Optimal. Leaf size=52 \[ -\frac {4 b \left (b x+c x^2\right )^{3/2}}{15 c^2 x^{3/2}}+\frac {2 \left (b x+c x^2\right )^{3/2}}{5 c \sqrt {x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {670, 662}
\begin {gather*} \frac {2 \left (b x+c x^2\right )^{3/2}}{5 c \sqrt {x}}-\frac {4 b \left (b x+c x^2\right )^{3/2}}{15 c^2 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rule 670
Rubi steps
\begin {align*} \int \sqrt {x} \sqrt {b x+c x^2} \, dx &=\frac {2 \left (b x+c x^2\right )^{3/2}}{5 c \sqrt {x}}-\frac {(2 b) \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx}{5 c}\\ &=-\frac {4 b \left (b x+c x^2\right )^{3/2}}{15 c^2 x^{3/2}}+\frac {2 \left (b x+c x^2\right )^{3/2}}{5 c \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 41, normalized size = 0.79 \begin {gather*} \frac {2 \sqrt {x (b+c x)} \left (-2 b^2+b c x+3 c^2 x^2\right )}{15 c^2 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 31, normalized size = 0.60
method | result | size |
default | \(-\frac {2 \left (c x +b \right ) \left (-3 c x +2 b \right ) \sqrt {x \left (c x +b \right )}}{15 c^{2} \sqrt {x}}\) | \(31\) |
gosper | \(-\frac {2 \left (c x +b \right ) \left (-3 c x +2 b \right ) \sqrt {c \,x^{2}+b x}}{15 c^{2} \sqrt {x}}\) | \(33\) |
risch | \(-\frac {2 \left (c x +b \right ) \sqrt {x}\, \left (-3 c^{2} x^{2}-b c x +2 b^{2}\right )}{15 \sqrt {x \left (c x +b \right )}\, c^{2}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 30, normalized size = 0.58 \begin {gather*} \frac {2 \, {\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} \sqrt {c x + b}}{15 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.67, size = 37, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} \sqrt {c x^{2} + b x}}{15 \, c^{2} \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 \sqrt {x} \sqrt {x \left (b + c x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 34, normalized size = 0.65 \begin {gather*} \frac {4 \, b^{\frac {5}{2}}}{15 \, c^{2}} + \frac {2 \, {\left (3 \, {\left (c x + b\right )}^{\frac {5}{2}} - 5 \, {\left (c x + b\right )}^{\frac {3}{2}} b\right )}}{15 \, c^{2}} \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 = -0.02 \begin {gather*} \int \sqrt {x}\,\sqrt {c\,x^2+b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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