Optimal. Leaf size=71 \[ 5 a^{3/2} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )+x \left (a+\frac {b}{x}\right )^{5/2}-\frac {5}{3} b \left (a+\frac {b}{x}\right )^{3/2}-5 a b \sqrt {a+\frac {b}{x}} \]
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Rubi [A] time = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {242, 47, 50, 63, 208} \[ 5 a^{3/2} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )+x \left (a+\frac {b}{x}\right )^{5/2}-\frac {5}{3} b \left (a+\frac {b}{x}\right )^{3/2}-5 a b \sqrt {a+\frac {b}{x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rule 242
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^{5/2} \, dx &=-\operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\left (a+\frac {b}{x}\right )^{5/2} x-\frac {1}{2} (5 b) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {5}{3} b \left (a+\frac {b}{x}\right )^{3/2}+\left (a+\frac {b}{x}\right )^{5/2} x-\frac {1}{2} (5 a b) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x}\right )\\ &=-5 a b \sqrt {a+\frac {b}{x}}-\frac {5}{3} b \left (a+\frac {b}{x}\right )^{3/2}+\left (a+\frac {b}{x}\right )^{5/2} x-\frac {1}{2} \left (5 a^2 b\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )\\ &=-5 a b \sqrt {a+\frac {b}{x}}-\frac {5}{3} b \left (a+\frac {b}{x}\right )^{3/2}+\left (a+\frac {b}{x}\right )^{5/2} x-\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )\\ &=-5 a b \sqrt {a+\frac {b}{x}}-\frac {5}{3} b \left (a+\frac {b}{x}\right )^{3/2}+\left (a+\frac {b}{x}\right )^{5/2} x+5 a^{3/2} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 64, normalized size = 0.90 \[ 5 a^{3/2} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )+\frac {\sqrt {a+\frac {b}{x}} \left (3 a^2 x^2-14 a b x-2 b^2\right )}{3 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 139, normalized size = 1.96 \[ \left [\frac {15 \, a^{\frac {3}{2}} b x \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (3 \, a^{2} x^{2} - 14 \, a b x - 2 \, b^{2}\right )} \sqrt {\frac {a x + b}{x}}}{6 \, x}, -\frac {15 \, \sqrt {-a} a b x \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) - {\left (3 \, a^{2} x^{2} - 14 \, a b x - 2 \, b^{2}\right )} \sqrt {\frac {a x + b}{x}}}{3 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 120, normalized size = 1.69 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (-15 a^{2} b \,x^{3} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-30 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} x^{3}+24 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} x +4 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b \right )}{6 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}\, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.43, size = 78, normalized size = 1.10 \[ \sqrt {a + \frac {b}{x}} a^{2} x - \frac {5}{2} \, a^{\frac {3}{2}} b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) - \frac {2}{3} \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b - 4 \, \sqrt {a + \frac {b}{x}} a b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 34, normalized size = 0.48 \[ -\frac {2\,x\,{\left (a+\frac {b}{x}\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{2},-\frac {3}{2};\ -\frac {1}{2};\ -\frac {a\,x}{b}\right )}{3\,{\left (\frac {a\,x}{b}+1\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.90, size = 99, normalized size = 1.39 \[ a^{\frac {5}{2}} x \sqrt {1 + \frac {b}{a x}} - \frac {14 a^{\frac {3}{2}} b \sqrt {1 + \frac {b}{a x}}}{3} - \frac {5 a^{\frac {3}{2}} b \log {\left (\frac {b}{a x} \right )}}{2} + 5 a^{\frac {3}{2}} b \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )} - \frac {2 \sqrt {a} b^{2} \sqrt {1 + \frac {b}{a x}}}{3 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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