Optimal. Leaf size=75 \[ \frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}+\frac {2}{b^2 \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b^{5/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {344, 294, 223,
212} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{b^{5/2}}+\frac {2}{b^2 \sqrt {x} \sqrt {a+\frac {b}{x}}}+\frac {2}{3 b x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 294
Rule 344
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx &=-\left (2 \text {Subst}\left (\int \frac {x^4}{\left (a+b x^2\right )^{5/2}} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}-\frac {2 \text {Subst}\left (\int \frac {x^2}{\left (a+b x^2\right )^{3/2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{b}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}+\frac {2}{b^2 \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{b^2}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}+\frac {2}{b^2 \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b^2}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}+\frac {2}{b^2 \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 10.14, size = 94, normalized size = 1.25 \begin {gather*} \frac {2 \left (\sqrt {b} \sqrt {x} (4 b+3 a x)-3 \sqrt {a} \sqrt {1+\frac {b}{a x}} x (b+a x) \sinh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )\right )}{3 b^{5/2} \sqrt {a+\frac {b}{x}} x (b+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 85, normalized size = 1.13
method | result | size |
default | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (-3 \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {a x +b}\, a x +4 b^{\frac {3}{2}}+3 a x \sqrt {b}-3 \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) b \sqrt {a x +b}\right )}{3 b^{\frac {5}{2}} \left (a x +b \right )^{2}}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 74, normalized size = 0.99 \begin {gather*} \frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{b^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, {\left (a + \frac {b}{x}\right )} x + b\right )}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 205, normalized size = 2.73 \begin {gather*} \left [\frac {3 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {b} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, {\left (3 \, a b x + 4 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}}, \frac {2 \, {\left (3 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (3 \, a b x + 4 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}\right )}}{3 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\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. 697 vs.
\(2 (63) = 126\).
time = 31.27, size = 697, normalized size = 9.29 \begin {gather*} \frac {3 a^{3} b^{4} x^{3} \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {6 a^{3} b^{4} x^{3} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {6 a^{2} b^{5} x^{2} \sqrt {\frac {a x}{b} + 1}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {9 a^{2} b^{5} x^{2} \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {18 a^{2} b^{5} x^{2} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {14 a b^{6} x \sqrt {\frac {a x}{b} + 1}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {9 a b^{6} x \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {18 a b^{6} x \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {8 b^{7} \sqrt {\frac {a x}{b} + 1}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {3 b^{7} \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {6 b^{7} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 45, normalized size = 0.60 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} + \frac {2 \, {\left (3 \, a x + 4 \, b\right )}}{3 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{2}} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{7/2}\,{\left (a+\frac {b}{x}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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