Optimal. Leaf size=60 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {2}{a^2 \sqrt {a+\frac {b}{x}}}-\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ -\frac {2}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2}{a^2 \sqrt {a+\frac {b}{x}}}-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{a^2}\\ &=-\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2}{a^2 \sqrt {a+\frac {b}{x}}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{a^2 b}\\ &=-\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 36, normalized size = 0.60 \[ -\frac {2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b}{a x}+1\right )}{3 a \left (a+\frac {b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.04, size = 197, normalized size = 3.28 \[ \left [\frac {3 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (4 \, a^{2} x^{2} + 3 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}}, -\frac {2 \, {\left (3 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (4 \, a^{2} x^{2} + 3 \, a b x\right )} \sqrt {\frac {a x + b}{x}}\right )}}{3 \, {\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 65, normalized size = 1.08 \[ -\frac {2 \, {\left (a + \frac {3 \, {\left (a x + b\right )}}{x}\right )} x}{3 \, {\left (a x + b\right )} a^{2} \sqrt {\frac {a x + b}{x}}} - \frac {2 \, \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 274, normalized size = 4.57 \[ \frac {\sqrt {\frac {a x +b}{x}}\, \left (3 a^{3} b \,x^{3} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+9 a^{2} b^{2} x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-6 \sqrt {\left (a x +b \right ) x}\, a^{\frac {7}{2}} x^{3}+9 a \,b^{3} x \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-18 \sqrt {\left (a x +b \right ) x}\, a^{\frac {5}{2}} b \,x^{2}+3 b^{4} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-18 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}} b^{2} x +6 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {5}{2}} x -6 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}\, b^{3}+4 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b \right ) x}{3 \sqrt {\left (a x +b \right ) x}\, \left (a x +b \right )^{3} a^{\frac {5}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.21, size = 62, normalized size = 1.03 \[ -\frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} - \frac {2 \, {\left (4 \, a + \frac {3 \, b}{x}\right )}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.58, size = 49, normalized size = 0.82 \[ \frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\frac {2\,\left (a+\frac {b}{x}\right )}{a^2}+\frac {2}{3\,a}}{{\left (a+\frac {b}{x}\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.56, size = 700, normalized size = 11.67 \[ - \frac {8 a^{7} x^{3} \sqrt {1 + \frac {b}{a x}}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} - \frac {3 a^{7} x^{3} \log {\left (\frac {b}{a x} \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} + \frac {6 a^{7} x^{3} \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} - \frac {14 a^{6} b x^{2} \sqrt {1 + \frac {b}{a x}}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} - \frac {9 a^{6} b x^{2} \log {\left (\frac {b}{a x} \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} + \frac {18 a^{6} b x^{2} \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} - \frac {6 a^{5} b^{2} x \sqrt {1 + \frac {b}{a x}}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} - \frac {9 a^{5} b^{2} x \log {\left (\frac {b}{a x} \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} + \frac {18 a^{5} b^{2} x \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} - \frac {3 a^{4} b^{3} \log {\left (\frac {b}{a x} \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} + \frac {6 a^{4} b^{3} \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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