Optimal. Leaf size=39 \[ \frac {2}{5} x^{5/2} (A c+b B)+\frac {2}{3} A b x^{3/2}+\frac {2}{7} B c x^{7/2} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} \frac {2}{5} x^{5/2} (A c+b B)+\frac {2}{3} A b x^{3/2}+\frac {2}{7} B c x^{7/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )}{\sqrt {x}} \, dx &=\int \left (A b \sqrt {x}+(b B+A c) x^{3/2}+B c x^{5/2}\right ) \, dx\\ &=\frac {2}{3} A b x^{3/2}+\frac {2}{5} (b B+A c) x^{5/2}+\frac {2}{7} B c x^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.85 \begin {gather*} \frac {2}{105} x^{3/2} (7 A (5 b+3 c x)+3 B x (7 b+5 c x)) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 41, normalized size = 1.05 \begin {gather*} \frac {2}{105} \left (35 A b x^{3/2}+21 A c x^{5/2}+21 b B x^{5/2}+15 B c x^{7/2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 30, normalized size = 0.77 \begin {gather*} \frac {2}{105} \, {\left (15 \, B c x^{3} + 35 \, A b x + 21 \, {\left (B b + A c\right )} x^{2}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 29, normalized size = 0.74 \begin {gather*} \frac {2}{7} \, B c x^{\frac {7}{2}} + \frac {2}{5} \, B b x^{\frac {5}{2}} + \frac {2}{5} \, A c x^{\frac {5}{2}} + \frac {2}{3} \, A b x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 28, normalized size = 0.72 \begin {gather*} \frac {2 \left (15 B c \,x^{2}+21 A c x +21 B b x +35 A b \right ) x^{\frac {3}{2}}}{105} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 27, normalized size = 0.69 \begin {gather*} \frac {2}{7} \, B c x^{\frac {7}{2}} + \frac {2}{3} \, A b x^{\frac {3}{2}} + \frac {2}{5} \, {\left (B b + A c\right )} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 27, normalized size = 0.69 \begin {gather*} \frac {2\,x^{3/2}\,\left (35\,A\,b+21\,A\,c\,x+21\,B\,b\,x+15\,B\,c\,x^2\right )}{105} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 46, normalized size = 1.18 \begin {gather*} \frac {2 A b x^{\frac {3}{2}}}{3} + \frac {2 A c x^{\frac {5}{2}}}{5} + \frac {2 B b x^{\frac {5}{2}}}{5} + \frac {2 B c x^{\frac {7}{2}}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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