Optimal. Leaf size=72 \[ \frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} \sqrt{d}}+\frac{\sqrt{a+b x} \sqrt{c+d x}}{b} \]
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Rubi [A] time = 0.0797769, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} \sqrt{d}}+\frac{\sqrt{a+b x} \sqrt{c+d x}}{b} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x]/Sqrt[a + b*x],x]
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Rubi in Sympy [A] time = 9.44763, size = 63, normalized size = 0.88 \[ \frac{\sqrt{a + b x} \sqrt{c + d x}}{b} - \frac{\left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{b^{\frac{3}{2}} \sqrt{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(1/2)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0750408, size = 88, normalized size = 1.22 \[ \frac{(b c-a d) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 b^{3/2} \sqrt{d}}+\frac{\sqrt{a+b x} \sqrt{c+d x}}{b} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c + d*x]/Sqrt[a + b*x],x]
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Maple [A] time = 0.005, size = 107, normalized size = 1.5 \[{\frac{1}{b}\sqrt{bx+a}\sqrt{dx+c}}-{\frac{ad-bc}{2\,b}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({1 \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(1/2)/(b*x+a)^(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(sqrt(d*x + c)/sqrt(b*x + a),x, algorithm="maxima")
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Fricas [A] time = 0.235577, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b c - a d\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right ) - 4 \, \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c}}{4 \, \sqrt{b d} b}, \frac{{\left (b c - a d\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right ) + 2 \, \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c}}{2 \, \sqrt{-b d} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/sqrt(b*x + a),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x}}{\sqrt{a + b x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**(1/2)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.227675, size = 126, normalized size = 1.75 \[ -\frac{{\left (\frac{{\left (b^{2} c - a b d\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d}} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}\right )}{\left | b \right |}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/sqrt(b*x + a),x, algorithm="giac")
[Out]