Optimal. Leaf size=129 \[ \frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}}+\frac {14}{3 b^2 \sqrt {a+\frac {b}{x}} x^{5/2}}-\frac {35 \sqrt {a+\frac {b}{x}}}{6 b^3 x^{3/2}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{4 b^4 \sqrt {x}}-\frac {35 a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{4 b^{9/2}} \]
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Rubi [A]
time = 0.05, 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 = {344, 294, 327,
223, 212} \begin {gather*} -\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}} \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 )^{5/2} x^{11/2}} \, dx &=-\left (2 \text {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 \text {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 \text {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) \text {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 ) \text {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 ) \text {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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 10.02, size = 56, normalized size = 0.43 \begin {gather*} -\frac {2 \sqrt {1+\frac {b}{a x}} \, _2F_1\left (\frac {5}{2},\frac {9}{2};\frac {11}{2};-\frac {b}{a x}\right )}{9 a^2 \sqrt {a+\frac {b}{x}} x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 117, normalized size = 0.91
method | result | size |
risch | \(\frac {\left (a x +b \right ) \left (11 a x -2 b \right )}{4 b^{4} x^{\frac {5}{2}} \sqrt {\frac {a x +b}{x}}}+\frac {a^{2} \left (-\frac {70 \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )}{\sqrt {b}}+\frac {48}{\sqrt {a x +b}}+\frac {16 b}{3 \left (a x +b \right )^{\frac {3}{2}}}\right ) \sqrt {a x +b}}{8 b^{4} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}\) | \(100\) |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (105 \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {a x +b}\, a^{3} x^{3}+105 \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) a^{2} b \,x^{2} \sqrt {a x +b}-105 a^{3} x^{3} \sqrt {b}-140 b^{\frac {3}{2}} a^{2} x^{2}-21 b^{\frac {5}{2}} a x +6 b^{\frac {7}{2}}\right )}{12 x^{\frac {3}{2}} \left (a x +b \right )^{2} b^{\frac {9}{2}}}\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 163, normalized size = 1.26 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 283, normalized size = 2.19 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.62, size = 93, normalized size = 0.72 \begin {gather*} \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}} \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^{11/2}\,{\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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