Optimal. Leaf size=50 \[ \frac {2 (A b-a B)}{a b \sqrt {a+b x}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 50, 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} \begin {gather*} \frac {2 (A b-a B)}{a b \sqrt {a+b x}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {A+B x}{x (a+b x)^{3/2}} \, dx &=\frac {2 (A b-a B)}{a b \sqrt {a+b x}}+\frac {A \int \frac {1}{x \sqrt {a+b x}} \, dx}{a}\\ &=\frac {2 (A b-a B)}{a b \sqrt {a+b x}}+\frac {(2 A) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{a b}\\ &=\frac {2 (A b-a B)}{a b \sqrt {a+b x}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 50, normalized size = 1.00 \begin {gather*} \frac {2 (A b-a B)}{a b \sqrt {a+b x}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 50, normalized size = 1.00 \begin {gather*} -\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2 (a B-A b)}{a b \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.36, size = 151, normalized size = 3.02 \begin {gather*} \left [\frac {{\left (A b^{2} x + A a b\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (B a^{2} - A a b\right )} \sqrt {b x + a}}{a^{2} b^{2} x + a^{3} b}, \frac {2 \, {\left ({\left (A b^{2} x + A a b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (B a^{2} - A a b\right )} \sqrt {b x + a}\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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giac [A] time = 1.26, size = 49, normalized size = 0.98 \begin {gather*} \frac {2 \, A \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {2 \, {\left (B a - A b\right )}}{\sqrt {b x + a} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 46, normalized size = 0.92 \begin {gather*} \frac {-\frac {2 A b \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}-\frac {2 \left (-A b +B a \right )}{\sqrt {b x +a}\, a}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.92, size = 57, normalized size = 1.14 \begin {gather*} \frac {A \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {2 \, {\left (B a - A b\right )}}{\sqrt {b x + a} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 42, normalized size = 0.84 \begin {gather*} \frac {2\,\left (A\,b-B\,a\right )}{a\,b\,\sqrt {a+b\,x}}-\frac {2\,A\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.88, size = 49, normalized size = 0.98 \begin {gather*} \frac {2 A \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{a \sqrt {- a}} - \frac {2 \left (- A b + B a\right )}{a b \sqrt {a + b x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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