Optimal. Leaf size=119 \[ \frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} f}-\frac{2 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{a+b x} \sqrt{d e-c f}}{\sqrt{c+d x} \sqrt{b e-a f}}\right )}{f \sqrt{b e-a f}} \]
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Rubi [A] time = 0.248899, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} f}-\frac{2 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{a+b x} \sqrt{d e-c f}}{\sqrt{c+d x} \sqrt{b e-a f}}\right )}{f \sqrt{b e-a f}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x]/(Sqrt[a + b*x]*(e + f*x)),x]
[Out]
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Rubi in Sympy [A] time = 27.0143, size = 104, normalized size = 0.87 \[ - \frac{2 \sqrt{c f - d e} \operatorname{atanh}{\left (\frac{\sqrt{a + b x} \sqrt{c f - d e}}{\sqrt{c + d x} \sqrt{a f - b e}} \right )}}{f \sqrt{a f - b e}} + \frac{2 \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{\sqrt{b} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(1/2)/(f*x+e)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.246668, size = 193, normalized size = 1.62 \[ \frac{-\frac{\sqrt{d e-c f} \log (e+f x)}{\sqrt{b e-a f}}+\frac{\sqrt{d e-c f} \log \left (2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{b e-a f} \sqrt{d e-c f}+2 a c f-a d e+a d f x-b c e+b c f x-2 b d e x\right )}{\sqrt{b e-a f}}+\frac{\sqrt{d} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{b}}}{f} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c + d*x]/(Sqrt[a + b*x]*(e + f*x)),x]
[Out]
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Maple [B] time = 0.034, size = 300, normalized size = 2.5 \[{\frac{1}{{f}^{2}} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) df\sqrt{{\frac{ \left ( cf-de \right ) \left ( af-be \right ) }{{f}^{2}}}}-\ln \left ({\frac{1}{fx+e} \left ( adfx+bcfx-2\,bdex+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{{\frac{ \left ( cf-de \right ) \left ( af-be \right ) }{{f}^{2}}}}f+2\,acf-ade-bce \right ) } \right ) cf\sqrt{bd}+\ln \left ({\frac{1}{fx+e} \left ( adfx+bcfx-2\,bdex+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{{\frac{ \left ( cf-de \right ) \left ( af-be \right ) }{{f}^{2}}}}f+2\,acf-ade-bce \right ) } \right ) de\sqrt{bd} \right ) \sqrt{bx+a}\sqrt{dx+c}{\frac{1}{\sqrt{{\frac{ \left ( cf-de \right ) \left ( af-be \right ) }{{f}^{2}}}}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(1/2)/(f*x+e)/(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)*(f*x + e)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 5.77307, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/(sqrt(b*x + a)*(f*x + e)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x}}{\sqrt{a + b x} \left (e + f x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**(1/2)/(f*x+e)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.265072, size = 321, normalized size = 2.7 \[ -\frac{{\left (\frac{\sqrt{b d}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{f} + \frac{2 \,{\left (\sqrt{b d} b^{2} c f - \sqrt{b d} b^{2} d e\right )} \arctan \left (-\frac{b^{2} c f + a b d f - 2 \, b^{2} d e -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} f}{2 \, \sqrt{-a b c d f^{2} + b^{2} c d f e + a b d^{2} f e - b^{2} d^{2} e^{2}} b}\right )}{\sqrt{-a b c d f^{2} + b^{2} c d f e + a b d^{2} f e - b^{2} d^{2} e^{2}} b f}\right )}{\left | b \right |}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/(sqrt(b*x + a)*(f*x + e)),x, algorithm="giac")
[Out]