### 3.238 $$\int \frac{d+e x+f x^2}{(g+h x)^2 (a+b x+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=421 $-\frac{2 \left (c x \left (2 a^2 f h^2-c \left (2 a \left (d h^2-2 e g h+f g^2\right )+b g (2 d h+e g)\right )-a b h (e h+2 f g)+b^2 \left (d h^2+f g^2\right )+2 c^2 d g^2\right )+b \left (a^2 f h^2+a c \left (-3 d h^2+2 e g h+f g^2\right )+c^2 d g^2\right )-b^2 h (a e h+2 c d g)-2 a c (a h (2 f g-e h)+c g (e g-2 d h))+b^3 d h^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a h^2-b g h+c g^2\right )^2}-\frac{h \sqrt{a+b x+c x^2} \left (f g^2-h (e g-d h)\right )}{(g+h x) \left (a h^2-b g h+c g^2\right )^2}+\frac{\tanh ^{-1}\left (\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right ) \left (2 c g \left (f g^2-h (2 e g-3 d h)\right )-h \left (2 a h (2 f g-e h)-b \left (-3 d h^2+e g h+f g^2\right )\right )\right )}{2 \left (a h^2-b g h+c g^2\right )^{5/2}}$

[Out]

(-2*(b^3*d*h^2 - b^2*h*(2*c*d*g + a*e*h) - 2*a*c*(c*g*(e*g - 2*d*h) + a*h*(2*f*g - e*h)) + b*(c^2*d*g^2 + a^2*
f*h^2 + a*c*(f*g^2 + 2*e*g*h - 3*d*h^2)) + c*(2*c^2*d*g^2 + 2*a^2*f*h^2 - a*b*h*(2*f*g + e*h) + b^2*(f*g^2 + d
*h^2) - c*(b*g*(e*g + 2*d*h) + 2*a*(f*g^2 - 2*e*g*h + d*h^2)))*x))/((b^2 - 4*a*c)*(c*g^2 - b*g*h + a*h^2)^2*Sq
rt[a + b*x + c*x^2]) - (h*(f*g^2 - h*(e*g - d*h))*Sqrt[a + b*x + c*x^2])/((c*g^2 - b*g*h + a*h^2)^2*(g + h*x))
+ ((2*c*g*(f*g^2 - h*(2*e*g - 3*d*h)) - h*(2*a*h*(2*f*g - e*h) - b*(f*g^2 + e*g*h - 3*d*h^2)))*ArcTanh[(b*g -
2*a*h + (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/(2*(c*g^2 - b*g*h + a*h^2)^(
5/2))

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Rubi [A]  time = 0.796687, antiderivative size = 418, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 32, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {1646, 806, 724, 206} $-\frac{2 \left (c x \left (2 a^2 f h^2-c \left (2 a \left (d h^2-2 e g h+f g^2\right )+b g (2 d h+e g)\right )-a b h (e h+2 f g)+b^2 \left (d h^2+f g^2\right )+2 c^2 d g^2\right )+b \left (a^2 f h^2+a c \left (-3 d h^2+2 e g h+f g^2\right )+c^2 d g^2\right )-b^2 h (a e h+2 c d g)-2 a c (a h (2 f g-e h)+c g (e g-2 d h))+b^3 d h^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a h^2-b g h+c g^2\right )^2}-\frac{h \sqrt{a+b x+c x^2} \left (f g^2-h (e g-d h)\right )}{(g+h x) \left (a h^2-b g h+c g^2\right )^2}+\frac{\tanh ^{-1}\left (\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right ) \left (h \left (-2 a h (2 f g-e h)+b h (e g-3 d h)+b f g^2\right )+2 c \left (f g^3-g h (2 e g-3 d h)\right )\right )}{2 \left (a h^2-b g h+c g^2\right )^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(b^3*d*h^2 - b^2*h*(2*c*d*g + a*e*h) - 2*a*c*(c*g*(e*g - 2*d*h) + a*h*(2*f*g - e*h)) + b*(c^2*d*g^2 + a^2*
f*h^2 + a*c*(f*g^2 + 2*e*g*h - 3*d*h^2)) + c*(2*c^2*d*g^2 + 2*a^2*f*h^2 - a*b*h*(2*f*g + e*h) + b^2*(f*g^2 + d
*h^2) - c*(b*g*(e*g + 2*d*h) + 2*a*(f*g^2 - 2*e*g*h + d*h^2)))*x))/((b^2 - 4*a*c)*(c*g^2 - b*g*h + a*h^2)^2*Sq
rt[a + b*x + c*x^2]) - (h*(f*g^2 - h*(e*g - d*h))*Sqrt[a + b*x + c*x^2])/((c*g^2 - b*g*h + a*h^2)^2*(g + h*x))
+ ((2*c*(f*g^3 - g*h*(2*e*g - 3*d*h)) + h*(b*f*g^2 + b*h*(e*g - 3*d*h) - 2*a*h*(2*f*g - e*h)))*ArcTanh[(b*g -
2*a*h + (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/(2*(c*g^2 - b*g*h + a*h^2)^(
5/2))

Rule 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{d+e x+f x^2}{(g+h x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 \left (b^3 d h^2-b^2 h (2 c d g+a e h)-2 a c (c g (e g-2 d h)+a h (2 f g-e h))+b \left (c^2 d g^2+a^2 f h^2+a c \left (f g^2+2 e g h-3 d h^2\right )\right )+c \left (2 c^2 d g^2+2 a^2 f h^2-a b h (2 f g+e h)+b^2 \left (f g^2+d h^2\right )-c \left (b g (e g+2 d h)+2 a \left (f g^2-2 e g h+d h^2\right )\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (c g^2-b g h+a h^2\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{-\frac{\left (b^2-4 a c\right ) \left (c \left (f g^4-g^2 h (2 e g-3 d h)\right )-h^2 \left (a f g^2-a d h^2-b g (e g-2 d h)\right )\right )}{2 \left (c g^2-b g h+a h^2\right )^2}+\frac{\left (b^2-4 a c\right ) h^2 \left (c g (e g-2 d h)+a h (2 f g-e h)-b \left (f g^2-d h^2\right )\right ) x}{2 \left (c g^2-b g h+a h^2\right )^2}}{(g+h x)^2 \sqrt{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac{2 \left (b^3 d h^2-b^2 h (2 c d g+a e h)-2 a c (c g (e g-2 d h)+a h (2 f g-e h))+b \left (c^2 d g^2+a^2 f h^2+a c \left (f g^2+2 e g h-3 d h^2\right )\right )+c \left (2 c^2 d g^2+2 a^2 f h^2-a b h (2 f g+e h)+b^2 \left (f g^2+d h^2\right )-c \left (b g (e g+2 d h)+2 a \left (f g^2-2 e g h+d h^2\right )\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (c g^2-b g h+a h^2\right )^2 \sqrt{a+b x+c x^2}}-\frac{h \left (f g^2-h (e g-d h)\right ) \sqrt{a+b x+c x^2}}{\left (c g^2-b g h+a h^2\right )^2 (g+h x)}+\frac{\left (2 c \left (f g^3-g h (2 e g-3 d h)\right )+h \left (b f g^2+b h (e g-3 d h)-2 a h (2 f g-e h)\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+b x+c x^2}} \, dx}{2 \left (c g^2-b g h+a h^2\right )^2}\\ &=-\frac{2 \left (b^3 d h^2-b^2 h (2 c d g+a e h)-2 a c (c g (e g-2 d h)+a h (2 f g-e h))+b \left (c^2 d g^2+a^2 f h^2+a c \left (f g^2+2 e g h-3 d h^2\right )\right )+c \left (2 c^2 d g^2+2 a^2 f h^2-a b h (2 f g+e h)+b^2 \left (f g^2+d h^2\right )-c \left (b g (e g+2 d h)+2 a \left (f g^2-2 e g h+d h^2\right )\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (c g^2-b g h+a h^2\right )^2 \sqrt{a+b x+c x^2}}-\frac{h \left (f g^2-h (e g-d h)\right ) \sqrt{a+b x+c x^2}}{\left (c g^2-b g h+a h^2\right )^2 (g+h x)}-\frac{\left (2 c \left (f g^3-g h (2 e g-3 d h)\right )+h \left (b f g^2+b h (e g-3 d h)-2 a h (2 f g-e h)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c g^2-4 b g h+4 a h^2-x^2} \, dx,x,\frac{-b g+2 a h-(2 c g-b h) x}{\sqrt{a+b x+c x^2}}\right )}{\left (c g^2-b g h+a h^2\right )^2}\\ &=-\frac{2 \left (b^3 d h^2-b^2 h (2 c d g+a e h)-2 a c (c g (e g-2 d h)+a h (2 f g-e h))+b \left (c^2 d g^2+a^2 f h^2+a c \left (f g^2+2 e g h-3 d h^2\right )\right )+c \left (2 c^2 d g^2+2 a^2 f h^2-a b h (2 f g+e h)+b^2 \left (f g^2+d h^2\right )-c \left (b g (e g+2 d h)+2 a \left (f g^2-2 e g h+d h^2\right )\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (c g^2-b g h+a h^2\right )^2 \sqrt{a+b x+c x^2}}-\frac{h \left (f g^2-h (e g-d h)\right ) \sqrt{a+b x+c x^2}}{\left (c g^2-b g h+a h^2\right )^2 (g+h x)}+\frac{\left (2 c \left (f g^3-g h (2 e g-3 d h)\right )+h \left (b f g^2+b h (e g-3 d h)-2 a h (2 f g-e h)\right )\right ) \tanh ^{-1}\left (\frac{b g-2 a h+(2 c g-b h) x}{2 \sqrt{c g^2-b g h+a h^2} \sqrt{a+b x+c x^2}}\right )}{2 \left (c g^2-b g h+a h^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 3.26168, size = 487, normalized size = 1.16 $\frac{\frac{4 c \left (-a^2 f h^2+a c (2 h (d h-e g+e h x)+f g (g-2 h x))+2 c^2 d g h x\right )-b^2 \left (a f h^2+4 c d h^2+c f g (g-4 h x)\right )-4 b c h (a h (f x-e)+c (-d g+d h x+e g x))+b^3 f g h}{\left (b^2-4 a c\right ) (g+h x) \sqrt{a+x (b+c x)} \left (h (b g-a h)-c g^2\right )}+\frac{c h \left (\frac{\left (4 a c-b^2\right ) \tanh ^{-1}\left (\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right ) \left (h \left (2 a h (e h-2 f g)+b h (e g-3 d h)+b f g^2\right )+2 c \left (g h (3 d h-2 e g)+f g^3\right )\right )}{\left (h (a h-b g)+c g^2\right )^{5/2}}-\frac{2 h \sqrt{a+x (b+c x)} \left (4 a^2 f h^2-2 c \left (2 a h (2 d h-3 e g)+4 a f g^2+b g (2 d h+e g)\right )-2 a b h (e h+2 f g)+b^2 \left (h (3 d h-e g)+3 f g^2\right )+4 c^2 d g^2\right )}{(g+h x) \left (h (a h-b g)+c g^2\right )^2}\right )}{b^2-4 a c}-\frac{f}{(g+h x) \sqrt{a+x (b+c x)}}}{2 c h}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-(f/((g + h*x)*Sqrt[a + x*(b + c*x)])) + (b^3*f*g*h - b^2*(4*c*d*h^2 + a*f*h^2 + c*f*g*(g - 4*h*x)) - 4*b*c*h
*(a*h*(-e + f*x) + c*(-(d*g) + e*g*x + d*h*x)) + 4*c*(-(a^2*f*h^2) + 2*c^2*d*g*h*x + a*c*(f*g*(g - 2*h*x) + 2*
h*(-(e*g) + d*h + e*h*x))))/((b^2 - 4*a*c)*(-(c*g^2) + h*(b*g - a*h))*(g + h*x)*Sqrt[a + x*(b + c*x)]) + (c*h*
((-2*h*(4*c^2*d*g^2 + 4*a^2*f*h^2 - 2*a*b*h*(2*f*g + e*h) - 2*c*(4*a*f*g^2 + 2*a*h*(-3*e*g + 2*d*h) + b*g*(e*g
+ 2*d*h)) + b^2*(3*f*g^2 + h*(-(e*g) + 3*d*h)))*Sqrt[a + x*(b + c*x)])/((c*g^2 + h*(-(b*g) + a*h))^2*(g + h*x
)) + ((-b^2 + 4*a*c)*(2*c*(f*g^3 + g*h*(-2*e*g + 3*d*h)) + h*(b*f*g^2 + b*h*(e*g - 3*d*h) + 2*a*h*(-2*f*g + e*
h)))*ArcTanh[(-(b*g) + 2*a*h - 2*c*g*x + b*h*x)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/(c*
g^2 + h*(-(b*g) + a*h))^(5/2)))/(b^2 - 4*a*c))/(2*c*h)

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Maple [B]  time = 0.301, size = 4930, normalized size = 11.7 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^2+e*x+d)/(h*x+g)^2/(c*x^2+b*x+a)^(3/2),x)

[Out]

-3/(a*h^2-b*g*h+c*g^2)^2/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*c*g^2*e+2*f/h^2*(2*
c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-2/h/(a*h^2-b*g*h+c*g^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h
+c*g^2)/h^2)^(1/2)*f*g-1/(a*h^2-b*g*h+c*g^2)/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2
)/h^2)^(1/2)*b^2*e-1/(a*h^2-b*g*h+c*g^2)/((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2-b*g*h+c*g^2)/h^2+(b*h-2*
c*g)/h*(x+g/h)+2*((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(
1/2))/(x+g/h))*e-1/(a*h^2-b*g*h+c*g^2)/(x+g/h)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/
2)*d-3/2*h^2/(a*h^2-b*g*h+c*g^2)^2/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*b*d-3/2/(
a*h^2-b*g*h+c*g^2)^2/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*b*f*g^2+1/(a*h^2-b*g*h+
c*g^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*e+6/h^2/(a*h^2-b*g*h+c*g^2)^2/(4*a*c-
b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*b*c^2*g^4*f+3/(a*h^2-b*g*h+c*g^2)^2/(4*
a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*x*b^2*c*f*g^2+12/(a*h^2-b*g*h+c*g^2
)^2/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*x*b*c^2*g^2*e-6*h/(a*h^2-b*g
*h+c*g^2)^2/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*b^2*c*g*d-6/h/(a*h^2
-b*g*h+c*g^2)^2/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*b^2*c*g^3*f-6/h/
(a*h^2-b*g*h+c*g^2)^2/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*b*c^2*g^3*
e-12/h/(a*h^2-b*g*h+c*g^2)^2/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*x*c
^3*g^3*e+3*h^2/(a*h^2-b*g*h+c*g^2)^2/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(
1/2)*x*b^2*c*d+12/h/(a*h^2-b*g*h+c*g^2)/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2
)^(1/2)*x*c^2*g*e+12/h^2/(a*h^2-b*g*h+c*g^2)^2/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g
^2)/h^2)^(1/2)*x*c^3*g^4*f+6/h/(a*h^2-b*g*h+c*g^2)/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h
+c*g^2)/h^2)^(1/2)*b*c*g*e-8/h^2/(a*h^2-b*g*h+c*g^2)/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g
*h+c*g^2)/h^2)^(1/2)*b*c*g^2*f-16/h^2/(a*h^2-b*g*h+c*g^2)/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^
2-b*g*h+c*g^2)/h^2)^(1/2)*x*c^2*g^2*f+4/h/(a*h^2-b*g*h+c*g^2)/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(
a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*x*b*c*f*g-3*h/(a*h^2-b*g*h+c*g^2)^2/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/
h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*x*b^2*c*e*g-12*h/(a*h^2-b*g*h+c*g^2)^2/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)/
h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*x*b*c^2*g*d-12/h/(a*h^2-b*g*h+c*g^2)^2/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-
2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*x*b*c^2*g^3*f-8*c^2/(a*h^2-b*g*h+c*g^2)/(4*a*c-b^2)/((x+g/h)^2
*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*x*d+3/(a*h^2-b*g*h+c*g^2)^2/((a*h^2-b*g*h+c*g^2)/h^2)^
(1/2)*ln((2*(a*h^2-b*g*h+c*g^2)/h^2+(b*h-2*c*g)/h*(x+g/h)+2*((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c+(b*h-
2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2))/(x+g/h))*c*g^2*e+3/2*h^2/(a*h^2-b*g*h+c*g^2)^2/((a*h^2-b*g*h+
c*g^2)/h^2)^(1/2)*ln((2*(a*h^2-b*g*h+c*g^2)/h^2+(b*h-2*c*g)/h*(x+g/h)+2*((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*((x+g/
h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2))/(x+g/h))*b*d+1/h/(a*h^2-b*g*h+c*g^2)/(x+g/h)/((x+
g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*e*g-1/h^2/(a*h^2-b*g*h+c*g^2)/(x+g/h)/((x+g/h)^2
*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*f*g^2+3/2*h/(a*h^2-b*g*h+c*g^2)^2/((x+g/h)^2*c+(b*h-2*
c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*b*e*g-4*c/(a*h^2-b*g*h+c*g^2)/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*
g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*b*d+3/2/(a*h^2-b*g*h+c*g^2)^2/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)
/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*b^3*f*g^2+3/2/(a*h^2-b*g*h+c*g^2)^2/((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*
ln((2*(a*h^2-b*g*h+c*g^2)/h^2+(b*h-2*c*g)/h*(x+g/h)+2*((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c+(b*h-2*c*g)
/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2))/(x+g/h))*b*f*g^2+2/h/(a*h^2-b*g*h+c*g^2)/((a*h^2-b*g*h+c*g^2)/h^2)^
(1/2)*ln((2*(a*h^2-b*g*h+c*g^2)/h^2+(b*h-2*c*g)/h*(x+g/h)+2*((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c+(b*h-
2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2))/(x+g/h))*f*g+3/h/(a*h^2-b*g*h+c*g^2)^2/((x+g/h)^2*c+(b*h-2*c*
g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*c*g^3*f+3/2*h^2/(a*h^2-b*g*h+c*g^2)^2/(4*a*c-b^2)/((x+g/h)^2*c+(b*
h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*b^3*d+3*h/(a*h^2-b*g*h+c*g^2)^2/((x+g/h)^2*c+(b*h-2*c*g)/h*(
x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*c*g*d-3/h/(a*h^2-b*g*h+c*g^2)^2/((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*ln((2*(a
*h^2-b*g*h+c*g^2)/h^2+(b*h-2*c*g)/h*(x+g/h)+2*((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/
h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2))/(x+g/h))*c*g^3*f-2/(a*h^2-b*g*h+c*g^2)/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)
/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*x*b*c*e+2/h/(a*h^2-b*g*h+c*g^2)/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g)
/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*b^2*f*g+12/(a*h^2-b*g*h+c*g^2)^2/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2*c*g
)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*x*c^3*g^2*d+6/(a*h^2-b*g*h+c*g^2)^2/(4*a*c-b^2)/((x+g/h)^2*c+(b*h-2
*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*b*c^2*g^2*d-3/2*h/(a*h^2-b*g*h+c*g^2)^2/((a*h^2-b*g*h+c*g^2)/h^
2)^(1/2)*ln((2*(a*h^2-b*g*h+c*g^2)/h^2+(b*h-2*c*g)/h*(x+g/h)+2*((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c+(b
*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2))/(x+g/h))*b*e*g-3/2*h/(a*h^2-b*g*h+c*g^2)^2/(4*a*c-b^2)/((x
+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*b^3*e*g-3*h/(a*h^2-b*g*h+c*g^2)^2/((a*h^2-b*g*h
+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2-b*g*h+c*g^2)/h^2+(b*h-2*c*g)/h*(x+g/h)+2*((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*((x+g
/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2))/(x+g/h))*c*g*d+6/(a*h^2-b*g*h+c*g^2)^2/(4*a*c-b^
2)/((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*b^2*c*g^2*e

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)/(h*x+g)^2/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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((f*x^2+e*x+d)/(h*x+g)^2/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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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((f*x**2+e*x+d)/(h*x+g)**2/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x^{2} + e x + d}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (h x + g\right )}^{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)/(h*x+g)^2/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

integrate((f*x^2 + e*x + d)/((c*x^2 + b*x + a)^(3/2)*(h*x + g)^2), x)