Optimal. Leaf size=85 \[ \frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}}-\frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.156154, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}}-\frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x]/(x*Sqrt[a + b*x]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.3273, size = 80, normalized size = 0.94 \[ \frac{2 \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{\sqrt{b}} - \frac{2 \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(1/2)/x/(b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0836855, size = 124, normalized size = 1.46 \[ -\frac{\sqrt{c} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a}}+\frac{\sqrt{d} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{b}}+\frac{\sqrt{c} \log (x)}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c + d*x]/(x*Sqrt[a + b*x]),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.027, size = 133, normalized size = 1.6 \[{1\sqrt{bx+a}\sqrt{dx+c} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) d\sqrt{ac}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) c\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(1/2)/x/(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(sqrt(d*x + c)/(sqrt(b*x + a)*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.346295, size = 1, normalized size = 0.01 \[ \left [\frac{1}{2} \, \sqrt{\frac{d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{d}{b}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) + \frac{1}{2} \, \sqrt{\frac{c}{a}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a^{2} c +{\left (a b c + a^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{c}{a}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ), \sqrt{-\frac{d}{b}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} b \sqrt{-\frac{d}{b}}}\right ) + \frac{1}{2} \, \sqrt{\frac{c}{a}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a^{2} c +{\left (a b c + a^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{c}{a}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ), -\sqrt{-\frac{c}{a}} \arctan \left (\frac{2 \, a c +{\left (b c + a d\right )} x}{2 \, \sqrt{b x + a} \sqrt{d x + c} a \sqrt{-\frac{c}{a}}}\right ) + \frac{1}{2} \, \sqrt{\frac{d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{d}{b}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ), -\sqrt{-\frac{c}{a}} \arctan \left (\frac{2 \, a c +{\left (b c + a d\right )} x}{2 \, \sqrt{b x + a} \sqrt{d x + c} a \sqrt{-\frac{c}{a}}}\right ) + \sqrt{-\frac{d}{b}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} b \sqrt{-\frac{d}{b}}}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/(sqrt(b*x + a)*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x}}{x \sqrt{a + b x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**(1/2)/x/(b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.237753, size = 184, normalized size = 2.16 \[ -\frac{{\left (\frac{2 \, \sqrt{b d} b c \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d}} + \sqrt{b d}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )\right )}{\left | b \right |}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/(sqrt(b*x + a)*x),x, algorithm="giac")
[Out]