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

Optimal. Leaf size=891 \[ \frac{x \sqrt{c x^2+b x+a} f^3}{2 c^3}+\frac{(12 c e-11 b f) \sqrt{c x^2+b x+a} f^2}{4 c^4}+\frac{\left (24 \left (e^2+d f\right ) c^2-20 f (3 b e+a f) c+35 b^2 f^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) f}{8 c^{9/2}}-\frac{2 \left (-f^3 b^7+3 c e f^2 b^6+3 c f \left (6 a f^2-c \left (e^2+d f\right )\right ) b^5-c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right ) b^4-3 c^2 \left (29 a^2 f^3-10 a c \left (e^2+d f\right ) f+c^2 d \left (e^2+d f\right )\right ) b^3+6 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right ) b^2-4 c^3 \left (2 c^3 d^3+3 a c^2 \left (e^2+d f\right ) d-29 a^3 f^3+24 a^2 c f \left (e^2+d f\right )\right ) b-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-c \left (-10 f^3 b^6+3 c f^2 (7 b e+26 a f) b^4-6 c^2 f \left (2 \left (e^2+d f\right ) b^2+25 a e f b+27 a^2 f^2\right ) b^2+16 c^6 d^3-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )+6 c^4 \left (-16 f \left (e^2+d f\right ) a^2-2 b e \left (e^2+6 d f\right ) a+b^2 d \left (e^2+d f\right )\right )+c^3 \left (\left (e^3+6 d f e\right ) b^3+84 a f \left (e^2+d f\right ) b^2+240 a^2 e f^2 b+56 a^3 f^3\right )\right ) x\right )}{3 c^5 \left (b^2-4 a c\right )^2 \sqrt{c x^2+b x+a}}+\frac{2 \left (-a f^3 b^5+3 a c e f^2 b^4+a c f \left (5 a f^2-3 c \left (e^2+d f\right )\right ) b^3-a c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right ) b^2-c^2 \left (c^3 d^3+3 a c^2 \left (e^2+d f\right ) d+5 a^3 f^3-9 a^2 c f \left (e^2+d f\right )\right ) b+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (f b^2-c e b+2 c^2 d-2 a c f\right ) \left (f^2 b^4-2 c e f b^3+c^2 e^2 b^2-4 a c f^2 b^2+c^2 d f b^2-c^3 d e b+7 a c^2 e f b+c^4 d^2-3 a c^3 e^2+a^2 c^2 f^2-2 a c^3 d f\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}} \]

[Out]

(2*(3*a*b^4*c*e*f^2 - a*b^5*f^3 + a*b^3*c*f*(5*a*f^2 - 3*c*(e^2 + d*f)) - b*c^2*(c^3*d^3 + 5*a^3*f^3 + 3*a*c^2
*d*(e^2 + d*f) - 9*a^2*c*f*(e^2 + d*f)) - a*b^2*c^2*e*(12*a*f^2 - c*(e^2 + 6*d*f)) + 2*a*c^3*e*(3*c^2*d^2 + 3*
a^2*f^2 - a*c*(e^2 + 6*d*f)) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*(c^4*d^2 - b*c^3*d*e + b^2*c^2*e^2 - 3*a*c^
3*e^2 + b^2*c^2*d*f - 2*a*c^3*d*f - 2*b^3*c*e*f + 7*a*b*c^2*e*f + b^4*f^2 - 4*a*b^2*c*f^2 + a^2*c^2*f^2)*x))/(
3*c^5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) - (2*(3*b^6*c*e*f^2 - b^7*f^3 + 3*b^5*c*f*(6*a*f^2 - c*(e^2 + d*f
)) - 3*b^3*c^2*(29*a^2*f^3 + c^2*d*(e^2 + d*f) - 10*a*c*f*(e^2 + d*f)) - 4*b*c^3*(2*c^3*d^3 - 29*a^3*f^3 + 3*a
*c^2*d*(e^2 + d*f) + 24*a^2*c*f*(e^2 + d*f)) - 24*a^2*c^4*e*(6*a*f^2 - c*(e^2 + 6*d*f)) - b^4*c^2*e*(42*a*f^2
- c*(e^2 + 6*d*f)) + 6*b^2*c^3*e*(2*c^2*d^2 + 28*a^2*f^2 - a*c*(e^2 + 6*d*f)) - c*(16*c^6*d^3 - 10*b^6*f^3 + 3
*b^4*c*f^2*(7*b*e + 26*a*f) - 24*c^5*d*(b*d*e - a*(e^2 + d*f)) - 6*b^2*c^2*f*(25*a*b*e*f + 27*a^2*f^2 + 2*b^2*
(e^2 + d*f)) + 6*c^4*(b^2*d*(e^2 + d*f) - 16*a^2*f*(e^2 + d*f) - 2*a*b*e*(e^2 + 6*d*f)) + c^3*(240*a^2*b*e*f^2
+ 56*a^3*f^3 + 84*a*b^2*f*(e^2 + d*f) + b^3*(e^3 + 6*d*e*f)))*x))/(3*c^5*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2
]) + (f^2*(12*c*e - 11*b*f)*Sqrt[a + b*x + c*x^2])/(4*c^4) + (f^3*x*Sqrt[a + b*x + c*x^2])/(2*c^3) + (f*(35*b^
2*f^2 - 20*c*f*(3*b*e + a*f) + 24*c^2*(e^2 + d*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*
c^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 1.76833, antiderivative size = 891, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1660, 1661, 640, 621, 206} \[ \frac{x \sqrt{c x^2+b x+a} f^3}{2 c^3}+\frac{(12 c e-11 b f) \sqrt{c x^2+b x+a} f^2}{4 c^4}+\frac{\left (24 \left (e^2+d f\right ) c^2-20 f (3 b e+a f) c+35 b^2 f^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) f}{8 c^{9/2}}-\frac{2 \left (-f^3 b^7+3 c e f^2 b^6+3 c f \left (6 a f^2-c \left (e^2+d f\right )\right ) b^5-c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right ) b^4-3 c^2 \left (29 a^2 f^3-10 a c \left (e^2+d f\right ) f+c^2 d \left (e^2+d f\right )\right ) b^3+6 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right ) b^2-4 c^3 \left (2 c^3 d^3+3 a c^2 \left (e^2+d f\right ) d-29 a^3 f^3+24 a^2 c f \left (e^2+d f\right )\right ) b-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-c \left (-10 f^3 b^6+3 c f^2 (7 b e+26 a f) b^4-6 c^2 f \left (2 \left (e^2+d f\right ) b^2+25 a e f b+27 a^2 f^2\right ) b^2+16 c^6 d^3-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )+6 c^4 \left (-16 f \left (e^2+d f\right ) a^2-2 b e \left (e^2+6 d f\right ) a+b^2 d \left (e^2+d f\right )\right )+c^3 \left (\left (e^3+6 d f e\right ) b^3+84 a f \left (e^2+d f\right ) b^2+240 a^2 e f^2 b+56 a^3 f^3\right )\right ) x\right )}{3 c^5 \left (b^2-4 a c\right )^2 \sqrt{c x^2+b x+a}}+\frac{2 \left (-a f^3 b^5+3 a c e f^2 b^4+a c f \left (5 a f^2-3 c \left (e^2+d f\right )\right ) b^3-a c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right ) b^2-c^2 \left (c^3 d^3+3 a c^2 \left (e^2+d f\right ) d+5 a^3 f^3-9 a^2 c f \left (e^2+d f\right )\right ) b+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (f b^2-c e b+2 c^2 d-2 a c f\right ) \left (f^2 b^4-2 c e f b^3+c^2 e^2 b^2-4 a c f^2 b^2+c^2 d f b^2-c^3 d e b+7 a c^2 e f b+c^4 d^2-3 a c^3 e^2+a^2 c^2 f^2-2 a c^3 d f\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(3*a*b^4*c*e*f^2 - a*b^5*f^3 + a*b^3*c*f*(5*a*f^2 - 3*c*(e^2 + d*f)) - b*c^2*(c^3*d^3 + 5*a^3*f^3 + 3*a*c^2
*d*(e^2 + d*f) - 9*a^2*c*f*(e^2 + d*f)) - a*b^2*c^2*e*(12*a*f^2 - c*(e^2 + 6*d*f)) + 2*a*c^3*e*(3*c^2*d^2 + 3*
a^2*f^2 - a*c*(e^2 + 6*d*f)) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*(c^4*d^2 - b*c^3*d*e + b^2*c^2*e^2 - 3*a*c^
3*e^2 + b^2*c^2*d*f - 2*a*c^3*d*f - 2*b^3*c*e*f + 7*a*b*c^2*e*f + b^4*f^2 - 4*a*b^2*c*f^2 + a^2*c^2*f^2)*x))/(
3*c^5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) - (2*(3*b^6*c*e*f^2 - b^7*f^3 + 3*b^5*c*f*(6*a*f^2 - c*(e^2 + d*f
)) - 3*b^3*c^2*(29*a^2*f^3 + c^2*d*(e^2 + d*f) - 10*a*c*f*(e^2 + d*f)) - 4*b*c^3*(2*c^3*d^3 - 29*a^3*f^3 + 3*a
*c^2*d*(e^2 + d*f) + 24*a^2*c*f*(e^2 + d*f)) - 24*a^2*c^4*e*(6*a*f^2 - c*(e^2 + 6*d*f)) - b^4*c^2*e*(42*a*f^2
- c*(e^2 + 6*d*f)) + 6*b^2*c^3*e*(2*c^2*d^2 + 28*a^2*f^2 - a*c*(e^2 + 6*d*f)) - c*(16*c^6*d^3 - 10*b^6*f^3 + 3
*b^4*c*f^2*(7*b*e + 26*a*f) - 24*c^5*d*(b*d*e - a*(e^2 + d*f)) - 6*b^2*c^2*f*(25*a*b*e*f + 27*a^2*f^2 + 2*b^2*
(e^2 + d*f)) + 6*c^4*(b^2*d*(e^2 + d*f) - 16*a^2*f*(e^2 + d*f) - 2*a*b*e*(e^2 + 6*d*f)) + c^3*(240*a^2*b*e*f^2
+ 56*a^3*f^3 + 84*a*b^2*f*(e^2 + d*f) + b^3*(e^3 + 6*d*e*f)))*x))/(3*c^5*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2
]) + (f^2*(12*c*e - 11*b*f)*Sqrt[a + b*x + c*x^2])/(4*c^4) + (f^3*x*Sqrt[a + b*x + c*x^2])/(2*c^3) + (f*(35*b^
2*f^2 - 20*c*f*(3*b*e + a*f) + 24*c^2*(e^2 + d*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*
c^(9/2))

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 1661

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[(e*x^(q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(q + 2*p + 1)), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[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{\left (d+e x+f x^2\right )^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=\frac{2 \left (3 a b^4 c e f^2-a b^5 f^3+a b^3 c f \left (5 a f^2-3 c \left (e^2+d f\right )\right )-b c^2 \left (c^3 d^3+5 a^3 f^3+3 a c^2 d \left (e^2+d f\right )-9 a^2 c f \left (e^2+d f\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right )+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \left (c^4 d^2-b c^3 d e+b^2 c^2 e^2-3 a c^3 e^2+b^2 c^2 d f-2 a c^3 d f-2 b^3 c e f+7 a b c^2 e f+b^4 f^2-4 a b^2 c f^2+a^2 c^2 f^2\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \int \frac{\frac{8 c^6 d^3+b^6 f^3-3 b^4 c f^2 (b e+a f)+3 c^4 \left (b^2 d-4 a^2 f\right ) \left (e^2+d f\right )-12 c^5 d \left (b d e-a \left (e^2+d f\right )\right )+3 b^2 c^2 f \left (2 a b e f-a^2 f^2+b^2 \left (e^2+d f\right )\right )+c^3 \left (12 a^2 b e f^2+4 a^3 f^3-3 a b^2 f \left (e^2+d f\right )-b^3 \left (e^3+6 d e f\right )\right )}{2 c^5}+\frac{3 \left (b^2-4 a c\right ) \left (b^3 f^3-b c f^2 (3 b e+2 a f)-c^3 \left (e^3+6 d e f\right )+3 c^2 f \left (a e f+b \left (e^2+d f\right )\right )\right ) x}{2 c^4}-\frac{3 \left (b^2-4 a c\right ) f \left (b^2 f^2-c f (3 b e+a f)+3 c^2 \left (e^2+d f\right )\right ) x^2}{2 c^3}-\frac{3 \left (b^2-4 a c\right ) f^2 (3 c e-b f) x^3}{2 c^2}+\frac{3}{2} \left (4 a-\frac{b^2}{c}\right ) f^3 x^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=\frac{2 \left (3 a b^4 c e f^2-a b^5 f^3+a b^3 c f \left (5 a f^2-3 c \left (e^2+d f\right )\right )-b c^2 \left (c^3 d^3+5 a^3 f^3+3 a c^2 d \left (e^2+d f\right )-9 a^2 c f \left (e^2+d f\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right )+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \left (c^4 d^2-b c^3 d e+b^2 c^2 e^2-3 a c^3 e^2+b^2 c^2 d f-2 a c^3 d f-2 b^3 c e f+7 a b c^2 e f+b^4 f^2-4 a b^2 c f^2+a^2 c^2 f^2\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (3 b^6 c e f^2-b^7 f^3+3 b^5 c f \left (6 a f^2-c \left (e^2+d f\right )\right )-3 b^3 c^2 \left (29 a^2 f^3+c^2 d \left (e^2+d f\right )-10 a c f \left (e^2+d f\right )\right )-4 b c^3 \left (2 c^3 d^3-29 a^3 f^3+3 a c^2 d \left (e^2+d f\right )+24 a^2 c f \left (e^2+d f\right )\right )-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-b^4 c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right )+6 b^2 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right )-c \left (16 c^6 d^3-10 b^6 f^3+3 b^4 c f^2 (7 b e+26 a f)-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )-6 b^2 c^2 f \left (25 a b e f+27 a^2 f^2+2 b^2 \left (e^2+d f\right )\right )+6 c^4 \left (b^2 d \left (e^2+d f\right )-16 a^2 f \left (e^2+d f\right )-2 a b e \left (e^2+6 d f\right )\right )+c^3 \left (240 a^2 b e f^2+56 a^3 f^3+84 a b^2 f \left (e^2+d f\right )+b^3 \left (e^3+6 d e f\right )\right )\right ) x\right )}{3 c^5 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{4 \int \frac{\frac{3 \left (b^2-4 a c\right )^2 f \left (3 b^2 f^2-2 c f (3 b e+a f)+3 c^2 \left (e^2+d f\right )\right )}{4 c^4}+\frac{3 \left (b^2-4 a c\right )^2 f^2 (3 c e-2 b f) x}{4 c^3}+\frac{3 \left (b^2-4 a c\right )^2 f^3 x^2}{4 c^2}}{\sqrt{a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2}\\ &=\frac{2 \left (3 a b^4 c e f^2-a b^5 f^3+a b^3 c f \left (5 a f^2-3 c \left (e^2+d f\right )\right )-b c^2 \left (c^3 d^3+5 a^3 f^3+3 a c^2 d \left (e^2+d f\right )-9 a^2 c f \left (e^2+d f\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right )+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \left (c^4 d^2-b c^3 d e+b^2 c^2 e^2-3 a c^3 e^2+b^2 c^2 d f-2 a c^3 d f-2 b^3 c e f+7 a b c^2 e f+b^4 f^2-4 a b^2 c f^2+a^2 c^2 f^2\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (3 b^6 c e f^2-b^7 f^3+3 b^5 c f \left (6 a f^2-c \left (e^2+d f\right )\right )-3 b^3 c^2 \left (29 a^2 f^3+c^2 d \left (e^2+d f\right )-10 a c f \left (e^2+d f\right )\right )-4 b c^3 \left (2 c^3 d^3-29 a^3 f^3+3 a c^2 d \left (e^2+d f\right )+24 a^2 c f \left (e^2+d f\right )\right )-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-b^4 c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right )+6 b^2 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right )-c \left (16 c^6 d^3-10 b^6 f^3+3 b^4 c f^2 (7 b e+26 a f)-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )-6 b^2 c^2 f \left (25 a b e f+27 a^2 f^2+2 b^2 \left (e^2+d f\right )\right )+6 c^4 \left (b^2 d \left (e^2+d f\right )-16 a^2 f \left (e^2+d f\right )-2 a b e \left (e^2+6 d f\right )\right )+c^3 \left (240 a^2 b e f^2+56 a^3 f^3+84 a b^2 f \left (e^2+d f\right )+b^3 \left (e^3+6 d e f\right )\right )\right ) x\right )}{3 c^5 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{f^3 x \sqrt{a+b x+c x^2}}{2 c^3}+\frac{2 \int \frac{\frac{3 \left (b^2-4 a c\right )^2 f \left (6 b^2 f^2-c f (12 b e+5 a f)+6 c^2 \left (e^2+d f\right )\right )}{4 c^3}+\frac{3 \left (b^2-4 a c\right )^2 f^2 (12 c e-11 b f) x}{8 c^2}}{\sqrt{a+b x+c x^2}} \, dx}{3 c \left (b^2-4 a c\right )^2}\\ &=\frac{2 \left (3 a b^4 c e f^2-a b^5 f^3+a b^3 c f \left (5 a f^2-3 c \left (e^2+d f\right )\right )-b c^2 \left (c^3 d^3+5 a^3 f^3+3 a c^2 d \left (e^2+d f\right )-9 a^2 c f \left (e^2+d f\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right )+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \left (c^4 d^2-b c^3 d e+b^2 c^2 e^2-3 a c^3 e^2+b^2 c^2 d f-2 a c^3 d f-2 b^3 c e f+7 a b c^2 e f+b^4 f^2-4 a b^2 c f^2+a^2 c^2 f^2\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (3 b^6 c e f^2-b^7 f^3+3 b^5 c f \left (6 a f^2-c \left (e^2+d f\right )\right )-3 b^3 c^2 \left (29 a^2 f^3+c^2 d \left (e^2+d f\right )-10 a c f \left (e^2+d f\right )\right )-4 b c^3 \left (2 c^3 d^3-29 a^3 f^3+3 a c^2 d \left (e^2+d f\right )+24 a^2 c f \left (e^2+d f\right )\right )-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-b^4 c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right )+6 b^2 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right )-c \left (16 c^6 d^3-10 b^6 f^3+3 b^4 c f^2 (7 b e+26 a f)-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )-6 b^2 c^2 f \left (25 a b e f+27 a^2 f^2+2 b^2 \left (e^2+d f\right )\right )+6 c^4 \left (b^2 d \left (e^2+d f\right )-16 a^2 f \left (e^2+d f\right )-2 a b e \left (e^2+6 d f\right )\right )+c^3 \left (240 a^2 b e f^2+56 a^3 f^3+84 a b^2 f \left (e^2+d f\right )+b^3 \left (e^3+6 d e f\right )\right )\right ) x\right )}{3 c^5 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{f^2 (12 c e-11 b f) \sqrt{a+b x+c x^2}}{4 c^4}+\frac{f^3 x \sqrt{a+b x+c x^2}}{2 c^3}+\frac{\left (f \left (35 b^2 f^2-20 c f (3 b e+a f)+24 c^2 \left (e^2+d f\right )\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{8 c^4}\\ &=\frac{2 \left (3 a b^4 c e f^2-a b^5 f^3+a b^3 c f \left (5 a f^2-3 c \left (e^2+d f\right )\right )-b c^2 \left (c^3 d^3+5 a^3 f^3+3 a c^2 d \left (e^2+d f\right )-9 a^2 c f \left (e^2+d f\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right )+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \left (c^4 d^2-b c^3 d e+b^2 c^2 e^2-3 a c^3 e^2+b^2 c^2 d f-2 a c^3 d f-2 b^3 c e f+7 a b c^2 e f+b^4 f^2-4 a b^2 c f^2+a^2 c^2 f^2\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (3 b^6 c e f^2-b^7 f^3+3 b^5 c f \left (6 a f^2-c \left (e^2+d f\right )\right )-3 b^3 c^2 \left (29 a^2 f^3+c^2 d \left (e^2+d f\right )-10 a c f \left (e^2+d f\right )\right )-4 b c^3 \left (2 c^3 d^3-29 a^3 f^3+3 a c^2 d \left (e^2+d f\right )+24 a^2 c f \left (e^2+d f\right )\right )-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-b^4 c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right )+6 b^2 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right )-c \left (16 c^6 d^3-10 b^6 f^3+3 b^4 c f^2 (7 b e+26 a f)-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )-6 b^2 c^2 f \left (25 a b e f+27 a^2 f^2+2 b^2 \left (e^2+d f\right )\right )+6 c^4 \left (b^2 d \left (e^2+d f\right )-16 a^2 f \left (e^2+d f\right )-2 a b e \left (e^2+6 d f\right )\right )+c^3 \left (240 a^2 b e f^2+56 a^3 f^3+84 a b^2 f \left (e^2+d f\right )+b^3 \left (e^3+6 d e f\right )\right )\right ) x\right )}{3 c^5 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{f^2 (12 c e-11 b f) \sqrt{a+b x+c x^2}}{4 c^4}+\frac{f^3 x \sqrt{a+b x+c x^2}}{2 c^3}+\frac{\left (f \left (35 b^2 f^2-20 c f (3 b e+a f)+24 c^2 \left (e^2+d f\right )\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 )}{4 c^4}\\ &=\frac{2 \left (3 a b^4 c e f^2-a b^5 f^3+a b^3 c f \left (5 a f^2-3 c \left (e^2+d f\right )\right )-b c^2 \left (c^3 d^3+5 a^3 f^3+3 a c^2 d \left (e^2+d f\right )-9 a^2 c f \left (e^2+d f\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (e^2+6 d f\right )\right )+2 a c^3 e \left (3 c^2 d^2+3 a^2 f^2-a c \left (e^2+6 d f\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \left (c^4 d^2-b c^3 d e+b^2 c^2 e^2-3 a c^3 e^2+b^2 c^2 d f-2 a c^3 d f-2 b^3 c e f+7 a b c^2 e f+b^4 f^2-4 a b^2 c f^2+a^2 c^2 f^2\right ) x\right )}{3 c^5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (3 b^6 c e f^2-b^7 f^3+3 b^5 c f \left (6 a f^2-c \left (e^2+d f\right )\right )-3 b^3 c^2 \left (29 a^2 f^3+c^2 d \left (e^2+d f\right )-10 a c f \left (e^2+d f\right )\right )-4 b c^3 \left (2 c^3 d^3-29 a^3 f^3+3 a c^2 d \left (e^2+d f\right )+24 a^2 c f \left (e^2+d f\right )\right )-24 a^2 c^4 e \left (6 a f^2-c \left (e^2+6 d f\right )\right )-b^4 c^2 e \left (42 a f^2-c \left (e^2+6 d f\right )\right )+6 b^2 c^3 e \left (2 c^2 d^2+28 a^2 f^2-a c \left (e^2+6 d f\right )\right )-c \left (16 c^6 d^3-10 b^6 f^3+3 b^4 c f^2 (7 b e+26 a f)-24 c^5 d \left (b d e-a \left (e^2+d f\right )\right )-6 b^2 c^2 f \left (25 a b e f+27 a^2 f^2+2 b^2 \left (e^2+d f\right )\right )+6 c^4 \left (b^2 d \left (e^2+d f\right )-16 a^2 f \left (e^2+d f\right )-2 a b e \left (e^2+6 d f\right )\right )+c^3 \left (240 a^2 b e f^2+56 a^3 f^3+84 a b^2 f \left (e^2+d f\right )+b^3 \left (e^3+6 d e f\right )\right )\right ) x\right )}{3 c^5 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{f^2 (12 c e-11 b f) \sqrt{a+b x+c x^2}}{4 c^4}+\frac{f^3 x \sqrt{a+b x+c x^2}}{2 c^3}+\frac{f \left (35 b^2 f^2-20 c f (3 b e+a f)+24 c^2 \left (e^2+d f\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 2.37583, size = 872, normalized size = 0.98 \[ \frac{-105 f^3 x^2 b^7-10 f^2 x (21 a f+2 c x (7 f x-9 e)) b^6-3 f \left (35 a^2 f^2-10 a c x (12 e+23 f x) f+c^2 x^2 \left (24 e^2-80 f x e+7 f^2 x^2+24 d f\right )\right ) b^5+6 c f \left (c^2 \left (-16 e^2+6 f x e+f^2 x^2-16 d f\right ) x^3-6 a c \left (4 e^2+30 f x e-31 f^2 x^2+4 d f\right ) x+5 a^2 f (6 e+53 f x)\right ) b^4-8 c \left (\left (d^3+9 x (e-f x) d^2-3 e x^2 (3 e+2 f x) d-e^3 x^3\right ) c^3-3 a f x^2 \left (18 e^2-74 f x e+f \left (7 f x^2+18 d\right )\right ) c^2+3 a^2 f \left (3 e^2+105 f x e+f \left (29 f x^2+3 d\right )\right ) c-95 a^3 f^3\right ) b^3-48 c^2 \left (f^2 (25 e+63 f x) a^3+c f x \left (-21 e^2-12 f x e+7 f \left (7 f x^2-3 d\right )\right ) a^2+c^2 \left ((e-6 f x) d^2-2 x \left (3 e^2-3 f x e+7 f^2 x^2\right ) d+x^2 \left (e^3-14 f x e^2+6 f^2 x^2 e+f^3 x^3\right )\right ) a-c^3 d x \left (d^2+x (f x-6 e) d+e^2 x^2\right )\right ) b^2-48 c^2 \left (27 f^3 a^4-2 c f \left (5 e^2+39 f x e+f \left (5 d-14 f x^2\right )\right ) a^3+c^2 \left (7 f^3 x^4-64 e f^2 x^3+4 e^3 x-4 d^2 f-4 d e (e-6 f x)\right ) a^2-2 c^3 \left (d^3+3 x (f x-e) d^2+3 e x^2 (e-2 f x) d-e^3 x^3\right ) a-4 c^4 d^2 x^2 (d-e x)\right ) b+32 c^3 \left (3 f^2 (16 e+5 f x) a^4-2 c \left (2 e^3+9 f x e^2+12 f \left (d-3 f x^2\right ) e+f^2 x \left (9 d-10 f x^2\right )\right ) a^3-3 c^2 \left (2 e d^2+4 f x^2 (3 e+2 f x) d+x^2 \left (2 e^3+8 f x e^2-6 f^2 x^2 e-f^3 x^3\right )\right ) a^2+6 c^3 d x \left (d^2+f x^2 d+e^2 x^2\right ) a+4 c^4 d^3 x^3\right )}{12 c^4 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}+\frac{f \left (24 \left (e^2+d f\right ) c^2-20 f (3 b e+a f) c+35 b^2 f^2\right ) \log \left (b+2 c x+2 \sqrt{c} \sqrt{a+x (b+c x)}\right )}{8 c^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-105*b^7*f^3*x^2 - 10*b^6*f^2*x*(21*a*f + 2*c*x*(-9*e + 7*f*x)) + 6*b^4*c*f*(5*a^2*f*(6*e + 53*f*x) - 6*a*c*x
*(4*e^2 + 4*d*f + 30*e*f*x - 31*f^2*x^2) + c^2*x^3*(-16*e^2 - 16*d*f + 6*e*f*x + f^2*x^2)) - 3*b^5*f*(35*a^2*f
^2 - 10*a*c*f*x*(12*e + 23*f*x) + c^2*x^2*(24*e^2 + 24*d*f - 80*e*f*x + 7*f^2*x^2)) - 48*b*c^2*(27*a^4*f^3 - 4
*c^4*d^2*x^2*(d - e*x) + a^2*c^2*(-4*d^2*f + 4*e^3*x - 64*e*f^2*x^3 + 7*f^3*x^4 - 4*d*e*(e - 6*f*x)) - 2*a*c^3
*(d^3 - e^3*x^3 + 3*d*e*x^2*(e - 2*f*x) + 3*d^2*x*(-e + f*x)) - 2*a^3*c*f*(5*e^2 + 39*e*f*x + f*(5*d - 14*f*x^
2))) - 8*b^3*c*(-95*a^3*f^3 + c^3*(d^3 - e^3*x^3 + 9*d^2*x*(e - f*x) - 3*d*e*x^2*(3*e + 2*f*x)) - 3*a*c^2*f*x^
2*(18*e^2 - 74*e*f*x + f*(18*d + 7*f*x^2)) + 3*a^2*c*f*(3*e^2 + 105*e*f*x + f*(3*d + 29*f*x^2))) + 32*c^3*(4*c
^4*d^3*x^3 + 3*a^4*f^2*(16*e + 5*f*x) + 6*a*c^3*d*x*(d^2 + e^2*x^2 + d*f*x^2) - 2*a^3*c*(2*e^3 + 9*e^2*f*x + f
^2*x*(9*d - 10*f*x^2) + 12*e*f*(d - 3*f*x^2)) - 3*a^2*c^2*(2*d^2*e + 4*d*f*x^2*(3*e + 2*f*x) + x^2*(2*e^3 + 8*
e^2*f*x - 6*e*f^2*x^2 - f^3*x^3))) - 48*b^2*c^2*(a^3*f^2*(25*e + 63*f*x) - c^3*d*x*(d^2 + e^2*x^2 + d*x*(-6*e
+ f*x)) + a^2*c*f*x*(-21*e^2 - 12*e*f*x + 7*f*(-3*d + 7*f*x^2)) + a*c^2*(d^2*(e - 6*f*x) - 2*d*x*(3*e^2 - 3*e*
f*x + 7*f^2*x^2) + x^2*(e^3 - 14*e^2*f*x + 6*e*f^2*x^2 + f^3*x^3))))/(12*c^4*(b^2 - 4*a*c)^2*(a + x*(b + c*x))
^(3/2)) + (f*(35*b^2*f^2 - 20*c*f*(3*b*e + a*f) + 24*c^2*(e^2 + d*f))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b
+ c*x)]])/(8*c^(9/2))

________________________________________________________________________________________

Maple [B]  time = 0.071, size = 4635, normalized size = 5.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*e^2*f+3/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d*f^2+3/c^2*b^
2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*e^2*f+1/4*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*d*e*f-6*b/c*a/(4*a*c-b^2
)/(c*x^2+b*x+a)^(3/2)*x*d*e*f+8*e*f^2*a^2/c^3/(c*x^2+b*x+a)^(3/2)+3/2*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2
)*x*d*f^2+3/2*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*e^2*f+96*e*f^2*a^2/c*b/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(
1/2)*x-19/4*e*f^2*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+12*e*f^2*a^2/c^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/
2)*x+4*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*d*e*f+12*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*d*f^2+12
*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*e^2*f-38*e*f^2*b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-48*b
*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*d*e*f-24*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*d*e*f-3*b^2/c^2*a/(4
*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*d*e*f+1/2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*d*e*f+3/4*b^3/c^3*a/(4*a*c-b
^2)/(c*x^2+b*x+a)^(3/2)*e^2*f+6*b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*d*f^2-8*d^2*e*b^2/(4*a*c-b^2)^2/(c
*x^2+b*x+a)^(1/2)+3*e*f^2*x^4/c/(c*x^2+b*x+a)^(3/2)-15/2*e*f^2*b/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^
(1/2))-15/4*e*f^2*b^2/c^4/(c*x^2+b*x+a)^(1/2)+5/32*e*f^2*b^4/c^5/(c*x^2+b*x+a)^(3/2)-3/2*b/c^2*x/(c*x^2+b*x+a)
^(3/2)*d*e*f-16*d^2*e*b*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-b/c*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*e^3+5/16
*e*f^2*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+5/2*e*f^2*b^5/c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+3*e*f^2
*a/c^3*b*x/(c*x^2+b*x+a)^(3/2)+23*f^3*b^4/c^3*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+3/c^(5/2)*ln((1/2*b+c*x)/c
^(1/2)+(c*x^2+b*x+a)^(1/2))*d*f^2+3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^2*f-x^2/c/(c*x^2+b*x
+a)^(3/2)*e^3+1/24*b^2/c^3/(c*x^2+b*x+a)^(3/2)*e^3-2/3*a/c^2/(c*x^2+b*x+a)^(3/2)*e^3-d^2*e/c/(c*x^2+b*x+a)^(3/
2)+2/3*d^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b+35/8*f^3*b^2/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+
1/2*f^3*x^5/c/(c*x^2+b*x+a)^(3/2)-5/2*f^3*a/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+35/16*f^3*b^3/
c^5/(c*x^2+b*x+a)^(1/2)-35/384*f^3*b^5/c^6/(c*x^2+b*x+a)^(3/2)+16/3*d^3*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b-
3/2*x/c/(c*x^2+b*x+a)^(3/2)*f*d^2-3/2*x/c/(c*x^2+b*x+a)^(3/2)*d*e^2+1/4*b/c^2/(c*x^2+b*x+a)^(3/2)*f*d^2+1/4*b/
c^2/(c*x^2+b*x+a)^(3/2)*d*e^2+173/96*f^3*b^3/c^5*a/(c*x^2+b*x+a)^(3/2)-35/8*f^3*b^2/c^4*x/(c*x^2+b*x+a)^(1/2)+
5/2*f^3*a/c^3*x/(c*x^2+b*x+a)^(1/2)-5/4*f^3*a/c^4*b/(c*x^2+b*x+a)^(1/2)+5/6*f^3*a/c^2*x^3/(c*x^2+b*x+a)^(3/2)-
11/2*f^3*b/c^4*a^2/(c*x^2+b*x+a)^(3/2)+3/2/c^3*b/(c*x^2+b*x+a)^(1/2)*e^2*f-x^3/c/(c*x^2+b*x+a)^(3/2)*d*f^2-x^3
/c/(c*x^2+b*x+a)^(3/2)*e^2*f-1/16*b^3/c^4/(c*x^2+b*x+a)^(3/2)*d*f^2-1/16*b^3/c^4/(c*x^2+b*x+a)^(3/2)*e^2*f-3/c
^2*x/(c*x^2+b*x+a)^(1/2)*d*f^2-3/c^2*x/(c*x^2+b*x+a)^(1/2)*e^2*f+3/2/c^3*b/(c*x^2+b*x+a)^(1/2)*d*f^2-7/4*f^3*b
/c^2*x^4/(c*x^2+b*x+a)^(3/2)+35/16*f^3*b^5/c^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-35/24*f^3*b^2/c^3*x^3/(c*x^2+b*
x+a)^(3/2)+35/16*f^3*b^3/c^4*x^2/(c*x^2+b*x+a)^(3/2)+35/64*f^3*b^4/c^5*x/(c*x^2+b*x+a)^(3/2)-35/384*f^3*b^7/c^
6/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-35/48*f^3*b^7/c^5/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-1/4*b/c^2*x/(c*x^2+b*x+a
)^(3/2)*e^3+1/24*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*e^3+1/3*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*e^3
+4/3*d^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*c+32/3*d^3*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+1/2*b^2/c/(4*a*c
-b^2)/(c*x^2+b*x+a)^(3/2)*x*d*e^2+a/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b*f*d^2+2*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b
*x+a)^(1/2)*d*e*f-5/2*f^3*a/c^3*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+1/2*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2
)*x*f*d^2+a/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b*d*e^2-33/4*f^3*b^2/c^3*a^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-6
6*f^3*b^2/c^2*a^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+23/8*f^3*b^4/c^4*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+6*e
*f^2*a^2/c^3*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+48*e*f^2*a^2/c^2*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-15/2*e
*f^2*b^3/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-1/8*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*d*f^2-1/8*b^4/c^3
/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*e^2*f-b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*d*f^2-b^4/c^2/(4*a*c-b^2)
^2/(c*x^2+b*x+a)^(1/2)*x*e^2*f+3/4*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*d*f^2-19/8*e*f^2*b^4/c^4*a/(4*a*c
-b^2)/(c*x^2+b*x+a)^(3/2)-19*e*f^2*b^4/c^3*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+16*a*c/(4*a*c-b^2)^2/(c*x^2+b*x
+a)^(1/2)*x*f*d^2+16*a*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*d*e^2+3/2*b/c^2*x^2/(c*x^2+b*x+a)^(3/2)*d*f^2+3/2
*b/c^2*x^2/(c*x^2+b*x+a)^(3/2)*e^2*f+3/8*b^2/c^3*x/(c*x^2+b*x+a)^(3/2)*d*f^2+3/8*b^2/c^3*x/(c*x^2+b*x+a)^(3/2)
*e^2*f+5/4*e*f^2*b^6/c^4/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-3*e*f^2*b^2/c^4*a/(c*x^2+b*x+a)^(3/2)+15/2*e*f^2*b/
c^3*x/(c*x^2+b*x+a)^(1/2)+35/8*f^3*b^4/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-1/2*b^5/c^3/(4*a*c-b^2)^2/(c*x^2+
b*x+a)^(1/2)*d*f^2-1/2*b^5/c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*e^2*f+b/c^3*a/(c*x^2+b*x+a)^(3/2)*d*f^2+b/c^3
*a/(c*x^2+b*x+a)^(3/2)*e^2*f+3/2/c^3*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d*f^2+3/2/c^3*b^3/(4*a*c-b^2)/(c*x^2+
b*x+a)^(1/2)*e^2*f-6*x^2/c/(c*x^2+b*x+a)^(3/2)*d*e*f+1/4*b^2/c^3/(c*x^2+b*x+a)^(3/2)*d*e*f+1/12*b^3/c^2/(4*a*c
-b^2)/(c*x^2+b*x+a)^(3/2)*x*e^3+2/3*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*e^3-1/2*b^2/c^2*a/(4*a*c-b^2)/(c
*x^2+b*x+a)^(3/2)*e^3-8*b*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*e^3-4*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2
)*e^3-4*a/c^2/(c*x^2+b*x+a)^(3/2)*d*e*f-15/4*e*f^2*b^4/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+12*e*f^2*a/c^2*x^2/
(c*x^2+b*x+a)^(3/2)+5/2*e*f^2*b/c^2*x^3/(c*x^2+b*x+a)^(3/2)-15/4*e*f^2*b^2/c^3*x^2/(c*x^2+b*x+a)^(3/2)-15/16*e
*f^2*b^3/c^4*x/(c*x^2+b*x+a)^(3/2)+5/32*e*f^2*b^6/c^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-d^2*e*b^2/c/(4*a*c-b^2)/
(c*x^2+b*x+a)^(3/2)+23/2*f^3*b^5/c^4*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-33/8*f^3*b^3/c^4*a^2/(4*a*c-b^2)/(c*x
^2+b*x+a)^(3/2)-33*f^3*b^3/c^3*a^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-5/4*f^3*a/c^4*b^3/(4*a*c-b^2)/(c*x^2+b*x+
a)^(1/2)-33/16*f^3*b^2/c^4*a*x/(c*x^2+b*x+a)^(3/2)-1/16*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*d*f^2-1/16*b^5
/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*e^2*f-33/4*f^3*b/c^3*a*x^2/(c*x^2+b*x+a)^(3/2)-35/192*f^3*b^6/c^5/(4*a*c-
b^2)/(c*x^2+b*x+a)^(3/2)*x-35/24*f^3*b^6/c^4/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+23/16*f^3*b^5/c^5*a/(4*a*c-b^
2)/(c*x^2+b*x+a)^(3/2)+1/4*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*f*d^2+1/4*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)
^(3/2)*d*e^2+4*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*f*d^2+4*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*d*e^2+2
*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*f*d^2+2*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*d*e^2+2*a/(4*a*c-b^2)
/(c*x^2+b*x+a)^(3/2)*x*f*d^2+2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*d*e^2+8*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)
*b*f*d^2+8*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b*d*e^2-2*d^2*e*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)^3/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 48.3546, size = 8338, normalized size = 9.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)^3/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/48*(3*((24*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*e^2*f + 5*(7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b^2*c^4 - 64
*a^3*c^5)*f^3 + 12*(2*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d - 5*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*e)*f^2
)*x^4 + 24*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*e^2*f + 5*(7*a^2*b^6 - 60*a^3*b^4*c + 144*a^4*b^2*c^2 -
64*a^5*c^3)*f^3 + 2*(24*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*e^2*f + 5*(7*b^7*c - 60*a*b^5*c^2 + 144*a^2*b^3
*c^3 - 64*a^3*b*c^4)*f^3 + 12*(2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d - 5*(b^6*c^2 - 8*a*b^4*c^3 + 16*a^2*
b^2*c^4)*e)*f^2)*x^3 + 12*(2*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d - 5*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*
a^4*b*c^3)*e)*f^2 + (24*(b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*e^2*f + 5*(7*b^8 - 46*a*b^6*c + 24*a^2*b^4*c^2 +
224*a^3*b^2*c^3 - 128*a^4*c^4)*f^3 + 12*(2*(b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d - 5*(b^7*c - 6*a*b^5*c^2 + 3
2*a^3*b*c^4)*e)*f^2)*x^2 + 2*(24*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*e^2*f + 5*(7*a*b^7 - 60*a^2*b^5*c
+ 144*a^3*b^3*c^2 - 64*a^4*b*c^3)*f^3 + 12*(2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d - 5*(a*b^6*c - 8*a^
2*b^4*c^2 + 16*a^3*b^2*c^3)*e)*f^2)*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*(192*a^2*b*c^5*d*e^2 - 128*a^3*c^5*e^3 + 6*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*f^3
*x^5 + 3*(12*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*e*f^2 - 7*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*f^3)*x^4 -
8*(b^3*c^5 - 12*a*b*c^6)*d^3 - 48*(a*b^2*c^5 + 4*a^2*c^6)*d^2*e - (105*a^2*b^5*c - 760*a^3*b^3*c^2 + 1296*a^4*
b*c^3)*f^3 + 4*(32*c^8*d^3 - 48*b*c^7*d^2*e + 12*(b^2*c^6 + 4*a*c^7)*d*e^2 + 2*(b^3*c^5 - 12*a*b*c^6)*e^3 - (3
5*b^6*c^2 - 279*a*b^4*c^3 + 588*a^2*b^2*c^4 - 160*a^3*c^5)*f^3 - 12*(2*(b^4*c^4 - 7*a*b^2*c^5 + 8*a^2*c^6)*d -
(5*b^5*c^3 - 37*a*b^3*c^4 + 64*a^2*b*c^5)*e)*f^2 + 12*((b^2*c^6 + 4*a*c^7)*d^2 + (b^3*c^5 - 12*a*b*c^6)*d*e -
2*(b^4*c^4 - 7*a*b^2*c^5 + 8*a^2*c^6)*e^2)*f)*x^3 - 12*(2*(3*a^2*b^3*c^3 - 20*a^3*b*c^4)*d - (15*a^2*b^4*c^2
- 100*a^3*b^2*c^3 + 128*a^4*c^4)*e)*f^2 + 3*(64*b*c^7*d^3 - 96*b^2*c^6*d^2*e + 24*(b^3*c^5 + 4*a*b*c^6)*d*e^2
- 16*(a*b^2*c^5 + 4*a^2*c^6)*e^3 - (35*b^7*c - 230*a*b^5*c^2 + 232*a^2*b^3*c^3 + 448*a^3*b*c^4)*f^3 - 12*(2*(b
^5*c^3 - 6*a*b^3*c^4)*d - (5*b^6*c^2 - 30*a*b^4*c^3 + 16*a^2*b^2*c^4 + 64*a^3*c^5)*e)*f^2 + 24*((b^3*c^5 + 4*a
*b*c^6)*d^2 - 4*(a*b^2*c^5 + 4*a^2*c^6)*d*e - (b^5*c^3 - 6*a*b^3*c^4)*e^2)*f)*x^2 + 24*(8*a^2*b*c^5*d^2 - 32*a
^3*c^5*d*e - (3*a^2*b^3*c^3 - 20*a^3*b*c^4)*e^2)*f + 6*(48*a*b^2*c^5*d*e^2 - 32*a^2*b*c^5*e^3 + 8*(b^2*c^6 + 4
*a*c^7)*d^3 - 12*(b^3*c^5 + 4*a*b*c^6)*d^2*e - (35*a*b^6*c - 265*a^2*b^4*c^2 + 504*a^3*b^2*c^3 - 80*a^4*c^4)*f
^3 - 12*(2*(a*b^4*c^3 - 7*a^2*b^2*c^4 + 4*a^3*c^5)*d - (5*a*b^5*c^2 - 35*a^2*b^3*c^3 + 52*a^3*b*c^4)*e)*f^2 +
24*(2*a*b^2*c^5*d^2 - 8*a^2*b*c^5*d*e - (a*b^4*c^3 - 7*a^2*b^2*c^4 + 4*a^3*c^5)*e^2)*f)*x)*sqrt(c*x^2 + b*x +
a))/(a^2*b^4*c^5 - 8*a^3*b^2*c^6 + 16*a^4*c^7 + (b^4*c^7 - 8*a*b^2*c^8 + 16*a^2*c^9)*x^4 + 2*(b^5*c^6 - 8*a*b^
3*c^7 + 16*a^2*b*c^8)*x^3 + (b^6*c^5 - 6*a*b^4*c^6 + 32*a^3*c^8)*x^2 + 2*(a*b^5*c^5 - 8*a^2*b^3*c^6 + 16*a^3*b
*c^7)*x), -1/24*(3*((24*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*e^2*f + 5*(7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b^2
*c^4 - 64*a^3*c^5)*f^3 + 12*(2*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d - 5*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^
5)*e)*f^2)*x^4 + 24*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*e^2*f + 5*(7*a^2*b^6 - 60*a^3*b^4*c + 144*a^4*b
^2*c^2 - 64*a^5*c^3)*f^3 + 2*(24*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*e^2*f + 5*(7*b^7*c - 60*a*b^5*c^2 + 14
4*a^2*b^3*c^3 - 64*a^3*b*c^4)*f^3 + 12*(2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d - 5*(b^6*c^2 - 8*a*b^4*c^3
+ 16*a^2*b^2*c^4)*e)*f^2)*x^3 + 12*(2*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d - 5*(a^2*b^5*c - 8*a^3*b^3*
c^2 + 16*a^4*b*c^3)*e)*f^2 + (24*(b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*e^2*f + 5*(7*b^8 - 46*a*b^6*c + 24*a^2*b
^4*c^2 + 224*a^3*b^2*c^3 - 128*a^4*c^4)*f^3 + 12*(2*(b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d - 5*(b^7*c - 6*a*b^
5*c^2 + 32*a^3*b*c^4)*e)*f^2)*x^2 + 2*(24*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*e^2*f + 5*(7*a*b^7 - 60*a
^2*b^5*c + 144*a^3*b^3*c^2 - 64*a^4*b*c^3)*f^3 + 12*(2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d - 5*(a*b^6
*c - 8*a^2*b^4*c^2 + 16*a^3*b^2*c^3)*e)*f^2)*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*(192*a^2*b*c^5*d*e^2 - 128*a^3*c^5*e^3 + 6*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*
f^3*x^5 + 3*(12*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*e*f^2 - 7*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*f^3)*x^4
- 8*(b^3*c^5 - 12*a*b*c^6)*d^3 - 48*(a*b^2*c^5 + 4*a^2*c^6)*d^2*e - (105*a^2*b^5*c - 760*a^3*b^3*c^2 + 1296*a
^4*b*c^3)*f^3 + 4*(32*c^8*d^3 - 48*b*c^7*d^2*e + 12*(b^2*c^6 + 4*a*c^7)*d*e^2 + 2*(b^3*c^5 - 12*a*b*c^6)*e^3 -
(35*b^6*c^2 - 279*a*b^4*c^3 + 588*a^2*b^2*c^4 - 160*a^3*c^5)*f^3 - 12*(2*(b^4*c^4 - 7*a*b^2*c^5 + 8*a^2*c^6)*
d - (5*b^5*c^3 - 37*a*b^3*c^4 + 64*a^2*b*c^5)*e)*f^2 + 12*((b^2*c^6 + 4*a*c^7)*d^2 + (b^3*c^5 - 12*a*b*c^6)*d*
e - 2*(b^4*c^4 - 7*a*b^2*c^5 + 8*a^2*c^6)*e^2)*f)*x^3 - 12*(2*(3*a^2*b^3*c^3 - 20*a^3*b*c^4)*d - (15*a^2*b^4*c
^2 - 100*a^3*b^2*c^3 + 128*a^4*c^4)*e)*f^2 + 3*(64*b*c^7*d^3 - 96*b^2*c^6*d^2*e + 24*(b^3*c^5 + 4*a*b*c^6)*d*e
^2 - 16*(a*b^2*c^5 + 4*a^2*c^6)*e^3 - (35*b^7*c - 230*a*b^5*c^2 + 232*a^2*b^3*c^3 + 448*a^3*b*c^4)*f^3 - 12*(2
*(b^5*c^3 - 6*a*b^3*c^4)*d - (5*b^6*c^2 - 30*a*b^4*c^3 + 16*a^2*b^2*c^4 + 64*a^3*c^5)*e)*f^2 + 24*((b^3*c^5 +
4*a*b*c^6)*d^2 - 4*(a*b^2*c^5 + 4*a^2*c^6)*d*e - (b^5*c^3 - 6*a*b^3*c^4)*e^2)*f)*x^2 + 24*(8*a^2*b*c^5*d^2 - 3
2*a^3*c^5*d*e - (3*a^2*b^3*c^3 - 20*a^3*b*c^4)*e^2)*f + 6*(48*a*b^2*c^5*d*e^2 - 32*a^2*b*c^5*e^3 + 8*(b^2*c^6
+ 4*a*c^7)*d^3 - 12*(b^3*c^5 + 4*a*b*c^6)*d^2*e - (35*a*b^6*c - 265*a^2*b^4*c^2 + 504*a^3*b^2*c^3 - 80*a^4*c^4
)*f^3 - 12*(2*(a*b^4*c^3 - 7*a^2*b^2*c^4 + 4*a^3*c^5)*d - (5*a*b^5*c^2 - 35*a^2*b^3*c^3 + 52*a^3*b*c^4)*e)*f^2
+ 24*(2*a*b^2*c^5*d^2 - 8*a^2*b*c^5*d*e - (a*b^4*c^3 - 7*a^2*b^2*c^4 + 4*a^3*c^5)*e^2)*f)*x)*sqrt(c*x^2 + b*x
+ a))/(a^2*b^4*c^5 - 8*a^3*b^2*c^6 + 16*a^4*c^7 + (b^4*c^7 - 8*a*b^2*c^8 + 16*a^2*c^9)*x^4 + 2*(b^5*c^6 - 8*a
*b^3*c^7 + 16*a^2*b*c^8)*x^3 + (b^6*c^5 - 6*a*b^4*c^6 + 32*a^3*c^8)*x^2 + 2*(a*b^5*c^5 - 8*a^2*b^3*c^6 + 16*a^
3*b*c^7)*x)]

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

[Out]

Timed out

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Giac [A]  time = 1.54095, size = 1891, normalized size = 2.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*((((3*(2*(b^4*c^3*f^3 - 8*a*b^2*c^4*f^3 + 16*a^2*c^5*f^3)*x/(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6) - (7*b^5
*c^2*f^3 - 56*a*b^3*c^3*f^3 + 112*a^2*b*c^4*f^3 - 12*b^4*c^3*f^2*e + 96*a*b^2*c^4*f^2*e - 192*a^2*c^5*f^2*e)/(
b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6))*x + 4*(32*c^7*d^3 + 12*b^2*c^5*d^2*f + 48*a*c^6*d^2*f - 24*b^4*c^3*d*f^2
+ 168*a*b^2*c^4*d*f^2 - 192*a^2*c^5*d*f^2 - 35*b^6*c*f^3 + 279*a*b^4*c^2*f^3 - 588*a^2*b^2*c^3*f^3 + 160*a^3*c
^4*f^3 - 48*b*c^6*d^2*e + 12*b^3*c^4*d*f*e - 144*a*b*c^5*d*f*e + 60*b^5*c^2*f^2*e - 444*a*b^3*c^3*f^2*e + 768*
a^2*b*c^4*f^2*e + 12*b^2*c^5*d*e^2 + 48*a*c^6*d*e^2 - 24*b^4*c^3*f*e^2 + 168*a*b^2*c^4*f*e^2 - 192*a^2*c^5*f*e
^2 + 2*b^3*c^4*e^3 - 24*a*b*c^5*e^3)/(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6))*x + 3*(64*b*c^6*d^3 + 24*b^3*c^4*d^
2*f + 96*a*b*c^5*d^2*f - 24*b^5*c^2*d*f^2 + 144*a*b^3*c^3*d*f^2 - 35*b^7*f^3 + 230*a*b^5*c*f^3 - 232*a^2*b^3*c
^2*f^3 - 448*a^3*b*c^3*f^3 - 96*b^2*c^5*d^2*e - 96*a*b^2*c^4*d*f*e - 384*a^2*c^5*d*f*e + 60*b^6*c*f^2*e - 360*
a*b^4*c^2*f^2*e + 192*a^2*b^2*c^3*f^2*e + 768*a^3*c^4*f^2*e + 24*b^3*c^4*d*e^2 + 96*a*b*c^5*d*e^2 - 24*b^5*c^2
*f*e^2 + 144*a*b^3*c^3*f*e^2 - 16*a*b^2*c^4*e^3 - 64*a^2*c^5*e^3)/(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6))*x + 6*
(8*b^2*c^5*d^3 + 32*a*c^6*d^3 + 48*a*b^2*c^4*d^2*f - 24*a*b^4*c^2*d*f^2 + 168*a^2*b^2*c^3*d*f^2 - 96*a^3*c^4*d
*f^2 - 35*a*b^6*f^3 + 265*a^2*b^4*c*f^3 - 504*a^3*b^2*c^2*f^3 + 80*a^4*c^3*f^3 - 12*b^3*c^4*d^2*e - 48*a*b*c^5
*d^2*e - 192*a^2*b*c^4*d*f*e + 60*a*b^5*c*f^2*e - 420*a^2*b^3*c^2*f^2*e + 624*a^3*b*c^3*f^2*e + 48*a*b^2*c^4*d
*e^2 - 24*a*b^4*c^2*f*e^2 + 168*a^2*b^2*c^3*f*e^2 - 96*a^3*c^4*f*e^2 - 32*a^2*b*c^4*e^3)/(b^4*c^4 - 8*a*b^2*c^
5 + 16*a^2*c^6))*x - (8*b^3*c^4*d^3 - 96*a*b*c^5*d^3 - 192*a^2*b*c^4*d^2*f + 72*a^2*b^3*c^2*d*f^2 - 480*a^3*b*
c^3*d*f^2 + 105*a^2*b^5*f^3 - 760*a^3*b^3*c*f^3 + 1296*a^4*b*c^2*f^3 + 48*a*b^2*c^4*d^2*e + 192*a^2*c^5*d^2*e
+ 768*a^3*c^4*d*f*e - 180*a^2*b^4*c*f^2*e + 1200*a^3*b^2*c^2*f^2*e - 1536*a^4*c^3*f^2*e - 192*a^2*b*c^4*d*e^2
+ 72*a^2*b^3*c^2*f*e^2 - 480*a^3*b*c^3*f*e^2 + 128*a^3*c^4*e^3)/(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6))/(c*x^2 +
b*x + a)^(3/2) - 1/8*(24*c^2*d*f^2 + 35*b^2*f^3 - 20*a*c*f^3 - 60*b*c*f^2*e + 24*c^2*f*e^2)*log(abs(-2*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(9/2)