Optimal. Leaf size=32 \[ \frac {a^2 x^2}{2}+\frac {4}{5} a b x^{5/2}+\frac {b^2 x^3}{3} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {a^2 x^2}{2}+\frac {4}{5} a b x^{5/2}+\frac {b^2 x^3}{3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \left (a+b \sqrt {x}\right )^2 x \, dx &=2 \operatorname {Subst}\left (\int x^3 (a+b x)^2 \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (a^2 x^3+2 a b x^4+b^2 x^5\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {a^2 x^2}{2}+\frac {4}{5} a b x^{5/2}+\frac {b^2 x^3}{3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.88 \[ \frac {1}{30} x^2 \left (15 a^2+24 a b \sqrt {x}+10 b^2 x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 24, normalized size = 0.75 \[ \frac {1}{3} \, b^{2} x^{3} + \frac {4}{5} \, a b x^{\frac {5}{2}} + \frac {1}{2} \, a^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 24, normalized size = 0.75 \[ \frac {1}{3} \, b^{2} x^{3} + \frac {4}{5} \, a b x^{\frac {5}{2}} + \frac {1}{2} \, a^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.78 \[ \frac {b^{2} x^{3}}{3}+\frac {4 a b \,x^{\frac {5}{2}}}{5}+\frac {a^{2} x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.90, size = 64, normalized size = 2.00 \[ \frac {{\left (b \sqrt {x} + a\right )}^{6}}{3 \, b^{4}} - \frac {6 \, {\left (b \sqrt {x} + a\right )}^{5} a}{5 \, b^{4}} + \frac {3 \, {\left (b \sqrt {x} + a\right )}^{4} a^{2}}{2 \, b^{4}} - \frac {2 \, {\left (b \sqrt {x} + a\right )}^{3} a^{3}}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 24, normalized size = 0.75 \[ \frac {a^2\,x^2}{2}+\frac {b^2\,x^3}{3}+\frac {4\,a\,b\,x^{5/2}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 27, normalized size = 0.84 \[ \frac {a^{2} x^{2}}{2} + \frac {4 a b x^{\frac {5}{2}}}{5} + \frac {b^{2} x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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