3.36 \(\int \frac{(g+h x) \sqrt{a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^2} \, dx\)

Optimal. Leaf size=48 \[ -\frac{2 (-2 a h+x (2 c g-b h)+b g)}{d^2 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.148237, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{2 (-2 a h+x (2 c g-b h)+b g)}{d^2 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((g + h*x)*Sqrt[a + b*x + c*x^2])/(a*d + b*d*x + c*d*x^2)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 33.2541, size = 46, normalized size = 0.96 \[ \frac{4 a h - 2 b g + x \left (2 b h - 4 c g\right )}{d^{2} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((h*x+g)*(c*x**2+b*x+a)**(1/2)/(c*d*x**2+b*d*x+a*d)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0680517, size = 46, normalized size = 0.96 \[ \frac{4 a h-2 b g+2 b h x-4 c g x}{d^2 \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[((g + h*x)*Sqrt[a + b*x + c*x^2])/(a*d + b*d*x + c*d*x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.007, size = 48, normalized size = 1. \[ -2\,{\frac{bhx-2\,cgx+2\,ah-bg}{\sqrt{c{x}^{2}+bx+a}{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((h*x+g)*(c*x^2+b*x+a)^(1/2)/(c*d*x^2+b*d*x+a*d)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x + a}{\left (h x + g\right )}}{{\left (c d x^{2} + b d x + a d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(h*x + g)/(c*d*x^2 + b*d*x + a*d)^2,x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x + a)*(h*x + g)/(c*d*x^2 + b*d*x + a*d)^2, x)

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Fricas [A]  time = 0.456367, size = 115, normalized size = 2.4 \[ -\frac{2 \, \sqrt{c x^{2} + b x + a}{\left (b g - 2 \, a h +{\left (2 \, c g - b h\right )} x\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x^{2} +{\left (b^{3} - 4 \, a b c\right )} d^{2} x +{\left (a b^{2} - 4 \, a^{2} c\right )} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(h*x + g)/(c*d*x^2 + b*d*x + a*d)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x+g)*(c*x**2+b*x+a)**(1/2)/(c*d*x**2+b*d*x+a*d)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275177, size = 109, normalized size = 2.27 \[ -\frac{2 \,{\left (\frac{{\left (2 \, c d^{2} g - b d^{2} h\right )} x}{b^{2} d^{4} - 4 \, a c d^{4}} + \frac{b d^{2} g - 2 \, a d^{2} h}{b^{2} d^{4} - 4 \, a c d^{4}}\right )}}{\sqrt{c x^{2} + b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(h*x + g)/(c*d*x^2 + b*d*x + a*d)^2,x, algorithm="giac")

[Out]

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