Optimal. Leaf size=130 \[ \frac {35 a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{8 b^{9/2}}-\frac {35 a^2 \sqrt {a+\frac {b}{x}}}{8 b^4 \sqrt {x}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{12 b^3 x^{3/2}}-\frac {7 \sqrt {a+\frac {b}{x}}}{3 b^2 x^{5/2}}+\frac {2}{b x^{7/2} \sqrt {a+\frac {b}{x}}} \]
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Rubi [A] time = 0.07, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {35 a^2 \sqrt {a+\frac {b}{x}}}{8 b^4 \sqrt {x}}+\frac {35 a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{8 b^{9/2}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{12 b^3 x^{3/2}}-\frac {7 \sqrt {a+\frac {b}{x}}}{3 b^2 x^{5/2}}+\frac {2}{b x^{7/2} \sqrt {a+\frac {b}{x}}} \]
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^{11/2}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {x^8}{\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^{7/2}}-\frac {14 \operatorname {Subst}\left (\int \frac {x^6}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{b}\\ &=\frac {2}{b \sqrt {a+\frac {b}{x}} x^{7/2}}-\frac {7 \sqrt {a+\frac {b}{x}}}{3 b^2 x^{5/2}}+\frac {(35 a) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{3 b^2}\\ &=\frac {2}{b \sqrt {a+\frac {b}{x}} x^{7/2}}-\frac {7 \sqrt {a+\frac {b}{x}}}{3 b^2 x^{5/2}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{12 b^3 x^{3/2}}-\frac {\left (35 a^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{4 b^3}\\ &=\frac {2}{b \sqrt {a+\frac {b}{x}} x^{7/2}}-\frac {7 \sqrt {a+\frac {b}{x}}}{3 b^2 x^{5/2}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{12 b^3 x^{3/2}}-\frac {35 a^2 \sqrt {a+\frac {b}{x}}}{8 b^4 \sqrt {x}}+\frac {\left (35 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{8 b^4}\\ &=\frac {2}{b \sqrt {a+\frac {b}{x}} x^{7/2}}-\frac {7 \sqrt {a+\frac {b}{x}}}{3 b^2 x^{5/2}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{12 b^3 x^{3/2}}-\frac {35 a^2 \sqrt {a+\frac {b}{x}}}{8 b^4 \sqrt {x}}+\frac {\left (35 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{8 b^4}\\ &=\frac {2}{b \sqrt {a+\frac {b}{x}} x^{7/2}}-\frac {7 \sqrt {a+\frac {b}{x}}}{3 b^2 x^{5/2}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{12 b^3 x^{3/2}}-\frac {35 a^2 \sqrt {a+\frac {b}{x}}}{8 b^4 \sqrt {x}}+\frac {35 a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{8 b^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 56, normalized size = 0.43 \[ -\frac {2 \sqrt {\frac {b}{a x}+1} \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};-\frac {b}{a x}\right )}{9 a x^{9/2} \sqrt {a+\frac {b}{x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 239, normalized size = 1.84 \[ \left [\frac {105 \, {\left (a^{4} x^{4} + a^{3} b x^{3}\right )} \sqrt {b} \log \left (\frac {a x + 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) - 2 \, {\left (105 \, a^{3} b x^{3} + 35 \, a^{2} b^{2} x^{2} - 14 \, a b^{3} x + 8 \, b^{4}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{48 \, {\left (a b^{5} x^{4} + b^{6} x^{3}\right )}}, -\frac {105 \, {\left (a^{4} x^{4} + a^{3} b x^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (105 \, a^{3} b x^{3} + 35 \, a^{2} b^{2} x^{2} - 14 \, a b^{3} x + 8 \, b^{4}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{24 \, {\left (a b^{5} x^{4} + b^{6} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 95, normalized size = 0.73 \[ -\frac {35 \, a^{3} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{8 \, \sqrt {-b} b^{4}} - \frac {2 \, a^{3}}{\sqrt {a x + b} b^{4}} - \frac {57 \, {\left (a x + b\right )}^{\frac {5}{2}} a^{3} - 136 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{3} b + 87 \, \sqrt {a x + b} a^{3} b^{2}}{24 \, a^{3} b^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 89, normalized size = 0.68 \[ \frac {\sqrt {\frac {a x +b}{x}}\, \left (105 \sqrt {a x +b}\, a^{3} x^{3} \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )-105 a^{3} \sqrt {b}\, x^{3}-35 a^{2} b^{\frac {3}{2}} x^{2}+14 a \,b^{\frac {5}{2}} x -8 b^{\frac {7}{2}}\right )}{24 \left (a x +b \right ) b^{\frac {9}{2}} x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.26, size = 181, normalized size = 1.39 \[ -\frac {105 \, {\left (a + \frac {b}{x}\right )}^{3} a^{3} x^{3} - 280 \, {\left (a + \frac {b}{x}\right )}^{2} a^{3} b x^{2} + 231 \, {\left (a + \frac {b}{x}\right )} a^{3} b^{2} x - 48 \, a^{3} b^{3}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} b^{4} x^{\frac {7}{2}} - 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b^{5} x^{\frac {5}{2}} + 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{6} x^{\frac {3}{2}} - \sqrt {a + \frac {b}{x}} b^{7} \sqrt {x}\right )}} - \frac {35 \, a^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{16 \, b^{\frac {9}{2}}} \]
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 \[ \int \frac {1}{x^{11/2}\,{\left (a+\frac {b}{x}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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