3.207 \(\int \frac {(a+b x+c x^2)^{3/2} (d+e x+f x^2)}{(g+h x)^8} \, dx\)

Optimal. Leaf size=1062 \[ \frac {\left (4 c^2 \left (5 f g^2+h (2 e g-51 d h)\right ) g^2-7 h^2 \left (\left (5 f g^2+5 e h g+9 d h^2\right ) b^2-2 a h (10 f g+7 e h) b+24 a^2 f h^2\right )-2 c h \left (3 b g \left (8 f g^2-15 e h g-34 d h^2\right )-2 a h \left (26 f g^2-61 e h g+12 d h^2\right )\right )\right ) \left (c x^2+b x+a\right )^{5/2}}{840 h \left (c g^2-b h g+a h^2\right )^3 (g+h x)^5}+\frac {\left (2 c g \left (5 f g^2+h (2 e g-9 d h)\right )+h \left (14 a h (2 f g-e h)-b \left (19 f g^2-5 e h g-9 d h^2\right )\right )\right ) \left (c x^2+b x+a\right )^{5/2}}{84 h \left (c g^2-b h g+a h^2\right )^2 (g+h x)^6}-\frac {\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{7 h \left (c g^2-b h g+a h^2\right ) (g+h x)^7}+\frac {\left (48 c^3 d g^3-8 c^2 \left (3 b g (e g+3 d h)+a \left (f g^2-8 e h g+3 d h^2\right )\right ) g-b h \left (\left (5 f g^2+5 e h g+9 d h^2\right ) b^2-2 a h (10 f g+7 e h) b+24 a^2 f h^2\right )+2 c \left (g \left (7 f g^2+10 e h g+21 d h^2\right ) b^2-2 a h \left (13 f g^2+13 e h g-3 d h^2\right ) b+4 a^2 h^2 (8 f g-e h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \left (c x^2+b x+a\right )^{3/2}}{384 \left (c g^2-b h g+a h^2\right )^4 (g+h x)^4}-\frac {\left (b^2-4 a c\right ) \left (48 c^3 d g^3-8 c^2 \left (3 b g (e g+3 d h)+a \left (f g^2-8 e h g+3 d h^2\right )\right ) g-b h \left (\left (5 f g^2+5 e h g+9 d h^2\right ) b^2-2 a h (10 f g+7 e h) b+24 a^2 f h^2\right )+2 c \left (g \left (7 f g^2+10 e h g+21 d h^2\right ) b^2-2 a h \left (13 f g^2+13 e h g-3 d h^2\right ) b+4 a^2 h^2 (8 f g-e h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \sqrt {c x^2+b x+a}}{1024 \left (c g^2-b h g+a h^2\right )^5 (g+h x)^2}+\frac {\left (b^2-4 a c\right )^2 \left (48 c^3 d g^3-8 c^2 \left (3 b g (e g+3 d h)+a \left (f g^2-8 e h g+3 d h^2\right )\right ) g-b h \left (\left (5 f g^2+5 e h g+9 d h^2\right ) b^2-2 a h (10 f g+7 e h) b+24 a^2 f h^2\right )+2 c \left (g \left (7 f g^2+10 e h g+21 d h^2\right ) b^2-2 a h \left (13 f g^2+13 e h g-3 d h^2\right ) b+4 a^2 h^2 (8 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 h g+a h^2} \sqrt {c x^2+b x+a}}\right )}{2048 \left (c g^2-b h g+a h^2\right )^{11/2}} \]

[Out]

1/384*(48*c^3*d*g^3-8*c^2*g*(3*b*g*(3*d*h+e*g)+a*(3*d*h^2-8*e*g*h+f*g^2))-b*h*(24*a^2*f*h^2-2*a*b*h*(7*e*h+10*
f*g)+b^2*(9*d*h^2+5*e*g*h+5*f*g^2))+2*c*(4*a^2*h^2*(-e*h+8*f*g)-2*a*b*h*(-3*d*h^2+13*e*g*h+13*f*g^2)+b^2*g*(21
*d*h^2+10*e*g*h+7*f*g^2)))*(b*g-2*a*h+(-b*h+2*c*g)*x)*(c*x^2+b*x+a)^(3/2)/(a*h^2-b*g*h+c*g^2)^4/(h*x+g)^4-1/7*
(f*g^2-h*(-d*h+e*g))*(c*x^2+b*x+a)^(5/2)/h/(a*h^2-b*g*h+c*g^2)/(h*x+g)^7+1/84*(2*c*g*(5*f*g^2+h*(-9*d*h+2*e*g)
)+h*(14*a*h*(-e*h+2*f*g)-b*(-9*d*h^2-5*e*g*h+19*f*g^2)))*(c*x^2+b*x+a)^(5/2)/h/(a*h^2-b*g*h+c*g^2)^2/(h*x+g)^6
+1/840*(4*c^2*g^2*(5*f*g^2+h*(-51*d*h+2*e*g))-7*h^2*(24*a^2*f*h^2-2*a*b*h*(7*e*h+10*f*g)+b^2*(9*d*h^2+5*e*g*h+
5*f*g^2))-2*c*h*(3*b*g*(-34*d*h^2-15*e*g*h+8*f*g^2)-2*a*h*(12*d*h^2-61*e*g*h+26*f*g^2)))*(c*x^2+b*x+a)^(5/2)/h
/(a*h^2-b*g*h+c*g^2)^3/(h*x+g)^5+1/2048*(-4*a*c+b^2)^2*(48*c^3*d*g^3-8*c^2*g*(3*b*g*(3*d*h+e*g)+a*(3*d*h^2-8*e
*g*h+f*g^2))-b*h*(24*a^2*f*h^2-2*a*b*h*(7*e*h+10*f*g)+b^2*(9*d*h^2+5*e*g*h+5*f*g^2))+2*c*(4*a^2*h^2*(-e*h+8*f*
g)-2*a*b*h*(-3*d*h^2+13*e*g*h+13*f*g^2)+b^2*g*(21*d*h^2+10*e*g*h+7*f*g^2)))*arctanh(1/2*(b*g-2*a*h+(-b*h+2*c*g
)*x)/(a*h^2-b*g*h+c*g^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*h^2-b*g*h+c*g^2)^(11/2)-1/1024*(-4*a*c+b^2)*(48*c^3*d*g
^3-8*c^2*g*(3*b*g*(3*d*h+e*g)+a*(3*d*h^2-8*e*g*h+f*g^2))-b*h*(24*a^2*f*h^2-2*a*b*h*(7*e*h+10*f*g)+b^2*(9*d*h^2
+5*e*g*h+5*f*g^2))+2*c*(4*a^2*h^2*(-e*h+8*f*g)-2*a*b*h*(-3*d*h^2+13*e*g*h+13*f*g^2)+b^2*g*(21*d*h^2+10*e*g*h+7
*f*g^2)))*(b*g-2*a*h+(-b*h+2*c*g)*x)*(c*x^2+b*x+a)^(1/2)/(a*h^2-b*g*h+c*g^2)^5/(h*x+g)^2

________________________________________________________________________________________

Rubi [A]  time = 3.00, antiderivative size = 1062, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1650, 834, 806, 720, 724, 206} \[ \frac {\left (4 \left (5 f g^4+h (2 e g-51 d h) g^2\right ) c^2-2 h \left (3 b g \left (8 f g^2-15 e h g-34 d h^2\right )-2 a h \left (26 f g^2-61 e h g+12 d h^2\right )\right ) c-7 h^2 \left (\left (5 f g^2+5 e h g+9 d h^2\right ) b^2-2 a h (10 f g+7 e h) b+24 a^2 f h^2\right )\right ) \left (c x^2+b x+a\right )^{5/2}}{840 h \left (c g^2-b h g+a h^2\right )^3 (g+h x)^5}+\frac {\left (2 c \left (5 f g^3+h (2 e g-9 d h) g\right )-h \left (19 b f g^2-b h (5 e g+9 d h)-14 a h (2 f g-e h)\right )\right ) \left (c x^2+b x+a\right )^{5/2}}{84 h \left (c g^2-b h g+a h^2\right )^2 (g+h x)^6}-\frac {\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{7 h \left (c g^2-b h g+a h^2\right ) (g+h x)^7}+\frac {\left (48 c^3 d g^3-8 c^2 \left (a f g^2+3 b (e g+3 d h) g-a h (8 e g-3 d h)\right ) g-b h \left (\left (5 f g^2+h (5 e g+9 d h)\right ) b^2-2 a h (10 f g+7 e h) b+24 a^2 f h^2\right )+2 c \left (\left (7 f g^3+h (10 e g+21 d h) g\right ) b^2-2 a h \left (13 f g^2+h (13 e g-3 d h)\right ) b+4 a^2 h^2 (8 f g-e h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \left (c x^2+b x+a\right )^{3/2}}{384 \left (c g^2-b h g+a h^2\right )^4 (g+h x)^4}-\frac {\left (b^2-4 a c\right ) \left (48 c^3 d g^3-8 c^2 \left (a f g^2+3 b (e g+3 d h) g-a h (8 e g-3 d h)\right ) g-b h \left (\left (5 f g^2+h (5 e g+9 d h)\right ) b^2-2 a h (10 f g+7 e h) b+24 a^2 f h^2\right )+2 c \left (\left (7 f g^3+h (10 e g+21 d h) g\right ) b^2-2 a h \left (13 f g^2+h (13 e g-3 d h)\right ) b+4 a^2 h^2 (8 f g-e h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \sqrt {c x^2+b x+a}}{1024 \left (c g^2-b h g+a h^2\right )^5 (g+h x)^2}+\frac {\left (b^2-4 a c\right )^2 \left (48 c^3 d g^3-8 c^2 \left (a f g^2+3 b (e g+3 d h) g-a h (8 e g-3 d h)\right ) g-b h \left (\left (5 f g^2+h (5 e g+9 d h)\right ) b^2-2 a h (10 f g+7 e h) b+24 a^2 f h^2\right )+2 c \left (\left (7 f g^3+h (10 e g+21 d h) g\right ) b^2-2 a h \left (13 f g^2+h (13 e g-3 d h)\right ) b+4 a^2 h^2 (8 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 h g+a h^2} \sqrt {c x^2+b x+a}}\right )}{2048 \left (c g^2-b h g+a h^2\right )^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^8,x]

[Out]

-((b^2 - 4*a*c)*(48*c^3*d*g^3 - 8*c^2*g*(a*f*g^2 - a*h*(8*e*g - 3*d*h) + 3*b*g*(e*g + 3*d*h)) - b*h*(24*a^2*f*
h^2 - 2*a*b*h*(10*f*g + 7*e*h) + b^2*(5*f*g^2 + h*(5*e*g + 9*d*h))) + 2*c*(4*a^2*h^2*(8*f*g - e*h) - 2*a*b*h*(
13*f*g^2 + h*(13*e*g - 3*d*h)) + b^2*(7*f*g^3 + g*h*(10*e*g + 21*d*h))))*(b*g - 2*a*h + (2*c*g - b*h)*x)*Sqrt[
a + b*x + c*x^2])/(1024*(c*g^2 - b*g*h + a*h^2)^5*(g + h*x)^2) + ((48*c^3*d*g^3 - 8*c^2*g*(a*f*g^2 - a*h*(8*e*
g - 3*d*h) + 3*b*g*(e*g + 3*d*h)) - b*h*(24*a^2*f*h^2 - 2*a*b*h*(10*f*g + 7*e*h) + b^2*(5*f*g^2 + h*(5*e*g + 9
*d*h))) + 2*c*(4*a^2*h^2*(8*f*g - e*h) - 2*a*b*h*(13*f*g^2 + h*(13*e*g - 3*d*h)) + b^2*(7*f*g^3 + g*h*(10*e*g
+ 21*d*h))))*(b*g - 2*a*h + (2*c*g - b*h)*x)*(a + b*x + c*x^2)^(3/2))/(384*(c*g^2 - b*g*h + a*h^2)^4*(g + h*x)
^4) - ((f*g^2 - h*(e*g - d*h))*(a + b*x + c*x^2)^(5/2))/(7*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^7) + ((2*c*(5*f
*g^3 + g*h*(2*e*g - 9*d*h)) - h*(19*b*f*g^2 - b*h*(5*e*g + 9*d*h) - 14*a*h*(2*f*g - e*h)))*(a + b*x + c*x^2)^(
5/2))/(84*h*(c*g^2 - b*g*h + a*h^2)^2*(g + h*x)^6) + ((4*c^2*(5*f*g^4 + g^2*h*(2*e*g - 51*d*h)) - 7*h^2*(24*a^
2*f*h^2 - 2*a*b*h*(10*f*g + 7*e*h) + b^2*(5*f*g^2 + 5*e*g*h + 9*d*h^2)) - 2*c*h*(3*b*g*(8*f*g^2 - 15*e*g*h - 3
4*d*h^2) - 2*a*h*(26*f*g^2 - 61*e*g*h + 12*d*h^2)))*(a + b*x + c*x^2)^(5/2))/(840*h*(c*g^2 - b*g*h + a*h^2)^3*
(g + h*x)^5) + ((b^2 - 4*a*c)^2*(48*c^3*d*g^3 - 8*c^2*g*(a*f*g^2 - a*h*(8*e*g - 3*d*h) + 3*b*g*(e*g + 3*d*h))
- b*h*(24*a^2*f*h^2 - 2*a*b*h*(10*f*g + 7*e*h) + b^2*(5*f*g^2 + h*(5*e*g + 9*d*h))) + 2*c*(4*a^2*h^2*(8*f*g -
e*h) - 2*a*b*h*(13*f*g^2 + h*(13*e*g - 3*d*h)) + b^2*(7*f*g^3 + g*h*(10*e*g + 21*d*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])])/(2048*(c*g^2 - b*g*h + a*h^2)^(11/2
))

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])

Rule 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, 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] && EqQ[m +
2*p + 2, 0] && GtQ[p, 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 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 834

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

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^8} \, dx &=-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{7 h \left (c g^2-b g h+a h^2\right ) (g+h x)^7}-\frac {\int \frac {\left (\frac {1}{2} \left (-14 c d g+5 b e g+14 a f g-\frac {5 b f g^2}{h}+9 b d h-14 a e h\right )-\left (2 c e g-7 b f g+\frac {5 c f g^2}{h}-2 c d h+7 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{(g+h x)^7} \, dx}{7 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{7 h \left (c g^2-b g h+a h^2\right ) (g+h x)^7}+\frac {\left (2 c \left (5 f g^3+g h (2 e g-9 d h)\right )-h \left (19 b f g^2-b h (5 e g+9 d h)-14 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{84 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^6}+\frac {\int \frac {\left (\frac {1}{4} \left (168 c^2 d g^2+168 a^2 f h^2-2 b c g \left (40 e g-\frac {5 f g^2}{h}+93 d h\right )-14 a b h (10 f g+7 e h)-24 a c \left (2 f g^2-h (9 e g-2 d h)\right )+7 b^2 \left (5 f g^2+h (5 e g+9 d h)\right )\right )+\frac {c \left (2 c \left (5 f g^3+g h (2 e g-9 d h)\right )-h \left (19 b f g^2-b h (5 e g+9 d h)-14 a h (2 f g-e h)\right )\right ) x}{2 h}\right ) \left (a+b x+c x^2\right )^{3/2}}{(g+h x)^6} \, dx}{42 \left (c g^2-b g h+a h^2\right )^2}\\ &=-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{7 h \left (c g^2-b g h+a h^2\right ) (g+h x)^7}+\frac {\left (2 c \left (5 f g^3+g h (2 e g-9 d h)\right )-h \left (19 b f g^2-b h (5 e g+9 d h)-14 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{84 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^6}+\frac {\left (4 c^2 \left (5 f g^4+g^2 h (2 e g-51 d h)\right )-7 h^2 \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+5 e g h+9 d h^2\right )\right )-2 c h \left (3 b g \left (8 f g^2-15 e g h-34 d h^2\right )-2 a h \left (26 f g^2-61 e g h+12 d h^2\right )\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{840 h \left (c g^2-b g h+a h^2\right )^3 (g+h x)^5}+\frac {\left (48 c^3 d g^3-8 c^2 g \left (a f g^2-a h (8 e g-3 d h)+3 b g (e g+3 d h)\right )-b h \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+h (5 e g+9 d h)\right )\right )+2 c \left (4 a^2 h^2 (8 f g-e h)-2 a b h \left (13 f g^2+h (13 e g-3 d h)\right )+b^2 \left (7 f g^3+g h (10 e g+21 d h)\right )\right )\right ) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(g+h x)^5} \, dx}{48 \left (c g^2-b g h+a h^2\right )^3}\\ &=\frac {\left (48 c^3 d g^3-8 c^2 g \left (a f g^2-a h (8 e g-3 d h)+3 b g (e g+3 d h)\right )-b h \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+h (5 e g+9 d h)\right )\right )+2 c \left (4 a^2 h^2 (8 f g-e h)-2 a b h \left (13 f g^2+h (13 e g-3 d h)\right )+b^2 \left (7 f g^3+g h (10 e g+21 d h)\right )\right )\right ) (b g-2 a h+(2 c g-b h) x) \left (a+b x+c x^2\right )^{3/2}}{384 \left (c g^2-b g h+a h^2\right )^4 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{7 h \left (c g^2-b g h+a h^2\right ) (g+h x)^7}+\frac {\left (2 c \left (5 f g^3+g h (2 e g-9 d h)\right )-h \left (19 b f g^2-b h (5 e g+9 d h)-14 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{84 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^6}+\frac {\left (4 c^2 \left (5 f g^4+g^2 h (2 e g-51 d h)\right )-7 h^2 \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+5 e g h+9 d h^2\right )\right )-2 c h \left (3 b g \left (8 f g^2-15 e g h-34 d h^2\right )-2 a h \left (26 f g^2-61 e g h+12 d h^2\right )\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{840 h \left (c g^2-b g h+a h^2\right )^3 (g+h x)^5}-\frac {\left (\left (b^2-4 a c\right ) \left (48 c^3 d g^3-8 c^2 g \left (a f g^2-a h (8 e g-3 d h)+3 b g (e g+3 d h)\right )-b h \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+h (5 e g+9 d h)\right )\right )+2 c \left (4 a^2 h^2 (8 f g-e h)-2 a b h \left (13 f g^2+h (13 e g-3 d h)\right )+b^2 \left (7 f g^3+g h (10 e g+21 d h)\right )\right )\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{(g+h x)^3} \, dx}{256 \left (c g^2-b g h+a h^2\right )^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (48 c^3 d g^3-8 c^2 g \left (a f g^2-a h (8 e g-3 d h)+3 b g (e g+3 d h)\right )-b h \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+h (5 e g+9 d h)\right )\right )+2 c \left (4 a^2 h^2 (8 f g-e h)-2 a b h \left (13 f g^2+h (13 e g-3 d h)\right )+b^2 \left (7 f g^3+g h (10 e g+21 d h)\right )\right )\right ) (b g-2 a h+(2 c g-b h) x) \sqrt {a+b x+c x^2}}{1024 \left (c g^2-b g h+a h^2\right )^5 (g+h x)^2}+\frac {\left (48 c^3 d g^3-8 c^2 g \left (a f g^2-a h (8 e g-3 d h)+3 b g (e g+3 d h)\right )-b h \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+h (5 e g+9 d h)\right )\right )+2 c \left (4 a^2 h^2 (8 f g-e h)-2 a b h \left (13 f g^2+h (13 e g-3 d h)\right )+b^2 \left (7 f g^3+g h (10 e g+21 d h)\right )\right )\right ) (b g-2 a h+(2 c g-b h) x) \left (a+b x+c x^2\right )^{3/2}}{384 \left (c g^2-b g h+a h^2\right )^4 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{7 h \left (c g^2-b g h+a h^2\right ) (g+h x)^7}+\frac {\left (2 c \left (5 f g^3+g h (2 e g-9 d h)\right )-h \left (19 b f g^2-b h (5 e g+9 d h)-14 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{84 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^6}+\frac {\left (4 c^2 \left (5 f g^4+g^2 h (2 e g-51 d h)\right )-7 h^2 \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+5 e g h+9 d h^2\right )\right )-2 c h \left (3 b g \left (8 f g^2-15 e g h-34 d h^2\right )-2 a h \left (26 f g^2-61 e g h+12 d h^2\right )\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{840 h \left (c g^2-b g h+a h^2\right )^3 (g+h x)^5}+\frac {\left (\left (b^2-4 a c\right )^2 \left (48 c^3 d g^3-8 c^2 g \left (a f g^2-a h (8 e g-3 d h)+3 b g (e g+3 d h)\right )-b h \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+h (5 e g+9 d h)\right )\right )+2 c \left (4 a^2 h^2 (8 f g-e h)-2 a b h \left (13 f g^2+h (13 e g-3 d h)\right )+b^2 \left (7 f g^3+g h (10 e g+21 d h)\right )\right )\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+b x+c x^2}} \, dx}{2048 \left (c g^2-b g h+a h^2\right )^5}\\ &=-\frac {\left (b^2-4 a c\right ) \left (48 c^3 d g^3-8 c^2 g \left (a f g^2-a h (8 e g-3 d h)+3 b g (e g+3 d h)\right )-b h \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+h (5 e g+9 d h)\right )\right )+2 c \left (4 a^2 h^2 (8 f g-e h)-2 a b h \left (13 f g^2+h (13 e g-3 d h)\right )+b^2 \left (7 f g^3+g h (10 e g+21 d h)\right )\right )\right ) (b g-2 a h+(2 c g-b h) x) \sqrt {a+b x+c x^2}}{1024 \left (c g^2-b g h+a h^2\right )^5 (g+h x)^2}+\frac {\left (48 c^3 d g^3-8 c^2 g \left (a f g^2-a h (8 e g-3 d h)+3 b g (e g+3 d h)\right )-b h \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+h (5 e g+9 d h)\right )\right )+2 c \left (4 a^2 h^2 (8 f g-e h)-2 a b h \left (13 f g^2+h (13 e g-3 d h)\right )+b^2 \left (7 f g^3+g h (10 e g+21 d h)\right )\right )\right ) (b g-2 a h+(2 c g-b h) x) \left (a+b x+c x^2\right )^{3/2}}{384 \left (c g^2-b g h+a h^2\right )^4 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{7 h \left (c g^2-b g h+a h^2\right ) (g+h x)^7}+\frac {\left (2 c \left (5 f g^3+g h (2 e g-9 d h)\right )-h \left (19 b f g^2-b h (5 e g+9 d h)-14 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{84 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^6}+\frac {\left (4 c^2 \left (5 f g^4+g^2 h (2 e g-51 d h)\right )-7 h^2 \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+5 e g h+9 d h^2\right )\right )-2 c h \left (3 b g \left (8 f g^2-15 e g h-34 d h^2\right )-2 a h \left (26 f g^2-61 e g h+12 d h^2\right )\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{840 h \left (c g^2-b g h+a h^2\right )^3 (g+h x)^5}-\frac {\left (\left (b^2-4 a c\right )^2 \left (48 c^3 d g^3-8 c^2 g \left (a f g^2-a h (8 e g-3 d h)+3 b g (e g+3 d h)\right )-b h \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+h (5 e g+9 d h)\right )\right )+2 c \left (4 a^2 h^2 (8 f g-e h)-2 a b h \left (13 f g^2+h (13 e g-3 d h)\right )+b^2 \left (7 f g^3+g h (10 e g+21 d h)\right )\right )\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 )}{1024 \left (c g^2-b g h+a h^2\right )^5}\\ &=-\frac {\left (b^2-4 a c\right ) \left (48 c^3 d g^3-8 c^2 g \left (a f g^2-a h (8 e g-3 d h)+3 b g (e g+3 d h)\right )-b h \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+h (5 e g+9 d h)\right )\right )+2 c \left (4 a^2 h^2 (8 f g-e h)-2 a b h \left (13 f g^2+h (13 e g-3 d h)\right )+b^2 \left (7 f g^3+g h (10 e g+21 d h)\right )\right )\right ) (b g-2 a h+(2 c g-b h) x) \sqrt {a+b x+c x^2}}{1024 \left (c g^2-b g h+a h^2\right )^5 (g+h x)^2}+\frac {\left (48 c^3 d g^3-8 c^2 g \left (a f g^2-a h (8 e g-3 d h)+3 b g (e g+3 d h)\right )-b h \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+h (5 e g+9 d h)\right )\right )+2 c \left (4 a^2 h^2 (8 f g-e h)-2 a b h \left (13 f g^2+h (13 e g-3 d h)\right )+b^2 \left (7 f g^3+g h (10 e g+21 d h)\right )\right )\right ) (b g-2 a h+(2 c g-b h) x) \left (a+b x+c x^2\right )^{3/2}}{384 \left (c g^2-b g h+a h^2\right )^4 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{7 h \left (c g^2-b g h+a h^2\right ) (g+h x)^7}+\frac {\left (2 c \left (5 f g^3+g h (2 e g-9 d h)\right )-h \left (19 b f g^2-b h (5 e g+9 d h)-14 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{84 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^6}+\frac {\left (4 c^2 \left (5 f g^4+g^2 h (2 e g-51 d h)\right )-7 h^2 \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+5 e g h+9 d h^2\right )\right )-2 c h \left (3 b g \left (8 f g^2-15 e g h-34 d h^2\right )-2 a h \left (26 f g^2-61 e g h+12 d h^2\right )\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{840 h \left (c g^2-b g h+a h^2\right )^3 (g+h x)^5}+\frac {\left (b^2-4 a c\right )^2 \left (48 c^3 d g^3-8 c^2 g \left (a f g^2-a h (8 e g-3 d h)+3 b g (e g+3 d h)\right )-b h \left (24 a^2 f h^2-2 a b h (10 f g+7 e h)+b^2 \left (5 f g^2+h (5 e g+9 d h)\right )\right )+2 c \left (4 a^2 h^2 (8 f g-e h)-2 a b h \left (13 f g^2+h (13 e g-3 d h)\right )+b^2 \left (7 f g^3+g h (10 e g+21 d h)\right )\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 )}{2048 \left (c g^2-b g h+a h^2\right )^{11/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 6.42, size = 1221, normalized size = 1.15 \[ \frac {(a+x (b+c x))^{3/2} \left (-\frac {\left (\frac {1}{2} h (5 b f g+4 c d h-14 a f h)-\frac {1}{2} g (10 c f g+4 c e h-9 b f h)\right ) \left (c x^2+b x+a\right )^{5/2}}{7 \left (c g^2-b h g+a h^2\right ) (g+h x)^7}-\frac {-\frac {\left (2 c g \left (5 c f g^2-7 f h (b g-a h)+2 c h (e g-d h)\right )-c h \left (5 b f g^2-b h (5 e g+9 d h)+14 h (c d g-a f g+a e h)\right )\right ) \left (c x^2+b x+a\right )^{5/2}}{6 \left (c g^2-b h g+a h^2\right ) (g+h x)^6}-\frac {\frac {\left (c^2 g \left (2 c \left (5 f g^3+h (2 e g-9 d h) g\right )-h \left (19 b f g^2-b h (5 e g+9 d h)-14 a h (2 f g-e h)\right )\right )-\frac {1}{2} c h \left (7 h \left (5 f g^2+h (5 e g+9 d h)\right ) b^2+2 \left (5 c f g^3-c h (40 e g+93 d h) g-7 a h^2 (10 f g+7 e h)\right ) b+24 h \left (7 c^2 d g^2+7 a^2 f h^2-a c \left (2 f g^2-h (9 e g-2 d h)\right )\right )\right )\right ) \left (c x^2+b x+a\right )^{5/2}}{5 \left (c g^2-b h g+a h^2\right ) (g+h x)^5}-\frac {\left (b \left (g \left (2 c \left (5 f g^3+h (2 e g-9 d h) g\right )-h \left (19 b f g^2-b h (5 e g+9 d h)-14 a h (2 f g-e h)\right )\right ) c^2+\frac {1}{2} h \left (7 h \left (5 f g^2+h (5 e g+9 d h)\right ) b^2+2 \left (5 c f g^3-c h (40 e g+93 d h) g-7 a h^2 (10 f g+7 e h)\right ) b+24 h \left (7 c^2 d g^2+7 a^2 f h^2-a c \left (2 f g^2-h (9 e g-2 d h)\right )\right )\right ) c\right )-2 \left (a h \left (2 c \left (5 f g^3+h (2 e g-9 d h) g\right )-h \left (19 b f g^2-b h (5 e g+9 d h)-14 a h (2 f g-e h)\right )\right ) c^2+\frac {1}{2} g \left (7 h \left (5 f g^2+h (5 e g+9 d h)\right ) b^2+2 \left (5 c f g^3-c h (40 e g+93 d h) g-7 a h^2 (10 f g+7 e h)\right ) b+24 h \left (7 c^2 d g^2+7 a^2 f h^2-a c \left (2 f g^2-h (9 e g-2 d h)\right )\right )\right ) c^2\right )\right ) \left (\frac {(b g-2 a h+(2 c g-b h) x) \left (c x^2+b x+a\right )^{3/2}}{8 \left (c g^2-b h g+a h^2\right ) (g+h x)^4}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {c x^2+b x+a} (b g-2 a h+(2 c g-b h) x)}{4 \left (c g^2-b h g+a h^2\right ) (g+h x)^2}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {-b g+2 a h-(2 c g-b h) x}{2 \sqrt {c g^2-b h g+a h^2} \sqrt {c x^2+b x+a}}\right )}{2 \sqrt {c g^2-b h g+a h^2} \left (4 c g^2-4 b h g+4 a h^2\right )}\right )}{16 \left (c g^2-b h g+a h^2\right )}\right )}{2 \left (c g^2-b h g+a h^2\right )}}{6 \left (c g^2-b h g+a h^2\right )}}{7 \left (c g^2-b h g+a h^2\right )}\right )}{2 c h \left (c x^2+b x+a\right )^{3/2}}-\frac {f \left (c x^2+b x+a\right ) (a+x (b+c x))^{3/2}}{2 c h (g+h x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^8,x]

[Out]

-1/2*(f*(a + b*x + c*x^2)*(a + x*(b + c*x))^(3/2))/(c*h*(g + h*x)^7) + ((a + x*(b + c*x))^(3/2)*(-1/7*(((h*(5*
b*f*g + 4*c*d*h - 14*a*f*h))/2 - (g*(10*c*f*g + 4*c*e*h - 9*b*f*h))/2)*(a + b*x + c*x^2)^(5/2))/((c*g^2 - b*g*
h + a*h^2)*(g + h*x)^7) - (-1/6*((2*c*g*(5*c*f*g^2 - 7*f*h*(b*g - a*h) + 2*c*h*(e*g - d*h)) - c*h*(5*b*f*g^2 -
b*h*(5*e*g + 9*d*h) + 14*h*(c*d*g - a*f*g + a*e*h)))*(a + b*x + c*x^2)^(5/2))/((c*g^2 - b*g*h + a*h^2)*(g + h
*x)^6) - (((c^2*g*(2*c*(5*f*g^3 + g*h*(2*e*g - 9*d*h)) - h*(19*b*f*g^2 - b*h*(5*e*g + 9*d*h) - 14*a*h*(2*f*g -
e*h))) - (c*h*(7*b^2*h*(5*f*g^2 + h*(5*e*g + 9*d*h)) + 2*b*(5*c*f*g^3 - c*g*h*(40*e*g + 93*d*h) - 7*a*h^2*(10
*f*g + 7*e*h)) + 24*h*(7*c^2*d*g^2 + 7*a^2*f*h^2 - a*c*(2*f*g^2 - h*(9*e*g - 2*d*h)))))/2)*(a + b*x + c*x^2)^(
5/2))/(5*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^5) - ((-2*(a*c^2*h*(2*c*(5*f*g^3 + g*h*(2*e*g - 9*d*h)) - h*(19*b*f
*g^2 - b*h*(5*e*g + 9*d*h) - 14*a*h*(2*f*g - e*h))) + (c^2*g*(7*b^2*h*(5*f*g^2 + h*(5*e*g + 9*d*h)) + 2*b*(5*c
*f*g^3 - c*g*h*(40*e*g + 93*d*h) - 7*a*h^2*(10*f*g + 7*e*h)) + 24*h*(7*c^2*d*g^2 + 7*a^2*f*h^2 - a*c*(2*f*g^2
- h*(9*e*g - 2*d*h)))))/2) + b*(c^2*g*(2*c*(5*f*g^3 + g*h*(2*e*g - 9*d*h)) - h*(19*b*f*g^2 - b*h*(5*e*g + 9*d*
h) - 14*a*h*(2*f*g - e*h))) + (c*h*(7*b^2*h*(5*f*g^2 + h*(5*e*g + 9*d*h)) + 2*b*(5*c*f*g^3 - c*g*h*(40*e*g + 9
3*d*h) - 7*a*h^2*(10*f*g + 7*e*h)) + 24*h*(7*c^2*d*g^2 + 7*a^2*f*h^2 - a*c*(2*f*g^2 - h*(9*e*g - 2*d*h)))))/2)
)*(((b*g - 2*a*h + (2*c*g - b*h)*x)*(a + b*x + c*x^2)^(3/2))/(8*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^4) - (3*(b^2
- 4*a*c)*(((b*g - 2*a*h + (2*c*g - b*h)*x)*Sqrt[a + b*x + c*x^2])/(4*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) + (
(b^2 - 4*a*c)*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*Sqrt[c*g^2 - b*g*h + a*h^2]*(4*c*g^2 - 4*b*g*h + 4*a*h^2))))/(16*(c*g^2 - b*g*h + a*h^2))))/(2*(c*g^2 -
b*g*h + a*h^2)))/(6*(c*g^2 - b*g*h + a*h^2)))/(7*(c*g^2 - b*g*h + a*h^2))))/(2*c*h*(a + b*x + c*x^2)^(3/2))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^8,x, algorithm="fricas")

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^8,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.18, size = 126612, normalized size = 119.22 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^8,x)

[Out]

result too large to display

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*h^2-b*g*h>0)', see `assume?`
for more details)Is a*h^2-b*g*h                            +c*g^2 zero or nonzero?

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^8,x)

[Out]

\text{Hanged}

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(3/2)*(f*x**2+e*x+d)/(h*x+g)**8,x)

[Out]

Timed out

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