Optimal. Leaf size=25 \[ \frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 b x^3} \]
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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 = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2014} \[ \frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 b x^3} \]
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin {align*} \int \frac {\sqrt {a x^2+b x^3}}{x} \, dx &=\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 b x^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.92 \[ \frac {2 \left (x^2 (a+b x)\right )^{3/2}}{3 b x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 26, normalized size = 1.04 \[ \frac {2 \, \sqrt {b x^{3} + a x^{2}} {\left (b x + a\right )}}{3 \, b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 50, normalized size = 2.00 \[ -\frac {2 \, a^{\frac {3}{2}} \mathrm {sgn}\relax (x)}{3 \, b} + \frac {2 \, {\left (3 \, \sqrt {b x + a} a \mathrm {sgn}\relax (x) + {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} \mathrm {sgn}\relax (x)\right )}}{3 \, b} \]
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 \[ \frac {2 \left (b x +a \right ) \sqrt {b \,x^{3}+a \,x^{2}}}{3 b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 12, normalized size = 0.48 \[ \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}}}{3 \, b} \]
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 \[ \int \frac {\sqrt {b\,x^3+a\,x^2}}{x} \,d x \]
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 \[ \int \frac {\sqrt {x^{2} \left (a + b x\right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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