3.232 \(\int \frac{d+e x+f x^2}{(g+h x)^3 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=336 \[ \frac{\tanh ^{-1}\left (\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right ) \left (8 a^2 f h^2-4 c \left (a \left (d h^2-3 e g h+f g^2\right )+b g (2 d h+e g)\right )-4 a b h (e h+2 f g)+b^2 \left (3 d h^2+e g h+3 f g^2\right )+8 c^2 d g^2\right )}{8 \left (a h^2-b g h+c g^2\right )^{5/2}}-\frac{\sqrt{a+b x+c x^2} \left (f g^2-h (e g-d h)\right )}{2 h (g+h x)^2 \left (a h^2-b g h+c g^2\right )}+\frac{\sqrt{a+b x+c x^2} \left (h \left (4 a h (2 f g-e h)-b \left (-3 d h^2-e g h+5 f g^2\right )\right )+2 c g \left (h (e g-3 d h)+f g^2\right )\right )}{4 h (g+h x) \left (a h^2-b g h+c g^2\right )^2} \]

[Out]

-((f*g^2 - h*(e*g - d*h))*Sqrt[a + b*x + c*x^2])/(2*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) + ((2*c*g*(f*g^2 +
h*(e*g - 3*d*h)) + h*(4*a*h*(2*f*g - e*h) - b*(5*f*g^2 - e*g*h - 3*d*h^2)))*Sqrt[a + b*x + c*x^2])/(4*h*(c*g^2
- b*g*h + a*h^2)^2*(g + h*x)) + ((8*c^2*d*g^2 + 8*a^2*f*h^2 - 4*a*b*h*(2*f*g + e*h) + b^2*(3*f*g^2 + e*g*h +
3*d*h^2) - 4*c*(b*g*(e*g + 2*d*h) + a*(f*g^2 - 3*e*g*h + d*h^2)))*ArcTanh[(b*g - 2*a*h + (2*c*g - b*h)*x)/(2*S
qrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/(8*(c*g^2 - b*g*h + a*h^2)^(5/2))

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Rubi [A]  time = 0.656221, antiderivative size = 336, normalized size of antiderivative = 1., 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 = {1650, 806, 724, 206} \[ \frac{\tanh ^{-1}\left (\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right ) \left (8 a^2 f h^2-4 c \left (-a h (3 e g-d h)+a f g^2+b g (2 d h+e g)\right )-4 a b h (e h+2 f g)+b^2 \left (h (3 d h+e g)+3 f g^2\right )+8 c^2 d g^2\right )}{8 \left (a h^2-b g h+c g^2\right )^{5/2}}-\frac{\sqrt{a+b x+c x^2} \left (f g^2-h (e g-d h)\right )}{2 h (g+h x)^2 \left (a h^2-b g h+c g^2\right )}+\frac{\sqrt{a+b x+c x^2} \left (2 c \left (g h (e g-3 d h)+f g^3\right )-h \left (-4 a h (2 f g-e h)-b h (3 d h+e g)+5 b f g^2\right )\right )}{4 h (g+h x) \left (a h^2-b g h+c g^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((f*g^2 - h*(e*g - d*h))*Sqrt[a + b*x + c*x^2])/(2*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) + ((2*c*(f*g^3 + g*
h*(e*g - 3*d*h)) - h*(5*b*f*g^2 - b*h*(e*g + 3*d*h) - 4*a*h*(2*f*g - e*h)))*Sqrt[a + b*x + c*x^2])/(4*h*(c*g^2
- b*g*h + a*h^2)^2*(g + h*x)) + ((8*c^2*d*g^2 + 8*a^2*f*h^2 - 4*a*b*h*(2*f*g + e*h) - 4*c*(a*f*g^2 - a*h*(3*e
*g - d*h) + b*g*(e*g + 2*d*h)) + b^2*(3*f*g^2 + h*(e*g + 3*d*h)))*ArcTanh[(b*g - 2*a*h + (2*c*g - b*h)*x)/(2*S
qrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/(8*(c*g^2 - b*g*h + a*h^2)^(5/2))

Rule 1650

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

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)^3 \sqrt{a+b x+c x^2}} \, dx &=-\frac{\left (f g^2-h (e g-d h)\right ) \sqrt{a+b x+c x^2}}{2 h \left (c g^2-b g h+a h^2\right ) (g+h x)^2}-\frac{\int \frac{\frac{1}{2} \left (-4 c d g+b e g+4 a f g-\frac{b f g^2}{h}+3 b d h-4 a e h\right )-\left (c e g-2 b f g+\frac{c f g^2}{h}-c d h+2 a f h\right ) x}{(g+h x)^2 \sqrt{a+b x+c x^2}} \, dx}{2 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac{\left (f g^2-h (e g-d h)\right ) \sqrt{a+b x+c x^2}}{2 h \left (c g^2-b g h+a h^2\right ) (g+h x)^2}+\frac{\left (2 c \left (f g^3+g h (e g-3 d h)\right )-h \left (5 b f g^2-b h (e g+3 d h)-4 a h (2 f g-e h)\right )\right ) \sqrt{a+b x+c x^2}}{4 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)}+\frac{\left (8 c^2 d g^2+8 a^2 f h^2-4 a b h (2 f g+e h)-4 c \left (a f g^2-a h (3 e g-d h)+b g (e g+2 d h)\right )+b^2 \left (3 f g^2+h (e g+3 d h)\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+b x+c x^2}} \, dx}{8 \left (c g^2-b g h+a h^2\right )^2}\\ &=-\frac{\left (f g^2-h (e g-d h)\right ) \sqrt{a+b x+c x^2}}{2 h \left (c g^2-b g h+a h^2\right ) (g+h x)^2}+\frac{\left (2 c \left (f g^3+g h (e g-3 d h)\right )-h \left (5 b f g^2-b h (e g+3 d h)-4 a h (2 f g-e h)\right )\right ) \sqrt{a+b x+c x^2}}{4 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)}-\frac{\left (8 c^2 d g^2+8 a^2 f h^2-4 a b h (2 f g+e h)-4 c \left (a f g^2-a h (3 e g-d h)+b g (e g+2 d h)\right )+b^2 \left (3 f g^2+h (e g+3 d 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 )}{4 \left (c g^2-b g h+a h^2\right )^2}\\ &=-\frac{\left (f g^2-h (e g-d h)\right ) \sqrt{a+b x+c x^2}}{2 h \left (c g^2-b g h+a h^2\right ) (g+h x)^2}+\frac{\left (2 c \left (f g^3+g h (e g-3 d h)\right )-h \left (5 b f g^2-b h (e g+3 d h)-4 a h (2 f g-e h)\right )\right ) \sqrt{a+b x+c x^2}}{4 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)}+\frac{\left (8 c^2 d g^2+8 a^2 f h^2-4 a b h (2 f g+e h)-4 c \left (a f g^2-a h (3 e g-d h)+b g (e g+2 d h)\right )+b^2 \left (3 f g^2+h (e g+3 d 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 )}{8 \left (c g^2-b g h+a h^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 1.34882, size = 367, normalized size = 1.09 \[ \frac{-\frac{c h \tanh ^{-1}\left (\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right ) \left (8 a^2 f h^2-4 c \left (a h (d h-3 e g)+a f g^2+b g (2 d h+e g)\right )-4 a b h (e h+2 f g)+b^2 \left (h (3 d h+e g)+3 f g^2\right )+8 c^2 d g^2\right )}{8 \left (h (a h-b g)+c g^2\right )^{5/2}}+\frac{c \sqrt{a+x (b+c x)} \left (h \left (-4 a h (e h-2 f g)+b h (3 d h+e g)-5 b f g^2\right )+2 c \left (g h (e g-3 d h)+f g^3\right )\right )}{4 (g+h x) \left (h (a h-b g)+c g^2\right )^2}+\frac{\sqrt{a+x (b+c x)} \left (2 f h (a h-b g)+c h (e g-d h)+c f g^2\right )}{2 (g+h x)^2 \left (h (a h-b g)+c g^2\right )}-\frac{f \sqrt{a+x (b+c x)}}{(g+h x)^2}}{c h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.285, size = 3615, normalized size = 10.8 \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)^3/(c*x^2+b*x+a)^(1/2),x)

[Out]

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*c*g^3*f-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*c*g^2*e-f/h^3/((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))-1/2/h/(a*h^2-b*g*h+c*g^2)/(x+g/h)^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)*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-3/2/h^2/(a
*h^2-b*g*h+c*g^2)^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)*c*g^3*f-3/8/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^2*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*c*g*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))*c^2*g^2*d+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))*c^2*g^3*e-3/2/h^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^2*g^4*f+3/4/h/(a*h^2-b*g*h+c*g^2)^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)*b*f*g^2-1/h^2/(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))*b*f*g-3/2/h^2/(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))*c*g*e+5/2/h^3/(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))*c*g^2*f+3/2/h/(a*h^2-b*g*h+c*g^2)^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)*c*g^2*e+2/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+1/2/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))*b*e-1/2/h^3/(a*h^2-b*g*h+c*g^2)/(x+g/h)^
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^2+3/4*h/(a*h^2-b*g*h+c*g^2)^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)*b*d-3/4/(a*h^2-b*g*h+c*g^2)^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)*b*e*g+1/2/h^2/(a*h^2-b*g*h+c*g^2)/(x+g/h)^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*g-3/2/(a*h^2-b*g*h+c*g^2)^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)*c*g*d-3/8*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^2*d+3/8/(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^2*e*g+1/2/h*c/(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))*d

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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)^3/(c*x^2+b*x+a)^(1/2),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)^3/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**2+e*x+d)/(h*x+g)**3/(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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Giac [B]  time = 1.36301, size = 3114, normalized size = 9.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)/(h*x+g)^3/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/4*(8*c^2*d*g^2 + 3*b^2*f*g^2 - 4*a*c*f*g^2 - 8*b*c*d*g*h - 8*a*b*f*g*h + 3*b^2*d*h^2 - 4*a*c*d*h^2 + 8*a^2*f
*h^2 - 4*b*c*g^2*e + b^2*g*h*e + 12*a*c*g*h*e - 4*a*b*h^2*e)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*h +
sqrt(c)*g)/sqrt(-c*g^2 + b*g*h - a*h^2))/((c^2*g^4 - 2*b*c*g^3*h + b^2*g^2*h^2 + 2*a*c*g^2*h^2 - 2*a*b*g*h^3 +
a^2*h^4)*sqrt(-c*g^2 + b*g*h - a*h^2)) + 1/4*(8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^2*f*g^4*h - 16*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^3*b*c*f*g^3*h^2 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^2*d*g^2*h^3 + 5*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*f*g^2*h^3 + 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c*f*g^2*h^3 + 8*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c*d*g*h^4 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*f*g*h^4 - 3*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*d*h^5 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c*d*h^5 + 4*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^3*b*c*g^2*h^3*e - (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*g*h^4*e - 12*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*a*c*g*h^4*e + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*h^5*e + 8*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^2*c^(5/2)*f*g^5 - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(3/2)*f*g^4*h - 24*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^2*c^(5/2)*d*g^3*h^2 - (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*sqrt(c)*f*g^3*h^
2 + 28*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(3/2)*f*g^3*h^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*
c^(3/2)*d*g^2*h^3 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*sqrt(c)*f*g^2*h^3 - 9*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^2*b^2*sqrt(c)*d*g*h^4 + 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(3/2)*d*g*h^4 - 16*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2*a^2*sqrt(c)*f*g*h^4 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(5/2)*g^4*h*e - 4*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(3/2)*g^3*h^2*e + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*sqrt(c
)*g^2*h^3*e - 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(3/2)*g^2*h^3*e - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^2*a*b*sqrt(c)*g*h^4*e + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*sqrt(c)*h^5*e + 8*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))*b*c^2*f*g^5 - 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c*f*g^4*h - 8*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))*a*c^2*f*g^4*h - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^2*d*g^3*h^2 + 3*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))*b^3*f*g^3*h^2 + 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c*f*g^3*h^2 + 20*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))*b^2*c*d*g^2*h^3 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*c^2*d*g^2*h^3 - 11*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))*a*b^2*f*g^2*h^3 - 44*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c*f*g^2*h^3 - 5*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*b^3*d*g*h^4 - 28*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c*d*g*h^4 + 8*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))*a^2*b*f*g*h^4 + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*d*h^5 + 4*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))*a^2*c*d*h^5 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^2*g^4*h*e - 16*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))*a*c^2*g^3*h^2*e + (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*g^2*h^3*e - 16*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*a*b*c*g^2*h^3*e + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*g*h^4*e + 20*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))*a^2*c*g*h^4*e - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*h^5*e + 2*b^2*c^(3/2)*f*g^5 - 5*b^3*
sqrt(c)*f*g^4*h - 4*a*b*c^(3/2)*f*g^4*h - 6*b^2*c^(3/2)*d*g^3*h^2 + 21*a*b^2*sqrt(c)*f*g^3*h^2 + 4*a^2*c^(3/2)
*f*g^3*h^2 + 3*b^3*sqrt(c)*d*g^2*h^3 + 20*a*b*c^(3/2)*d*g^2*h^3 - 32*a^2*b*sqrt(c)*f*g^2*h^3 - 11*a*b^2*sqrt(c
)*d*g*h^4 - 12*a^2*c^(3/2)*d*g*h^4 + 16*a^3*sqrt(c)*f*g*h^4 + 8*a^2*b*sqrt(c)*d*h^5 + 2*b^2*c^(3/2)*g^4*h*e +
b^3*sqrt(c)*g^3*h^2*e - 8*a*b*c^(3/2)*g^3*h^2*e - 5*a*b^2*sqrt(c)*g^2*h^3*e + 4*a^2*c^(3/2)*g^2*h^3*e + 12*a^2
*b*sqrt(c)*g*h^4*e - 8*a^3*sqrt(c)*h^5*e)/((c^2*g^4*h^2 - 2*b*c*g^3*h^3 + b^2*g^2*h^4 + 2*a*c*g^2*h^4 - 2*a*b*
g*h^5 + a^2*h^6)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*h + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c)*g +
b*g - a*h)^2)