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

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])

________________________________________________________________________________________

Rubi [A]  time = 0.0498577, 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, Rules used = {998, 636} \[ -\frac{2 (-2 a h+x (2 c g-b h)+b g)}{d^2 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

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])

Rule 998

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_)
, x_Symbol] :> Dist[(c/f)^p, Int[(g + h*x)^m*(d + e*x + f*x^2)^(p + q), x], x] /; FreeQ[{a, b, c, d, e, f, g,
h, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && (IntegerQ[p] || GtQ[c/f, 0]) && ( !IntegerQ[q] || Le
afCount[d + e*x + f*x^2] <= LeafCount[a + b*x + c*x^2])

Rule 636

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-2*(b*d - 2*a*e + (2*c*
d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \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 &=\frac{\int \frac{g+h x}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{d^2}\\ &=-\frac{2 (b g-2 a h+(2 c g-b h) x)}{\left (b^2-4 a c\right ) d^2 \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.198364, 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)])

________________________________________________________________________________________

Maple [A]  time = 0.044, size = 48, normalized size = 1. \begin{align*} -2\,{\frac{bhx-2\,cgx+2\,ah-bg}{\sqrt{c{x}^{2}+bx+a}{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }} \end{align*}

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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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, 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)

________________________________________________________________________________________

Fricas [A]  time = 3.7852, size = 181, normalized size = 3.77 \begin{align*} -\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}} \end{align*}

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, 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)

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Giac [A]  time = 1.16985, size = 109, normalized size = 2.27 \begin{align*} -\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}} \end{align*}

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, 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)