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 = 0.0619321, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{2 \sqrt{x} (A b-a B)}{a b \sqrt{a+b x}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(Sqrt[x]*(a + b*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 6.4396, size = 53, normalized size = 0.88 \[ \frac{2 B \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{b^{\frac{3}{2}}} + \frac{2 \sqrt{x} \left (A b - B a\right )}{a b \sqrt{a + b x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(b*x+a)**(3/2)/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.0810571, size = 63, normalized size = 1.05 \[ \frac{2 B \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{b^{3/2}}-\frac{2 \sqrt{x} (a B-A b)}{a b \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(Sqrt[x]*(a + b*x)^(3/2)),x]
[Out]
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Maple [B] time = 0.025, size = 121, normalized size = 2. \[{\frac{1}{a} \left ( B\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ) xab+2\,A{b}^{3/2}\sqrt{x \left ( bx+a \right ) }+B\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ){a}^{2}-2\,Ba\sqrt{b}\sqrt{x \left ( bx+a \right ) } \right ) \sqrt{x}{b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{\frac{1}{\sqrt{bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(b*x+a)^(3/2)/x^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^(3/2)*sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.24141, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b x + a} B a \sqrt{x} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) - 2 \,{\left (B a - A b\right )} \sqrt{b} x}{\sqrt{b x + a} a b^{\frac{3}{2}} \sqrt{x}}, \frac{2 \,{\left (\sqrt{b x + a} B a \sqrt{x} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (B a - A b\right )} \sqrt{-b} x\right )}}{\sqrt{b x + a} a \sqrt{-b} b \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^(3/2)*sqrt(x)),x, algorithm="fricas")
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Sympy [A] time = 37.2562, 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.
[In] integrate((B*x+A)/(b*x+a)**(3/2)/x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.234905, size = 131, normalized size = 2.18 \[ -\frac{B{\rm ln}\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{\sqrt{b}{\left | b \right |}} - \frac{4 \,{\left (B a \sqrt{b} - A b^{\frac{3}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^(3/2)*sqrt(x)),x, algorithm="giac")
[Out]