3.829 \(\int \frac{a+b x+c x^2}{(f+g x)^{3/2}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{2 \left (a g^2-b f g+c f^2\right )}{g^3 \sqrt{f+g x}}-\frac{2 \sqrt{f+g x} (2 c f-b g)}{g^3}+\frac{2 c (f+g x)^{3/2}}{3 g^3} \]

[Out]

(-2*(c*f^2 - b*f*g + a*g^2))/(g^3*Sqrt[f + g*x]) - (2*(2*c*f - b*g)*Sqrt[f + g*x
])/g^3 + (2*c*(f + g*x)^(3/2))/(3*g^3)

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Rubi [A]  time = 0.0884247, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{2 \left (a g^2-b f g+c f^2\right )}{g^3 \sqrt{f+g x}}-\frac{2 \sqrt{f+g x} (2 c f-b g)}{g^3}+\frac{2 c (f+g x)^{3/2}}{3 g^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)/(f + g*x)^(3/2),x]

[Out]

(-2*(c*f^2 - b*f*g + a*g^2))/(g^3*Sqrt[f + g*x]) - (2*(2*c*f - b*g)*Sqrt[f + g*x
])/g^3 + (2*c*(f + g*x)^(3/2))/(3*g^3)

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Rubi in Sympy [A]  time = 14.5306, size = 70, normalized size = 0.99 \[ \frac{2 c \left (f + g x\right )^{\frac{3}{2}}}{3 g^{3}} + \frac{2 \sqrt{f + g x} \left (b g - 2 c f\right )}{g^{3}} - \frac{2 \left (a g^{2} - b f g + c f^{2}\right )}{g^{3} \sqrt{f + g x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)/(g*x+f)**(3/2),x)

[Out]

2*c*(f + g*x)**(3/2)/(3*g**3) + 2*sqrt(f + g*x)*(b*g - 2*c*f)/g**3 - 2*(a*g**2 -
b*f*g + c*f**2)/(g**3*sqrt(f + g*x))

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Mathematica [A]  time = 0.0538819, size = 54, normalized size = 0.76 \[ \frac{6 g (-a g+2 b f+b g x)+2 c \left (-8 f^2-4 f g x+g^2 x^2\right )}{3 g^3 \sqrt{f+g x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)/(f + g*x)^(3/2),x]

[Out]

(6*g*(2*b*f - a*g + b*g*x) + 2*c*(-8*f^2 - 4*f*g*x + g^2*x^2))/(3*g^3*Sqrt[f + g
*x])

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Maple [A]  time = 0.005, size = 53, normalized size = 0.8 \[ -{\frac{-2\,c{x}^{2}{g}^{2}-6\,b{g}^{2}x+8\,cfgx+6\,a{g}^{2}-12\,bfg+16\,c{f}^{2}}{3\,{g}^{3}}{\frac{1}{\sqrt{gx+f}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)/(g*x+f)^(3/2),x)

[Out]

-2/3/(g*x+f)^(1/2)*(-c*g^2*x^2-3*b*g^2*x+4*c*f*g*x+3*a*g^2-6*b*f*g+8*c*f^2)/g^3

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Maxima [A]  time = 0.684049, size = 89, normalized size = 1.25 \[ \frac{2 \,{\left (\frac{{\left (g x + f\right )}^{\frac{3}{2}} c - 3 \,{\left (2 \, c f - b g\right )} \sqrt{g x + f}}{g^{2}} - \frac{3 \,{\left (c f^{2} - b f g + a g^{2}\right )}}{\sqrt{g x + f} g^{2}}\right )}}{3 \, g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(g*x + f)^(3/2),x, algorithm="maxima")

[Out]

2/3*(((g*x + f)^(3/2)*c - 3*(2*c*f - b*g)*sqrt(g*x + f))/g^2 - 3*(c*f^2 - b*f*g
+ a*g^2)/(sqrt(g*x + f)*g^2))/g

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Fricas [A]  time = 0.269362, size = 72, normalized size = 1.01 \[ \frac{2 \,{\left (c g^{2} x^{2} - 8 \, c f^{2} + 6 \, b f g - 3 \, a g^{2} -{\left (4 \, c f g - 3 \, b g^{2}\right )} x\right )}}{3 \, \sqrt{g x + f} g^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(g*x + f)^(3/2),x, algorithm="fricas")

[Out]

2/3*(c*g^2*x^2 - 8*c*f^2 + 6*b*f*g - 3*a*g^2 - (4*c*f*g - 3*b*g^2)*x)/(sqrt(g*x
+ f)*g^3)

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Sympy [A]  time = 11.0852, size = 590, normalized size = 8.31 \[ - \frac{2 a}{g \sqrt{f + g x}} + b \left (\begin{cases} \frac{4 f}{g^{2} \sqrt{f + g x}} + \frac{2 x}{g \sqrt{f + g x}} & \text{for}\: g \neq 0 \\\frac{x^{2}}{2 f^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + c \left (- \frac{16 f^{\frac{19}{2}} \sqrt{1 + \frac{g x}{f}}}{3 f^{8} g^{3} + 9 f^{7} g^{4} x + 9 f^{6} g^{5} x^{2} + 3 f^{5} g^{6} x^{3}} + \frac{16 f^{\frac{19}{2}}}{3 f^{8} g^{3} + 9 f^{7} g^{4} x + 9 f^{6} g^{5} x^{2} + 3 f^{5} g^{6} x^{3}} - \frac{40 f^{\frac{17}{2}} g x \sqrt{1 + \frac{g x}{f}}}{3 f^{8} g^{3} + 9 f^{7} g^{4} x + 9 f^{6} g^{5} x^{2} + 3 f^{5} g^{6} x^{3}} + \frac{48 f^{\frac{17}{2}} g x}{3 f^{8} g^{3} + 9 f^{7} g^{4} x + 9 f^{6} g^{5} x^{2} + 3 f^{5} g^{6} x^{3}} - \frac{30 f^{\frac{15}{2}} g^{2} x^{2} \sqrt{1 + \frac{g x}{f}}}{3 f^{8} g^{3} + 9 f^{7} g^{4} x + 9 f^{6} g^{5} x^{2} + 3 f^{5} g^{6} x^{3}} + \frac{48 f^{\frac{15}{2}} g^{2} x^{2}}{3 f^{8} g^{3} + 9 f^{7} g^{4} x + 9 f^{6} g^{5} x^{2} + 3 f^{5} g^{6} x^{3}} - \frac{4 f^{\frac{13}{2}} g^{3} x^{3} \sqrt{1 + \frac{g x}{f}}}{3 f^{8} g^{3} + 9 f^{7} g^{4} x + 9 f^{6} g^{5} x^{2} + 3 f^{5} g^{6} x^{3}} + \frac{16 f^{\frac{13}{2}} g^{3} x^{3}}{3 f^{8} g^{3} + 9 f^{7} g^{4} x + 9 f^{6} g^{5} x^{2} + 3 f^{5} g^{6} x^{3}} + \frac{2 f^{\frac{11}{2}} g^{4} x^{4} \sqrt{1 + \frac{g x}{f}}}{3 f^{8} g^{3} + 9 f^{7} g^{4} x + 9 f^{6} g^{5} x^{2} + 3 f^{5} g^{6} x^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)/(g*x+f)**(3/2),x)

[Out]

-2*a/(g*sqrt(f + g*x)) + b*Piecewise((4*f/(g**2*sqrt(f + g*x)) + 2*x/(g*sqrt(f +
g*x)), Ne(g, 0)), (x**2/(2*f**(3/2)), True)) + c*(-16*f**(19/2)*sqrt(1 + g*x/f)
/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f**5*g**6*x**3) + 16*f**(19
/2)/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f**5*g**6*x**3) - 40*f**
(17/2)*g*x*sqrt(1 + g*x/f)/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f
**5*g**6*x**3) + 48*f**(17/2)*g*x/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*x**
2 + 3*f**5*g**6*x**3) - 30*f**(15/2)*g**2*x**2*sqrt(1 + g*x/f)/(3*f**8*g**3 + 9*
f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f**5*g**6*x**3) + 48*f**(15/2)*g**2*x**2/(3*f
**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f**5*g**6*x**3) - 4*f**(13/2)*g*
*3*x**3*sqrt(1 + g*x/f)/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f**5
*g**6*x**3) + 16*f**(13/2)*g**3*x**3/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*
x**2 + 3*f**5*g**6*x**3) + 2*f**(11/2)*g**4*x**4*sqrt(1 + g*x/f)/(3*f**8*g**3 +
9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f**5*g**6*x**3))

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GIAC/XCAS [A]  time = 0.264497, size = 100, normalized size = 1.41 \[ -\frac{2 \,{\left (c f^{2} - b f g + a g^{2}\right )}}{\sqrt{g x + f} g^{3}} + \frac{2 \,{\left ({\left (g x + f\right )}^{\frac{3}{2}} c g^{6} - 6 \, \sqrt{g x + f} c f g^{6} + 3 \, \sqrt{g x + f} b g^{7}\right )}}{3 \, g^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(g*x + f)^(3/2),x, algorithm="giac")

[Out]

-2*(c*f^2 - b*f*g + a*g^2)/(sqrt(g*x + f)*g^3) + 2/3*((g*x + f)^(3/2)*c*g^6 - 6*
sqrt(g*x + f)*c*f*g^6 + 3*sqrt(g*x + f)*b*g^7)/g^9