Optimal. Leaf size=47 \[ \frac {\sqrt {x}}{b (a-b x)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 47, 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 = {47, 63, 208} \begin {gather*} \frac {\sqrt {x}}{b (a-b x)}-\frac {\tanh ^{-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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Rule 47
Rule 63
Rule 208
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 {\operatorname {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{b}\\ &=\frac {\sqrt {x}}{b (a-b x)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 1.30 \begin {gather*} \frac {\sqrt {a} \sqrt {b} \sqrt {x}+(b x-a) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2} (a-b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 49, normalized size = 1.04 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}-\frac {\sqrt {x}}{b (b x-a)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 123, normalized size = 2.62 \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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giac [A] time = 1.06, size = 40, normalized size = 0.85 \begin {gather*} \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} b} - \frac {\sqrt {x}}{{\left (b x - a\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 40, normalized size = 0.85 \begin {gather*} -\frac {\arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}-\frac {\sqrt {x}}{\left (b x -a \right ) b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 56, normalized size = 1.19 \begin {gather*} -\frac {\sqrt {x}}{b^{2} x - a b} + \frac {\log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{2 \, \sqrt {a b} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 35, normalized size = 0.74 \begin {gather*} \frac {\sqrt {x}}{b\,\left (a-b\,x\right )}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a}\,b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.43, size = 311, normalized size = 6.62 \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 {2 \sqrt {a} b \sqrt {x} \sqrt {\frac {1}{b}}}{- 2 a^{\frac {3}{2}} b^{2} \sqrt {\frac {1}{b}} + 2 \sqrt {a} b^{3} x \sqrt {\frac {1}{b}}} - \frac {a \log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{- 2 a^{\frac {3}{2}} b^{2} \sqrt {\frac {1}{b}} + 2 \sqrt {a} b^{3} x \sqrt {\frac {1}{b}}} + \frac {a \log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{- 2 a^{\frac {3}{2}} b^{2} \sqrt {\frac {1}{b}} + 2 \sqrt {a} b^{3} x \sqrt {\frac {1}{b}}} + \frac {b x \log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{- 2 a^{\frac {3}{2}} b^{2} \sqrt {\frac {1}{b}} + 2 \sqrt {a} b^{3} x \sqrt {\frac {1}{b}}} - \frac {b x \log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{- 2 a^{\frac {3}{2}} b^{2} \sqrt {\frac {1}{b}} + 2 \sqrt {a} b^{3} x \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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