Optimal. Leaf size=25 \[ \frac {2 \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \]
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Rubi [A]
time = 0.00, 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 = {662}
\begin {gather*} \frac {2 \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rubi steps
\begin {align*} \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx &=\frac {2 \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 0.92 \begin {gather*} \frac {2 (x (b+c x))^{3/2}}{3 c x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 23, normalized size = 0.92
method | result | size |
default | \(\frac {2 \left (c x +b \right ) \sqrt {x \left (c x +b \right )}}{3 c \sqrt {x}}\) | \(23\) |
gosper | \(\frac {2 \left (c x +b \right ) \sqrt {c \,x^{2}+b x}}{3 c \sqrt {x}}\) | \(25\) |
risch | \(\frac {2 \left (c x +b \right )^{2} \sqrt {x}}{3 \sqrt {x \left (c x +b \right )}\, c}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 12, normalized size = 0.48 \begin {gather*} \frac {2 \, {\left (c x + b\right )}^{\frac {3}{2}}}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.67, size = 24, normalized size = 0.96 \begin {gather*} \frac {2 \, \sqrt {c x^{2} + b x} {\left (c x + b\right )}}{3 \, c \sqrt {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 {\sqrt {x \left (b + c x\right )}}{\sqrt {x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.77, size = 21, normalized size = 0.84 \begin {gather*} \frac {2 \, {\left (c x + b\right )}^{\frac {3}{2}}}{3 \, c} - \frac {2 \, b^{\frac {3}{2}}}{3 \, c} \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 {\sqrt {c\,x^2+b\,x}}{\sqrt {x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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