3.822 \(\int \frac{a+b x+c x^2}{\sqrt{f+g x}} \, dx\)

Optimal. Leaf size=73 \[ \frac{2 \sqrt{f+g x} \left (a g^2-b f g+c f^2\right )}{g^3}-\frac{2 (f+g x)^{3/2} (2 c f-b g)}{3 g^3}+\frac{2 c (f+g x)^{5/2}}{5 g^3} \]

[Out]

(2*(c*f^2 - b*f*g + a*g^2)*Sqrt[f + g*x])/g^3 - (2*(2*c*f - b*g)*(f + g*x)^(3/2))/(3*g^3) + (2*c*(f + g*x)^(5/
2))/(5*g^3)

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Rubi [A]  time = 0.042033, antiderivative size = 73, 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} \[ \frac{2 \sqrt{f+g x} \left (a g^2-b f g+c f^2\right )}{g^3}-\frac{2 (f+g x)^{3/2} (2 c f-b g)}{3 g^3}+\frac{2 c (f+g x)^{5/2}}{5 g^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*f^2 - b*f*g + a*g^2)*Sqrt[f + g*x])/g^3 - (2*(2*c*f - b*g)*(f + g*x)^(3/2))/(3*g^3) + (2*c*(f + g*x)^(5/
2))/(5*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}{\sqrt{f+g x}} \, dx &=\int \left (\frac{c f^2-b f g+a g^2}{g^2 \sqrt{f+g x}}+\frac{(-2 c f+b g) \sqrt{f+g x}}{g^2}+\frac{c (f+g x)^{3/2}}{g^2}\right ) \, dx\\ &=\frac{2 \left (c f^2-b f g+a g^2\right ) \sqrt{f+g x}}{g^3}-\frac{2 (2 c f-b g) (f+g x)^{3/2}}{3 g^3}+\frac{2 c (f+g x)^{5/2}}{5 g^3}\\ \end{align*}

Mathematica [A]  time = 0.0490043, size = 54, normalized size = 0.74 \[ \frac{2 \sqrt{f+g x} \left (5 g (3 a g-2 b f+b g x)+c \left (8 f^2-4 f g x+3 g^2 x^2\right )\right )}{15 g^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[f + g*x]*(5*g*(-2*b*f + 3*a*g + b*g*x) + c*(8*f^2 - 4*f*g*x + 3*g^2*x^2)))/(15*g^3)

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Maple [A]  time = 0.046, size = 53, normalized size = 0.7 \begin{align*}{\frac{6\,c{x}^{2}{g}^{2}+10\,b{g}^{2}x-8\,cfgx+30\,a{g}^{2}-20\,bfg+16\,c{f}^{2}}{15\,{g}^{3}}\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)^(1/2),x)

[Out]

2/15*(g*x+f)^(1/2)*(3*c*g^2*x^2+5*b*g^2*x-4*c*f*g*x+15*a*g^2-10*b*f*g+8*c*f^2)/g^3

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Maxima [A]  time = 0.956599, size = 104, normalized size = 1.42 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{g x + f} a + \frac{5 \,{\left ({\left (g x + f\right )}^{\frac{3}{2}} - 3 \, \sqrt{g x + f} f\right )} b}{g} + \frac{{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f + 15 \, \sqrt{g x + f} f^{2}\right )} c}{g^{2}}\right )}}{15 \, g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(g*x+f)^(1/2),x, algorithm="maxima")

[Out]

2/15*(15*sqrt(g*x + f)*a + 5*((g*x + f)^(3/2) - 3*sqrt(g*x + f)*f)*b/g + (3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/
2)*f + 15*sqrt(g*x + f)*f^2)*c/g^2)/g

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Fricas [A]  time = 1.71499, size = 127, normalized size = 1.74 \begin{align*} \frac{2 \,{\left (3 \, c g^{2} x^{2} + 8 \, c f^{2} - 10 \, b f g + 15 \, a g^{2} -{\left (4 \, c f g - 5 \, b g^{2}\right )} x\right )} \sqrt{g x + f}}{15 \, g^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(g*x+f)^(1/2),x, algorithm="fricas")

[Out]

2/15*(3*c*g^2*x^2 + 8*c*f^2 - 10*b*f*g + 15*a*g^2 - (4*c*f*g - 5*b*g^2)*x)*sqrt(g*x + f)/g^3

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Sympy [A]  time = 10.5038, size = 223, normalized size = 3.05 \begin{align*} \begin{cases} - \frac{\frac{2 a f}{\sqrt{f + g x}} + 2 a \left (- \frac{f}{\sqrt{f + g x}} - \sqrt{f + g x}\right ) + \frac{2 b f \left (- \frac{f}{\sqrt{f + g x}} - \sqrt{f + g x}\right )}{g} + \frac{2 b \left (\frac{f^{2}}{\sqrt{f + g x}} + 2 f \sqrt{f + g x} - \frac{\left (f + g x\right )^{\frac{3}{2}}}{3}\right )}{g} + \frac{2 c f \left (\frac{f^{2}}{\sqrt{f + g x}} + 2 f \sqrt{f + g x} - \frac{\left (f + g x\right )^{\frac{3}{2}}}{3}\right )}{g^{2}} + \frac{2 c \left (- \frac{f^{3}}{\sqrt{f + g x}} - 3 f^{2} \sqrt{f + g x} + f \left (f + g x\right )^{\frac{3}{2}} - \frac{\left (f + g x\right )^{\frac{5}{2}}}{5}\right )}{g^{2}}}{g} & \text{for}\: g \neq 0 \\\frac{a x + \frac{b x^{2}}{2} + \frac{c x^{3}}{3}}{\sqrt{f}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)/(g*x+f)**(1/2),x)

[Out]

Piecewise((-(2*a*f/sqrt(f + g*x) + 2*a*(-f/sqrt(f + g*x) - sqrt(f + g*x)) + 2*b*f*(-f/sqrt(f + g*x) - sqrt(f +
g*x))/g + 2*b*(f**2/sqrt(f + g*x) + 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g + 2*c*f*(f**2/sqrt(f + g*x) + 2
*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g**2 + 2*c*(-f**3/sqrt(f + g*x) - 3*f**2*sqrt(f + g*x) + f*(f + g*x)**(
3/2) - (f + g*x)**(5/2)/5)/g**2)/g, Ne(g, 0)), ((a*x + b*x**2/2 + c*x**3/3)/sqrt(f), True))

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Giac [A]  time = 1.16665, size = 104, normalized size = 1.42 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{g x + f} a + \frac{5 \,{\left ({\left (g x + f\right )}^{\frac{3}{2}} - 3 \, \sqrt{g x + f} f\right )} b}{g} + \frac{{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f + 15 \, \sqrt{g x + f} f^{2}\right )} c}{g^{2}}\right )}}{15 \, g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(g*x+f)^(1/2),x, algorithm="giac")

[Out]

2/15*(15*sqrt(g*x + f)*a + 5*((g*x + f)^(3/2) - 3*sqrt(g*x + f)*f)*b/g + (3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/
2)*f + 15*sqrt(g*x + f)*f^2)*c/g^2)/g