Optimal. Leaf size=126 \[ -\frac {5 a^2 \sqrt {a+\frac {b}{x}}}{32 x^{3/2}}-\frac {5 a \left (a+\frac {b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{4 x^{3/2}}-\frac {5 a^3 \sqrt {a+\frac {b}{x}}}{64 b \sqrt {x}}+\frac {5 a^4 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{64 b^{3/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 126, 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, 285, 327,
223, 212} \begin {gather*} \frac {5 a^4 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{64 b^{3/2}}-\frac {5 a^3 \sqrt {a+\frac {b}{x}}}{64 b \sqrt {x}}-\frac {5 a^2 \sqrt {a+\frac {b}{x}}}{32 x^{3/2}}-\frac {5 a \left (a+\frac {b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{4 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 285
Rule 327
Rule 344
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{x^{5/2}} \, dx &=-\left (2 \text {Subst}\left (\int x^2 \left (a+b x^2\right )^{5/2} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{4 x^{3/2}}-\frac {1}{4} (5 a) \text {Subst}\left (\int x^2 \left (a+b x^2\right )^{3/2} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=-\frac {5 a \left (a+\frac {b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{4 x^{3/2}}-\frac {1}{8} \left (5 a^2\right ) \text {Subst}\left (\int x^2 \sqrt {a+b x^2} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=-\frac {5 a^2 \sqrt {a+\frac {b}{x}}}{32 x^{3/2}}-\frac {5 a \left (a+\frac {b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{4 x^{3/2}}-\frac {1}{32} \left (5 a^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=-\frac {5 a^2 \sqrt {a+\frac {b}{x}}}{32 x^{3/2}}-\frac {5 a \left (a+\frac {b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{4 x^{3/2}}-\frac {5 a^3 \sqrt {a+\frac {b}{x}}}{64 b \sqrt {x}}+\frac {\left (5 a^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{64 b}\\ &=-\frac {5 a^2 \sqrt {a+\frac {b}{x}}}{32 x^{3/2}}-\frac {5 a \left (a+\frac {b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{4 x^{3/2}}-\frac {5 a^3 \sqrt {a+\frac {b}{x}}}{64 b \sqrt {x}}+\frac {\left (5 a^4\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{64 b}\\ &=-\frac {5 a^2 \sqrt {a+\frac {b}{x}}}{32 x^{3/2}}-\frac {5 a \left (a+\frac {b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{4 x^{3/2}}-\frac {5 a^3 \sqrt {a+\frac {b}{x}}}{64 b \sqrt {x}}+\frac {5 a^4 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{64 b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 5.82, size = 104, normalized size = 0.83 \begin {gather*} \frac {\sqrt {a+\frac {b}{x}} \sqrt {x} \left (\frac {\sqrt {b+a x} \left (-48 b^3-136 a b^2 x-118 a^2 b x^2-15 a^3 x^3\right )}{192 b x^4}+\frac {5 a^4 \tanh ^{-1}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )}{64 b^{3/2}}\right )}{\sqrt {b+a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 110, normalized size = 0.87
method | result | size |
risch | \(-\frac {\left (15 a^{3} x^{3}+118 a^{2} b \,x^{2}+136 a \,b^{2} x +48 b^{3}\right ) \sqrt {\frac {a x +b}{x}}}{192 x^{\frac {7}{2}} b}+\frac {5 a^{4} \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}{64 b^{\frac {3}{2}} \sqrt {a x +b}}\) | \(92\) |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (-15 \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) a^{4} x^{4}+48 \sqrt {a x +b}\, b^{\frac {7}{2}}+136 a \,b^{\frac {5}{2}} x \sqrt {a x +b}+118 a^{2} b^{\frac {3}{2}} x^{2} \sqrt {a x +b}+15 a^{3} x^{3} \sqrt {a x +b}\, \sqrt {b}\right )}{192 x^{\frac {7}{2}} b^{\frac {3}{2}} \sqrt {a x +b}}\) | \(110\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 194 vs.
\(2 (92) = 184\).
time = 0.50, size = 194, normalized size = 1.54 \begin {gather*} -\frac {5 \, a^{4} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{128 \, b^{\frac {3}{2}}} - \frac {15 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{4} x^{\frac {7}{2}} + 73 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{4} b x^{\frac {5}{2}} - 55 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{4} b^{2} x^{\frac {3}{2}} + 15 \, \sqrt {a + \frac {b}{x}} a^{4} b^{3} \sqrt {x}}{192 \, {\left ({\left (a + \frac {b}{x}\right )}^{4} b x^{4} - 4 \, {\left (a + \frac {b}{x}\right )}^{3} b^{2} x^{3} + 6 \, {\left (a + \frac {b}{x}\right )}^{2} b^{3} x^{2} - 4 \, {\left (a + \frac {b}{x}\right )} b^{4} x + b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 195, normalized size = 1.55 \begin {gather*} \left [\frac {15 \, a^{4} \sqrt {b} x^{4} \log \left (\frac {a x + 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) - 2 \, {\left (15 \, a^{3} b x^{3} + 118 \, a^{2} b^{2} x^{2} + 136 \, a b^{3} x + 48 \, b^{4}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{384 \, b^{2} x^{4}}, -\frac {15 \, a^{4} \sqrt {-b} x^{4} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (15 \, a^{3} b x^{3} + 118 \, a^{2} b^{2} x^{2} + 136 \, a b^{3} x + 48 \, b^{4}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{192 \, b^{2} x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 12.02, size = 155, normalized size = 1.23 \begin {gather*} - \frac {5 a^{\frac {7}{2}}}{64 b \sqrt {x} \sqrt {1 + \frac {b}{a x}}} - \frac {133 a^{\frac {5}{2}}}{192 x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}} - \frac {127 a^{\frac {3}{2}} b}{96 x^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x}}} - \frac {23 \sqrt {a} b^{2}}{24 x^{\frac {7}{2}} \sqrt {1 + \frac {b}{a x}}} + \frac {5 a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{64 b^{\frac {3}{2}}} - \frac {b^{3}}{4 \sqrt {a} x^{\frac {9}{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 = 0.57, size = 99, normalized size = 0.79 \begin {gather*} -\frac {\frac {15 \, a^{5} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} + \frac {15 \, {\left (a x + b\right )}^{\frac {7}{2}} a^{5} + 73 \, {\left (a x + b\right )}^{\frac {5}{2}} a^{5} b - 55 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{5} b^{2} + 15 \, \sqrt {a x + b} a^{5} b^{3}}{a^{4} b x^{4}}}{192 \, a} \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 {{\left (a+\frac {b}{x}\right )}^{5/2}}{x^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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