Optimal. Leaf size=129 \[ -\frac {35 a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{4 b^{9/2}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{4 b^4 \sqrt {x}}-\frac {35 \sqrt {a+\frac {b}{x}}}{6 b^3 x^{3/2}}+\frac {14}{3 b^2 x^{5/2} \sqrt {a+\frac {b}{x}}}+\frac {2}{3 b x^{7/2} \left (a+\frac {b}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 129, 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} \[ -\frac {35 a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{4 b^{9/2}}-\frac {35 \sqrt {a+\frac {b}{x}}}{6 b^3 x^{3/2}}+\frac {14}{3 b^2 x^{5/2} \sqrt {a+\frac {b}{x}}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{4 b^4 \sqrt {x}}+\frac {2}{3 b x^{7/2} \left (a+\frac {b}{x}\right )^{3/2}} \]
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 )^{5/2} x^{11/2}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {x^8}{\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^{7/2}}-\frac {14 \operatorname {Subst}\left (\int \frac {x^6}{\left (a+b x^2\right )^{3/2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{3 b}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}}+\frac {14}{3 b^2 \sqrt {a+\frac {b}{x}} x^{5/2}}-\frac {70 \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{3 b^2}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}}+\frac {14}{3 b^2 \sqrt {a+\frac {b}{x}} x^{5/2}}-\frac {35 \sqrt {a+\frac {b}{x}}}{6 b^3 x^{3/2}}+\frac {(35 a) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{2 b^3}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}}+\frac {14}{3 b^2 \sqrt {a+\frac {b}{x}} x^{5/2}}-\frac {35 \sqrt {a+\frac {b}{x}}}{6 b^3 x^{3/2}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{4 b^4 \sqrt {x}}-\frac {\left (35 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{4 b^4}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}}+\frac {14}{3 b^2 \sqrt {a+\frac {b}{x}} x^{5/2}}-\frac {35 \sqrt {a+\frac {b}{x}}}{6 b^3 x^{3/2}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{4 b^4 \sqrt {x}}-\frac {\left (35 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{4 b^4}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}}+\frac {14}{3 b^2 \sqrt {a+\frac {b}{x}} x^{5/2}}-\frac {35 \sqrt {a+\frac {b}{x}}}{6 b^3 x^{3/2}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{4 b^4 \sqrt {x}}-\frac {35 a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{4 b^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 56, normalized size = 0.43 \[ -\frac {2 \sqrt {\frac {b}{a x}+1} \, _2F_1\left (\frac {5}{2},\frac {9}{2};\frac {11}{2};-\frac {b}{a x}\right )}{9 a^2 x^{9/2} \sqrt {a+\frac {b}{x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 283, normalized size = 2.19 \[ \left [\frac {105 \, {\left (a^{4} x^{4} + 2 \, a^{3} b x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {b} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, {\left (105 \, a^{3} b x^{3} + 140 \, a^{2} b^{2} x^{2} + 21 \, a b^{3} x - 6 \, b^{4}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{24 \, {\left (a^{2} b^{5} x^{4} + 2 \, a b^{6} x^{3} + b^{7} x^{2}\right )}}, \frac {105 \, {\left (a^{4} x^{4} + 2 \, a^{3} b x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (105 \, a^{3} b x^{3} + 140 \, a^{2} b^{2} x^{2} + 21 \, a b^{3} x - 6 \, b^{4}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{12 \, {\left (a^{2} b^{5} x^{4} + 2 \, a b^{6} x^{3} + b^{7} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 93, normalized size = 0.72 \[ \frac {35 \, a^{2} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{4 \, \sqrt {-b} b^{4}} + \frac {2 \, {\left (9 \, {\left (a x + b\right )} a^{2} + a^{2} b\right )}}{3 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{4}} + \frac {11 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{2} - 13 \, \sqrt {a x + b} a^{2} b}{4 \, a^{2} b^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 117, normalized size = 0.91 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (105 \sqrt {a x +b}\, a^{3} x^{3} \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )-105 a^{3} \sqrt {b}\, x^{3}+105 \sqrt {a x +b}\, a^{2} b \,x^{2} \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )-140 a^{2} b^{\frac {3}{2}} x^{2}-21 a \,b^{\frac {5}{2}} x +6 b^{\frac {7}{2}}\right )}{12 \left (a x +b \right )^{2} b^{\frac {9}{2}} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.50, size = 163, normalized size = 1.26 \[ \frac {105 \, {\left (a + \frac {b}{x}\right )}^{3} a^{2} x^{3} - 175 \, {\left (a + \frac {b}{x}\right )}^{2} a^{2} b x^{2} + 56 \, {\left (a + \frac {b}{x}\right )} a^{2} b^{2} x + 8 \, a^{2} b^{3}}{12 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} b^{4} x^{\frac {7}{2}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b^{5} x^{\frac {5}{2}} + {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{6} x^{\frac {3}{2}}\right )}} + \frac {35 \, a^{2} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{8 \, b^{\frac {9}{2}}} \]
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 \[ \int \frac {1}{x^{11/2}\,{\left (a+\frac {b}{x}\right )}^{5/2}} \,d x \]
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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