Optimal. Leaf size=46 \[ \frac {\sqrt {x}}{a (a-b x)}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {44, 65, 214}
\begin {gather*} \frac {\tanh ^{-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 214
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 {\tanh ^{-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 = 46, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x}}{a (a-b x)}+\frac {\tanh ^{-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 = 4.19, size = 266, normalized size = 5.78 \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 = 0.11, size = 37, normalized size = 0.80
method | result | size |
derivativedivides | \(\frac {\sqrt {x}}{a \left (-b x +a \right )}+\frac {\arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}\) | \(37\) |
default | \(\frac {\sqrt {x}}{a \left (-b x +a \right )}+\frac {\arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 56, normalized size = 1.22 \begin {gather*} -\frac {\sqrt {x}}{a b x - a^{2}} - \frac {\log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{2 \, \sqrt {a b} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 122, normalized size = 2.65 \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 = 2.74, size = 252, normalized size = 5.48 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 \sqrt {x}}{a^{2}} & \text {for}\: b = 0 \\- \frac {2}{3 b^{2} x^{\frac {3}{2}}} & \text {for}\: a = 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 = 53, normalized size = 1.15 \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 = 0.05, size = 34, normalized size = 0.74 \begin {gather*} \frac {\sqrt {x}}{a\,\left (a-b\,x\right )}+\frac {\mathrm {atanh}\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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