Optimal. Leaf size=96 \[ \frac{2 (b+2 c x) \log (b+2 c x)}{\sqrt{\frac{b^2}{c}+4 b x+4 c x^2} (2 c d-b e)}-\frac{2 (b+2 c x) \log (d+e x)}{\sqrt{\frac{b^2}{c}+4 b x+4 c x^2} (2 c d-b e)} \]
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Rubi [A] time = 0.119437, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{2 (b+2 c x) \log (b+2 c x)}{\sqrt{\frac{b^2}{c}+4 b x+4 c x^2} (2 c d-b e)}-\frac{2 (b+2 c x) \log (d+e x)}{\sqrt{\frac{b^2}{c}+4 b x+4 c x^2} (2 c d-b e)} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)*Sqrt[b^2/(4*c) + b*x + c*x^2]),x]
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Rubi in Sympy [A] time = 22.2524, size = 95, normalized size = 0.99 \[ - \frac{4 c \sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2}} \log{\left (b + 2 c x \right )}}{\left (2 b + 4 c x\right ) \left (b e - 2 c d\right )} + \frac{4 c \sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2}} \log{\left (d + e x \right )}}{\left (2 b + 4 c x\right ) \left (b e - 2 c d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(2/(e*x+d)/(1/c*b**2+4*b*x+4*c*x**2)**(1/2),x)
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Mathematica [A] time = 0.0657165, size = 51, normalized size = 0.53 \[ \frac{2 (b+2 c x) (\log (b+2 c x)-\log (d+e x))}{\sqrt{\frac{(b+2 c x)^2}{c}} (2 c d-b e)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)*Sqrt[b^2/(4*c) + b*x + c*x^2]),x]
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Maple [A] time = 0.029, size = 58, normalized size = 0.6 \[ -2\,{\frac{ \left ( 2\,cx+b \right ) \left ( \ln \left ( 2\,cx+b \right ) -\ln \left ( ex+d \right ) \right ) }{be-2\,cd}{\frac{1}{\sqrt{{\frac{4\,{c}^{2}{x}^{2}+4\,bxc+{b}^{2}}{c}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(2/(e*x+d)/(b^2/c+4*b*x+4*c*x^2)^(1/2),x)
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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(2/(sqrt(4*c*x^2 + 4*b*x + b^2/c)*(e*x + d)),x, algorithm="maxima")
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Fricas [A] time = 0.235788, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{c} \log \left (\frac{{\left (4 \, b c^{2} d^{2} - b^{3} e^{2} + 8 \,{\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} x^{2} + 2 \,{\left (4 \, c^{3} d^{2} + 4 \, b c^{2} d e - 3 \, b^{2} c e^{2}\right )} x\right )} \sqrt{c} +{\left (8 \, c^{3} e^{2} x^{2} + 4 \, c^{3} d^{2} + b^{2} c e^{2} + 4 \,{\left (2 \, c^{3} d e + b c^{2} e^{2}\right )} x\right )} \sqrt{\frac{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}}{4 \, c^{2} e x^{3} + b^{2} d + 4 \,{\left (c^{2} d + b c e\right )} x^{2} +{\left (4 \, b c d + b^{2} e\right )} x}\right )}{2 \, c d - b e}, \frac{4 \, \sqrt{-c} \arctan \left (-\frac{{\left (4 \, c^{2} e x + 2 \, c^{2} d + b c e\right )} \sqrt{\frac{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}}{{\left (2 \, b c d - b^{2} e + 2 \,{\left (2 \, c^{2} d - b c e\right )} x\right )} \sqrt{-c}}\right )}{2 \, c d - b e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(2/(sqrt(4*c*x^2 + 4*b*x + b^2/c)*(e*x + d)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ 2 \int \frac{1}{d \sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2}} + e x \sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(2/(e*x+d)/(1/c*b**2+4*b*x+4*c*x**2)**(1/2),x)
[Out]
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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(2/(sqrt(4*c*x^2 + 4*b*x + b^2/c)*(e*x + d)),x, algorithm="giac")
[Out]