Optimal. Leaf size=82 \[ \frac {(2 A b+a B) \sqrt {x} \sqrt {a+b x}}{a}-\frac {2 A (a+b x)^{3/2}}{a \sqrt {x}}+\frac {(2 A b+a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{\sqrt {b}} \]
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Rubi [A]
time = 0.03, 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 = {79, 52, 65, 223,
212} \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 52
Rule 65
Rule 79
Rule 212
Rule 223
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) \text {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) \text {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 = 0.13, size = 61, normalized size = 0.74 \begin {gather*} \frac {\sqrt {a+b x} (-2 A+B x)}{\sqrt {x}}+\frac {(-2 A b-a B) \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 118, normalized size = 1.44
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-B x +2 A \right )}{\sqrt {x}}+\frac {\left (\ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {b}\, A +\frac {\ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) B a}{2 \sqrt {b}}\right ) \sqrt {\left (b x +a \right ) x}}{\sqrt {b x +a}\, \sqrt {x}}\) | \(103\) |
default | \(\frac {\sqrt {b x +a}\, \left (2 A \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) b x +B \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a x +2 B x \sqrt {\left (b x +a \right ) x}\, \sqrt {b}-4 A \sqrt {b}\, \sqrt {\left (b x +a \right ) x}\right )}{2 \sqrt {x}\, \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, 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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Fricas [A]
time = 1.18, 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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Sympy [A]
time = 4.76, 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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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \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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