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

Optimal. Leaf size=504 \[ \frac{h \sqrt{a+b x+c x^2} \left (8 c^2 \left (32 a^2 f h^2+39 a b h (e h+3 f g)+b^2 \left (9 h (d h+3 e g)+20 f g^2\right )\right )+2 c h x \left (-8 c^2 (9 a e h+11 a f g+3 b d h+3 b e g)+2 b c (58 a f h+15 b e h+17 b f g)-35 b^3 f h+48 c^3 d g\right )-10 b^2 c h (46 a f h+9 b (e h+3 f g))-16 c^3 \left (4 a \left (3 d h^2+9 e g h+7 f g^2\right )+3 b g (3 d h+2 e g)\right )+105 b^4 f h^2+192 c^4 d g^2\right )}{24 c^4 \left (b^2-4 a c\right )}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (24 c^2 h \left (a h (e h+3 f g)+b \left (d h^2+3 e g h+3 f g^2\right )\right )-30 b c h^2 (2 a f h+b e h+3 b f g)+35 b^3 f h^3-16 c^3 g \left (3 h (d h+e g)+f g^2\right )\right )}{16 c^{9/2}}+\frac{2 (g+h x)^3 \left (c \left (2 a e-b \left (\frac{a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{h (g+h x)^2 \sqrt{a+b x+c x^2} \left (-16 a c f+7 b^2 f-6 b c e+12 c^2 d\right )}{3 c^2 \left (b^2-4 a c\right )} \]

[Out]

(2*(c*(2*a*e - b*(d + (a*f)/c)) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x)*(g + h*x)^3)/(c*(b^2 - 4*a*c)*Sqrt[a
+ b*x + c*x^2]) + ((12*c^2*d - 6*b*c*e + 7*b^2*f - 16*a*c*f)*h*(g + h*x)^2*Sqrt[a + b*x + c*x^2])/(3*c^2*(b^2
- 4*a*c)) + (h*(192*c^4*d*g^2 + 105*b^4*f*h^2 - 10*b^2*c*h*(46*a*f*h + 9*b*(3*f*g + e*h)) - 16*c^3*(3*b*g*(2*e
*g + 3*d*h) + 4*a*(7*f*g^2 + 9*e*g*h + 3*d*h^2)) + 8*c^2*(32*a^2*f*h^2 + 39*a*b*h*(3*f*g + e*h) + b^2*(20*f*g^
2 + 9*h*(3*e*g + d*h))) + 2*c*h*(48*c^3*d*g - 35*b^3*f*h - 8*c^2*(3*b*e*g + 11*a*f*g + 3*b*d*h + 9*a*e*h) + 2*
b*c*(17*b*f*g + 15*b*e*h + 58*a*f*h))*x)*Sqrt[a + b*x + c*x^2])/(24*c^4*(b^2 - 4*a*c)) - ((35*b^3*f*h^3 - 30*b
*c*h^2*(3*b*f*g + b*e*h + 2*a*f*h) - 16*c^3*g*(f*g^2 + 3*h*(e*g + d*h)) + 24*c^2*h*(a*h*(3*f*g + e*h) + b*(3*f
*g^2 + 3*e*g*h + d*h^2)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 1.1756, antiderivative size = 502, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1644, 832, 779, 621, 206} \[ \frac{h \sqrt{a+b x+c x^2} \left (8 c \left (32 a^2 f h^2+39 a b h (e h+3 f g)+b^2 \left (9 h (d h+3 e g)+20 f g^2\right )\right )+2 h x \left (-8 c^2 (9 a e h+11 a f g+3 b d h+3 b e g)+2 b c (58 a f h+15 b e h+17 b f g)-35 b^3 f h+48 c^3 d g\right )-10 b^2 h (46 a f h+9 b (e h+3 f g))-16 c^2 \left (4 a \left (3 d h^2+9 e g h+7 f g^2\right )+3 b g (3 d h+2 e g)\right )+\frac{105 b^4 f h^2}{c}+192 c^3 d g^2\right )}{24 c^3 \left (b^2-4 a c\right )}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (24 c^2 h \left (a h (e h+3 f g)+b h (d h+3 e g)+3 b f g^2\right )-30 b c h^2 (2 a f h+b e h+3 b f g)+35 b^3 f h^3-16 c^3 \left (3 g h (d h+e g)+f g^3\right )\right )}{16 c^{9/2}}+\frac{2 (g+h x)^3 \left (c \left (2 a e-b \left (\frac{a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{h (g+h x)^2 \sqrt{a+b x+c x^2} \left (-16 a c f+7 b^2 f-6 b c e+12 c^2 d\right )}{3 c^2 \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*(2*a*e - b*(d + (a*f)/c)) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x)*(g + h*x)^3)/(c*(b^2 - 4*a*c)*Sqrt[a
+ b*x + c*x^2]) + ((12*c^2*d - 6*b*c*e + 7*b^2*f - 16*a*c*f)*h*(g + h*x)^2*Sqrt[a + b*x + c*x^2])/(3*c^2*(b^2
- 4*a*c)) + (h*(192*c^3*d*g^2 + (105*b^4*f*h^2)/c - 10*b^2*h*(46*a*f*h + 9*b*(3*f*g + e*h)) - 16*c^2*(3*b*g*(2
*e*g + 3*d*h) + 4*a*(7*f*g^2 + 9*e*g*h + 3*d*h^2)) + 8*c*(32*a^2*f*h^2 + 39*a*b*h*(3*f*g + e*h) + b^2*(20*f*g^
2 + 9*h*(3*e*g + d*h))) + 2*h*(48*c^3*d*g - 35*b^3*f*h - 8*c^2*(3*b*e*g + 11*a*f*g + 3*b*d*h + 9*a*e*h) + 2*b*
c*(17*b*f*g + 15*b*e*h + 58*a*f*h))*x)*Sqrt[a + b*x + c*x^2])/(24*c^3*(b^2 - 4*a*c)) - ((35*b^3*f*h^3 - 30*b*c
*h^2*(3*b*f*g + b*e*h + 2*a*f*h) - 16*c^3*(f*g^3 + 3*g*h*(e*g + d*h)) + 24*c^2*h*(3*b*f*g^2 + b*h*(3*e*g + d*h
) + a*h*(3*f*g + e*h)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(9/2))

Rule 1644

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

Rule 832

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

Rule 779

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

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])

Rubi steps

\begin{align*} \int \frac{(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac{2 \left (c \left (2 a e-b \left (d+\frac{a f}{c}\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right ) (g+h x)^3}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{(g+h x)^2 \left (-\frac{b^2 f g+6 b (c d+a f) h-4 a c (f g+3 e h)}{2 c}-\frac{1}{2} \left (12 c d-6 b e-16 a f+\frac{7 b^2 f}{c}\right ) h x\right )}{\sqrt{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=\frac{2 \left (c \left (2 a e-b \left (d+\frac{a f}{c}\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right ) (g+h x)^3}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{\left (12 c^2 d+7 b^2 f-2 c (3 b e+8 a f)\right ) h (g+h x)^2 \sqrt{a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}-\frac{2 \int \frac{(g+h x) \left (\frac{7 b^3 f g h-4 b c h (6 c d g+13 a f g+6 a e h)+b^2 \left (28 a f h^2-6 c g (f g+e h)\right )-8 a c \left (8 a f h^2-3 c \left (f g^2+3 e g h+2 d h^2\right )\right )}{4 c}-\frac{h \left (48 c^3 d g-35 b^3 f h-8 c^2 (3 b e g+11 a f g+3 b d h+9 a e h)+2 b c (17 b f g+15 b e h+58 a f h)\right ) x}{4 c}\right )}{\sqrt{a+b x+c x^2}} \, dx}{3 c \left (b^2-4 a c\right )}\\ &=\frac{2 \left (c \left (2 a e-b \left (d+\frac{a f}{c}\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right ) (g+h x)^3}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{\left (12 c^2 d+7 b^2 f-2 c (3 b e+8 a f)\right ) h (g+h x)^2 \sqrt{a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac{h \left (192 c^3 d g^2+\frac{105 b^4 f h^2}{c}-10 b^2 h (46 a f h+9 b (3 f g+e h))-16 c^2 \left (3 b g (2 e g+3 d h)+4 a \left (7 f g^2+9 e g h+3 d h^2\right )\right )+8 c \left (32 a^2 f h^2+39 a b h (3 f g+e h)+b^2 \left (20 f g^2+9 h (3 e g+d h)\right )\right )+2 h \left (48 c^3 d g-35 b^3 f h-8 c^2 (3 b e g+11 a f g+3 b d h+9 a e h)+2 b c (17 b f g+15 b e h+58 a f h)\right ) x\right ) \sqrt{a+b x+c x^2}}{24 c^3 \left (b^2-4 a c\right )}-\frac{\left (35 b^3 f h^3-30 b c h^2 (3 b f g+b e h+2 a f h)-16 c^3 \left (f g^3+3 g h (e g+d h)\right )+24 c^2 h \left (3 b f g^2+b h (3 e g+d h)+a h (3 f g+e h)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{16 c^4}\\ &=\frac{2 \left (c \left (2 a e-b \left (d+\frac{a f}{c}\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right ) (g+h x)^3}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{\left (12 c^2 d+7 b^2 f-2 c (3 b e+8 a f)\right ) h (g+h x)^2 \sqrt{a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac{h \left (192 c^3 d g^2+\frac{105 b^4 f h^2}{c}-10 b^2 h (46 a f h+9 b (3 f g+e h))-16 c^2 \left (3 b g (2 e g+3 d h)+4 a \left (7 f g^2+9 e g h+3 d h^2\right )\right )+8 c \left (32 a^2 f h^2+39 a b h (3 f g+e h)+b^2 \left (20 f g^2+9 h (3 e g+d h)\right )\right )+2 h \left (48 c^3 d g-35 b^3 f h-8 c^2 (3 b e g+11 a f g+3 b d h+9 a e h)+2 b c (17 b f g+15 b e h+58 a f h)\right ) x\right ) \sqrt{a+b x+c x^2}}{24 c^3 \left (b^2-4 a c\right )}-\frac{\left (35 b^3 f h^3-30 b c h^2 (3 b f g+b e h+2 a f h)-16 c^3 \left (f g^3+3 g h (e g+d h)\right )+24 c^2 h \left (3 b f g^2+b h (3 e g+d h)+a h (3 f g+e h)\right )\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 )}{8 c^4}\\ &=\frac{2 \left (c \left (2 a e-b \left (d+\frac{a f}{c}\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right ) (g+h x)^3}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{\left (12 c^2 d+7 b^2 f-2 c (3 b e+8 a f)\right ) h (g+h x)^2 \sqrt{a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac{h \left (192 c^3 d g^2+\frac{105 b^4 f h^2}{c}-10 b^2 h (46 a f h+9 b (3 f g+e h))-16 c^2 \left (3 b g (2 e g+3 d h)+4 a \left (7 f g^2+9 e g h+3 d h^2\right )\right )+8 c \left (32 a^2 f h^2+39 a b h (3 f g+e h)+b^2 \left (20 f g^2+9 h (3 e g+d h)\right )\right )+2 h \left (48 c^3 d g-35 b^3 f h-8 c^2 (3 b e g+11 a f g+3 b d h+9 a e h)+2 b c (17 b f g+15 b e h+58 a f h)\right ) x\right ) \sqrt{a+b x+c x^2}}{24 c^3 \left (b^2-4 a c\right )}-\frac{\left (35 b^3 f h^3-30 b c h^2 (3 b f g+b e h+2 a f h)-16 c^3 \left (f g^3+3 g h (e g+d h)\right )+24 c^2 h \left (3 b f g^2+b h (3 e g+d h)+a h (3 f g+e h)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 1.64845, size = 715, normalized size = 1.42 \[ \frac{3 \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \left (24 c^2 h \left (a h (e h+3 f g)+b h (d h+3 e g)+3 b f g^2\right )-30 b c h^2 (2 a f h+b e h+3 b f g)+35 b^3 f h^3-16 c^3 g \left (3 h (d h+e g)+f g^2\right )\right )-2 \sqrt{c} \left (4 b^2 c \left (-115 a^2 f h^3+a c h \left (3 h (6 d h+18 e g+31 e h x)+f \left (54 g^2+279 g h x-43 h^2 x^2\right )\right )+c^2 x \left (3 h \left (2 d h (h x-6 g)+e \left (-12 g^2+6 g h x+h^2 x^2\right )\right )+f \left (18 g^2 h x-12 g^3+9 g h^2 x^2+2 h^3 x^3\right )\right )\right )-8 b c^2 \left (-a^2 h^2 (39 e h+117 f g+61 f h x)+a c \left (3 h \left (2 d h (3 g+5 h x)+e \left (6 g^2+30 g h x-5 h^2 x^2\right )\right )+f \left (90 g^2 h x+6 g^3-45 g h^2 x^2-7 h^3 x^3\right )\right )-6 c^2 g^2 (-d g+3 d h x+e g x)\right )-16 c^2 \left (a^2 c h \left (3 h (4 d h+3 e (4 g+h x))+f \left (36 g^2+27 g h x-8 h^2 x^2\right )\right )-16 a^3 f h^3+a c^2 \left (6 d h \left (-3 g^2-3 g h x+h^2 x^2\right )-3 e \left (6 g^2 h x+2 g^3-6 g h^2 x^2-h^3 x^3\right )+f x \left (18 g^2 h x-6 g^3+9 g h^2 x^2+2 h^3 x^3\right )\right )+6 c^3 d g^3 x\right )-2 b^3 c h \left (5 a h (9 e h+27 f g+53 f h x)+c x \left (3 h (-12 d h-36 e g+5 e h x)+f \left (-108 g^2+45 g h x+7 h^2 x^2\right )\right )\right )+5 b^4 h^2 (21 a f h+c x (-18 e h-54 f g+7 f h x))+105 b^5 f h^3 x\right )}{48 c^{9/2} \left (4 a c-b^2\right ) \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[c]*(105*b^5*f*h^3*x + 5*b^4*h^2*(21*a*f*h + c*x*(-54*f*g - 18*e*h + 7*f*h*x)) - 2*b^3*c*h*(5*a*h*(27*
f*g + 9*e*h + 53*f*h*x) + c*x*(3*h*(-36*e*g - 12*d*h + 5*e*h*x) + f*(-108*g^2 + 45*g*h*x + 7*h^2*x^2))) - 16*c
^2*(-16*a^3*f*h^3 + 6*c^3*d*g^3*x + a*c^2*(6*d*h*(-3*g^2 - 3*g*h*x + h^2*x^2) - 3*e*(2*g^3 + 6*g^2*h*x - 6*g*h
^2*x^2 - h^3*x^3) + f*x*(-6*g^3 + 18*g^2*h*x + 9*g*h^2*x^2 + 2*h^3*x^3)) + a^2*c*h*(f*(36*g^2 + 27*g*h*x - 8*h
^2*x^2) + 3*h*(4*d*h + 3*e*(4*g + h*x)))) - 8*b*c^2*(-6*c^2*g^2*(-(d*g) + e*g*x + 3*d*h*x) - a^2*h^2*(117*f*g
+ 39*e*h + 61*f*h*x) + a*c*(f*(6*g^3 + 90*g^2*h*x - 45*g*h^2*x^2 - 7*h^3*x^3) + 3*h*(2*d*h*(3*g + 5*h*x) + e*(
6*g^2 + 30*g*h*x - 5*h^2*x^2)))) + 4*b^2*c*(-115*a^2*f*h^3 + a*c*h*(3*h*(18*e*g + 6*d*h + 31*e*h*x) + f*(54*g^
2 + 279*g*h*x - 43*h^2*x^2)) + c^2*x*(f*(-12*g^3 + 18*g^2*h*x + 9*g*h^2*x^2 + 2*h^3*x^3) + 3*h*(2*d*h*(-6*g +
h*x) + e*(-12*g^2 + 6*g*h*x + h^2*x^2))))) + 3*(b^2 - 4*a*c)*(35*b^3*f*h^3 - 30*b*c*h^2*(3*b*f*g + b*e*h + 2*a
*f*h) - 16*c^3*g*(f*g^2 + 3*h*(e*g + d*h)) + 24*c^2*h*(3*b*f*g^2 + b*h*(3*e*g + d*h) + a*h*(3*f*g + e*h)))*Sqr
t[a + x*(b + c*x)]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(48*c^(9/2)*(-b^2 + 4*a*c)*Sqrt[a +
x*(b + c*x)])

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Maple [B]  time = 0.064, size = 2780, normalized size = 5.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-39/4*b/c^3*a/(c*x^2+b*x+a)^(1/2)*g*h^2*f-13/4*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*h^3*e-15/4*b/c^2*x^2/
(c*x^2+b*x+a)^(1/2)*g*h^2*f-15/4*h^3*f*b/c^3*a*x/(c*x^2+b*x+a)^(1/2)-8/3*h^3*f*a^2/c^3*b^2/(4*a*c-b^2)/(c*x^2+
b*x+a)^(1/2)+9/2*b/c^2*x/(c*x^2+b*x+a)^(1/2)*g*h^2*e+9/2*b/c^2*x/(c*x^2+b*x+a)^(1/2)*g^2*h*f-3/2*b^3/c^2/(4*a*
c-b^2)/(c*x^2+b*x+a)^(1/2)*x*h^3*d-9/4*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*g^2*h*f-9/4*b^4/c^3/(4*a*c-b^2)
/(c*x^2+b*x+a)^(1/2)*g*h^2*e+2*a/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*h^3*d+x^2/c/(c*x^2+b*x+a)^(1/2)*h^3*d
-3/4*b^2/c^3/(c*x^2+b*x+a)^(1/2)*h^3*d-3/2*b/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*h^3*d+1/2*x^3
/c/(c*x^2+b*x+a)^(1/2)*h^3*e+15/16*b^3/c^4/(c*x^2+b*x+a)^(1/2)*h^3*e+15/8*b^2/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))*h^3*e-3/2*a/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*h^3*e+3/c^(3/2)*ln((1/2*b+
c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*g^2*h*e-3/c/(c*x^2+b*x+a)^(1/2)*g^2*h*d+2*g^3*d*(2*c*x+b)/(4*a*c-b^2)/(c*x^2
+b*x+a)^(1/2)+4*a/c*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*h^3*d-16/3*h^3*f*a^2/c^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(
1/2)*x+45/8*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*g*h^2*f-13/2*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x
*h^3*e-39/4*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*g*h^2*f+6*a/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*g*h^
2*e+6*a/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*g^2*h*f-9/2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*g*h^2*e-
35/16*h^3*f*b^3/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-8/3*h^3*f*a^2/c^3/(c*x^2+b*x+a)^(1/2)-x/c/
(c*x^2+b*x+a)^(1/2)*g^3*f+1/2*b/c^2/(c*x^2+b*x+a)^(1/2)*g^3*f+3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(
1/2))*g*h^2*d+2*a/c^2/(c*x^2+b*x+a)^(1/2)*h^3*d-9/2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*g^2*h*f+3*b^2/c/
(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*g*h^2*d+3*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*g^2*h*e+115/12*h^3*f*b^3/c
^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*g^3*f-1/c/(c*x^2+
b*x+a)^(1/2)*g^3*e-39/2*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*g*h^2*f+12*a/c*b/(4*a*c-b^2)/(c*x^2+b*x+a)
^(1/2)*x*g*h^2*e+12*a/c*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*g^2*h*f+1/3*h^3*f*x^4/c/(c*x^2+b*x+a)^(1/2)-35/32*
h^3*f*b^4/c^5/(c*x^2+b*x+a)^(1/2)+115/24*h^3*f*b^4/c^4*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-35/16*h^3*f*b^5/c^4/(
4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*g^3*f+9/2*a/c^2*x/(c*x^2+b*x+a)^(1/2)
*g*h^2*f+3/2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*g*h^2*d+3/2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*g^2*h
*e-45/8*b^2/c^3*x/(c*x^2+b*x+a)^(1/2)*g*h^2*f+15/8*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*h^3*e+45/16*b^5/c
^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*g*h^2*f+3/2*b/c^2*x/(c*x^2+b*x+a)^(1/2)*h^3*d-9/4*b^2/c^3/(c*x^2+b*x+a)^(1/
2)*g*h^2*e-9/4*b^2/c^3/(c*x^2+b*x+a)^(1/2)*g^2*h*f-3/4*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*h^3*d-9/2*b/c^(
5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*g*h^2*e-9/2*b/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1
/2))*g^2*h*f+6*a/c^2/(c*x^2+b*x+a)^(1/2)*g*h^2*e+6*a/c^2/(c*x^2+b*x+a)^(1/2)*g^2*h*f+3/2*x^3/c/(c*x^2+b*x+a)^(
1/2)*g*h^2*f+3/2*b/c^2/(c*x^2+b*x+a)^(1/2)*g^2*h*e+1/2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*g^3*f-2*b/(4*a*
c-b^2)/(c*x^2+b*x+a)^(1/2)*x*g^3*e-b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*g^3*e-4/3*h^3*f*a/c^2*x^2/(c*x^2+b*x+
a)^(1/2)+35/16*h^3*f*b^3/c^4*x/(c*x^2+b*x+a)^(1/2)-35/32*h^3*f*b^6/c^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+115/24*
h^3*f*b^2/c^4*a/(c*x^2+b*x+a)^(1/2)+15/4*h^3*f*b/c^(7/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-7/12*h^
3*f*b/c^2*x^3/(c*x^2+b*x+a)^(1/2)+35/24*h^3*f*b^2/c^3*x^2/(c*x^2+b*x+a)^(1/2)-3*x/c/(c*x^2+b*x+a)^(1/2)*g*h^2*
d-3*x/c/(c*x^2+b*x+a)^(1/2)*g^2*h*e+3/2*b/c^2/(c*x^2+b*x+a)^(1/2)*g*h^2*d-5/4*b/c^2*x^2/(c*x^2+b*x+a)^(1/2)*h^
3*e-15/8*b^2/c^3*x/(c*x^2+b*x+a)^(1/2)*h^3*e+45/16*b^3/c^4/(c*x^2+b*x+a)^(1/2)*g*h^2*f+15/16*b^5/c^4/(4*a*c-b^
2)/(c*x^2+b*x+a)^(1/2)*h^3*e+45/8*b^2/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*g*h^2*f-13/4*b/c^3*a
/(c*x^2+b*x+a)^(1/2)*h^3*e+3/2*a/c^2*x/(c*x^2+b*x+a)^(1/2)*h^3*e-9/2*a/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))*g*h^2*f+3*x^2/c/(c*x^2+b*x+a)^(1/2)*g^2*h*f+3*x^2/c/(c*x^2+b*x+a)^(1/2)*g*h^2*e-6*b/(4*a*c-b^2)/(
c*x^2+b*x+a)^(1/2)*x*g^2*h*d-3*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*g^2*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((h*x+g)^3*(f*x^2+e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 132.617, size = 6238, normalized size = 12.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/96*(3*(16*(a*b^2*c^3 - 4*a^2*c^4)*f*g^3 + 24*(2*(a*b^2*c^3 - 4*a^2*c^4)*e - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*f)*
g^2*h + 6*(8*(a*b^2*c^3 - 4*a^2*c^4)*d - 12*(a*b^3*c^2 - 4*a^2*b*c^3)*e + 3*(5*a*b^4*c - 24*a^2*b^2*c^2 + 16*a
^3*c^3)*f)*g*h^2 - (24*(a*b^3*c^2 - 4*a^2*b*c^3)*d - 6*(5*a*b^4*c - 24*a^2*b^2*c^2 + 16*a^3*c^3)*e + 5*(7*a*b^
5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*f)*h^3 + (16*(b^2*c^4 - 4*a*c^5)*f*g^3 + 24*(2*(b^2*c^4 - 4*a*c^5)*e - 3*(b^3
*c^3 - 4*a*b*c^4)*f)*g^2*h + 6*(8*(b^2*c^4 - 4*a*c^5)*d - 12*(b^3*c^3 - 4*a*b*c^4)*e + 3*(5*b^4*c^2 - 24*a*b^2
*c^3 + 16*a^2*c^4)*f)*g*h^2 - (24*(b^3*c^3 - 4*a*b*c^4)*d - 6*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*e + 5*(7
*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*f)*h^3)*x^2 + (16*(b^3*c^3 - 4*a*b*c^4)*f*g^3 + 24*(2*(b^3*c^3 - 4*a*b*c
^4)*e - 3*(b^4*c^2 - 4*a*b^2*c^3)*f)*g^2*h + 6*(8*(b^3*c^3 - 4*a*b*c^4)*d - 12*(b^4*c^2 - 4*a*b^2*c^3)*e + 3*(
5*b^5*c - 24*a*b^3*c^2 + 16*a^2*b*c^3)*f)*g*h^2 - (24*(b^4*c^2 - 4*a*b^2*c^3)*d - 6*(5*b^5*c - 24*a*b^3*c^2 +
16*a^2*b*c^3)*e + 5*(7*b^6 - 40*a*b^4*c + 48*a^2*b^2*c^2)*f)*h^3)*x)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 -
4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*(8*(b^2*c^4 - 4*a*c^5)*f*h^3*x^4 - 48*(b*c^5*d - 2*a*
c^5*e + a*b*c^4*f)*g^3 + 72*(4*a*c^5*d - 2*a*b*c^4*e + (3*a*b^2*c^3 - 8*a^2*c^4)*f)*g^2*h - 18*(8*a*b*c^4*d -
4*(3*a*b^2*c^3 - 8*a^2*c^4)*e + (15*a*b^3*c^2 - 52*a^2*b*c^3)*f)*g*h^2 + (24*(3*a*b^2*c^3 - 8*a^2*c^4)*d - 6*(
15*a*b^3*c^2 - 52*a^2*b*c^3)*e + (105*a*b^4*c - 460*a^2*b^2*c^2 + 256*a^3*c^3)*f)*h^3 + 2*(18*(b^2*c^4 - 4*a*c
^5)*f*g*h^2 + (6*(b^2*c^4 - 4*a*c^5)*e - 7*(b^3*c^3 - 4*a*b*c^4)*f)*h^3)*x^3 + (72*(b^2*c^4 - 4*a*c^5)*f*g^2*h
+ 18*(4*(b^2*c^4 - 4*a*c^5)*e - 5*(b^3*c^3 - 4*a*b*c^4)*f)*g*h^2 + (24*(b^2*c^4 - 4*a*c^5)*d - 30*(b^3*c^3 -
4*a*b*c^4)*e + (35*b^4*c^2 - 172*a*b^2*c^3 + 128*a^2*c^4)*f)*h^3)*x^2 - (48*(2*c^6*d - b*c^5*e + (b^2*c^4 - 2*
a*c^5)*f)*g^3 - 72*(2*b*c^5*d - 2*(b^2*c^4 - 2*a*c^5)*e + (3*b^3*c^3 - 10*a*b*c^4)*f)*g^2*h + 18*(8*(b^2*c^4 -
2*a*c^5)*d - 4*(3*b^3*c^3 - 10*a*b*c^4)*e + (15*b^4*c^2 - 62*a*b^2*c^3 + 24*a^2*c^4)*f)*g*h^2 - (24*(3*b^3*c^
3 - 10*a*b*c^4)*d - 6*(15*b^4*c^2 - 62*a*b^2*c^3 + 24*a^2*c^4)*e + (105*b^5*c - 530*a*b^3*c^2 + 488*a^2*b*c^3)
*f)*h^3)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^5 - 4*a^2*c^6 + (b^2*c^6 - 4*a*c^7)*x^2 + (b^3*c^5 - 4*a*b*c^6)*x)
, -1/48*(3*(16*(a*b^2*c^3 - 4*a^2*c^4)*f*g^3 + 24*(2*(a*b^2*c^3 - 4*a^2*c^4)*e - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*f
)*g^2*h + 6*(8*(a*b^2*c^3 - 4*a^2*c^4)*d - 12*(a*b^3*c^2 - 4*a^2*b*c^3)*e + 3*(5*a*b^4*c - 24*a^2*b^2*c^2 + 16
*a^3*c^3)*f)*g*h^2 - (24*(a*b^3*c^2 - 4*a^2*b*c^3)*d - 6*(5*a*b^4*c - 24*a^2*b^2*c^2 + 16*a^3*c^3)*e + 5*(7*a*
b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*f)*h^3 + (16*(b^2*c^4 - 4*a*c^5)*f*g^3 + 24*(2*(b^2*c^4 - 4*a*c^5)*e - 3*(b
^3*c^3 - 4*a*b*c^4)*f)*g^2*h + 6*(8*(b^2*c^4 - 4*a*c^5)*d - 12*(b^3*c^3 - 4*a*b*c^4)*e + 3*(5*b^4*c^2 - 24*a*b
^2*c^3 + 16*a^2*c^4)*f)*g*h^2 - (24*(b^3*c^3 - 4*a*b*c^4)*d - 6*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*e + 5*
(7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*f)*h^3)*x^2 + (16*(b^3*c^3 - 4*a*b*c^4)*f*g^3 + 24*(2*(b^3*c^3 - 4*a*b
*c^4)*e - 3*(b^4*c^2 - 4*a*b^2*c^3)*f)*g^2*h + 6*(8*(b^3*c^3 - 4*a*b*c^4)*d - 12*(b^4*c^2 - 4*a*b^2*c^3)*e + 3
*(5*b^5*c - 24*a*b^3*c^2 + 16*a^2*b*c^3)*f)*g*h^2 - (24*(b^4*c^2 - 4*a*b^2*c^3)*d - 6*(5*b^5*c - 24*a*b^3*c^2
+ 16*a^2*b*c^3)*e + 5*(7*b^6 - 40*a*b^4*c + 48*a^2*b^2*c^2)*f)*h^3)*x)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x +
a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - 2*(8*(b^2*c^4 - 4*a*c^5)*f*h^3*x^4 - 48*(b*c^5*d - 2*a*c^5*
e + a*b*c^4*f)*g^3 + 72*(4*a*c^5*d - 2*a*b*c^4*e + (3*a*b^2*c^3 - 8*a^2*c^4)*f)*g^2*h - 18*(8*a*b*c^4*d - 4*(3
*a*b^2*c^3 - 8*a^2*c^4)*e + (15*a*b^3*c^2 - 52*a^2*b*c^3)*f)*g*h^2 + (24*(3*a*b^2*c^3 - 8*a^2*c^4)*d - 6*(15*a
*b^3*c^2 - 52*a^2*b*c^3)*e + (105*a*b^4*c - 460*a^2*b^2*c^2 + 256*a^3*c^3)*f)*h^3 + 2*(18*(b^2*c^4 - 4*a*c^5)*
f*g*h^2 + (6*(b^2*c^4 - 4*a*c^5)*e - 7*(b^3*c^3 - 4*a*b*c^4)*f)*h^3)*x^3 + (72*(b^2*c^4 - 4*a*c^5)*f*g^2*h + 1
8*(4*(b^2*c^4 - 4*a*c^5)*e - 5*(b^3*c^3 - 4*a*b*c^4)*f)*g*h^2 + (24*(b^2*c^4 - 4*a*c^5)*d - 30*(b^3*c^3 - 4*a*
b*c^4)*e + (35*b^4*c^2 - 172*a*b^2*c^3 + 128*a^2*c^4)*f)*h^3)*x^2 - (48*(2*c^6*d - b*c^5*e + (b^2*c^4 - 2*a*c^
5)*f)*g^3 - 72*(2*b*c^5*d - 2*(b^2*c^4 - 2*a*c^5)*e + (3*b^3*c^3 - 10*a*b*c^4)*f)*g^2*h + 18*(8*(b^2*c^4 - 2*a
*c^5)*d - 4*(3*b^3*c^3 - 10*a*b*c^4)*e + (15*b^4*c^2 - 62*a*b^2*c^3 + 24*a^2*c^4)*f)*g*h^2 - (24*(3*b^3*c^3 -
10*a*b*c^4)*d - 6*(15*b^4*c^2 - 62*a*b^2*c^3 + 24*a^2*c^4)*e + (105*b^5*c - 530*a*b^3*c^2 + 488*a^2*b*c^3)*f)*
h^3)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^5 - 4*a^2*c^6 + (b^2*c^6 - 4*a*c^7)*x^2 + (b^3*c^5 - 4*a*b*c^6)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.24518, size = 1423, normalized size = 2.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(((2*(4*(b^2*c^3*f*h^3 - 4*a*c^4*f*h^3)*x/(b^2*c^4 - 4*a*c^5) + (18*b^2*c^3*f*g*h^2 - 72*a*c^4*f*g*h^2 -
7*b^3*c^2*f*h^3 + 28*a*b*c^3*f*h^3 + 6*b^2*c^3*h^3*e - 24*a*c^4*h^3*e)/(b^2*c^4 - 4*a*c^5))*x + (72*b^2*c^3*f*
g^2*h - 288*a*c^4*f*g^2*h - 90*b^3*c^2*f*g*h^2 + 360*a*b*c^3*f*g*h^2 + 24*b^2*c^3*d*h^3 - 96*a*c^4*d*h^3 + 35*
b^4*c*f*h^3 - 172*a*b^2*c^2*f*h^3 + 128*a^2*c^3*f*h^3 + 72*b^2*c^3*g*h^2*e - 288*a*c^4*g*h^2*e - 30*b^3*c^2*h^
3*e + 120*a*b*c^3*h^3*e)/(b^2*c^4 - 4*a*c^5))*x - (96*c^5*d*g^3 + 48*b^2*c^3*f*g^3 - 96*a*c^4*f*g^3 - 144*b*c^
4*d*g^2*h - 216*b^3*c^2*f*g^2*h + 720*a*b*c^3*f*g^2*h + 144*b^2*c^3*d*g*h^2 - 288*a*c^4*d*g*h^2 + 270*b^4*c*f*
g*h^2 - 1116*a*b^2*c^2*f*g*h^2 + 432*a^2*c^3*f*g*h^2 - 72*b^3*c^2*d*h^3 + 240*a*b*c^3*d*h^3 - 105*b^5*f*h^3 +
530*a*b^3*c*f*h^3 - 488*a^2*b*c^2*f*h^3 - 48*b*c^4*g^3*e + 144*b^2*c^3*g^2*h*e - 288*a*c^4*g^2*h*e - 216*b^3*c
^2*g*h^2*e + 720*a*b*c^3*g*h^2*e + 90*b^4*c*h^3*e - 372*a*b^2*c^2*h^3*e + 144*a^2*c^3*h^3*e)/(b^2*c^4 - 4*a*c^
5))*x - (48*b*c^4*d*g^3 + 48*a*b*c^3*f*g^3 - 288*a*c^4*d*g^2*h - 216*a*b^2*c^2*f*g^2*h + 576*a^2*c^3*f*g^2*h +
144*a*b*c^3*d*g*h^2 + 270*a*b^3*c*f*g*h^2 - 936*a^2*b*c^2*f*g*h^2 - 72*a*b^2*c^2*d*h^3 + 192*a^2*c^3*d*h^3 -
105*a*b^4*f*h^3 + 460*a^2*b^2*c*f*h^3 - 256*a^3*c^2*f*h^3 - 96*a*c^4*g^3*e + 144*a*b*c^3*g^2*h*e - 216*a*b^2*c
^2*g*h^2*e + 576*a^2*c^3*g*h^2*e + 90*a*b^3*c*h^3*e - 312*a^2*b*c^2*h^3*e)/(b^2*c^4 - 4*a*c^5))/sqrt(c*x^2 + b
*x + a) - 1/16*(16*c^3*f*g^3 - 72*b*c^2*f*g^2*h + 48*c^3*d*g*h^2 + 90*b^2*c*f*g*h^2 - 72*a*c^2*f*g*h^2 - 24*b*
c^2*d*h^3 - 35*b^3*f*h^3 + 60*a*b*c*f*h^3 + 48*c^3*g^2*h*e - 72*b*c^2*g*h^2*e + 30*b^2*c*h^3*e - 24*a*c^2*h^3*
e)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(9/2)