Optimal. Leaf size=38 \[ \frac {2 \left (a+b \sqrt {x}\right )^5}{5 b^2}-\frac {a \left (a+b \sqrt {x}\right )^4}{2 b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac {2 \left (a+b \sqrt {x}\right )^5}{5 b^2}-\frac {a \left (a+b \sqrt {x}\right )^4}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rubi steps
\begin {align*} \int \left (a+b \sqrt {x}\right )^3 \, dx &=2 \operatorname {Subst}\left (\int x (a+b x)^3 \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {a (a+b x)^3}{b}+\frac {(a+b x)^4}{b}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a \left (a+b \sqrt {x}\right )^4}{2 b^2}+\frac {2 \left (a+b \sqrt {x}\right )^5}{5 b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 0.74 \[ -\frac {\left (a-4 b \sqrt {x}\right ) \left (a+b \sqrt {x}\right )^4}{10 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 35, normalized size = 0.92 \[ \frac {3}{2} \, a b^{2} x^{2} + a^{3} x + \frac {2}{5} \, {\left (b^{3} x^{2} + 5 \, a^{2} b x\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 32, normalized size = 0.84 \[ \frac {2}{5} \, b^{3} x^{\frac {5}{2}} + \frac {3}{2} \, a b^{2} x^{2} + 2 \, a^{2} b x^{\frac {3}{2}} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 33, normalized size = 0.87 \[ \frac {2 b^{3} x^{\frac {5}{2}}}{5}+\frac {3 a \,b^{2} x^{2}}{2}+2 a^{2} b \,x^{\frac {3}{2}}+a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 32, normalized size = 0.84 \[ \frac {2}{5} \, b^{3} x^{\frac {5}{2}} + \frac {3}{2} \, a b^{2} x^{2} + 2 \, a^{2} b x^{\frac {3}{2}} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 32, normalized size = 0.84 \[ a^3\,x+\frac {2\,b^3\,x^{5/2}}{5}+\frac {3\,a\,b^2\,x^2}{2}+2\,a^2\,b\,x^{3/2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.75, size = 39, normalized size = 1.03 \[ a^{3} x + 2 a^{2} b x^{\frac {3}{2}} + \frac {3 a b^{2} x^{2}}{2} + \frac {2 b^{3} x^{\frac {5}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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