Optimal. Leaf size=47 \[ \frac {a^2 x^{m+1}}{m+1}+\frac {4 a b x^{m+\frac {3}{2}}}{2 m+3}+\frac {b^2 x^{m+2}}{m+2} \]
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Rubi [A] time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \[ \frac {a^2 x^{m+1}}{m+1}+\frac {4 a b x^{m+\frac {3}{2}}}{2 m+3}+\frac {b^2 x^{m+2}}{m+2} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int \left (a+b \sqrt {x}\right )^2 x^m \, dx &=\int \left (a^2 x^m+2 a b x^{\frac {1}{2}+m}+b^2 x^{1+m}\right ) \, dx\\ &=\frac {a^2 x^{1+m}}{1+m}+\frac {4 a b x^{\frac {3}{2}+m}}{3+2 m}+\frac {b^2 x^{2+m}}{2+m}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 42, normalized size = 0.89 \[ x^{m+1} \left (\frac {a^2}{m+1}+\frac {4 a b \sqrt {x}}{2 m+3}+\frac {b^2 x}{m+2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 89, normalized size = 1.89 \[ \frac {{\left ({\left (2 \, b^{2} m^{2} + 5 \, b^{2} m + 3 \, b^{2}\right )} x^{2} + 4 \, {\left (a b m^{2} + 3 \, a b m + 2 \, a b\right )} x^{\frac {3}{2}} + {\left (2 \, a^{2} m^{2} + 7 \, a^{2} m + 6 \, a^{2}\right )} x\right )} x^{m}}{2 \, m^{3} + 9 \, m^{2} + 13 \, m + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 58, normalized size = 1.23 \[ \frac {b^{2} x^{2} \sqrt {x}^{2 \, m}}{m + 2} + \frac {4 \, a b x^{\frac {3}{2}} \sqrt {x}^{2 \, m}}{2 \, m + 3} + \frac {a^{2} x \sqrt {x}^{2 \, m}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int \left (b \sqrt {x}+a \right )^{2} x^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 45, normalized size = 0.96 \[ \frac {b^{2} x^{m + 2}}{m + 2} + \frac {4 \, a b x^{m + \frac {3}{2}}}{2 \, m + 3} + \frac {a^{2} x^{m + 1}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 103, normalized size = 2.19 \[ x^m\,\left (\frac {b^2\,x^2\,\left (2\,m^2+5\,m+3\right )}{2\,m^3+9\,m^2+13\,m+6}+\frac {a^2\,x\,\left (2\,m^2+7\,m+6\right )}{2\,m^3+9\,m^2+13\,m+6}+\frac {4\,a\,b\,x^{3/2}\,\left (m^2+3\,m+2\right )}{2\,m^3+9\,m^2+13\,m+6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.20, size = 63, normalized size = 1.34 \[ a^{2} \left (\begin {cases} \frac {x^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) + 4 a b \left (\begin {cases} \frac {x^{\frac {3}{2}} x^{m}}{2 m + 3} & \text {for}\: m \neq - \frac {3}{2} \\\log {\left (\sqrt {x} \right )} & \text {otherwise} \end {cases}\right ) + b^{2} \left (\begin {cases} \frac {x^{2} x^{m}}{m + 2} & \text {for}\: m \neq -2 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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