Optimal. Leaf size=48 \[ \frac {2 x^{7/2} \left (a+\frac {b}{x}\right )^{5/2}}{7 a}-\frac {4 b x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}}{35 a^2} \]
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Rubi [A] time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac {2 x^{7/2} \left (a+\frac {b}{x}\right )^{5/2}}{7 a}-\frac {4 b x^{5/2} \left (a+\frac {b}{x}\right )^{5/2}}{35 a^2} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^{3/2} x^{5/2} \, dx &=\frac {2 \left (a+\frac {b}{x}\right )^{5/2} x^{7/2}}{7 a}-\frac {(2 b) \int \left (a+\frac {b}{x}\right )^{3/2} x^{3/2} \, dx}{7 a}\\ &=-\frac {4 b \left (a+\frac {b}{x}\right )^{5/2} x^{5/2}}{35 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{5/2} x^{7/2}}{7 a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 0.83 \[ \frac {2}{35} x^{5/2} \sqrt {a+\frac {b}{x}} \left (\frac {b}{a x}+1\right )^2 (5 a x-2 b) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 48, normalized size = 1.00 \[ \frac {2 \, {\left (5 \, a^{3} x^{3} + 8 \, a^{2} b x^{2} + a b^{2} x - 2 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{35 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 90, normalized size = 1.88 \[ \frac {2}{15} \, b {\left (\frac {2 \, b^{\frac {5}{2}}}{a^{2}} + \frac {3 \, {\left (a x + b\right )}^{\frac {5}{2}} - 5 \, {\left (a x + b\right )}^{\frac {3}{2}} b}{a^{2}}\right )} \mathrm {sgn}\relax (x) - \frac {2}{105} \, a {\left (\frac {8 \, b^{\frac {7}{2}}}{a^{3}} - \frac {15 \, {\left (a x + b\right )}^{\frac {7}{2}} - 42 \, {\left (a x + b\right )}^{\frac {5}{2}} b + 35 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{2}}{a^{3}}\right )} \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 33, normalized size = 0.69 \[ \frac {2 \left (a x +b \right ) \left (5 a x -2 b \right ) \left (\frac {a x +b}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}}}{35 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 35, normalized size = 0.73 \[ \frac {2 \, {\left (5 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} x^{\frac {7}{2}} - 7 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b x^{\frac {5}{2}}\right )}}{35 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 45, normalized size = 0.94 \[ \sqrt {a+\frac {b}{x}}\,\left (\frac {2\,a\,x^{7/2}}{7}+\frac {16\,b\,x^{5/2}}{35}+\frac {2\,b^2\,x^{3/2}}{35\,a}-\frac {4\,b^3\,\sqrt {x}}{35\,a^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 27.92, size = 88, normalized size = 1.83 \[ \frac {2 a \sqrt {b} x^{3} \sqrt {\frac {a x}{b} + 1}}{7} + \frac {16 b^{\frac {3}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{35} + \frac {2 b^{\frac {5}{2}} x \sqrt {\frac {a x}{b} + 1}}{35 a} - \frac {4 b^{\frac {7}{2}} \sqrt {\frac {a x}{b} + 1}}{35 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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