Optimal. Leaf size=37 \[ \frac{2 \sqrt{a+b x+c x^2}}{d^2 \left (b^2-4 a c\right ) (b+2 c x)} \]
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Rubi [A] time = 0.0558118, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{2 \sqrt{a+b x+c x^2}}{d^2 \left (b^2-4 a c\right ) (b+2 c x)} \]
Antiderivative was successfully verified.
[In] Int[1/((b*d + 2*c*d*x)^2*Sqrt[a + b*x + c*x^2]),x]
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Rubi in Sympy [A] time = 13.8769, size = 32, normalized size = 0.86 \[ \frac{2 \sqrt{a + b x + c x^{2}}}{d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2*c*d*x+b*d)**2/(c*x**2+b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0540458, size = 36, normalized size = 0.97 \[ \frac{2 \sqrt{a+x (b+c x)}}{d^2 \left (b^2-4 a c\right ) (b+2 c x)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((b*d + 2*c*d*x)^2*Sqrt[a + b*x + c*x^2]),x]
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Maple [A] time = 0.007, size = 38, normalized size = 1. \[ -2\,{\frac{\sqrt{c{x}^{2}+bx+a}}{ \left ( 2\,cx+b \right ){d}^{2} \left ( 4\,ac-{b}^{2} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2*c*d*x+b*d)^2/(c*x^2+b*x+a)^(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(1/((2*c*d*x + b*d)^2*sqrt(c*x^2 + b*x + a)),x, algorithm="maxima")
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Fricas [A] time = 0.256141, size = 65, normalized size = 1.76 \[ \frac{2 \, \sqrt{c x^{2} + b x + a}}{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x +{\left (b^{3} - 4 \, a b c\right )} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^2*sqrt(c*x^2 + b*x + a)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{1}{b^{2} \sqrt{a + b x + c x^{2}} + 4 b c x \sqrt{a + b x + c x^{2}} + 4 c^{2} x^{2} \sqrt{a + b x + c x^{2}}}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2*c*d*x+b*d)**2/(c*x**2+b*x+a)**(1/2),x)
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GIAC/XCAS [A] time = 0.226842, size = 194, normalized size = 5.24 \[ -\frac{\sqrt{c}{\rm sign}\left (\frac{1}{2 \, c d x + b d}\right ){\rm sign}\left (c\right ){\rm sign}\left (d\right )}{b^{2} c d^{2} - 4 \, a c^{2} d^{2}} + \frac{\sqrt{-\frac{b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac{4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} c^{2}}{b^{2} c^{3} d^{2}{\rm sign}\left (\frac{1}{2 \, c d x + b d}\right ){\rm sign}\left (c\right ){\rm sign}\left (d\right ) - 4 \, a c^{4} d^{2}{\rm sign}\left (\frac{1}{2 \, c d x + b d}\right ){\rm sign}\left (c\right ){\rm sign}\left (d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^2*sqrt(c*x^2 + b*x + a)),x, algorithm="giac")
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