Optimal. Leaf size=60 \[ \frac {2 \sqrt {x} (A b-a B)}{a b \sqrt {a+b x}}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 60, 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 = {78, 63, 217, 206} \[ \frac {2 \sqrt {x} (A b-a B)}{a b \sqrt {a+b x}}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {x} (a+b x)^{3/2}} \, dx &=\frac {2 (A b-a B) \sqrt {x}}{a b \sqrt {a+b x}}+\frac {B \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{b}\\ &=\frac {2 (A b-a B) \sqrt {x}}{a b \sqrt {a+b x}}+\frac {(2 B) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{b}\\ &=\frac {2 (A b-a B) \sqrt {x}}{a b \sqrt {a+b x}}+\frac {(2 B) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{b}\\ &=\frac {2 (A b-a B) \sqrt {x}}{a b \sqrt {a+b x}}+\frac {2 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 = 76, normalized size = 1.27 \[ \frac {2 a^{3/2} B \sqrt {\frac {b x}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+2 \sqrt {b} \sqrt {x} (A b-a B)}{a b^{3/2} \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 157, normalized size = 2.62 \[ \left [\frac {{\left (B a b x + B a^{2}\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (B a b - A b^{2}\right )} \sqrt {b x + a} \sqrt {x}}{a b^{3} x + a^{2} b^{2}}, -\frac {2 \, {\left ({\left (B a b x + B a^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (B a b - A b^{2}\right )} \sqrt {b x + a} \sqrt {x}\right )}}{a b^{3} x + a^{2} b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 86.16, size = 97, normalized size = 1.62 \[ -\frac {B \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{\sqrt {b} {\left | b \right |}} - \frac {4 \, {\left (B a \sqrt {b} - A b^{\frac {3}{2}}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 121, normalized size = 2.02 \[ \frac {\left (B a b x \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+B \,a^{2} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+2 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {3}{2}}-2 \sqrt {\left (b x +a \right ) x}\, B a \sqrt {b}\right ) \sqrt {x}}{\sqrt {\left (b x +a \right ) x}\, \sqrt {b x +a}\, a \,b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 81, normalized size = 1.35 \[ \frac {2 \, \sqrt {b x^{2} + a x} A}{a b x + a^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{b^{2} x + a b} + \frac {B \log \left (2 \, x + \frac {a}{b} + \frac {2 \, \sqrt {b x^{2} + a x}}{\sqrt {b}}\right )}{b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {A+B\,x}{\sqrt {x}\,{\left (a+b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.50, size = 68, normalized size = 1.13 \[ \frac {2 A}{a \sqrt {b} \sqrt {\frac {a}{b x} + 1}} + B \left (\frac {2 \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {2 \sqrt {x}}{\sqrt {a} b \sqrt {1 + \frac {b x}{a}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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