Optimal. Leaf size=87 \[ \frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 \sqrt {a}}+\frac {5}{8} b^2 x \sqrt {a+\frac {b}{x}}+\frac {1}{3} x^3 \left (a+\frac {b}{x}\right )^{5/2}+\frac {5}{12} b x^2 \left (a+\frac {b}{x}\right )^{3/2} \]
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Rubi [A] time = 0.04, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 47, 63, 208} \[ \frac {5}{8} b^2 x \sqrt {a+\frac {b}{x}}+\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 \sqrt {a}}+\frac {5}{12} b x^2 \left (a+\frac {b}{x}\right )^{3/2}+\frac {1}{3} x^3 \left (a+\frac {b}{x}\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^{5/2} x^2 \, dx &=-\operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} \left (a+\frac {b}{x}\right )^{5/2} x^3-\frac {1}{6} (5 b) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {5}{12} b \left (a+\frac {b}{x}\right )^{3/2} x^2+\frac {1}{3} \left (a+\frac {b}{x}\right )^{5/2} x^3-\frac {1}{8} \left (5 b^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {5}{8} b^2 \sqrt {a+\frac {b}{x}} x+\frac {5}{12} b \left (a+\frac {b}{x}\right )^{3/2} x^2+\frac {1}{3} \left (a+\frac {b}{x}\right )^{5/2} x^3-\frac {1}{16} \left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {5}{8} b^2 \sqrt {a+\frac {b}{x}} x+\frac {5}{12} b \left (a+\frac {b}{x}\right )^{3/2} x^2+\frac {1}{3} \left (a+\frac {b}{x}\right )^{5/2} x^3-\frac {1}{8} \left (5 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )\\ &=\frac {5}{8} b^2 \sqrt {a+\frac {b}{x}} x+\frac {5}{12} b \left (a+\frac {b}{x}\right )^{3/2} x^2+\frac {1}{3} \left (a+\frac {b}{x}\right )^{5/2} x^3+\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 87, normalized size = 1.00 \[ \frac {x \sqrt {a+\frac {b}{x}} \left (8 a^3 x^3+34 a^2 b x^2+15 b^3 \sqrt {\frac {b}{a x}+1} \tanh ^{-1}\left (\sqrt {\frac {b}{a x}+1}\right )+59 a b^2 x+33 b^3\right )}{24 (a x+b)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 152, normalized size = 1.75 \[ \left [\frac {15 \, \sqrt {a} b^{3} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (8 \, a^{3} x^{3} + 26 \, a^{2} b x^{2} + 33 \, a b^{2} x\right )} \sqrt {\frac {a x + b}{x}}}{48 \, a}, -\frac {15 \, \sqrt {-a} b^{3} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) - {\left (8 \, a^{3} x^{3} + 26 \, a^{2} b x^{2} + 33 \, a b^{2} x\right )} \sqrt {\frac {a x + b}{x}}}{24 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 93, normalized size = 1.07 \[ -\frac {5 \, b^{3} \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} - b \right |}\right ) \mathrm {sgn}\relax (x)}{16 \, \sqrt {a}} + \frac {5 \, b^{3} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\relax (x)}{16 \, \sqrt {a}} + \frac {1}{24} \, \sqrt {a x^{2} + b x} {\left (33 \, b^{2} \mathrm {sgn}\relax (x) + 2 \, {\left (4 \, a^{2} x \mathrm {sgn}\relax (x) + 13 \, a b \mathrm {sgn}\relax (x)\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 115, normalized size = 1.32 \[ \frac {\sqrt {\frac {a x +b}{x}}\, \left (15 a \,b^{3} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+36 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b x +66 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b^{2}+16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}}\right ) x}{48 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.31, size = 131, normalized size = 1.51 \[ -\frac {5 \, b^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{16 \, \sqrt {a}} + \frac {33 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b^{3} - 40 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a b^{3} + 15 \, \sqrt {a + \frac {b}{x}} a^{2} b^{3}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{3} - 3 \, {\left (a + \frac {b}{x}\right )}^{2} a + 3 \, {\left (a + \frac {b}{x}\right )} a^{2} - a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 72, normalized size = 0.83 \[ \frac {11\,x^3\,{\left (a+\frac {b}{x}\right )}^{5/2}}{8}-\frac {5\,a\,x^3\,{\left (a+\frac {b}{x}\right )}^{3/2}}{3}+\frac {5\,a^2\,x^3\,\sqrt {a+\frac {b}{x}}}{8}-\frac {b^3\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{8\,\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.58, size = 102, normalized size = 1.17 \[ \frac {a^{2} \sqrt {b} x^{\frac {5}{2}} \sqrt {\frac {a x}{b} + 1}}{3} + \frac {13 a b^{\frac {3}{2}} x^{\frac {3}{2}} \sqrt {\frac {a x}{b} + 1}}{12} + \frac {11 b^{\frac {5}{2}} \sqrt {x} \sqrt {\frac {a x}{b} + 1}}{8} + \frac {5 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{8 \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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