Optimal. Leaf size=34 \[ -\frac {2 b^2}{5 x^{5/2}}-\frac {4 a b}{3 x^{3/2}}-\frac {2 a^2}{\sqrt {x}} \]
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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 a^2}{\sqrt {x}}-\frac {4 a b}{3 x^{3/2}}-\frac {2 b^2}{5 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 269
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^2}{x^{3/2}} \, dx &=\int \frac {(b+a x)^2}{x^{7/2}} \, dx\\ &=\int \left (\frac {b^2}{x^{7/2}}+\frac {2 a b}{x^{5/2}}+\frac {a^2}{x^{3/2}}\right ) \, dx\\ &=-\frac {2 b^2}{5 x^{5/2}}-\frac {4 a b}{3 x^{3/2}}-\frac {2 a^2}{\sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.82 \begin {gather*} -\frac {2 \left (3 b^2+10 a b x+15 a^2 x^2\right )}{15 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 25, normalized size = 0.74
method | result | size |
gosper | \(-\frac {2 \left (15 a^{2} x^{2}+10 a b x +3 b^{2}\right )}{15 x^{\frac {5}{2}}}\) | \(25\) |
derivativedivides | \(-\frac {2 b^{2}}{5 x^{\frac {5}{2}}}-\frac {4 a b}{3 x^{\frac {3}{2}}}-\frac {2 a^{2}}{\sqrt {x}}\) | \(25\) |
default | \(-\frac {2 b^{2}}{5 x^{\frac {5}{2}}}-\frac {4 a b}{3 x^{\frac {3}{2}}}-\frac {2 a^{2}}{\sqrt {x}}\) | \(25\) |
trager | \(-\frac {2 \left (15 a^{2} x^{2}+10 a b x +3 b^{2}\right )}{15 x^{\frac {5}{2}}}\) | \(25\) |
risch | \(-\frac {2 \left (15 a^{2} x^{2}+10 a b x +3 b^{2}\right )}{15 x^{\frac {5}{2}}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 24, normalized size = 0.71 \begin {gather*} -\frac {2 \, a^{2}}{\sqrt {x}} - \frac {4 \, a b}{3 \, x^{\frac {3}{2}}} - \frac {2 \, b^{2}}{5 \, 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 \, {\left (15 \, a^{2} x^{2} + 10 \, a b x + 3 \, b^{2}\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 34, normalized size = 1.00 \begin {gather*} - \frac {2 a^{2}}{\sqrt {x}} - \frac {4 a b}{3 x^{\frac {3}{2}}} - \frac {2 b^{2}}{5 x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 24, normalized size = 0.71 \begin {gather*} -\frac {2 \, {\left (15 \, a^{2} x^{2} + 10 \, a b x + 3 \, b^{2}\right )}}{15 \, x^{\frac {5}{2}}} \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 = 0.71 \begin {gather*} -\frac {30\,a^2\,x^2+20\,a\,b\,x+6\,b^2}{15\,x^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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