Optimal. Leaf size=34 \[ -\frac {2 a^2}{\sqrt {x}}+\frac {4}{3} a b x^{3/2}+\frac {2}{7} b^2 x^{7/2} \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \[ -\frac {2 a^2}{\sqrt {x}}+\frac {4}{3} a b x^{3/2}+\frac {2}{7} b^2 x^{7/2} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^{3/2}} \, dx &=\int \left (\frac {a^2}{x^{3/2}}+2 a b \sqrt {x}+b^2 x^{5/2}\right ) \, dx\\ &=-\frac {2 a^2}{\sqrt {x}}+\frac {4}{3} a b x^{3/2}+\frac {2}{7} b^2 x^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.88 \[ \frac {2 \left (-21 a^2+14 a b x^2+3 b^2 x^4\right )}{21 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 26, normalized size = 0.76 \[ \frac {2 \, {\left (3 \, b^{2} x^{4} + 14 \, a b x^{2} - 21 \, a^{2}\right )}}{21 \, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 24, normalized size = 0.71 \[ \frac {2}{7} \, b^{2} x^{\frac {7}{2}} + \frac {4}{3} \, a b x^{\frac {3}{2}} - \frac {2 \, a^{2}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.79 \[ -\frac {2 \left (-3 b^{2} x^{4}-14 a b \,x^{2}+21 a^{2}\right )}{21 \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 24, normalized size = 0.71 \[ \frac {2}{7} \, b^{2} x^{\frac {7}{2}} + \frac {4}{3} \, a b x^{\frac {3}{2}} - \frac {2 \, a^{2}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 26, normalized size = 0.76 \[ \frac {-42\,a^2+28\,a\,b\,x^2+6\,b^2\,x^4}{21\,\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.94, size = 32, normalized size = 0.94 \[ - \frac {2 a^{2}}{\sqrt {x}} + \frac {4 a b x^{\frac {3}{2}}}{3} + \frac {2 b^{2} x^{\frac {7}{2}}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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