Optimal. Leaf size=74 \[ -\frac {16 b^2}{3 a^3 \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {8 b \sqrt {x}}{3 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{3/2}}{3 a \sqrt {a+\frac {b}{x}}} \]
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Rubi [A]
time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {277, 270}
\begin {gather*} -\frac {16 b^2}{3 a^3 \sqrt {x} \sqrt {a+\frac {b}{x}}}-\frac {8 b \sqrt {x}}{3 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{3/2}}{3 a \sqrt {a+\frac {b}{x}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx &=\frac {2 x^{3/2}}{3 a \sqrt {a+\frac {b}{x}}}-\frac {(4 b) \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \, dx}{3 a}\\ &=-\frac {8 b \sqrt {x}}{3 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{3/2}}{3 a \sqrt {a+\frac {b}{x}}}+\frac {\left (8 b^2\right ) \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^{3/2}} \, dx}{3 a^2}\\ &=-\frac {16 b^2}{3 a^3 \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {8 b \sqrt {x}}{3 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{3/2}}{3 a \sqrt {a+\frac {b}{x}}}\\ \end {align*}
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Mathematica [A]
time = 4.57, size = 48, normalized size = 0.65 \begin {gather*} \frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x} \left (-8 b^2-4 a b x+a^2 x^2\right )}{3 a^3 (b+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 45, normalized size = 0.61
method | result | size |
gosper | \(\frac {2 \left (a x +b \right ) \left (a^{2} x^{2}-4 a b x -8 b^{2}\right )}{3 a^{3} x^{\frac {3}{2}} \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}\) | \(43\) |
default | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (a^{2} x^{2}-4 a b x -8 b^{2}\right )}{3 \left (a x +b \right ) a^{3}}\) | \(45\) |
risch | \(\frac {2 \left (a x -5 b \right ) \left (a x +b \right )}{3 a^{3} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}-\frac {2 b^{2}}{a^{3} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 55, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} x^{\frac {3}{2}} - 6 \, \sqrt {a + \frac {b}{x}} b \sqrt {x}\right )}}{3 \, a^{3}} - \frac {2 \, b^{2}}{\sqrt {a + \frac {b}{x}} a^{3} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 47, normalized size = 0.64 \begin {gather*} \frac {2 \, {\left (a^{2} x^{2} - 4 \, a b x - 8 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{4} x + a^{3} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 206 vs.
\(2 (63) = 126\).
time = 1.11, size = 206, normalized size = 2.78 \begin {gather*} \frac {2 a^{3} b^{\frac {9}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} - \frac {6 a^{2} b^{\frac {11}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} - \frac {24 a b^{\frac {13}{2}} x \sqrt {\frac {a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} - \frac {16 b^{\frac {15}{2}} \sqrt {\frac {a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.57, size = 46, normalized size = 0.62 \begin {gather*} -\frac {2 \, b^{2}}{\sqrt {a x + b} a^{3}} + \frac {2 \, {\left ({\left (a x + b\right )}^{\frac {3}{2}} a^{6} - 6 \, \sqrt {a x + b} a^{6} b\right )}}{3 \, a^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.49, size = 49, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {a+\frac {b}{x}}\,\left (\frac {8\,b\,x^{3/2}}{3\,a^3}-\frac {2\,x^{5/2}}{3\,a^2}+\frac {16\,b^2\,\sqrt {x}}{3\,a^4}\right )}{x+\frac {b}{a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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