Optimal. Leaf size=45 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {a} b^{3/2}}+\frac {\sqrt {x}}{b (a x+b)} \]
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Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {263, 51, 63, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {a} b^{3/2}}+\frac {\sqrt {x}}{b (a x+b)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rule 263
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{5/2}} \, dx &=\int \frac {1}{\sqrt {x} (b+a x)^2} \, dx\\ &=\frac {\sqrt {x}}{b (b+a x)}+\frac {\int \frac {1}{\sqrt {x} (b+a x)} \, dx}{2 b}\\ &=\frac {\sqrt {x}}{b (b+a x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )}{b}\\ &=\frac {\sqrt {x}}{b (b+a x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {a} b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 45, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {a} b^{3/2}}+\frac {\sqrt {x}}{a b x+b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 116, normalized size = 2.58 \[ \left [\frac {2 \, a b \sqrt {x} - \sqrt {-a b} {\left (a x + b\right )} \log \left (\frac {a x - b - 2 \, \sqrt {-a b} \sqrt {x}}{a x + b}\right )}{2 \, {\left (a^{2} b^{2} x + a b^{3}\right )}}, \frac {a b \sqrt {x} - \sqrt {a b} {\left (a x + b\right )} \arctan \left (\frac {\sqrt {a b}}{a \sqrt {x}}\right )}{a^{2} b^{2} x + a b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 35, normalized size = 0.78 \[ \frac {\arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {\sqrt {x}}{{\left (a x + b\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.80 \[ \frac {\arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {\sqrt {x}}{\left (a x +b \right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.33, size = 39, normalized size = 0.87 \[ -\frac {\arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} b} + \frac {1}{{\left (a b + \frac {b^{2}}{x}\right )} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 33, normalized size = 0.73 \[ \frac {\sqrt {x}}{b\,\left (b+a\,x\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{\sqrt {a}\,b^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 102.66, size = 328, normalized size = 7.29 \[ \begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 \sqrt {x}}{b^{2}} & \text {for}\: a = 0 \\- \frac {2}{3 a^{2} x^{\frac {3}{2}}} & \text {for}\: b = 0 \\\frac {2 i a \sqrt {b} \sqrt {x} \sqrt {\frac {1}{a}}}{2 i a^{2} b^{\frac {3}{2}} x \sqrt {\frac {1}{a}} + 2 i a b^{\frac {5}{2}} \sqrt {\frac {1}{a}}} + \frac {a x \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{2 i a^{2} b^{\frac {3}{2}} x \sqrt {\frac {1}{a}} + 2 i a b^{\frac {5}{2}} \sqrt {\frac {1}{a}}} - \frac {a x \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{2 i a^{2} b^{\frac {3}{2}} x \sqrt {\frac {1}{a}} + 2 i a b^{\frac {5}{2}} \sqrt {\frac {1}{a}}} + \frac {b \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{2 i a^{2} b^{\frac {3}{2}} x \sqrt {\frac {1}{a}} + 2 i a b^{\frac {5}{2}} \sqrt {\frac {1}{a}}} - \frac {b \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{2 i a^{2} b^{\frac {3}{2}} x \sqrt {\frac {1}{a}} + 2 i a b^{\frac {5}{2}} \sqrt {\frac {1}{a}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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