Optimal. Leaf size=61 \[ \frac {2 B \left (b x+c x^2\right )^{3/2}}{5 c \sqrt {x}}-\frac {2 \left (b x+c x^2\right )^{3/2} (2 b B-5 A c)}{15 c^2 x^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {794, 648} \begin {gather*} \frac {2 B \left (b x+c x^2\right )^{3/2}}{5 c \sqrt {x}}-\frac {2 \left (b x+c x^2\right )^{3/2} (2 b B-5 A c)}{15 c^2 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {b x+c x^2}}{\sqrt {x}} \, dx &=\frac {2 B \left (b x+c x^2\right )^{3/2}}{5 c \sqrt {x}}+\frac {\left (2 \left (\frac {1}{2} (b B-A c)+\frac {3}{2} (-b B+2 A c)\right )\right ) \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx}{5 c}\\ &=-\frac {2 (2 b B-5 A c) \left (b x+c x^2\right )^{3/2}}{15 c^2 x^{3/2}}+\frac {2 B \left (b x+c x^2\right )^{3/2}}{5 c \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 37, normalized size = 0.61 \begin {gather*} \frac {2 (x (b+c x))^{3/2} (5 A c-2 b B+3 B c x)}{15 c^2 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 39, normalized size = 0.64 \begin {gather*} \frac {2 \left (b x+c x^2\right )^{3/2} (5 A c-2 b B+3 B c x)}{15 c^2 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 53, normalized size = 0.87 \begin {gather*} \frac {2 \, {\left (3 \, B c^{2} x^{2} - 2 \, B b^{2} + 5 \, A b c + {\left (B b c + 5 \, A c^{2}\right )} x\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 = 0.17, size = 60, normalized size = 0.98 \begin {gather*} \frac {2}{15} \, B {\left (\frac {2 \, b^{\frac {5}{2}}}{c^{2}} + \frac {3 \, {\left (c x + b\right )}^{\frac {5}{2}} - 5 \, {\left (c x + b\right )}^{\frac {3}{2}} b}{c^{2}}\right )} + \frac {2}{3} \, A {\left (\frac {{\left (c x + b\right )}^{\frac {3}{2}}}{c} - \frac {b^{\frac {3}{2}}}{c}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 39, normalized size = 0.64 \begin {gather*} \frac {2 \left (c x +b \right ) \left (3 B c x +5 A c -2 b 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 = 0.52, size = 45, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (c x + b\right )}^{\frac {3}{2}} A}{3 \, c} + \frac {2 \, {\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} \sqrt {c x + b} 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 \frac {\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right )}{\sqrt {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 \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{\sqrt {x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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