Optimal. Leaf size=45 \[ \frac {\sqrt {x}}{b (b+a x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {a} b^{3/2}} \]
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Rubi [A]
time = 0.01, 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 = {269, 44, 65,
211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {a} b^{3/2}}+\frac {\sqrt {x}}{b (a x+b)} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 211
Rule 269
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 {\text {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 = 0.05, size = 45, normalized size = 1.00 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 36, normalized size = 0.80
method | result | size |
derivativedivides | \(\frac {\sqrt {x}}{b \left (a x +b \right )}+\frac {\arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(36\) |
default | \(\frac {\sqrt {x}}{b \left (a x +b \right )}+\frac {\arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 39, normalized size = 0.87 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 116, normalized size = 2.58 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 277 vs.
\(2 (37) = 74\).
time = 16.40, size = 277, normalized size = 6.16 \begin {gather*} \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 a \sqrt {x} \sqrt {- \frac {b}{a}}}{2 a^{2} b x \sqrt {- \frac {b}{a}} + 2 a b^{2} \sqrt {- \frac {b}{a}}} + \frac {a x \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{2 a^{2} b x \sqrt {- \frac {b}{a}} + 2 a b^{2} \sqrt {- \frac {b}{a}}} - \frac {a x \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{2 a^{2} b x \sqrt {- \frac {b}{a}} + 2 a b^{2} \sqrt {- \frac {b}{a}}} + \frac {b \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{2 a^{2} b x \sqrt {- \frac {b}{a}} + 2 a b^{2} \sqrt {- \frac {b}{a}}} - \frac {b \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{2 a^{2} b x \sqrt {- \frac {b}{a}} + 2 a b^{2} \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.18, size = 35, normalized size = 0.78 \begin {gather*} \frac {\arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {\sqrt {x}}{{\left (a x + b\right )} b} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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