Optimal. Leaf size=45 \[ \frac {\sqrt {x}}{a (a+b x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 45, 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 = {44, 65, 211}
\begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}+\frac {\sqrt {x}}{a (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 211
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx &=\frac {\sqrt {x}}{a (a+b x)}+\frac {\int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 a}\\ &=\frac {\sqrt {x}}{a (a+b x)}+\frac {\text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a}\\ &=\frac {\sqrt {x}}{a (a+b x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 45, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x}}{a (a+b x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}} \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 = 4.13, size = 281, normalized size = 6.24 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\frac {1}{x^{\frac {3}{2}}}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {2 \sqrt {x}}{a^2},b\text {==}0\right \},\left \{\frac {-2}{3 b^2 x^{\frac {3}{2}}},a\text {==}0\right \}\right \},-\frac {a \text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{2 a^2 b \sqrt {-\frac {a}{b}}+2 a b^2 x \sqrt {-\frac {a}{b}}}+\frac {a \text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{2 a^2 b \sqrt {-\frac {a}{b}}+2 a b^2 x \sqrt {-\frac {a}{b}}}+\frac {2 b \sqrt {x} \sqrt {-\frac {a}{b}}}{2 a^2 b \sqrt {-\frac {a}{b}}+2 a b^2 x \sqrt {-\frac {a}{b}}}-\frac {b x \text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{2 a^2 b \sqrt {-\frac {a}{b}}+2 a b^2 x \sqrt {-\frac {a}{b}}}+\frac {b x \text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{2 a^2 b \sqrt {-\frac {a}{b}}+2 a b^2 x \sqrt {-\frac {a}{b}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 36, normalized size = 0.80
method | result | size |
derivativedivides | \(\frac {\sqrt {x}}{a \left (b x +a \right )}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}\) | \(36\) |
default | \(\frac {\sqrt {x}}{a \left (b x +a \right )}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 35, normalized size = 0.78 \begin {gather*} \frac {\sqrt {x}}{a b x + a^{2}} + \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 116, normalized size = 2.58 \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^{2} b^{2} x + a^{3} b\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^{2} b^{2} x + a^{3} b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.75, size = 277, normalized size = 6.16 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 b^{2} x^{\frac {3}{2}}} & \text {for}\: a = 0 \\\frac {2 \sqrt {x}}{a^{2}} & \text {for}\: b = 0 \\\frac {a \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{2} b \sqrt {- \frac {a}{b}} + 2 a b^{2} x \sqrt {- \frac {a}{b}}} - \frac {a \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{2} b \sqrt {- \frac {a}{b}} + 2 a b^{2} x \sqrt {- \frac {a}{b}}} + \frac {2 b \sqrt {x} \sqrt {- \frac {a}{b}}}{2 a^{2} b \sqrt {- \frac {a}{b}} + 2 a b^{2} x \sqrt {- \frac {a}{b}}} + \frac {b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{2} b \sqrt {- \frac {a}{b}} + 2 a b^{2} x \sqrt {- \frac {a}{b}}} - \frac {b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{2} b \sqrt {- \frac {a}{b}} + 2 a b^{2} 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 = 48, normalized size = 1.07 \begin {gather*} 2 \left (\frac {\sqrt {x}}{2 a \left (x b+a\right )}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a\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 = 0.09, size = 33, normalized size = 0.73 \begin {gather*} \frac {\sqrt {x}}{a\,\left (a+b\,x\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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