Optimal. Leaf size=74 \[ \frac {2}{b \sqrt {a+\frac {b}{x}} x^{3/2}}-\frac {3 \sqrt {a+\frac {b}{x}}}{b^2 \sqrt {x}}+\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b^{5/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {344, 294, 327,
223, 212} \begin {gather*} \frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{b^{5/2}}-\frac {3 \sqrt {a+\frac {b}{x}}}{b^2 \sqrt {x}}+\frac {2}{b x^{3/2} \sqrt {a+\frac {b}{x}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 294
Rule 327
Rule 344
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^{7/2}} \, dx &=-\left (2 \text {Subst}\left (\int \frac {x^4}{\left (a+b x^2\right )^{3/2}} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=\frac {2}{b \sqrt {a+\frac {b}{x}} x^{3/2}}-\frac {6 \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{b}\\ &=\frac {2}{b \sqrt {a+\frac {b}{x}} x^{3/2}}-\frac {3 \sqrt {a+\frac {b}{x}}}{b^2 \sqrt {x}}+\frac {(3 a) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{b^2}\\ &=\frac {2}{b \sqrt {a+\frac {b}{x}} x^{3/2}}-\frac {3 \sqrt {a+\frac {b}{x}}}{b^2 \sqrt {x}}+\frac {(3 a) \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}{b \sqrt {a+\frac {b}{x}} x^{3/2}}-\frac {3 \sqrt {a+\frac {b}{x}}}{b^2 \sqrt {x}}+\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 10.02, size = 56, normalized size = 0.76 \begin {gather*} -\frac {2 \sqrt {1+\frac {b}{a x}} \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};-\frac {b}{a x}\right )}{5 a \sqrt {a+\frac {b}{x}} x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 61, normalized size = 0.82
method | result | size |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (-3 \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {a x +b}\, a x +3 a x \sqrt {b}+b^{\frac {3}{2}}\right )}{\sqrt {x}\, \left (a x +b \right ) b^{\frac {5}{2}}}\) | \(61\) |
risch | \(-\frac {a x +b}{b^{2} x^{\frac {3}{2}} \sqrt {\frac {a x +b}{x}}}-\frac {a \left (-\frac {6 \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )}{\sqrt {b}}+\frac {4}{\sqrt {a x +b}}\right ) \sqrt {a x +b}}{2 b^{2} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 101, normalized size = 1.36 \begin {gather*} -\frac {3 \, {\left (a + \frac {b}{x}\right )} a x - 2 \, a b}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} x^{\frac {3}{2}} - \sqrt {a + \frac {b}{x}} b^{3} \sqrt {x}} - \frac {3 \, a \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{2 \, b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 179, normalized size = 2.42 \begin {gather*} \left [\frac {3 \, {\left (a^{2} x^{2} + a b x\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 + b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a b^{3} x^{2} + b^{4} x\right )}}, -\frac {3 \, {\left (a^{2} x^{2} + a b x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (3 \, a b x + b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a b^{3} x^{2} + b^{4} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 19.28, size = 73, normalized size = 0.99 \begin {gather*} - \frac {3 \sqrt {a}}{b^{2} \sqrt {x} \sqrt {1 + \frac {b}{a x}}} + \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{b^{\frac {5}{2}}} - \frac {1}{\sqrt {a} b x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.01, size = 64, normalized size = 0.86 \begin {gather*} -\frac {3 \, a \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} - \frac {3 \, {\left (a x + b\right )} a - 2 \, a b}{{\left ({\left (a x + b\right )}^{\frac {3}{2}} - \sqrt {a x + b} b\right )} 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 )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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