Optimal. Leaf size=77 \[ \frac{B \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c}}-\frac{A \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.140565, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{B \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c}}-\frac{A \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x*Sqrt[a + b*x + c*x^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 15.4184, size = 70, normalized size = 0.91 \[ - \frac{A \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{\sqrt{a}} + \frac{B \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{\sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x/(c*x**2+b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.105801, size = 80, normalized size = 1.04 \[ -\frac{A \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{\sqrt{a}}+\frac{A \log (x)}{\sqrt{a}}+\frac{B \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x*Sqrt[a + b*x + c*x^2]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 67, normalized size = 0.9 \[{B\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{A\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x/(c*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(c*x^2 + b*x + a)*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.380747, size = 1, normalized size = 0.01 \[ \left [\frac{B \sqrt{a} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) + A \sqrt{c} \log \left (\frac{4 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{2}}\right )}{2 \, \sqrt{a} \sqrt{c}}, \frac{2 \, B \sqrt{a} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right ) + A \sqrt{-c} \log \left (\frac{4 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{2}}\right )}{2 \, \sqrt{a} \sqrt{-c}}, -\frac{2 \, A \sqrt{c} \arctan \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{2} + b x + a} a}\right ) - B \sqrt{-a} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{2 \, \sqrt{-a} \sqrt{c}}, -\frac{A \sqrt{-c} \arctan \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{2} + b x + a} a}\right ) - B \sqrt{-a} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{\sqrt{-a} \sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(c*x^2 + b*x + a)*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{x \sqrt{a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x/(c*x**2+b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(c*x^2 + b*x + a)*x),x, algorithm="giac")
[Out]