3.1228 \(\int \frac{1}{(b d+2 c d x)^2 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=37 \[ \frac{2 \sqrt{a+b x+c x^2}}{d^2 \left (b^2-4 a c\right ) (b+2 c x)} \]

[Out]

(2*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*d^2*(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]

[Out]

(2*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*d^2*(b + 2*c*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]

2*sqrt(a + b*x + c*x**2)/(d**2*(b + 2*c*x)*(-4*a*c + b**2))

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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]

[Out]

(2*Sqrt[a + x*(b + c*x)])/((b^2 - 4*a*c)*d^2*(b + 2*c*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)

[Out]

-2*(c*x^2+b*x+a)^(1/2)/(2*c*x+b)/d^2/(4*a*c-b^2)

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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")

[Out]

Exception raised: ValueError

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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")

[Out]

2*sqrt(c*x^2 + b*x + a)/(2*(b^2*c - 4*a*c^2)*d^2*x + (b^3 - 4*a*b*c)*d^2)

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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)

[Out]

Integral(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)), x)/d**2

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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")

[Out]

-sqrt(c)*sign(1/(2*c*d*x + b*d))*sign(c)*sign(d)/(b^2*c*d^2 - 4*a*c^2*d^2) + sqr
t(-b^2*c*d^2/(2*c*d*x + b*d)^2 + 4*a*c^2*d^2/(2*c*d*x + b*d)^2 + c)*c^2/(b^2*c^3
*d^2*sign(1/(2*c*d*x + b*d))*sign(c)*sign(d) - 4*a*c^4*d^2*sign(1/(2*c*d*x + b*d
))*sign(c)*sign(d))