Optimal. Leaf size=46 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}+\frac {\sqrt {x}}{a (a-b x)} \]
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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 = {51, 63, 208} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}+\frac {\sqrt {x}}{a (a-b x)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
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 {\operatorname {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 = 0.02, size = 46, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}+\frac {\sqrt {x}}{a^2-a b x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 122, normalized size = 2.65 \[ \left [-\frac {2 \, a b \sqrt {x} - \sqrt {a b} {\left (b x - a\right )} \log \left (\frac {b x + a + 2 \, \sqrt {a b} \sqrt {x}}{b x - a}\right )}{2 \, {\left (a^{2} b^{2} x - a^{3} b\right )}}, -\frac {a b \sqrt {x} + \sqrt {-a b} {\left (b x - a\right )} \arctan \left (\frac {\sqrt {-a b}}{b \sqrt {x}}\right )}{a^{2} b^{2} x - a^{3} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 41, normalized size = 0.89 \[ -\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} a} - \frac {\sqrt {x}}{{\left (b x - a\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 39, normalized size = 0.85 \[ \frac {\arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}-\frac {\sqrt {x}}{\left (b x -a \right ) a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 56, normalized size = 1.22 \[ -\frac {\sqrt {x}}{a b x - a^{2}} - \frac {\log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{2 \, \sqrt {a b} a} \]
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 \[ \frac {\sqrt {x}}{a\,\left (a-b\,x\right )}+\frac {\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}\,\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.41, size = 303, normalized size = 6.59 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 \sqrt {x}}{a^{2}} & \text {for}\: b = 0 \\- \frac {2}{3 b^{2} x^{\frac {3}{2}}} & \text {for}\: a = 0 \\- \frac {2 \sqrt {a} b \sqrt {x} \sqrt {\frac {1}{b}}}{- 2 a^{\frac {5}{2}} b \sqrt {\frac {1}{b}} + 2 a^{\frac {3}{2}} b^{2} x \sqrt {\frac {1}{b}}} + \frac {a \log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{- 2 a^{\frac {5}{2}} b \sqrt {\frac {1}{b}} + 2 a^{\frac {3}{2}} b^{2} x \sqrt {\frac {1}{b}}} - \frac {a \log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{- 2 a^{\frac {5}{2}} b \sqrt {\frac {1}{b}} + 2 a^{\frac {3}{2}} b^{2} x \sqrt {\frac {1}{b}}} - \frac {b x \log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{- 2 a^{\frac {5}{2}} b \sqrt {\frac {1}{b}} + 2 a^{\frac {3}{2}} b^{2} x \sqrt {\frac {1}{b}}} + \frac {b x \log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{- 2 a^{\frac {5}{2}} b \sqrt {\frac {1}{b}} + 2 a^{\frac {3}{2}} b^{2} x \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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