Optimal. Leaf size=49 \[ \frac {(A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {A \sqrt {a+b x}}{a x} \]
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Rubi [A] time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {78, 63, 208} \[ \frac {(A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {A \sqrt {a+b x}}{a x} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {A+B x}{x^2 \sqrt {a+b x}} \, dx &=-\frac {A \sqrt {a+b x}}{a x}+\frac {\left (-\frac {A b}{2}+a B\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{a}\\ &=-\frac {A \sqrt {a+b x}}{a x}+\frac {\left (2 \left (-\frac {A b}{2}+a B\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{a b}\\ &=-\frac {A \sqrt {a+b x}}{a x}+\frac {(A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 1.00 \[ \frac {(A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {A \sqrt {a+b x}}{a x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 111, normalized size = 2.27 \[ \left [-\frac {{\left (2 \, B a - A b\right )} \sqrt {a} x \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, \sqrt {b x + a} A a}{2 \, a^{2} x}, \frac {{\left (2 \, B a - A b\right )} \sqrt {-a} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - \sqrt {b x + a} A a}{a^{2} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.28, size = 58, normalized size = 1.18 \[ -\frac {\frac {\sqrt {b x + a} A b}{a x} - \frac {{\left (2 \, B a b - A b^{2}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 42, normalized size = 0.86 \[ \frac {\left (A b -2 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}-\frac {\sqrt {b x +a}\, A}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.94, size = 74, normalized size = 1.51 \[ -\frac {1}{2} \, b {\left (\frac {2 \, \sqrt {b x + a} A}{{\left (b x + a\right )} a - a^{2}} - \frac {{\left (2 \, B a - A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}} b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 41, normalized size = 0.84 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (A\,b-2\,B\,a\right )}{a^{3/2}}-\frac {A\,\sqrt {a+b\,x}}{a\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 27.86, size = 82, normalized size = 1.67 \[ - \frac {A \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{a \sqrt {x}} + \frac {A b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{a^{\frac {3}{2}}} + \frac {2 B \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{a}} \sqrt {a + b x}} \right )}}{a \sqrt {- \frac {1}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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