Optimal. Leaf size=52 \[ \frac {2 \left (b x+c x^2\right )^{3/2}}{5 c \sqrt {x}}-\frac {4 b \left (b x+c x^2\right )^{3/2}}{15 c^2 x^{3/2}} \]
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Rubi [A] time = 0.02, 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 = {656, 648} \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 648
Rule 656
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 = 31, normalized size = 0.60 \begin {gather*} \frac {2 (x (b+c x))^{3/2} (3 c x-2 b)}{15 c^2 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 43, normalized size = 0.83 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \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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fricas [A] time = 0.41, 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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giac [A] time = 0.16, 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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maple [A] time = 0.16, size = 33, normalized size = 0.63 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.49, 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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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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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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