Optimal. Leaf size=56 \[ \frac {(2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}}+\frac {B \sqrt {x} \sqrt {a+b x}}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {80, 63, 217, 206} \[ \frac {(2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}}+\frac {B \sqrt {x} \sqrt {a+b x}}{b} \]
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {x} \sqrt {a+b x}} \, dx &=\frac {B \sqrt {x} \sqrt {a+b x}}{b}+\frac {\left (A b-\frac {a B}{2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{b}\\ &=\frac {B \sqrt {x} \sqrt {a+b x}}{b}+\frac {\left (2 \left (A b-\frac {a B}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{b}\\ &=\frac {B \sqrt {x} \sqrt {a+b x}}{b}+\frac {\left (2 \left (A b-\frac {a B}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{b}\\ &=\frac {B \sqrt {x} \sqrt {a+b x}}{b}+\frac {(2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 77, normalized size = 1.38 \[ \frac {\sqrt {b} B \sqrt {x} (a+b x)-\sqrt {a} \sqrt {\frac {b x}{a}+1} (a B-2 A b) \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2} \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 108, normalized size = 1.93 \[ \left [\frac {2 \, \sqrt {b x + a} B b \sqrt {x} - {\left (B a - 2 \, A b\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right )}{2 \, b^{2}}, \frac {\sqrt {b x + a} B b \sqrt {x} + {\left (B a - 2 \, A b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right )}{b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 101, normalized size = 1.80 \[ \frac {\sqrt {b x +a}\, \left (2 A b \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-B a \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+2 \sqrt {\left (b x +a \right ) x}\, B \sqrt {b}\right ) \sqrt {x}}{2 \sqrt {\left (b x +a \right ) x}\, b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 75, normalized size = 1.34 \[ -\frac {B a \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{2 \, b^{\frac {3}{2}}} + \frac {A \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{\sqrt {b}} + \frac {\sqrt {b x^{2} + a x} B}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 77, normalized size = 1.38 \[ \frac {B\,\sqrt {x}\,\sqrt {a+b\,x}}{b}-\frac {2\,B\,a\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a+b\,x}-\sqrt {a}}\right )}{b^{3/2}}-\frac {4\,A\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {-b}\,\sqrt {x}}\right )}{\sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.29, size = 73, normalized size = 1.30 \[ \frac {2 A \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}} + \frac {B \sqrt {a} \sqrt {x} \sqrt {1 + \frac {b x}{a}}}{b} - \frac {B a \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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