3.205 \(\int \frac{(a+b x+c x^2)^{3/2} (d+e x+f x^2)}{(g+h x)^6} \, dx\)

Optimal. Leaf size=1226 \[ \text{result too large to display} \]

[Out]

-((128*c^4*f*g^7 - 32*c^3*f*g^5*h*(11*b*g - 10*a*h) + 8*c^2*g*h^2*(38*b^2*f*g^4 + 2*a^2*h^2*(13*f*g^2 + 3*d*h^
2) - a*b*g*h*(65*f*g^2 + 3*d*h^2)) - 2*c*h^3*(8*a^3*h^3*(2*f*g - 3*e*h) - 2*a*b^2*g^2*h*(34*f*g + 3*e*h) + 4*a
^2*b*h^2*(5*f*g^2 + 6*e*g*h + 3*d*h^2) + b^3*(35*f*g^4 - 3*d*g^2*h^2)) - b*h^4*(b*g - 2*a*h)*(16*a^2*f*h^2 - 2
*a*b*h*(10*f*g + 3*e*h) + b^2*(7*f*g^2 + 3*h*(e*g + d*h))) + h*(128*c*f*(c*g^2 - h*(b*g - a*h))^3 + (2*c*g - b
*h)*(32*c^3*f*g^5 - 8*c^2*g*h*(10*b*f*g^3 - 11*a*f*g^2*h + 3*a*d*h^3) + 2*c*h^2*(4*a^2*h^2*(10*f*g - 3*e*h) -
6*a*b*h*(11*f*g^2 - e*g*h - d*h^2) + b^2*(29*f*g^3 + 3*d*g*h^2)) - b*h^3*(16*a^2*f*h^2 - 2*a*b*h*(10*f*g + 3*e
*h) + b^2*(7*f*g^2 + 3*h*(e*g + d*h)))))*x)*Sqrt[a + b*x + c*x^2])/(128*h^5*(c*g^2 - b*g*h + a*h^2)^3*(g + h*x
)^2) - ((16*c^2*f*g^5 - 2*c*g*h*(13*b*f*g^3 - 10*a*f*g^2*h + 3*b*d*g*h^2 - 6*a*d*h^3) - h^2*(4*a^2*h^2*(2*f*g
- 3*e*h) - b^2*g*(7*f*g^2 + 3*h*(e*g + d*h)) + 2*a*b*h*(f*g^2 + 3*h*(2*e*g + d*h))) + h*(4*c^2*(7*f*g^4 - 3*d*
g^2*h^2) + 2*c*g*h*(2*a*h*(14*f*g - 3*e*h) - b*(28*f*g^2 - 3*e*g*h - 6*d*h^2)) + h^2*(16*a^2*f*h^2 - 2*a*b*h*(
22*f*g - 3*e*h) + b^2*(25*f*g^2 - 3*h*(e*g + d*h))))*x)*(a + b*x + c*x^2)^(3/2))/(48*h^3*(c*g^2 - b*g*h + a*h^
2)^2*(g + h*x)^4) - ((f*g^2 - h*(e*g - d*h))*(a + b*x + c*x^2)^(5/2))/(5*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^5
) + (c^(3/2)*f*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/h^6 - ((256*c^5*f*g^7 - 896*c^4*f*g^5*h
*(b*g - a*h) + 32*c^3*g*h^2*(35*b^2*f*g^4 - 70*a*b*f*g^3*h + a^2*h^2*(35*f*g^2 - 3*d*h^2)) - 16*c^2*h^3*(35*b^
3*f*g^4 - 6*a^3*h^3*(6*f*g - e*h) + 3*a^2*b*h^2*(35*f*g^2 - e*g*h - d*h^2) - 3*a*b^2*g*h*(35*f*g^2 + d*h^2)) +
b^3*h^5*(16*a^2*f*h^2 - 2*a*b*h*(10*f*g + 3*e*h) + b^2*(7*f*g^2 + 3*h*(e*g + d*h))) - 2*b*c*h^4*(96*a^3*f*h^3
- 24*a^2*b*h^2*(8*f*g + e*h) - b^3*(35*f*g^3 - 3*d*g*h^2) + 4*a*b^2*h*(35*f*g^2 + 3*h*(e*g + 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])])/(256*h^6*(c*g^2 - b*g*
h + a*h^2)^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 3.99728, antiderivative size = 1223, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1650, 810, 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}}{5 h \left (c g^2-b h g+a h^2\right ) (g+h x)^5}-\frac{\left (16 c^2 f g^5-2 c h \left (13 b f g^3-10 a f h g^2+3 b d h^2 g-6 a d h^3\right ) g-h^2 \left (-g \left (7 f g^2+3 h (e g+d h)\right ) b^2+2 a h \left (f g^2+3 h (2 e g+d h)\right ) b+4 a^2 h^2 (2 f g-3 e h)\right )+h^2 \left (16 a^2 f h^3+4 a c g (14 f g-3 e h) h+b^2 \left (25 f g^2-3 h (e g+d h)\right ) h+c^2 \left (\frac{28 f g^4}{h}-12 d g^2 h\right )-b \left (56 c f g^3-6 c h (e g+2 d h) g+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{48 h^3 \left (c g^2-b h g+a h^2\right )^2 (g+h x)^4}-\frac{\left (\frac{128 c^4 f g^7}{h}-32 c^3 f (11 b g-10 a h) g^5+8 c^2 h \left (38 b^2 f g^4-a b h \left (65 f g^2+3 d h^2\right ) g+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )\right ) g-b h^3 (b g-2 a h) \left (\left (7 f g^2+3 h (e g+d h)\right ) b^2-2 a h (10 f g+3 e h) b+16 a^2 f h^2\right )-2 c h^2 \left (\left (35 f g^4-3 d g^2 h^2\right ) b^3-2 a g^2 h (34 f g+3 e h) b^2+4 a^2 h^2 \left (5 f g^2+3 h (2 e g+d h)\right ) b+8 a^3 h^3 (2 f g-3 e h)\right )+\left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 h \left (10 b f g^3-11 a f h g^2+3 a d h^3\right ) g+2 c h^2 \left (\left (29 f g^3+3 d h^2 g\right ) b^2-6 a h \left (11 f g^2-e h g-d h^2\right ) b+4 a^2 h^2 (10 f g-3 e h)\right )-b h^3 \left (\left (7 f g^2+3 h (e g+d h)\right ) b^2-2 a h (10 f g+3 e h) b+16 a^2 f h^2\right )\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{128 h^4 \left (c g^2-b h g+a h^2\right )^3 (g+h x)^2}+\frac{c^{3/2} f \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{h^6}-\frac{\left (256 c^5 f g^7-896 c^4 f h (b g-a h) g^5+32 c^3 h^2 \left (35 b^2 f g^4-70 a b f h g^3+a^2 h^2 \left (35 f g^2-3 d h^2\right )\right ) g-16 c^2 h^3 \left (35 b^3 f g^4-3 a b^2 h \left (35 f g^2+d h^2\right ) g-6 a^3 h^3 (6 f g-e h)+3 a^2 b h^2 \left (35 f g^2-e h g-d h^2\right )\right )+b^3 h^5 \left (\left (7 f g^2+3 h (e g+d h)\right ) b^2-2 a h (10 f g+3 e h) b+16 a^2 f h^2\right )-2 b c h^4 \left (-\left (35 f g^3-3 d g h^2\right ) b^3+4 a h \left (35 f g^2+3 h (e g+d h)\right ) b^2-24 a^2 h^2 (8 f g+e h) b+96 a^3 f h^3\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 )}{256 h^6 \left (c g^2-b h g+a h^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^6,x]

[Out]

-(((128*c^4*f*g^7)/h - 32*c^3*f*g^5*(11*b*g - 10*a*h) + 8*c^2*g*h*(38*b^2*f*g^4 + 2*a^2*h^2*(13*f*g^2 + 3*d*h^
2) - a*b*g*h*(65*f*g^2 + 3*d*h^2)) - b*h^3*(b*g - 2*a*h)*(16*a^2*f*h^2 - 2*a*b*h*(10*f*g + 3*e*h) + b^2*(7*f*g
^2 + 3*h*(e*g + d*h))) - 2*c*h^2*(8*a^3*h^3*(2*f*g - 3*e*h) - 2*a*b^2*g^2*h*(34*f*g + 3*e*h) + b^3*(35*f*g^4 -
3*d*g^2*h^2) + 4*a^2*b*h^2*(5*f*g^2 + 3*h*(2*e*g + d*h))) + (128*c*f*(c*g^2 - h*(b*g - a*h))^3 + (2*c*g - b*h
)*(32*c^3*f*g^5 - 8*c^2*g*h*(10*b*f*g^3 - 11*a*f*g^2*h + 3*a*d*h^3) + 2*c*h^2*(4*a^2*h^2*(10*f*g - 3*e*h) - 6*
a*b*h*(11*f*g^2 - e*g*h - d*h^2) + b^2*(29*f*g^3 + 3*d*g*h^2)) - b*h^3*(16*a^2*f*h^2 - 2*a*b*h*(10*f*g + 3*e*h
) + b^2*(7*f*g^2 + 3*h*(e*g + d*h)))))*x)*Sqrt[a + b*x + c*x^2])/(128*h^4*(c*g^2 - b*g*h + a*h^2)^3*(g + h*x)^
2) - ((16*c^2*f*g^5 - 2*c*g*h*(13*b*f*g^3 - 10*a*f*g^2*h + 3*b*d*g*h^2 - 6*a*d*h^3) - h^2*(4*a^2*h^2*(2*f*g -
3*e*h) - b^2*g*(7*f*g^2 + 3*h*(e*g + d*h)) + 2*a*b*h*(f*g^2 + 3*h*(2*e*g + d*h))) + h^2*(16*a^2*f*h^3 + 4*a*c*
g*h*(14*f*g - 3*e*h) + c^2*((28*f*g^4)/h - 12*d*g^2*h) + b^2*h*(25*f*g^2 - 3*h*(e*g + d*h)) - b*(56*c*f*g^3 -
6*c*g*h*(e*g + 2*d*h) + 2*a*h^2*(22*f*g - 3*e*h)))*x)*(a + b*x + c*x^2)^(3/2))/(48*h^3*(c*g^2 - b*g*h + a*h^2)
^2*(g + h*x)^4) - ((f*g^2 - h*(e*g - d*h))*(a + b*x + c*x^2)^(5/2))/(5*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^5)
+ (c^(3/2)*f*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/h^6 - ((256*c^5*f*g^7 - 896*c^4*f*g^5*h*(
b*g - a*h) + 32*c^3*g*h^2*(35*b^2*f*g^4 - 70*a*b*f*g^3*h + a^2*h^2*(35*f*g^2 - 3*d*h^2)) - 16*c^2*h^3*(35*b^3*
f*g^4 - 6*a^3*h^3*(6*f*g - e*h) + 3*a^2*b*h^2*(35*f*g^2 - e*g*h - d*h^2) - 3*a*b^2*g*h*(35*f*g^2 + d*h^2)) + b
^3*h^5*(16*a^2*f*h^2 - 2*a*b*h*(10*f*g + 3*e*h) + b^2*(7*f*g^2 + 3*h*(e*g + d*h))) - 2*b*c*h^4*(96*a^3*f*h^3 -
24*a^2*b*h^2*(8*f*g + e*h) - b^3*(35*f*g^3 - 3*d*g*h^2) + 4*a*b^2*h*(35*f*g^2 + 3*h*(e*g + 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])])/(256*h^6*(c*g^2 - b*g*h
+ a*h^2)^(7/2))

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 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)^6} \, dx &=-\frac{\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}-\frac{\int \frac{\left (-\frac{5}{2} \left (2 c d g-b e g-2 a f g+\frac{b f g^2}{h}-b d h+2 a e h\right )+5 f \left (b g-\frac{c g^2}{h}-a h\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{(g+h x)^5} \, dx}{5 \left (c g^2-b g h+a h^2\right )}\\ &=-\frac{\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h^2 \left (16 a^2 f h^3+4 a c g h (14 f g-3 e h)+c^2 \left (\frac{28 f g^4}{h}-12 d g^2 h\right )+b^2 h \left (25 f g^2-3 h (e g+d h)\right )-b \left (56 c f g^3-6 c g h (e g+2 d h)+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (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}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}+\frac{\int \frac{\left (\frac{5 \left (b^3 h^2 \left (7 f g^2+3 h (e g+d h)\right )-24 a c h \left (a h^2 (2 f g-e h)+c \left (f g^3-d g h^2\right )\right )-2 b^2 \left (a h^3 (10 f g+3 e h)+c \left (13 f g^3 h+3 d g h^3\right )\right )+4 b \left (4 c^2 f g^4+4 a^2 f h^4+a c h^2 \left (17 f g^2-3 h (e g+d h)\right )\right )\right )}{4 h}+\frac{40 c f \left (c g^2-b g h+a h^2\right )^2 x}{h}\right ) \sqrt{a+b x+c x^2}}{(g+h x)^3} \, dx}{40 h^2 \left (c g^2-b g h+a h^2\right )^2}\\ &=-\frac{\left (\frac{128 c^4 f g^7}{h}-32 c^3 f g^5 (11 b g-10 a h)+8 c^2 g h \left (38 b^2 f g^4+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )-a b g h \left (65 f g^2+3 d h^2\right )\right )-b h^3 (b g-2 a h) \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 c h^2 \left (8 a^3 h^3 (2 f g-3 e h)-2 a b^2 g^2 h (34 f g+3 e h)+b^3 \left (35 f g^4-3 d g^2 h^2\right )+4 a^2 b h^2 \left (5 f g^2+3 h (2 e g+d h)\right )\right )+\left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 g h \left (10 b f g^3-11 a f g^2 h+3 a d h^3\right )+2 c h^2 \left (4 a^2 h^2 (10 f g-3 e h)-6 a b h \left (11 f g^2-e g h-d h^2\right )+b^2 \left (29 f g^3+3 d g h^2\right )\right )-b h^3 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 h^4 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^2}-\frac{\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h^2 \left (16 a^2 f h^3+4 a c g h (14 f g-3 e h)+c^2 \left (\frac{28 f g^4}{h}-12 d g^2 h\right )+b^2 h \left (25 f g^2-3 h (e g+d h)\right )-b \left (56 c f g^3-6 c g h (e g+2 d h)+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (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}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}-\frac{\int \frac{\frac{5 \left (b^4 \left (70 c f g^3 h^3-6 c d g h^5-20 a f g h^5-6 a e h^6\right )+b^5 h^4 \left (7 f g^2+3 h (e g+d h)\right )-16 b c \left (8 c^3 f g^6+44 a c^2 f g^4 h^2+12 a^3 f h^6+3 a^2 c h^4 \left (19 f g^2-e g h-d h^2\right )\right )+16 b^2 c h \left (22 c^2 f g^5+3 a^2 h^4 (8 f g+e h)+3 a c g h^2 \left (19 f g^2+d h^2\right )\right )+32 a c^2 h \left (4 c^2 f g^5+a^2 h^4 (10 f g-3 e h)+a c \left (11 f g^3 h^2-3 d g h^4\right )\right )-8 b^3 \left (38 c^2 f g^4 h^2-2 a^2 f h^6+a c h^4 \left (35 f g^2+3 h (e g+d h)\right )\right )\right )}{8 h}-\frac{160 c^2 f \left (c g^2-b g h+a h^2\right )^3 x}{h}}{(g+h x) \sqrt{a+b x+c x^2}} \, dx}{160 h^4 \left (c g^2-b g h+a h^2\right )^3}\\ &=-\frac{\left (\frac{128 c^4 f g^7}{h}-32 c^3 f g^5 (11 b g-10 a h)+8 c^2 g h \left (38 b^2 f g^4+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )-a b g h \left (65 f g^2+3 d h^2\right )\right )-b h^3 (b g-2 a h) \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 c h^2 \left (8 a^3 h^3 (2 f g-3 e h)-2 a b^2 g^2 h (34 f g+3 e h)+b^3 \left (35 f g^4-3 d g^2 h^2\right )+4 a^2 b h^2 \left (5 f g^2+3 h (2 e g+d h)\right )\right )+\left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 g h \left (10 b f g^3-11 a f g^2 h+3 a d h^3\right )+2 c h^2 \left (4 a^2 h^2 (10 f g-3 e h)-6 a b h \left (11 f g^2-e g h-d h^2\right )+b^2 \left (29 f g^3+3 d g h^2\right )\right )-b h^3 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 h^4 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^2}-\frac{\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h^2 \left (16 a^2 f h^3+4 a c g h (14 f g-3 e h)+c^2 \left (\frac{28 f g^4}{h}-12 d g^2 h\right )+b^2 h \left (25 f g^2-3 h (e g+d h)\right )-b \left (56 c f g^3-6 c g h (e g+2 d h)+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (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}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}+\frac{\left (c^2 f\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{h^6}-\frac{\left (256 c^5 f g^7-896 c^4 f g^5 h (b g-a h)+32 c^3 g h^2 \left (35 b^2 f g^4-70 a b f g^3 h+a^2 h^2 \left (35 f g^2-3 d h^2\right )\right )-16 c^2 h^3 \left (35 b^3 f g^4-6 a^3 h^3 (6 f g-e h)+3 a^2 b h^2 \left (35 f g^2-e g h-d h^2\right )-3 a b^2 g h \left (35 f g^2+d h^2\right )\right )+b^3 h^5 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 b c h^4 \left (96 a^3 f h^3-24 a^2 b h^2 (8 f g+e h)-b^3 \left (35 f g^3-3 d g h^2\right )+4 a b^2 h \left (35 f g^2+3 h (e g+d h)\right )\right )\right ) \int \frac{1}{(g+h x) \sqrt{a+b x+c x^2}} \, dx}{256 h^6 \left (c g^2-b g h+a h^2\right )^3}\\ &=-\frac{\left (\frac{128 c^4 f g^7}{h}-32 c^3 f g^5 (11 b g-10 a h)+8 c^2 g h \left (38 b^2 f g^4+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )-a b g h \left (65 f g^2+3 d h^2\right )\right )-b h^3 (b g-2 a h) \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 c h^2 \left (8 a^3 h^3 (2 f g-3 e h)-2 a b^2 g^2 h (34 f g+3 e h)+b^3 \left (35 f g^4-3 d g^2 h^2\right )+4 a^2 b h^2 \left (5 f g^2+3 h (2 e g+d h)\right )\right )+\left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 g h \left (10 b f g^3-11 a f g^2 h+3 a d h^3\right )+2 c h^2 \left (4 a^2 h^2 (10 f g-3 e h)-6 a b h \left (11 f g^2-e g h-d h^2\right )+b^2 \left (29 f g^3+3 d g h^2\right )\right )-b h^3 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 h^4 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^2}-\frac{\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h^2 \left (16 a^2 f h^3+4 a c g h (14 f g-3 e h)+c^2 \left (\frac{28 f g^4}{h}-12 d g^2 h\right )+b^2 h \left (25 f g^2-3 h (e g+d h)\right )-b \left (56 c f g^3-6 c g h (e g+2 d h)+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (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}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}+\frac{\left (2 c^2 f\right ) \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 (256 c^5 f g^7-896 c^4 f g^5 h (b g-a h)+32 c^3 g h^2 \left (35 b^2 f g^4-70 a b f g^3 h+a^2 h^2 \left (35 f g^2-3 d h^2\right )\right )-16 c^2 h^3 \left (35 b^3 f g^4-6 a^3 h^3 (6 f g-e h)+3 a^2 b h^2 \left (35 f g^2-e g h-d h^2\right )-3 a b^2 g h \left (35 f g^2+d h^2\right )\right )+b^3 h^5 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 b c h^4 \left (96 a^3 f h^3-24 a^2 b h^2 (8 f g+e h)-b^3 \left (35 f g^3-3 d g h^2\right )+4 a b^2 h \left (35 f g^2+3 h (e g+d h)\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 )}{128 h^6 \left (c g^2-b g h+a h^2\right )^3}\\ &=-\frac{\left (\frac{128 c^4 f g^7}{h}-32 c^3 f g^5 (11 b g-10 a h)+8 c^2 g h \left (38 b^2 f g^4+2 a^2 h^2 \left (13 f g^2+3 d h^2\right )-a b g h \left (65 f g^2+3 d h^2\right )\right )-b h^3 (b g-2 a h) \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 c h^2 \left (8 a^3 h^3 (2 f g-3 e h)-2 a b^2 g^2 h (34 f g+3 e h)+b^3 \left (35 f g^4-3 d g^2 h^2\right )+4 a^2 b h^2 \left (5 f g^2+3 h (2 e g+d h)\right )\right )+\left (128 c f \left (c g^2-h (b g-a h)\right )^3+(2 c g-b h) \left (32 c^3 f g^5-8 c^2 g h \left (10 b f g^3-11 a f g^2 h+3 a d h^3\right )+2 c h^2 \left (4 a^2 h^2 (10 f g-3 e h)-6 a b h \left (11 f g^2-e g h-d h^2\right )+b^2 \left (29 f g^3+3 d g h^2\right )\right )-b h^3 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{128 h^4 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^2}-\frac{\left (16 c^2 f g^5-2 c g h \left (13 b f g^3-10 a f g^2 h+3 b d g h^2-6 a d h^3\right )-h^2 \left (4 a^2 h^2 (2 f g-3 e h)-b^2 g \left (7 f g^2+3 h (e g+d h)\right )+2 a b h \left (f g^2+3 h (2 e g+d h)\right )\right )+h^2 \left (16 a^2 f h^3+4 a c g h (14 f g-3 e h)+c^2 \left (\frac{28 f g^4}{h}-12 d g^2 h\right )+b^2 h \left (25 f g^2-3 h (e g+d h)\right )-b \left (56 c f g^3-6 c g h (e g+2 d h)+2 a h^2 (22 f g-3 e h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 h^3 \left (c g^2-b g h+a h^2\right )^2 (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}}{5 h \left (c g^2-b g h+a h^2\right ) (g+h x)^5}+\frac{c^{3/2} f \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{h^6}-\frac{\left (256 c^5 f g^7-896 c^4 f g^5 h (b g-a h)+32 c^3 g h^2 \left (35 b^2 f g^4-70 a b f g^3 h+a^2 h^2 \left (35 f g^2-3 d h^2\right )\right )-16 c^2 h^3 \left (35 b^3 f g^4-6 a^3 h^3 (6 f g-e h)+3 a^2 b h^2 \left (35 f g^2-e g h-d h^2\right )-3 a b^2 g h \left (35 f g^2+d h^2\right )\right )+b^3 h^5 \left (16 a^2 f h^2-2 a b h (10 f g+3 e h)+b^2 \left (7 f g^2+3 h (e g+d h)\right )\right )-2 b c h^4 \left (96 a^3 f h^3-24 a^2 b h^2 (8 f g+e h)-b^3 \left (35 f g^3-3 d g h^2\right )+4 a b^2 h \left (35 f g^2+3 h (e g+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 )}{256 h^6 \left (c g^2-b g h+a h^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 6.30515, size = 1111, normalized size = 0.91 \[ \frac{f (a+x (b+c x))^{3/2} \left (\frac{-\frac{(b h-2 c g) \left (c x^2+b x+a\right )^{3/2}}{2 \left (c g^2-b h g+a h^2\right ) (g+h x)^2}-\frac{\frac{\left (\frac{1}{2} h \left (h b^2+2 c g b-8 a c h\right )-c g (2 c g-b h)\right ) \left (c x^2+b x+a\right )^{3/2}}{\left (-c g^2+b h g-a h^2\right ) (g+h x)}+\frac{\frac{\left (h \left (4 c^2 g^2-b^2 h^2-4 c h (b g-2 a h)\right ) x c^2-\left (2 c g-\frac{b h}{2}\right ) \left (4 c^2 g^2-b^2 h^2-4 c h (b g-2 a h)\right ) c+\frac{1}{2} h \left (-h^2 b^3-10 c g h b^2+4 c \left (2 c g^2+3 a h^2\right ) b+8 a c^2 g h\right ) c\right ) \sqrt{c x^2+b x+a}}{2 c h^2}-\frac{-\frac{16 \left (c g^2-h (b g-a h)\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) c^{5/2}}{h}-\frac{4 \sqrt{c g^2-b h g+a h^2} \left (16 c^3 g \left (c g^2-h (b g-a h)\right )^2-c h \left (c g^2-b h g+a h^2\right ) \left (-h^2 b^3-6 c g h b^2+8 c^2 g^2 b+12 a c h^2 b-8 a c^2 g 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 )}{h \left (4 c g^2-4 b h g+4 a h^2\right )}}{4 c h^2}}{-c g^2+b h g-a h^2}}{2 \left (c g^2-b h g+a h^2\right )}}{2 h}-\frac{\left (c x^2+b x+a\right )^{3/2}}{3 h (g+h x)^3}\right )}{h^2 \left (c x^2+b x+a\right )^{3/2}}-\frac{(a+x (b+c x))^{3/2} \left (-\frac{\left (h \left (f g^2-d h^2\right )-g h (2 f g-e h)\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 h (2 f g-e h)+h \left (f g^2-d h^2\right )\right )-2 \left (a (2 f g-e h) h^2+c g \left (f g^2-d h^2\right )\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 )}\right )}{h^2 \left (c x^2+b x+a\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^6,x]

[Out]

-(((a + x*(b + c*x))^(3/2)*(-((-(g*h*(2*f*g - e*h)) + h*(f*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*h^2*(2*f*g - e*h) + c*g*(f*g^2 - d*h^2)) + b*(g*h*(2*f*g - e*h) + h*(f*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))))/(h^2*(a + b*x + c*x^2)^(3/2))) + (f*(a + x*(b + c*x))^(3/2)*(-(a + b*x + c*x^2)^
(3/2)/(3*h*(g + h*x)^3) + (-((-2*c*g + b*h)*(a + b*x + c*x^2)^(3/2))/(2*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) -
(((-(c*g*(2*c*g - b*h)) + (h*(2*b*c*g + b^2*h - 8*a*c*h))/2)*(a + b*x + c*x^2)^(3/2))/((-(c*g^2) + b*g*h - a*
h^2)*(g + h*x)) + (((-(c*(2*c*g - (b*h)/2)*(4*c^2*g^2 - b^2*h^2 - 4*c*h*(b*g - 2*a*h))) + (c*h*(-10*b^2*c*g*h
+ 8*a*c^2*g*h - b^3*h^2 + 4*b*c*(2*c*g^2 + 3*a*h^2)))/2 + c^2*h*(4*c^2*g^2 - b^2*h^2 - 4*c*h*(b*g - 2*a*h))*x)
*Sqrt[a + b*x + c*x^2])/(2*c*h^2) - ((-16*c^(5/2)*(c*g^2 - h*(b*g - a*h))^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqr
t[a + b*x + c*x^2])])/h - (4*Sqrt[c*g^2 - b*g*h + a*h^2]*(-(c*h*(c*g^2 - b*g*h + a*h^2)*(8*b*c^2*g^2 - 6*b^2*c
*g*h - 8*a*c^2*g*h - b^3*h^2 + 12*a*b*c*h^2)) + 16*c^3*g*(c*g^2 - h*(b*g - a*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])])/(h*(4*c*g^2 - 4*b*g*h + 4*a*h^2)))/(4
*c*h^2))/(-(c*g^2) + b*g*h - a*h^2))/(2*(c*g^2 - b*g*h + a*h^2)))/(2*h)))/(h^2*(a + b*x + c*x^2)^(3/2))

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Maple [B]  time = 0.288, size = 76693, normalized size = 62.6 \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)^6,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)^6,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)^6,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)**6,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)^6,x, algorithm="giac")

[Out]

Timed out