Optimal. Leaf size=46 \[ -\frac {\sqrt {x}}{b (a+b x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {43, 65, 211}
\begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}-\frac {\sqrt {x}}{b (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 211
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{(a+b x)^2} \, dx &=-\frac {\sqrt {x}}{b (a+b x)}+\frac {\int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 b}\\ &=-\frac {\sqrt {x}}{b (a+b x)}+\frac {\text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b}\\ &=-\frac {\sqrt {x}}{b (a+b x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 46, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {x}}{b (a+b x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 3.11, size = 276, normalized size = 6.00 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\frac {1}{\sqrt {x}}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {-2}{b^2 \sqrt {x}},a\text {==}0\right \},\left \{\frac {2 x^{\frac {3}{2}}}{3 a^2},b\text {==}0\right \}\right \},-\frac {a \text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{2 a b^2 \sqrt {-\frac {a}{b}}+2 b^3 x \sqrt {-\frac {a}{b}}}+\frac {a \text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{2 a b^2 \sqrt {-\frac {a}{b}}+2 b^3 x \sqrt {-\frac {a}{b}}}-\frac {2 b \sqrt {x} \sqrt {-\frac {a}{b}}}{2 a b^2 \sqrt {-\frac {a}{b}}+2 b^3 x \sqrt {-\frac {a}{b}}}-\frac {b x \text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{2 a b^2 \sqrt {-\frac {a}{b}}+2 b^3 x \sqrt {-\frac {a}{b}}}+\frac {b x \text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{2 a b^2 \sqrt {-\frac {a}{b}}+2 b^3 x \sqrt {-\frac {a}{b}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 37, normalized size = 0.80
method | result | size |
derivativedivides | \(-\frac {\sqrt {x}}{b \left (b x +a \right )}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(37\) |
default | \(-\frac {\sqrt {x}}{b \left (b x +a \right )}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 37, normalized size = 0.80 \begin {gather*} -\frac {\sqrt {x}}{b^{2} x + a b} + \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.63, size = 115, normalized size = 2.50 \begin {gather*} \left [-\frac {2 \, a b \sqrt {x} + \sqrt {-a b} {\left (b x + a\right )} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right )}{2 \, {\left (a b^{3} x + a^{2} b^{2}\right )}}, -\frac {a b \sqrt {x} + \sqrt {a b} {\left (b x + a\right )} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right )}{a b^{3} x + a^{2} b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.70, size = 269, normalized size = 5.85 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a^{2}} & \text {for}\: b = 0 \\- \frac {2}{b^{2} \sqrt {x}} & \text {for}\: a = 0 \\\frac {a \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a b^{2} \sqrt {- \frac {a}{b}} + 2 b^{3} x \sqrt {- \frac {a}{b}}} - \frac {a \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a b^{2} \sqrt {- \frac {a}{b}} + 2 b^{3} x \sqrt {- \frac {a}{b}}} - \frac {2 b \sqrt {x} \sqrt {- \frac {a}{b}}}{2 a b^{2} \sqrt {- \frac {a}{b}} + 2 b^{3} x \sqrt {- \frac {a}{b}}} + \frac {b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a b^{2} \sqrt {- \frac {a}{b}} + 2 b^{3} x \sqrt {- \frac {a}{b}}} - \frac {b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a b^{2} \sqrt {- \frac {a}{b}} + 2 b^{3} x \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 49, normalized size = 1.07 \begin {gather*} 2 \left (-\frac {\sqrt {x}}{2 b \left (x b+a\right )}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b\cdot 2 \sqrt {a b}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 34, normalized size = 0.74 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a}\,b^{3/2}}-\frac {\sqrt {x}}{b\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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