right )} \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 = -0.04 \begin {gather*} \int \frac {1}{\sqrt {x}\,\sqrt {b\,x^3+a\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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