Optimal. Leaf size=49 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{A \sqrt{a+b x}}{a x} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0742844, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{A \sqrt{a+b x}}{a x} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^2*Sqrt[a + b*x]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 6.97793, size = 42, normalized size = 0.86 \[ - \frac{A \sqrt{a + b x}}{a x} + \frac{2 \left (\frac{A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**2/(b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0657315, size = 51, normalized size = 1.04 \[ -\frac{(2 a B-A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{A \sqrt{a+b x}}{a x} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^2*Sqrt[a + b*x]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.016, size = 42, normalized size = 0.9 \[{(Ab-2\,Ba){\it Artanh} \left ({1\sqrt{bx+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}}-{\frac{A}{ax}\sqrt{bx+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^2/(b*x+a)^(1/2),x)
[Out]
_______________________________________________________________________________________
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)/(sqrt(b*x + a)*x^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.23151, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (2 \, B a - A b\right )} x \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \, \sqrt{b x + a} A \sqrt{a}}{2 \, a^{\frac{3}{2}} x}, \frac{{\left (2 \, B a - A b\right )} x \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) - \sqrt{b x + a} A \sqrt{-a}}{\sqrt{-a} a x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(b*x + a)*x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 26.1596, size = 165, normalized size = 3.37 \[ - \frac{A \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{a \sqrt{x}} + \frac{A b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{a^{\frac{3}{2}}} + 2 B \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{a}} \sqrt{a + b x}} \right )}}{a \sqrt{- \frac{1}{a}}} & \text{for}\: - \frac{1}{a} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{a + b x} \sqrt{\frac{1}{a}}} \right )}}{a \sqrt{\frac{1}{a}}} & \text{for}\: - \frac{1}{a} < 0 \wedge \frac{1}{a} < \frac{1}{a + b x} \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{a + b x} \sqrt{\frac{1}{a}}} \right )}}{a \sqrt{\frac{1}{a}}} & \text{for}\: \frac{1}{a} > \frac{1}{a + b x} \wedge - \frac{1}{a} < 0 \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x**2/(b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.213989, size = 78, normalized size = 1.59 \[ -\frac{\frac{\sqrt{b x + a} A b}{a x} - \frac{{\left (2 \, B a b - A b^{2}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(b*x + a)*x^2),x, algorithm="giac")
[Out]