Optimal. Leaf size=36 \[ -\frac {2 a^2}{3 x^{3/2}}-\frac {4 a b}{5 x^{5/2}}-\frac {2 b^2}{7 x^{7/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 36, 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 = {263, 43} \[ -\frac {2 a^2}{3 x^{3/2}}-\frac {4 a b}{5 x^{5/2}}-\frac {2 b^2}{7 x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^2}{x^{5/2}} \, dx &=\int \frac {(b+a x)^2}{x^{9/2}} \, dx\\ &=\int \left (\frac {b^2}{x^{9/2}}+\frac {2 a b}{x^{7/2}}+\frac {a^2}{x^{5/2}}\right ) \, dx\\ &=-\frac {2 b^2}{7 x^{7/2}}-\frac {4 a b}{5 x^{5/2}}-\frac {2 a^2}{3 x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.78 \[ -\frac {2 \left (35 a^2 x^2+42 a b x+15 b^2\right )}{105 x^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 24, normalized size = 0.67 \[ -\frac {2 \, {\left (35 \, a^{2} x^{2} + 42 \, a b x + 15 \, b^{2}\right )}}{105 \, x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 24, normalized size = 0.67 \[ -\frac {2 \, {\left (35 \, a^{2} x^{2} + 42 \, a b x + 15 \, b^{2}\right )}}{105 \, x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.69 \[ -\frac {2 \left (35 a^{2} x^{2}+42 a b x +15 b^{2}\right )}{105 x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 24, normalized size = 0.67 \[ -\frac {2 \, a^{2}}{3 \, x^{\frac {3}{2}}} - \frac {4 \, a b}{5 \, x^{\frac {5}{2}}} - \frac {2 \, b^{2}}{7 \, x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 24, normalized size = 0.67 \[ -\frac {70\,a^2\,x^2+84\,a\,b\,x+30\,b^2}{105\,x^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.48, size = 36, normalized size = 1.00 \[ - \frac {2 a^{2}}{3 x^{\frac {3}{2}}} - \frac {4 a b}{5 x^{\frac {5}{2}}} - \frac {2 b^{2}}{7 x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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