Optimal. Leaf size=60 \[ -\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \]
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Rubi [A]
time = 0.02, 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 = {272, 53, 65,
214} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 214
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x} \, dx &=-\text {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 {\text {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 {\text {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 \text {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 [A]
time = 0.08, size = 60, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {a+\frac {b}{x}} x (3 b+4 a x)}{3 a^2 (b+a x)^2}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(273\) vs.
\(2(48)=96\).
time = 0.04, size = 274, normalized size = 4.57
method | result | size |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \left (3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b \,x^{3}-6 \sqrt {x \left (a x +b \right )}\, a^{\frac {7}{2}} x^{3}+9 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{2} x^{2}+6 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {5}{2}} x -18 \sqrt {x \left (a x +b \right )}\, a^{\frac {5}{2}} b \,x^{2}+9 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3} x +4 b \,a^{\frac {3}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}}-18 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}} b^{2} x +3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{4}-6 \sqrt {x \left (a x +b \right )}\, \sqrt {a}\, b^{3}\right )}{3 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, \left (a x +b \right )^{3} b}\) | \(274\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 62, normalized size = 1.03 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 97 vs.
\(2 (48) = 96\).
time = 0.38, size = 197, normalized size = 3.28 \begin {gather*} \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 ] \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. 700 vs.
\(2 (48) = 96\).
time = 1.45, size = 700, normalized size = 11.67 \begin {gather*} - \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs.
\(2 (48) = 96\).
time = 0.56, size = 148, normalized size = 2.47 \begin {gather*} \frac {{\left (3 \, \log \left ({\left | b \right |}\right ) + 8\right )} \mathrm {sgn}\left (x\right )}{3 \, a^{\frac {5}{2}}} - \frac {\log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} - b \right |}\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} - \frac {2 \, {\left (6 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b + 9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{2} + 4 \, b^{3}\right )}}{3 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )}^{3} a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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