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 = 0.0422612, 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, Rules used = {698} \[ -\frac{2 \left (a g^2-b f g+c f^2\right )}{g^3 \sqrt{f+g x}}-\frac{2 \sqrt{f+g x} (2 c f-b g)}{g^3}+\frac{2 c (f+g x)^{3/2}}{3 g^3} \]

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)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{a+b x+c x^2}{(f+g x)^{3/2}} \, dx &=\int \left (\frac{c f^2-b f g+a g^2}{g^2 (f+g x)^{3/2}}+\frac{-2 c f+b g}{g^2 \sqrt{f+g x}}+\frac{c \sqrt{f+g x}}{g^2}\right ) \, dx\\ &=-\frac{2 \left (c f^2-b f g+a g^2\right )}{g^3 \sqrt{f+g x}}-\frac{2 (2 c f-b g) \sqrt{f+g x}}{g^3}+\frac{2 c (f+g x)^{3/2}}{3 g^3}\\ \end{align*}

Mathematica [A]  time = 0.0573995, 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 = 0.048, size = 53, normalized size = 0.8 \begin{align*} -{\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}}}} \end{align*}

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 = 0.959989, size = 89, normalized size = 1.25 \begin{align*} \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} \end{align*}

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 = 1.45533, size = 136, normalized size = 1.92 \begin{align*} \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 )} \sqrt{g x + f}}{3 \,{\left (g^{4} x + f g^{3}\right )}} \end{align*}

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^4*x + f*g^3)

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Sympy [A]  time = 10.0877, size = 70, normalized size = 0.99 \begin{align*} \frac{2 c \left (f + g x\right )^{\frac{3}{2}}}{3 g^{3}} + \frac{\sqrt{f + g x} \left (2 b g - 4 c f\right )}{g^{3}} - \frac{2 \left (a g^{2} - b f g + c f^{2}\right )}{g^{3} \sqrt{f + g x}} \end{align*}

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

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Giac [A]  time = 1.1739, size = 100, normalized size = 1.41 \begin{align*} -\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}} \end{align*}

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