Optimal. Leaf size=63 \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}+\frac{\sqrt{x} (A b-a B)}{a b (a+b x)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0775568, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}+\frac{\sqrt{x} (A b-a B)}{a b (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(Sqrt[x]*(a^2 + 2*a*b*x + b^2*x^2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 20.8778, size = 53, normalized size = 0.84 \[ \frac{\sqrt{x} \left (A b - B a\right )}{a b \left (a + b x\right )} + \frac{\left (A b + B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(b**2*x**2+2*a*b*x+a**2)/x**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.064859, size = 64, normalized size = 1.02 \[ \frac{(a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} b^{3/2}}-\frac{\sqrt{x} (a B-A b)}{a b (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(Sqrt[x]*(a^2 + 2*a*b*x + b^2*x^2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.02, size = 69, normalized size = 1.1 \[{\frac{Ab-Ba}{ab \left ( bx+a \right ) }\sqrt{x}}+{\frac{A}{a}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{b}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(b^2*x^2+2*a*b*x+a^2)/x^(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)/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(x)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.297203, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \,{\left (B a - A b\right )} \sqrt{-a b} \sqrt{x} -{\left (B a^{2} + A a b +{\left (B a b + A b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{2 \,{\left (a b^{2} x + a^{2} b\right )} \sqrt{-a b}}, -\frac{{\left (B a - A b\right )} \sqrt{a b} \sqrt{x} +{\left (B a^{2} + A a b +{\left (B a b + A b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{{\left (a b^{2} x + a^{2} b\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(x)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 39.6527, size = 532, normalized size = 8.44 \[ A \left (\begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 \sqrt{x}}{a^{2}} & \text{for}\: b = 0 \\- \frac{2}{3 b^{2} x^{\frac{3}{2}}} & \text{for}\: a = 0 \\\frac{2 i \sqrt{a} b \sqrt{x} \sqrt{\frac{1}{b}}}{2 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 i a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} + \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 i a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 i a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} + \frac{b x \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 i a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} - \frac{b x \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 2 i a^{\frac{3}{2}} b^{2} x \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases}\right ) - \frac{2 B a \sqrt{x}}{2 a^{2} b + 2 a b^{2} x} + \frac{B a \sqrt{- \frac{1}{a^{3} b}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} b}} + \sqrt{x} \right )}}{2 b} - \frac{B a \sqrt{- \frac{1}{a^{3} b}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} b}} + \sqrt{x} \right )}}{2 b} + \frac{2 B \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}} \right )}}{b \sqrt{\frac{a}{b}}} & \text{for}\: \frac{a}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{a}{b}}} \right )}}{b \sqrt{- \frac{a}{b}}} & \text{for}\: x > - \frac{a}{b} \wedge \frac{a}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{a}{b}}} \right )}}{b \sqrt{- \frac{a}{b}}} & \text{for}\: x < - \frac{a}{b} \wedge \frac{a}{b} < 0 \end{cases}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(b**2*x**2+2*a*b*x+a**2)/x**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.272159, size = 81, normalized size = 1.29 \[ \frac{{\left (B a + A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a b} - \frac{B a \sqrt{x} - A b \sqrt{x}}{{\left (b x + a\right )} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(x)),x, algorithm="giac")
[Out]