3.204 \(\int \frac{(a+b x+c x^2)^{3/2} (d+e x+f x^2)}{(g+h x)^5} \, dx\)

Optimal. Leaf size=1097 \[ \text{result too large to display} \]

[Out]

((64*c^3*g^4*(5*f*g - e*h) - 16*c^2*g^2*h*(b*g*(41*f*g - 7*e*h) - 8*a*h*(5*f*g - e*h)) + 4*c*h^2*(2*b^2*g^2*(4
6*f*g - 5*e*h) + 16*a^2*h^2*(5*f*g - e*h) - a*b*h*(173*f*g^2 - 25*e*g*h - 3*d*h^2)) - b*h^3*(48*a^2*f*h^2 - 8*
a*b*h*(10*f*g + e*h) + b^2*(35*f*g^2 + 5*e*g*h + 3*d*h^2)) + 2*c*h*(16*c^2*g^3*(5*f*g - e*h) - 4*c*h*(6*b*g^2*
(6*f*g - e*h) - a*h*(35*f*g^2 - h*(7*e*g - 3*d*h))) + h^2*(48*a^2*f*h^2 - 8*a*b*h*(14*f*g - e*h) + b^2*(61*f*g
^2 - h*(5*e*g + 3*d*h))))*x)*Sqrt[a + b*x + c*x^2])/(64*h^5*(c*g^2 - b*g*h + a*h^2)^2*(g + h*x)) - ((16*c^2*g^
4*(5*f*g - e*h) - h^2*(16*a^2*h^2*(f*g - 2*e*h) - b^2*g*(35*f*g^2 + 5*e*g*h + 3*d*h^2) + 4*a*b*h*(7*f*g^2 + 7*
e*g*h + 3*d*h^2)) - 4*c*g*h*(b*g*(31*f*g^2 - 5*e*g*h + 3*d*h^2) - a*h*(25*f*g^2 - 5*e*g*h + 9*d*h^2)) + 3*h*(8
*c^2*g^2*(5*f*g^2 - h*(e*g + d*h)) + h^2*(16*a^2*f*h^2 - 8*a*b*h*(6*f*g - e*h) + b^2*(29*f*g^2 - 5*e*g*h - 3*d
*h^2)) - 4*c*h*(2*b*g*(9*f*g^2 - 2*e*g*h - d*h^2) - a*h*(17*f*g^2 - 5*e*g*h + d*h^2)))*x)*(a + b*x + c*x^2)^(3
/2))/(96*h^3*(c*g^2 - b*g*h + a*h^2)^2*(g + h*x)^3) - ((f*g^2 - h*(e*g - d*h))*(a + b*x + c*x^2)^(5/2))/(4*h*(
c*g^2 - b*g*h + a*h^2)*(g + h*x)^4) - (Sqrt[c]*(10*c*f*g - 2*c*e*h - 3*b*f*h)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*S
qrt[a + b*x + c*x^2])])/(2*h^6) + ((128*c^4*g^5*(5*f*g - e*h) - 64*c^3*g^3*h*(b*g*(28*f*g - 5*e*h) - 5*a*h*(5*
f*g - e*h)) + 8*c*h^3*(24*a^3*f*h^3 - 12*a^2*b*h^2*(10*f*g - e*h) - 5*b^3*g^2*(14*f*g - e*h) + 3*a*b^2*h*(55*f
*g^2 - 5*e*g*h - d*h^2)) - 48*c^2*h^2*(10*a*b*g^2*h*(6*f*g - e*h) - 5*b^2*g^3*(7*f*g - e*h) - a^2*h^2*(25*f*g^
2 - 5*e*g*h + d*h^2)) + b^2*h^4*(48*a^2*f*h^2 - 8*a*b*h*(10*f*g + e*h) + b^2*(35*f*g^2 + 5*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])])/(128*h^6*(c*g^
2 - b*g*h + a*h^2)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 3.12273, antiderivative size = 1096, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {1650, 810, 812, 843, 621, 206, 724} \[ -\frac{\left (f g^2-h (e g-d h)\right ) \left (c x^2+b x+a\right )^{5/2}}{4 h \left (c g^2-b h g+a h^2\right ) (g+h x)^4}-\frac{\left (\frac{16 c^2 (5 f g-e h) g^4}{h}-4 c \left (b g \left (31 f g^2-5 e h g+3 d h^2\right )-a h \left (25 f g^2-5 e h g+9 d h^2\right )\right ) g-h \left (-g \left (35 f g^2+5 e h g+3 d h^2\right ) b^2+4 a h \left (7 f g^2+7 e h g+3 d h^2\right ) b+16 a^2 h^2 (f g-2 e h)\right )+3 h \left (\frac{40 c^2 f g^4}{h}-8 c^2 (e g+d h) g^2-8 b c \left (9 f g^2-h (2 e g+d h)\right ) g+16 a^2 f h^3-8 a b h^2 (6 f g-e h)+4 a c h \left (17 f g^2-h (5 e g-d h)\right )+b^2 h \left (29 f g^2-h (5 e g+3 d h)\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{96 h^2 \left (c g^2-b h g+a h^2\right )^2 (g+h x)^3}+\frac{\left (\frac{64 c^3 (5 f g-e h) g^4}{h}-16 c^2 (b g (41 f g-7 e h)-8 a h (5 f g-e h)) g^2+4 c h \left (2 b^2 (46 f g-5 e h) g^2+16 a^2 h^2 (5 f g-e h)-a b h \left (173 f g^2-25 e h g-3 d h^2\right )\right )-b h^2 \left (\left (35 f g^2+5 e h g+3 d h^2\right ) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\right )+2 c \left (16 c^2 (5 f g-e h) g^3-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-h (7 e g-3 d h)\right )\right )+h^2 \left (\left (61 f g^2-h (5 e g+3 d h)\right ) b^2-8 a h (14 f g-e h) b+48 a^2 f h^2\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{64 h^4 \left (c g^2-b h g+a h^2\right )^2 (g+h x)}-\frac{\sqrt{c} (10 c f g-2 c e h-3 b f h) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{2 h^6}+\frac{\left (128 c^4 (5 f g-e h) g^5-64 c^3 h (b g (28 f g-5 e h)-5 a h (5 f g-e h)) g^3+8 c h^3 \left (-5 g^2 (14 f g-e h) b^3+3 a h \left (55 f g^2-5 e h g-d h^2\right ) b^2-12 a^2 h^2 (10 f g-e h) b+24 a^3 f h^3\right )-48 c^2 h^2 \left (-5 b^2 (7 f g-e h) g^3+10 a b h (6 f g-e h) g^2-a^2 h^2 \left (25 f g^2-5 e h g+d h^2\right )\right )+b^2 h^4 \left (\left (35 f g^2+5 e h g+3 d h^2\right ) b^2-8 a h (10 f g+e h) b+48 a^2 f h^2\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 )}{128 h^6 \left (c g^2-b h g+a h^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^5,x]

[Out]

(((64*c^3*g^4*(5*f*g - e*h))/h - 16*c^2*g^2*(b*g*(41*f*g - 7*e*h) - 8*a*h*(5*f*g - e*h)) + 4*c*h*(2*b^2*g^2*(4
6*f*g - 5*e*h) + 16*a^2*h^2*(5*f*g - e*h) - a*b*h*(173*f*g^2 - 25*e*g*h - 3*d*h^2)) - b*h^2*(48*a^2*f*h^2 - 8*
a*b*h*(10*f*g + e*h) + b^2*(35*f*g^2 + 5*e*g*h + 3*d*h^2)) + 2*c*(16*c^2*g^3*(5*f*g - e*h) - 4*c*h*(6*b*g^2*(6
*f*g - e*h) - a*h*(35*f*g^2 - h*(7*e*g - 3*d*h))) + h^2*(48*a^2*f*h^2 - 8*a*b*h*(14*f*g - e*h) + b^2*(61*f*g^2
- h*(5*e*g + 3*d*h))))*x)*Sqrt[a + b*x + c*x^2])/(64*h^4*(c*g^2 - b*g*h + a*h^2)^2*(g + h*x)) - (((16*c^2*g^4
*(5*f*g - e*h))/h - h*(16*a^2*h^2*(f*g - 2*e*h) - b^2*g*(35*f*g^2 + 5*e*g*h + 3*d*h^2) + 4*a*b*h*(7*f*g^2 + 7*
e*g*h + 3*d*h^2)) - 4*c*g*(b*g*(31*f*g^2 - 5*e*g*h + 3*d*h^2) - a*h*(25*f*g^2 - 5*e*g*h + 9*d*h^2)) + 3*h*((40
*c^2*f*g^4)/h + 16*a^2*f*h^3 - 8*c^2*g^2*(e*g + d*h) - 8*a*b*h^2*(6*f*g - e*h) + 4*a*c*h*(17*f*g^2 - h*(5*e*g
- d*h)) - 8*b*c*g*(9*f*g^2 - h*(2*e*g + d*h)) + b^2*h*(29*f*g^2 - h*(5*e*g + 3*d*h)))*x)*(a + b*x + c*x^2)^(3/
2))/(96*h^2*(c*g^2 - b*g*h + a*h^2)^2*(g + h*x)^3) - ((f*g^2 - h*(e*g - d*h))*(a + b*x + c*x^2)^(5/2))/(4*h*(c
*g^2 - b*g*h + a*h^2)*(g + h*x)^4) - (Sqrt[c]*(10*c*f*g - 2*c*e*h - 3*b*f*h)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sq
rt[a + b*x + c*x^2])])/(2*h^6) + ((128*c^4*g^5*(5*f*g - e*h) - 64*c^3*g^3*h*(b*g*(28*f*g - 5*e*h) - 5*a*h*(5*f
*g - e*h)) + 8*c*h^3*(24*a^3*f*h^3 - 12*a^2*b*h^2*(10*f*g - e*h) - 5*b^3*g^2*(14*f*g - e*h) + 3*a*b^2*h*(55*f*
g^2 - 5*e*g*h - d*h^2)) - 48*c^2*h^2*(10*a*b*g^2*h*(6*f*g - e*h) - 5*b^2*g^3*(7*f*g - e*h) - a^2*h^2*(25*f*g^2
- 5*e*g*h + d*h^2)) + b^2*h^4*(48*a^2*f*h^2 - 8*a*b*h*(10*f*g + e*h) + b^2*(35*f*g^2 + 5*e*g*h + 3*d*h^2)))*A
rcTanh[(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])])/(128*h^6*(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 810

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

Rule 812

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

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

Rule 621

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

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]

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)^5} \, dx &=-\frac{\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{4 h \left (c g^2-b g h+a h^2\right ) (g+h x)^4}-\frac{\int \frac{\left (\frac{1}{2} \left (-8 c d g+5 b e g+8 a f g-\frac{5 b f g^2}{h}+3 b d h-8 a e h\right )+\left (c e g+4 b f g-\frac{5 c f g^2}{h}-c d h-4 a f h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{(g+h x)^4} \, dx}{4 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac{\left (\frac{16 c^2 g^4 (5 f g-e h)}{h}-h \left (16 a^2 h^2 (f g-2 e h)-b^2 g \left (35 f g^2+5 e g h+3 d h^2\right )+4 a b h \left (7 f g^2+7 e g h+3 d h^2\right )\right )-4 c g \left (b g \left (31 f g^2-5 e g h+3 d h^2\right )-a h \left (25 f g^2-5 e g h+9 d h^2\right )\right )+3 h \left (\frac{40 c^2 f g^4}{h}+16 a^2 f h^3-8 c^2 g^2 (e g+d h)-8 a b h^2 (6 f g-e h)+4 a c h \left (17 f g^2-h (5 e g-d h)\right )-8 b c g \left (9 f g^2-h (2 e g+d h)\right )+b^2 h \left (29 f g^2-h (5 e g+3 d h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 h^2 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^3}-\frac{\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{4 h \left (c g^2-b g h+a h^2\right ) (g+h x)^4}+\frac{\int \frac{\left (\frac{1}{4} \left (48 a^2 b f h^3+\frac{16 b c^2 g^3 (5 f g-e h)}{h}+b^3 h \left (35 f g^2+h (5 e g+3 d h)\right )+4 a b c h \left (61 f g^2-h (17 e g+3 d h)\right )-16 a c \left (5 c f g^3-c g h (e g+3 d h)+4 a h^2 (2 f g-e h)\right )-4 b^2 \left (31 c f g^3-c g h (5 e g-3 d h)+2 a h^2 (10 f g+e h)\right )\right )+\frac{c \left (16 c^2 g^3 (5 f g-e h)+h^2 \left (48 a^2 f h^2-8 a b h (14 f g-e h)+b^2 \left (61 f g^2-5 e g h-3 d h^2\right )\right )-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-7 e g h+3 d h^2\right )\right )\right ) x}{2 h}\right ) \sqrt{a+b x+c x^2}}{(g+h x)^2} \, dx}{16 h^2 \left (c g^2-b g h+a h^2\right )^2}\\ &=\frac{\left (\frac{64 c^3 g^4 (5 f g-e h)}{h}-16 c^2 g^2 (b g (41 f g-7 e h)-8 a h (5 f g-e h))+4 c h \left (2 b^2 g^2 (46 f g-5 e h)+16 a^2 h^2 (5 f g-e h)-a b h \left (173 f g^2-25 e g h-3 d h^2\right )\right )-b h^2 \left (48 a^2 f h^2-8 a b h (10 f g+e h)+b^2 \left (35 f g^2+5 e g h+3 d h^2\right )\right )+2 c \left (16 c^2 g^3 (5 f g-e h)-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-h (7 e g-3 d h)\right )\right )+h^2 \left (48 a^2 f h^2-8 a b h (14 f g-e h)+b^2 \left (61 f g^2-h (5 e g+3 d h)\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 h^4 \left (c g^2-b g h+a h^2\right )^2 (g+h x)}-\frac{\left (\frac{16 c^2 g^4 (5 f g-e h)}{h}-h \left (16 a^2 h^2 (f g-2 e h)-b^2 g \left (35 f g^2+5 e g h+3 d h^2\right )+4 a b h \left (7 f g^2+7 e g h+3 d h^2\right )\right )-4 c g \left (b g \left (31 f g^2-5 e g h+3 d h^2\right )-a h \left (25 f g^2-5 e g h+9 d h^2\right )\right )+3 h \left (\frac{40 c^2 f g^4}{h}+16 a^2 f h^3-8 c^2 g^2 (e g+d h)-8 a b h^2 (6 f g-e h)+4 a c h \left (17 f g^2-h (5 e g-d h)\right )-8 b c g \left (9 f g^2-h (2 e g+d h)\right )+b^2 h \left (29 f g^2-h (5 e g+3 d h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 h^2 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^3}-\frac{\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{4 h \left (c g^2-b g h+a h^2\right ) (g+h x)^4}-\frac{\int \frac{-\frac{b^4 h^3 \left (35 f g^2+h (5 e g+3 d h)\right )-32 b c \left (c g^2+3 a h^2\right ) \left (2 c g^2 (5 f g-e h)+a h^2 (8 f g-e h)\right )-8 b^3 \left (c g^2 h^2 (46 f g-5 e h)+a h^4 (10 f g+e h)\right )+8 b^2 \left (6 a^2 f h^5+2 c^2 g^3 h (41 f g-7 e h)+3 a c h^3 \left (39 f g^2-5 e g h-d h^2\right )\right )+16 a c h \left (12 a^2 f h^4+4 c^2 g^3 (5 f g-e h)+a c h^2 \left (35 f g^2-7 e g h+3 d h^2\right )\right )}{4 h}+\frac{16 c (10 c f g-2 c e h-3 b f h) \left (c g^2-b g h+a h^2\right )^2 x}{h}}{(g+h x) \sqrt{a+b x+c x^2}} \, dx}{32 h^4 \left (c g^2-b g h+a h^2\right )^2}\\ &=\frac{\left (\frac{64 c^3 g^4 (5 f g-e h)}{h}-16 c^2 g^2 (b g (41 f g-7 e h)-8 a h (5 f g-e h))+4 c h \left (2 b^2 g^2 (46 f g-5 e h)+16 a^2 h^2 (5 f g-e h)-a b h \left (173 f g^2-25 e g h-3 d h^2\right )\right )-b h^2 \left (48 a^2 f h^2-8 a b h (10 f g+e h)+b^2 \left (35 f g^2+5 e g h+3 d h^2\right )\right )+2 c \left (16 c^2 g^3 (5 f g-e h)-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-h (7 e g-3 d h)\right )\right )+h^2 \left (48 a^2 f h^2-8 a b h (14 f g-e h)+b^2 \left (61 f g^2-h (5 e g+3 d h)\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 h^4 \left (c g^2-b g h+a h^2\right )^2 (g+h x)}-\frac{\left (\frac{16 c^2 g^4 (5 f g-e h)}{h}-h \left (16 a^2 h^2 (f g-2 e h)-b^2 g \left (35 f g^2+5 e g h+3 d h^2\right )+4 a b h \left (7 f g^2+7 e g h+3 d h^2\right )\right )-4 c g \left (b g \left (31 f g^2-5 e g h+3 d h^2\right )-a h \left (25 f g^2-5 e g h+9 d h^2\right )\right )+3 h \left (\frac{40 c^2 f g^4}{h}+16 a^2 f h^3-8 c^2 g^2 (e g+d h)-8 a b h^2 (6 f g-e h)+4 a c h \left (17 f g^2-h (5 e g-d h)\right )-8 b c g \left (9 f g^2-h (2 e g+d h)\right )+b^2 h \left (29 f g^2-h (5 e g+3 d h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 h^2 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^3}-\frac{\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{4 h \left (c g^2-b g h+a h^2\right ) (g+h x)^4}-\frac{(c (10 c f g-2 c e h-3 b f h)) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 h^6}+\frac{\left (128 c^4 g^5 (5 f g-e h)-64 c^3 g^3 h (b g (28 f g-5 e h)-5 a h (5 f g-e h))+8 c h^3 \left (24 a^3 f h^3-12 a^2 b h^2 (10 f g-e h)-5 b^3 g^2 (14 f g-e h)+3 a b^2 h \left (55 f g^2-5 e g h-d h^2\right )\right )-48 c^2 h^2 \left (10 a b g^2 h (6 f g-e h)-5 b^2 g^3 (7 f g-e h)-a^2 h^2 \left (25 f g^2-5 e g h+d h^2\right )\right )+b^2 h^4 \left (48 a^2 f h^2-8 a b h (10 f g+e h)+b^2 \left (35 f g^2+5 e g h+3 d h^2\right )\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+b x+c x^2}} \, dx}{128 h^6 \left (c g^2-b g h+a h^2\right )^2}\\ &=\frac{\left (\frac{64 c^3 g^4 (5 f g-e h)}{h}-16 c^2 g^2 (b g (41 f g-7 e h)-8 a h (5 f g-e h))+4 c h \left (2 b^2 g^2 (46 f g-5 e h)+16 a^2 h^2 (5 f g-e h)-a b h \left (173 f g^2-25 e g h-3 d h^2\right )\right )-b h^2 \left (48 a^2 f h^2-8 a b h (10 f g+e h)+b^2 \left (35 f g^2+5 e g h+3 d h^2\right )\right )+2 c \left (16 c^2 g^3 (5 f g-e h)-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-h (7 e g-3 d h)\right )\right )+h^2 \left (48 a^2 f h^2-8 a b h (14 f g-e h)+b^2 \left (61 f g^2-h (5 e g+3 d h)\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 h^4 \left (c g^2-b g h+a h^2\right )^2 (g+h x)}-\frac{\left (\frac{16 c^2 g^4 (5 f g-e h)}{h}-h \left (16 a^2 h^2 (f g-2 e h)-b^2 g \left (35 f g^2+5 e g h+3 d h^2\right )+4 a b h \left (7 f g^2+7 e g h+3 d h^2\right )\right )-4 c g \left (b g \left (31 f g^2-5 e g h+3 d h^2\right )-a h \left (25 f g^2-5 e g h+9 d h^2\right )\right )+3 h \left (\frac{40 c^2 f g^4}{h}+16 a^2 f h^3-8 c^2 g^2 (e g+d h)-8 a b h^2 (6 f g-e h)+4 a c h \left (17 f g^2-h (5 e g-d h)\right )-8 b c g \left (9 f g^2-h (2 e g+d h)\right )+b^2 h \left (29 f g^2-h (5 e g+3 d h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 h^2 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^3}-\frac{\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{4 h \left (c g^2-b g h+a h^2\right ) (g+h x)^4}-\frac{(c (10 c f g-2 c e h-3 b f h)) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{h^6}-\frac{\left (128 c^4 g^5 (5 f g-e h)-64 c^3 g^3 h (b g (28 f g-5 e h)-5 a h (5 f g-e h))+8 c h^3 \left (24 a^3 f h^3-12 a^2 b h^2 (10 f g-e h)-5 b^3 g^2 (14 f g-e h)+3 a b^2 h \left (55 f g^2-5 e g h-d h^2\right )\right )-48 c^2 h^2 \left (10 a b g^2 h (6 f g-e h)-5 b^2 g^3 (7 f g-e h)-a^2 h^2 \left (25 f g^2-5 e g h+d h^2\right )\right )+b^2 h^4 \left (48 a^2 f h^2-8 a b h (10 f g+e h)+b^2 \left (35 f g^2+5 e g h+3 d h^2\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 )}{64 h^6 \left (c g^2-b g h+a h^2\right )^2}\\ &=\frac{\left (\frac{64 c^3 g^4 (5 f g-e h)}{h}-16 c^2 g^2 (b g (41 f g-7 e h)-8 a h (5 f g-e h))+4 c h \left (2 b^2 g^2 (46 f g-5 e h)+16 a^2 h^2 (5 f g-e h)-a b h \left (173 f g^2-25 e g h-3 d h^2\right )\right )-b h^2 \left (48 a^2 f h^2-8 a b h (10 f g+e h)+b^2 \left (35 f g^2+5 e g h+3 d h^2\right )\right )+2 c \left (16 c^2 g^3 (5 f g-e h)-4 c h \left (6 b g^2 (6 f g-e h)-a h \left (35 f g^2-h (7 e g-3 d h)\right )\right )+h^2 \left (48 a^2 f h^2-8 a b h (14 f g-e h)+b^2 \left (61 f g^2-h (5 e g+3 d h)\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 h^4 \left (c g^2-b g h+a h^2\right )^2 (g+h x)}-\frac{\left (\frac{16 c^2 g^4 (5 f g-e h)}{h}-h \left (16 a^2 h^2 (f g-2 e h)-b^2 g \left (35 f g^2+5 e g h+3 d h^2\right )+4 a b h \left (7 f g^2+7 e g h+3 d h^2\right )\right )-4 c g \left (b g \left (31 f g^2-5 e g h+3 d h^2\right )-a h \left (25 f g^2-5 e g h+9 d h^2\right )\right )+3 h \left (\frac{40 c^2 f g^4}{h}+16 a^2 f h^3-8 c^2 g^2 (e g+d h)-8 a b h^2 (6 f g-e h)+4 a c h \left (17 f g^2-h (5 e g-d h)\right )-8 b c g \left (9 f g^2-h (2 e g+d h)\right )+b^2 h \left (29 f g^2-h (5 e g+3 d h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 h^2 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^3}-\frac{\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{4 h \left (c g^2-b g h+a h^2\right ) (g+h x)^4}-\frac{\sqrt{c} (10 c f g-2 c e h-3 b f h) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 h^6}+\frac{\left (128 c^4 g^5 (5 f g-e h)-64 c^3 g^3 h (b g (28 f g-5 e h)-5 a h (5 f g-e h))+8 c h^3 \left (24 a^3 f h^3-12 a^2 b h^2 (10 f g-e h)-5 b^3 g^2 (14 f g-e h)+3 a b^2 h \left (55 f g^2-5 e g h-d h^2\right )\right )-48 c^2 h^2 \left (10 a b g^2 h (6 f g-e h)-5 b^2 g^3 (7 f g-e h)-a^2 h^2 \left (25 f g^2-5 e g h+d h^2\right )\right )+b^2 h^4 \left (48 a^2 f h^2-8 a b h (10 f g+e h)+b^2 \left (35 f g^2+5 e g h+3 d h^2\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 )}{128 h^6 \left (c g^2-b g h+a h^2\right )^{5/2}}\\ \end{align*}

Mathematica [B]  time = 6.63227, size = 46895, normalized size = 42.75 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^5,x]

[Out]

Result too large to show

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Maple [B]  time = 0.28, size = 57957, normalized size = 52.8 \begin{align*} \text{output too large to display} \end{align*}

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)^5,x)

[Out]

result too large to display

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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((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^5,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((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^5,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((c*x**2+b*x+a)**(3/2)*(f*x**2+e*x+d)/(h*x+g)**5,x)

[Out]

Timed out

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Giac [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((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^5,x, algorithm="giac")

[Out]

Timed out