Optimal. Leaf size=32 \[ -\frac {2 a^2}{\sqrt {x}}+4 a b \sqrt {x}+\frac {2}{3} b^2 x^{3/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45}
\begin {gather*} -\frac {2 a^2}{\sqrt {x}}+4 a b \sqrt {x}+\frac {2}{3} b^2 x^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{x^{3/2}} \, dx &=\int \left (\frac {a^2}{x^{3/2}}+\frac {2 a b}{\sqrt {x}}+b^2 \sqrt {x}\right ) \, dx\\ &=-\frac {2 a^2}{\sqrt {x}}+4 a b \sqrt {x}+\frac {2}{3} b^2 x^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.88 \begin {gather*} -\frac {2 \left (3 a^2-6 a b x-b^2 x^2\right )}{3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 25, normalized size = 0.78
method | result | size |
gosper | \(-\frac {2 \left (-x^{2} b^{2}-6 a b x +3 a^{2}\right )}{3 \sqrt {x}}\) | \(25\) |
derivativedivides | \(\frac {2 b^{2} x^{\frac {3}{2}}}{3}-\frac {2 a^{2}}{\sqrt {x}}+4 a b \sqrt {x}\) | \(25\) |
default | \(\frac {2 b^{2} x^{\frac {3}{2}}}{3}-\frac {2 a^{2}}{\sqrt {x}}+4 a b \sqrt {x}\) | \(25\) |
trager | \(-\frac {2 \left (-x^{2} b^{2}-6 a b x +3 a^{2}\right )}{3 \sqrt {x}}\) | \(25\) |
risch | \(-\frac {2 \left (-x^{2} b^{2}-6 a b x +3 a^{2}\right )}{3 \sqrt {x}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 24, normalized size = 0.75 \begin {gather*} \frac {2}{3} \, b^{2} x^{\frac {3}{2}} + 4 \, a b \sqrt {x} - \frac {2 \, a^{2}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 23, normalized size = 0.72 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2} + 6 \, a b x - 3 \, a^{2}\right )}}{3 \, \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 31, normalized size = 0.97 \begin {gather*} - \frac {2 a^{2}}{\sqrt {x}} + 4 a b \sqrt {x} + \frac {2 b^{2} x^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 24, normalized size = 0.75 \begin {gather*} \frac {2}{3} \, b^{2} x^{\frac {3}{2}} + 4 \, a b \sqrt {x} - \frac {2 \, a^{2}}{\sqrt {x}} \end {gather*}
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 \begin {gather*} \frac {-6\,a^2+12\,a\,b\,x+2\,b^2\,x^2}{3\,\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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