Optimal. Leaf size=82 \[ \frac {\sqrt {x} \sqrt {a+b x} (a B+2 A b)}{a}+\frac {(a B+2 A b) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{\sqrt {b}}-\frac {2 A (a+b x)^{3/2}}{a \sqrt {x}} \]
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Rubi [A] time = 0.04, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {78, 50, 63, 217, 206} \begin {gather*} \frac {\sqrt {x} \sqrt {a+b x} (a B+2 A b)}{a}+\frac {(a B+2 A b) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{\sqrt {b}}-\frac {2 A (a+b x)^{3/2}}{a \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{x^{3/2}} \, dx &=-\frac {2 A (a+b x)^{3/2}}{a \sqrt {x}}+\frac {\left (2 \left (A b+\frac {a B}{2}\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx}{a}\\ &=\frac {(2 A b+a B) \sqrt {x} \sqrt {a+b x}}{a}-\frac {2 A (a+b x)^{3/2}}{a \sqrt {x}}+\frac {1}{2} (2 A b+a B) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx\\ &=\frac {(2 A b+a B) \sqrt {x} \sqrt {a+b x}}{a}-\frac {2 A (a+b x)^{3/2}}{a \sqrt {x}}+(2 A b+a B) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {(2 A b+a B) \sqrt {x} \sqrt {a+b x}}{a}-\frac {2 A (a+b x)^{3/2}}{a \sqrt {x}}+(2 A b+a B) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )\\ &=\frac {(2 A b+a B) \sqrt {x} \sqrt {a+b x}}{a}-\frac {2 A (a+b x)^{3/2}}{a \sqrt {x}}+\frac {(2 A b+a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 72, normalized size = 0.88 \begin {gather*} \sqrt {a+b x} \left (\frac {(a B+2 A b) \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {\frac {b x}{a}+1}}+\frac {B x-2 A}{\sqrt {x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 61, normalized size = 0.74 \begin {gather*} \frac {\sqrt {a+b x} (B x-2 A)}{\sqrt {x}}+\frac {(-a B-2 A b) \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.56, size = 131, normalized size = 1.60 \begin {gather*} \left [\frac {{\left (B a + 2 \, A b\right )} \sqrt {b} x \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (B b x - 2 \, A b\right )} \sqrt {b x + a} \sqrt {x}}{2 \, b x}, -\frac {{\left (B a + 2 \, A b\right )} \sqrt {-b} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (B b x - 2 \, A b\right )} \sqrt {b x + a} \sqrt {x}}{b x}\right ] \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 118, normalized size = 1.44 \begin {gather*} \frac {\sqrt {b x +a}\, \left (2 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 x \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}\, x -4 \sqrt {\left (b x +a \right ) x}\, A \sqrt {b}\right )}{2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}\, \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 89, normalized size = 1.09 \begin {gather*} \frac {B a \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{2 \, \sqrt {b}} + A \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + \sqrt {b x^{2} + a x} B - \frac {2 \, \sqrt {b x^{2} + a x} A}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (A+B\,x\right )\,\sqrt {a+b\,x}}{x^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.63, size = 116, normalized size = 1.41 \begin {gather*} A \left (- \frac {2 \sqrt {a}}{\sqrt {x} \sqrt {1 + \frac {b x}{a}}} + 2 \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} - \frac {2 b \sqrt {x}}{\sqrt {a} \sqrt {1 + \frac {b x}{a}}}\right ) + B \left (\sqrt {a} \sqrt {x} \sqrt {1 + \frac {b x}{a}} + \frac {a \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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