Optimal. Leaf size=44 \[ \frac {b \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {\sqrt {b x-a}}{a x} \]
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Rubi [A] time = 0.01, antiderivative size = 44, 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, 205} \begin {gather*} \frac {b \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {\sqrt {b x-a}}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {-a+b x}} \, dx &=\frac {\sqrt {-a+b x}}{a x}+\frac {b \int \frac {1}{x \sqrt {-a+b x}} \, dx}{2 a}\\ &=\frac {\sqrt {-a+b x}}{a x}+\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{a}\\ &=\frac {\sqrt {-a+b x}}{a x}+\frac {b \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 53, normalized size = 1.20 \begin {gather*} \frac {b \sqrt {b x-a} \left (\frac {a}{b x}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {b x}{a}}\right )}{\sqrt {1-\frac {b x}{a}}}\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 44, normalized size = 1.00 \begin {gather*} \frac {b \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {\sqrt {b x-a}}{a x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 97, normalized size = 2.20 \begin {gather*} \left [-\frac {\sqrt {-a} b x \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) - 2 \, \sqrt {b x - a} a}{2 \, a^{2} x}, \frac {\sqrt {a} b x \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + \sqrt {b x - a} a}{a^{2} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 43, normalized size = 0.98 \begin {gather*} \frac {\frac {b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {\sqrt {b x - a} b}{a x}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 37, normalized size = 0.84 \begin {gather*} \frac {b \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}+\frac {\sqrt {b x -a}}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 46, normalized size = 1.05 \begin {gather*} \frac {\sqrt {b x - a} b}{{\left (b x - a\right )} a + a^{2}} + \frac {b \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 36, normalized size = 0.82 \begin {gather*} \frac {\sqrt {b\,x-a}}{a\,x}+\frac {b\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.46, size = 121, normalized size = 2.75 \begin {gather*} \begin {cases} \frac {i \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{a \sqrt {x}} + \frac {i b \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{a^{\frac {3}{2}}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {1}{\sqrt {b} x^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}} + \frac {\sqrt {b}}{a \sqrt {x} \sqrt {- \frac {a}{b x} + 1}} - \frac {b \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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