Optimal. Leaf size=34 \[ 2 b^2 \sqrt {x}+\frac {4}{3} a b x^{3/2}+\frac {2}{5} a^2 x^{5/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {269, 45}
\begin {gather*} \frac {2}{5} a^2 x^{5/2}+\frac {4}{3} a b x^{3/2}+2 b^2 \sqrt {x} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 269
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^2 x^{3/2} \, dx &=\int \frac {(b+a x)^2}{\sqrt {x}} \, dx\\ &=\int \left (\frac {b^2}{\sqrt {x}}+2 a b \sqrt {x}+a^2 x^{3/2}\right ) \, dx\\ &=2 b^2 \sqrt {x}+\frac {4}{3} a b x^{3/2}+\frac {2}{5} a^2 x^{5/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 0.82 \begin {gather*} \frac {2}{15} \sqrt {x} \left (15 b^2+10 a b x+3 a^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 25, normalized size = 0.74
method | result | size |
trager | \(\left (\frac {2}{5} a^{2} x^{2}+\frac {4}{3} a b x +2 b^{2}\right ) \sqrt {x}\) | \(24\) |
gosper | \(\frac {2 \left (3 a^{2} x^{2}+10 a b x +15 b^{2}\right ) \sqrt {x}}{15}\) | \(25\) |
derivativedivides | \(\frac {4 a b \,x^{\frac {3}{2}}}{3}+\frac {2 a^{2} x^{\frac {5}{2}}}{5}+2 b^{2} \sqrt {x}\) | \(25\) |
default | \(\frac {4 a b \,x^{\frac {3}{2}}}{3}+\frac {2 a^{2} x^{\frac {5}{2}}}{5}+2 b^{2} \sqrt {x}\) | \(25\) |
risch | \(\frac {2 \left (3 a^{2} x^{2}+10 a b x +15 b^{2}\right ) \sqrt {x}}{15}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 26, normalized size = 0.76 \begin {gather*} \frac {2}{15} \, {\left (3 \, a^{2} + \frac {10 \, a b}{x} + \frac {15 \, b^{2}}{x^{2}}\right )} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 24, normalized size = 0.71 \begin {gather*} \frac {2}{15} \, {\left (3 \, a^{2} x^{2} + 10 \, a b x + 15 \, b^{2}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 32, normalized size = 0.94 \begin {gather*} \frac {2 a^{2} x^{\frac {5}{2}}}{5} + \frac {4 a b x^{\frac {3}{2}}}{3} + 2 b^{2} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 24, normalized size = 0.71 \begin {gather*} \frac {2}{5} \, a^{2} x^{\frac {5}{2}} + \frac {4}{3} \, a b x^{\frac {3}{2}} + 2 \, b^{2} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 24, normalized size = 0.71 \begin {gather*} \frac {2\,\sqrt {x}\,\left (3\,a^2\,x^2+10\,a\,b\,x+15\,b^2\right )}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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