Optimal. Leaf size=103 \[ \frac {a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{8 b^{3/2}}-\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b \sqrt {x}}-\frac {a \sqrt {a+\frac {b}{x}}}{4 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{3/2}}{3 x^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {337, 279, 321, 217, 206} \[ \frac {a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{8 b^{3/2}}-\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b \sqrt {x}}-\frac {a \sqrt {a+\frac {b}{x}}}{4 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{3/2}}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 279
Rule 321
Rule 337
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^{5/2}} \, dx &=-\left (2 \operatorname {Subst}\left (\int x^2 \left (a+b x^2\right )^{3/2} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\frac {\left (a+\frac {b}{x}\right )^{3/2}}{3 x^{3/2}}-a \operatorname {Subst}\left (\int x^2 \sqrt {a+b x^2} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=-\frac {a \sqrt {a+\frac {b}{x}}}{4 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{3/2}}{3 x^{3/2}}-\frac {1}{4} a^2 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=-\frac {a \sqrt {a+\frac {b}{x}}}{4 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{3/2}}{3 x^{3/2}}-\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b \sqrt {x}}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{8 b}\\ &=-\frac {a \sqrt {a+\frac {b}{x}}}{4 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{3/2}}{3 x^{3/2}}-\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b \sqrt {x}}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{8 b}\\ &=-\frac {a \sqrt {a+\frac {b}{x}}}{4 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{3/2}}{3 x^{3/2}}-\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b \sqrt {x}}+\frac {a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{8 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 90, normalized size = 0.87 \[ \frac {\sqrt {a+\frac {b}{x}} \left (\frac {3 a^{5/2} \sinh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )}{\sqrt {\frac {b}{a x}+1}}-\frac {\sqrt {b} \left (3 a^2 x^2+14 a b x+8 b^2\right )}{x^{5/2}}\right )}{24 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 173, normalized size = 1.68 \[ \left [\frac {3 \, a^{3} \sqrt {b} x^{3} \log \left (\frac {a x + 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) - 2 \, {\left (3 \, a^{2} b x^{2} + 14 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{48 \, b^{2} x^{3}}, -\frac {3 \, a^{3} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (3 \, a^{2} b x^{2} + 14 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{24 \, b^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 84, normalized size = 0.82 \[ -\frac {\frac {3 \, a^{4} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} + \frac {3 \, {\left (a x + b\right )}^{\frac {5}{2}} a^{4} + 8 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{4} b - 3 \, \sqrt {a x + b} a^{4} b^{2}}{a^{3} b x^{3}}}{24 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 92, normalized size = 0.89 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (-3 a^{3} x^{3} \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )+3 \sqrt {a x +b}\, a^{2} \sqrt {b}\, x^{2}+14 \sqrt {a x +b}\, a \,b^{\frac {3}{2}} x +8 \sqrt {a x +b}\, b^{\frac {5}{2}}\right )}{24 \sqrt {a x +b}\, b^{\frac {3}{2}} x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.41, size = 159, normalized size = 1.54 \[ -\frac {a^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{16 \, b^{\frac {3}{2}}} - \frac {3 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3} x^{\frac {5}{2}} + 8 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{3} b x^{\frac {3}{2}} - 3 \, \sqrt {a + \frac {b}{x}} a^{3} b^{2} \sqrt {x}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{3} b x^{3} - 3 \, {\left (a + \frac {b}{x}\right )}^{2} b^{2} x^{2} + 3 \, {\left (a + \frac {b}{x}\right )} b^{3} x - b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+\frac {b}{x}\right )}^{3/2}}{x^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.32, size = 124, normalized size = 1.20 \[ - \frac {a^{\frac {5}{2}}}{8 b \sqrt {x} \sqrt {1 + \frac {b}{a x}}} - \frac {17 a^{\frac {3}{2}}}{24 x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}} - \frac {11 \sqrt {a} b}{12 x^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x}}} + \frac {a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{8 b^{\frac {3}{2}}} - \frac {b^{2}}{3 \sqrt {a} x^{\frac {7}{2}} \sqrt {1 + \frac {b}{a x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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