Optimal. Leaf size=73 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 a^{3/2} b^{3/2}}+\frac {\sqrt {x}}{4 a b (a x+b)}-\frac {\sqrt {x}}{2 a (a x+b)^2} \]
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Rubi [A] time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {263, 47, 51, 63, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 a^{3/2} b^{3/2}}+\frac {\sqrt {x}}{4 a b (a x+b)}-\frac {\sqrt {x}}{2 a (a x+b)^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 205
Rule 263
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^{5/2}} \, dx &=\int \frac {\sqrt {x}}{(b+a x)^3} \, dx\\ &=-\frac {\sqrt {x}}{2 a (b+a x)^2}+\frac {\int \frac {1}{\sqrt {x} (b+a x)^2} \, dx}{4 a}\\ &=-\frac {\sqrt {x}}{2 a (b+a x)^2}+\frac {\sqrt {x}}{4 a b (b+a x)}+\frac {\int \frac {1}{\sqrt {x} (b+a x)} \, dx}{8 a b}\\ &=-\frac {\sqrt {x}}{2 a (b+a x)^2}+\frac {\sqrt {x}}{4 a b (b+a x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )}{4 a b}\\ &=-\frac {\sqrt {x}}{2 a (b+a x)^2}+\frac {\sqrt {x}}{4 a b (b+a x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 a^{3/2} b^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 27, normalized size = 0.37 \[ \frac {2 x^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};-\frac {a x}{b}\right )}{3 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 186, normalized size = 2.55 \[ \left [-\frac {{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {-a b} \log \left (\frac {a x - b - 2 \, \sqrt {-a b} \sqrt {x}}{a x + b}\right ) - 2 \, {\left (a^{2} b x - a b^{2}\right )} \sqrt {x}}{8 \, {\left (a^{4} b^{2} x^{2} + 2 \, a^{3} b^{3} x + a^{2} b^{4}\right )}}, -\frac {{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{a \sqrt {x}}\right ) - {\left (a^{2} b x - a b^{2}\right )} \sqrt {x}}{4 \, {\left (a^{4} b^{2} x^{2} + 2 \, a^{3} b^{3} x + a^{2} b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 52, normalized size = 0.71 \[ \frac {\arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a b} + \frac {a x^{\frac {3}{2}} - b \sqrt {x}}{4 \, {\left (a x + b\right )}^{2} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.71 \[ \frac {\arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, a b}+\frac {\frac {x^{\frac {3}{2}}}{4 b}-\frac {\sqrt {x}}{4 a}}{\left (a x +b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.20, size = 66, normalized size = 0.90 \[ \frac {\frac {a}{\sqrt {x}} - \frac {b}{x^{\frac {3}{2}}}}{4 \, {\left (a^{3} b + \frac {2 \, a^{2} b^{2}}{x} + \frac {a b^{3}}{x^{2}}\right )}} - \frac {\arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{4 \, \sqrt {a b} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 57, normalized size = 0.78 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{4\,a^{3/2}\,b^{3/2}}-\frac {\frac {\sqrt {x}}{4\,a}-\frac {x^{3/2}}{4\,b}}{a^2\,x^2+2\,a\,b\,x+b^2} \]
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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