Optimal. Leaf size=74 \[ \frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{b^{5/2}}-\frac {3 \sqrt {a+\frac {b}{x}}}{b^2 \sqrt {x}}+\frac {2}{b x^{3/2} \sqrt {a+\frac {b}{x}}} \]
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Rubi [A] time = 0.04, 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 = {337, 288, 321, 217, 206} \[ -\frac {3 \sqrt {a+\frac {b}{x}}}{b^2 \sqrt {x}}+\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{b^{5/2}}+\frac {2}{b x^{3/2} \sqrt {a+\frac {b}{x}}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 288
Rule 321
Rule 337
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^{7/2}} \, dx &=-\left (2 \operatorname {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 \operatorname {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) \operatorname {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) \operatorname {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] time = 0.02, size = 56, normalized size = 0.76 \[ -\frac {2 \sqrt {\frac {b}{a x}+1} \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};-\frac {b}{a x}\right )}{5 a x^{5/2} \sqrt {a+\frac {b}{x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 179, normalized size = 2.42 \[ \left [\frac {3 \, {\left (a^{2} x^{2} + a b x\right )} \sqrt {b} \log \left (\frac {a x + 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) - 2 \, {\left (3 \, a b x + b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a b^{3} x^{2} + b^{4} x\right )}}, -\frac {3 \, {\left (a^{2} x^{2} + a b x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (3 \, a b x + b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a b^{3} x^{2} + b^{4} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 64, normalized size = 0.86 \[ -\frac {3 \, a \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} - \frac {3 \, {\left (a x + b\right )} a - 2 \, a b}{{\left ({\left (a x + b\right )}^{\frac {3}{2}} - \sqrt {a x + b} b\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 61, normalized size = 0.82 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (-3 \sqrt {a x +b}\, a x \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )+3 a \sqrt {b}\, x +b^{\frac {3}{2}}\right )}{\left (a x +b \right ) b^{\frac {5}{2}} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.40, size = 101, normalized size = 1.36 \[ -\frac {3 \, {\left (a + \frac {b}{x}\right )} a x - 2 \, a b}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} x^{\frac {3}{2}} - \sqrt {a + \frac {b}{x}} b^{3} \sqrt {x}} - \frac {3 \, a \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{2 \, b^{\frac {5}{2}}} \]
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 \[ \int \frac {1}{x^{7/2}\,{\left (a+\frac {b}{x}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 38.02, size = 73, normalized size = 0.99 \[ - \frac {3 \sqrt {a}}{b^{2} \sqrt {x} \sqrt {1 + \frac {b}{a x}}} + \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{b^{\frac {5}{2}}} - \frac {1}{\sqrt {a} b x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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