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

Optimal. Leaf size=649 \[ \frac{2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^2 c^2 d f+b^2 c^2 e^2-2 b^3 c e f+b^4 f^2-b c^3 d e+c^4 d^2\right )-b c^2 \left (-9 a^2 c f \left (d f+e^2\right )+5 a^3 f^3+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )+3 a b^4 c e f^2-a b^5 f^3\right )}{c^5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (80 c^2 f \left (a^2 f^2+6 a b e f+3 b^2 \left (d f+e^2\right )\right )-280 b^2 c f^2 (a f+b e)-64 c^3 \left (3 a f \left (d f+e^2\right )+b \left (6 d e f+e^3\right )\right )+105 b^4 f^3+128 c^4 d \left (d f+e^2\right )\right )}{128 c^{11/2}}+\frac{f x \sqrt{a+b x+c x^2} \left (-4 c f (7 a f+22 b e)+41 b^2 f^2+48 c^2 \left (d f+e^2\right )\right )}{32 c^4}-\frac{\sqrt{a+b x+c x^2} \left (16 c^2 f \left (20 a e f+21 b \left (d f+e^2\right )\right )-4 b c f^2 (73 a f+114 b e)+187 b^3 f^3-64 c^3 \left (6 d e f+e^3\right )\right )}{64 c^5}+\frac{f^2 x^2 \sqrt{a+b x+c x^2} (8 c e-5 b f)}{8 c^3}+\frac{f^3 x^3 \sqrt{a+b x+c x^2}}{4 c^2} \]

[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))/(
c^5*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) - ((187*b^3*f^3 - 4*b*c*f^2*(114*b*e + 73*a*f) - 64*c^3*(e^3 + 6*d*e*
f) + 16*c^2*f*(20*a*e*f + 21*b*(e^2 + d*f)))*Sqrt[a + b*x + c*x^2])/(64*c^5) + (f*(41*b^2*f^2 - 4*c*f*(22*b*e
+ 7*a*f) + 48*c^2*(e^2 + d*f))*x*Sqrt[a + b*x + c*x^2])/(32*c^4) + (f^2*(8*c*e - 5*b*f)*x^2*Sqrt[a + b*x + c*x
^2])/(8*c^3) + (f^3*x^3*Sqrt[a + b*x + c*x^2])/(4*c^2) + (3*(105*b^4*f^3 - 280*b^2*c*f^2*(b*e + a*f) + 128*c^4
*d*(e^2 + d*f) + 80*c^2*f*(6*a*b*e*f + a^2*f^2 + 3*b^2*(e^2 + d*f)) - 64*c^3*(3*a*f*(e^2 + d*f) + b*(e^3 + 6*d
*e*f)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 2.10557, antiderivative size = 649, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{2 \left (-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right ) \left (a^2 c^2 f^2-4 a b^2 c f^2+7 a b c^2 e f-2 a c^3 d f-3 a c^3 e^2+b^2 c^2 d f+b^2 c^2 e^2-2 b^3 c e f+b^4 f^2-b c^3 d e+c^4 d^2\right )-b c^2 \left (-9 a^2 c f \left (d f+e^2\right )+5 a^3 f^3+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )+2 a c^3 e \left (3 a^2 f^2-a c \left (6 d f+e^2\right )+3 c^2 d^2\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )+3 a b^4 c e f^2-a b^5 f^3\right )}{c^5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (80 c^2 f \left (a^2 f^2+6 a b e f+3 b^2 \left (d f+e^2\right )\right )-280 b^2 c f^2 (a f+b e)-64 c^3 \left (3 a f \left (d f+e^2\right )+b \left (6 d e f+e^3\right )\right )+105 b^4 f^3+128 c^4 d \left (d f+e^2\right )\right )}{128 c^{11/2}}+\frac{f x \sqrt{a+b x+c x^2} \left (-4 c f (7 a f+22 b e)+41 b^2 f^2+48 c^2 \left (d f+e^2\right )\right )}{32 c^4}-\frac{\sqrt{a+b x+c x^2} \left (16 c^2 f \left (20 a e f+21 b \left (d f+e^2\right )\right )-4 b c f^2 (73 a f+114 b e)+187 b^3 f^3-64 c^3 \left (6 d e f+e^3\right )\right )}{64 c^5}+\frac{f^2 x^2 \sqrt{a+b x+c x^2} (8 c e-5 b f)}{8 c^3}+\frac{f^3 x^3 \sqrt{a+b x+c x^2}}{4 c^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x + f*x^2)^3/(a + b*x + c*x^2)^(3/2),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))/(
c^5*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) - ((187*b^3*f^3 - 4*b*c*f^2*(114*b*e + 73*a*f) - 64*c^3*(e^3 + 6*d*e*
f) + 16*c^2*f*(20*a*e*f + 21*b*(e^2 + d*f)))*Sqrt[a + b*x + c*x^2])/(64*c^5) + (f*(41*b^2*f^2 - 4*c*f*(22*b*e
+ 7*a*f) + 48*c^2*(e^2 + d*f))*x*Sqrt[a + b*x + c*x^2])/(32*c^4) + (f^2*(8*c*e - 5*b*f)*x^2*Sqrt[a + b*x + c*x
^2])/(8*c^3) + (f^3*x^3*Sqrt[a + b*x + c*x^2])/(4*c^2) + (3*(105*b^4*f^3 - 280*b^2*c*f^2*(b*e + a*f) + 128*c^4
*d*(e^2 + d*f) + 80*c^2*f*(6*a*b*e*f + a^2*f^2 + 3*b^2*(e^2 + d*f)) - 64*c^3*(3*a*f*(e^2 + d*f) + b*(e^3 + 6*d
*e*f)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(11/2))

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 )^{3/2}} \, 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 )}{c^5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{-\frac{\left (b^2-4 a c\right ) \left (b^4 f^3-3 b^2 c f^2 (b e+a f)+3 c^4 d \left (e^2+d f\right )+c^2 f \left (6 a b e f+a^2 f^2+3 b^2 \left (e^2+d f\right )\right )-c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d e f\right )\right )\right )}{2 c^5}+\frac{\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{\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{\left (b^2-4 a c\right ) f^2 (3 c e-b f) x^3}{2 c^2}-\frac{\left (b^2-4 a c\right ) f^3 x^4}{2 c}}{\sqrt{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=\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 )}{c^5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f^3 x^3 \sqrt{a+b x+c x^2}}{4 c^2}-\frac{\int \frac{-\frac{2 \left (b^2-4 a c\right ) \left (b^4 f^3-3 b^2 c f^2 (b e+a f)+3 c^4 d \left (e^2+d f\right )+c^2 f \left (6 a b e f+a^2 f^2+3 b^2 \left (e^2+d f\right )\right )-c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d e f\right )\right )\right )}{c^4}+\frac{2 \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}{c^3}-\frac{\left (b^2-4 a c\right ) f \left (4 b^2 f^2-c f (12 b e+7 a f)+12 c^2 \left (e^2+d f\right )\right ) x^2}{2 c^2}-\frac{3 \left (b^2-4 a c\right ) f^2 (8 c e-5 b f) x^3}{4 c}}{\sqrt{a+b x+c x^2}} \, dx}{2 c \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 )}{c^5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f^2 (8 c e-5 b f) x^2 \sqrt{a+b x+c x^2}}{8 c^3}+\frac{f^3 x^3 \sqrt{a+b x+c x^2}}{4 c^2}-\frac{\int \frac{-\frac{6 \left (b^2-4 a c\right ) \left (b^4 f^3-3 b^2 c f^2 (b e+a f)+3 c^4 d \left (e^2+d f\right )+c^2 f \left (6 a b e f+a^2 f^2+3 b^2 \left (e^2+d f\right )\right )-c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d e f\right )\right )\right )}{c^3}+\frac{3 \left (b^2-4 a c\right ) \left (4 b^3 f^3-b c f^2 (12 b e+13 a f)-4 c^3 \left (e^3+6 d e f\right )+4 c^2 f \left (5 a e f+3 b \left (e^2+d f\right )\right )\right ) x}{2 c^2}-\frac{3 \left (b^2-4 a c\right ) f \left (41 b^2 f^2-4 c f (22 b e+7 a f)+48 c^2 \left (e^2+d f\right )\right ) x^2}{8 c}}{\sqrt{a+b x+c x^2}} \, dx}{6 c^2 \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 )}{c^5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f \left (41 b^2 f^2-4 c f (22 b e+7 a f)+48 c^2 \left (e^2+d f\right )\right ) x \sqrt{a+b x+c x^2}}{32 c^4}+\frac{f^2 (8 c e-5 b f) x^2 \sqrt{a+b x+c x^2}}{8 c^3}+\frac{f^3 x^3 \sqrt{a+b x+c x^2}}{4 c^2}-\frac{\int \frac{-\frac{3 \left (b^2-4 a c\right ) \left (32 b^4 f^3-b^2 c f^2 (96 b e+137 a f)+96 c^4 d \left (e^2+d f\right )+4 c^2 f \left (70 a b e f+15 a^2 f^2+24 b^2 \left (e^2+d f\right )\right )-16 c^3 \left (9 a f \left (e^2+d f\right )+2 b \left (e^3+6 d e f\right )\right )\right )}{8 c^2}+\frac{3 \left (b^2-4 a c\right ) \left (187 b^3 f^3-4 b c f^2 (114 b e+73 a f)-64 c^3 \left (e^3+6 d e f\right )+16 c^2 f \left (20 a e f+21 b \left (e^2+d f\right )\right )\right ) x}{16 c}}{\sqrt{a+b x+c x^2}} \, dx}{12 c^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 )}{c^5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{\left (187 b^3 f^3-4 b c f^2 (114 b e+73 a f)-64 c^3 \left (e^3+6 d e f\right )+16 c^2 f \left (20 a e f+21 b \left (e^2+d f\right )\right )\right ) \sqrt{a+b x+c x^2}}{64 c^5}+\frac{f \left (41 b^2 f^2-4 c f (22 b e+7 a f)+48 c^2 \left (e^2+d f\right )\right ) x \sqrt{a+b x+c x^2}}{32 c^4}+\frac{f^2 (8 c e-5 b f) x^2 \sqrt{a+b x+c x^2}}{8 c^3}+\frac{f^3 x^3 \sqrt{a+b x+c x^2}}{4 c^2}+\frac{\left (3 \left (105 b^4 f^3-280 b^2 c f^2 (b e+a f)+128 c^4 d \left (e^2+d f\right )+80 c^2 f \left (6 a b e f+a^2 f^2+3 b^2 \left (e^2+d f\right )\right )-64 c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d e f\right )\right )\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{128 c^5}\\ &=\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 )}{c^5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{\left (187 b^3 f^3-4 b c f^2 (114 b e+73 a f)-64 c^3 \left (e^3+6 d e f\right )+16 c^2 f \left (20 a e f+21 b \left (e^2+d f\right )\right )\right ) \sqrt{a+b x+c x^2}}{64 c^5}+\frac{f \left (41 b^2 f^2-4 c f (22 b e+7 a f)+48 c^2 \left (e^2+d f\right )\right ) x \sqrt{a+b x+c x^2}}{32 c^4}+\frac{f^2 (8 c e-5 b f) x^2 \sqrt{a+b x+c x^2}}{8 c^3}+\frac{f^3 x^3 \sqrt{a+b x+c x^2}}{4 c^2}+\frac{\left (3 \left (105 b^4 f^3-280 b^2 c f^2 (b e+a f)+128 c^4 d \left (e^2+d f\right )+80 c^2 f \left (6 a b e f+a^2 f^2+3 b^2 \left (e^2+d f\right )\right )-64 c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d e f\right )\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 )}{64 c^5}\\ &=\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 )}{c^5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{\left (187 b^3 f^3-4 b c f^2 (114 b e+73 a f)-64 c^3 \left (e^3+6 d e f\right )+16 c^2 f \left (20 a e f+21 b \left (e^2+d f\right )\right )\right ) \sqrt{a+b x+c x^2}}{64 c^5}+\frac{f \left (41 b^2 f^2-4 c f (22 b e+7 a f)+48 c^2 \left (e^2+d f\right )\right ) x \sqrt{a+b x+c x^2}}{32 c^4}+\frac{f^2 (8 c e-5 b f) x^2 \sqrt{a+b x+c x^2}}{8 c^3}+\frac{f^3 x^3 \sqrt{a+b x+c x^2}}{4 c^2}+\frac{3 \left (105 b^4 f^3-280 b^2 c f^2 (b e+a f)+128 c^4 d \left (e^2+d f\right )+80 c^2 f \left (6 a b e f+a^2 f^2+3 b^2 \left (e^2+d f\right )\right )-64 c^3 \left (3 a f \left (e^2+d f\right )+b \left (e^3+6 d e f\right )\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{11/2}}\\ \end{align*}

Mathematica [A]  time = 1.6847, size = 745, normalized size = 1.15 \[ \frac{-16 b^2 c^2 \left (-a^2 f^2 (230 e+169 f x)+a c \left (2 e f \left (36 d-43 f x^2\right )+f^2 x \left (186 d-13 f x^2\right )+186 e^2 f x+12 e^3\right )+c^2 x \left (-24 d^2 f+6 d \left (-4 e^2+4 e f x+f^2 x^2\right )+x \left (6 e^2 f x+4 e^3+4 e f^2 x^2+f^3 x^3\right )\right )\right )-8 b^3 c \left (210 a^2 f^3+a c f \left (f \left (77 f x^2-90 d\right )-90 e^2-530 e f x\right )-c^2 x \left (2 e f \left (7 f x^2-72 d\right )+3 f^2 x \left (10 d+f x^2\right )+30 e^2 f x-24 e^3\right )\right )+16 b c^2 \left (a^2 c f \left (f \left (49 f x^2-156 d\right )-156 e^2-244 e f x\right )+113 a^3 f^3+2 a c^2 \left (12 d^2 f+6 d \left (2 e^2+20 e f x-5 f^2 x^2\right )-x \left (30 e^2 f x-20 e^3+14 e f^2 x^2+3 f^3 x^3\right )\right )+8 c^3 d^2 (d-3 e x)\right )+32 c^3 \left (a^2 c \left (-32 e f \left (f x^2-3 d\right )+f^2 x \left (36 d-5 f x^2\right )+36 e^2 f x+16 e^3\right )+a^3 \left (-f^2\right ) (64 e+15 f x)+2 a c^2 \left (-12 d^2 (e+f x)+6 d x \left (-2 e^2+4 e f x+f^2 x^2\right )+x^2 \left (6 e^2 f x+4 e^3+4 e f^2 x^2+f^3 x^3\right )\right )+8 c^3 d^3 x\right )-2 b^4 c f \left (105 a f (4 e+9 f x)+c x \left (-360 d f-360 e^2+140 e f x+21 f^2 x^2\right )\right )+105 b^5 f^2 (3 a f+c x (f x-8 e))+315 b^6 f^3 x}{64 c^5 \left (4 a c-b^2\right ) \sqrt{a+x (b+c x)}}+\frac{3 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (80 c^2 f \left (a^2 f^2+6 a b e f+3 b^2 \left (d f+e^2\right )\right )-280 b^2 c f^2 (a f+b e)-64 c^3 \left (3 a f \left (d f+e^2\right )+b \left (6 d e f+e^3\right )\right )+105 b^4 f^3+128 c^4 d \left (d f+e^2\right )\right )}{128 c^{11/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(315*b^6*f^3*x + 105*b^5*f^2*(3*a*f + c*x*(-8*e + f*x)) - 2*b^4*c*f*(105*a*f*(4*e + 9*f*x) + c*x*(-360*e^2 - 3
60*d*f + 140*e*f*x + 21*f^2*x^2)) - 8*b^3*c*(210*a^2*f^3 - c^2*x*(-24*e^3 + 30*e^2*f*x + 3*f^2*x*(10*d + f*x^2
) + 2*e*f*(-72*d + 7*f*x^2)) + a*c*f*(-90*e^2 - 530*e*f*x + f*(-90*d + 77*f*x^2))) - 16*b^2*c^2*(-(a^2*f^2*(23
0*e + 169*f*x)) + a*c*(12*e^3 + 186*e^2*f*x + 2*e*f*(36*d - 43*f*x^2) + f^2*x*(186*d - 13*f*x^2)) + c^2*x*(-24
*d^2*f + 6*d*(-4*e^2 + 4*e*f*x + f^2*x^2) + x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))) + 32*c^3*(8*c^3*d^
3*x - a^3*f^2*(64*e + 15*f*x) + a^2*c*(16*e^3 + 36*e^2*f*x + f^2*x*(36*d - 5*f*x^2) - 32*e*f*(-3*d + f*x^2)) +
2*a*c^2*(-12*d^2*(e + f*x) + 6*d*x*(-2*e^2 + 4*e*f*x + f^2*x^2) + x^2*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*
x^3))) + 16*b*c^2*(113*a^3*f^3 + 8*c^3*d^2*(d - 3*e*x) + a^2*c*f*(-156*e^2 - 244*e*f*x + f*(-156*d + 49*f*x^2)
) + 2*a*c^2*(12*d^2*f + 6*d*(2*e^2 + 20*e*f*x - 5*f^2*x^2) - x*(-20*e^3 + 30*e^2*f*x + 14*e*f^2*x^2 + 3*f^3*x^
3))))/(64*c^5*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)]) + (3*(105*b^4*f^3 - 280*b^2*c*f^2*(b*e + a*f) + 128*c^4*d*
(e^2 + d*f) + 80*c^2*f*(6*a*b*e*f + a^2*f^2 + 3*b^2*(e^2 + d*f)) - 64*c^3*(3*a*f*(e^2 + d*f) + b*(e^3 + 6*d*e*
f)))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(128*c^(11/2))

________________________________________________________________________________________

Maple [B]  time = 0.069, size = 2827, normalized size = 4.4 \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)^(3/2),x)

[Out]

3/2*b/c^2*x/(c*x^2+b*x+a)^(1/2)*e^3-3/4*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*e^3+115/8*e*f^2*b^4/c^4*a/(4*a
*c-b^2)/(c*x^2+b*x+a)^(1/2)-45/4*e*f^2*b/c^3*a*x/(c*x^2+b*x+a)^(1/2)+113/8*f^3*b^2/c^3*a^2/(4*a*c-b^2)/(c*x^2+
b*x+a)^(1/2)*x+9*b/c^2*x/(c*x^2+b*x+a)^(1/2)*d*e*f-9/2*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d*e*f-8*e*f^2*a
^2/c^3*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+45/8*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d*f^2+45/8*b^4/c^3/(
4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*e^2*f-39/4*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d*f^2-39/4*b^3/c^3*a/(4*
a*c-b^2)/(c*x^2+b*x+a)^(1/2)*e^2*f+3*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d*e^2+4*a/c*b/(4*a*c-b^2)/(c*x^2+
b*x+a)^(1/2)*x*e^3+3*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*f*d^2+24*a/c*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*
d*e*f+6*x^2/c/(c*x^2+b*x+a)^(1/2)*d*e*f-9/2*b^2/c^3/(c*x^2+b*x+a)^(1/2)*d*e*f-3/2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b
*x+a)^(1/2)*x*e^3-9*b/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e*f+12*a/c^2/(c*x^2+b*x+a)^(1/2)*d
*e*f-39/2*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d*f^2-16*e*f^2*a^2/c^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)
*x+115/4*e*f^2*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-39/2*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*e^
2*f+12*a/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d*e*f-9*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d*e*f-105/3
2*e*f^2*b^4/c^5/(c*x^2+b*x+a)^(1/2)-105/16*e*f^2*b^3/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-8*e*f
^2*a^2/c^3/(c*x^2+b*x+a)^(1/2)-5/8*f^3*a/c^2*x^3/(c*x^2+b*x+a)^(1/2)-3/8*f^3*b/c^2*x^4/(c*x^2+b*x+a)^(1/2)+21/
32*f^3*b^2/c^3*x^3/(c*x^2+b*x+a)^(1/2)-105/64*f^3*b^3/c^4*x^2/(c*x^2+b*x+a)^(1/2)-315/128*f^3*b^4/c^5*x/(c*x^2
+b*x+a)^(1/2)+315/256*f^3*b^7/c^6/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-105/16*f^3*b^3/c^5*a/(c*x^2+b*x+a)^(1/2)+113
/16*f^3*b/c^4*a^2/(c*x^2+b*x+a)^(1/2)-105/16*f^3*b^2/c^(9/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-15/
8*f^3*a^2/c^3*x/(c*x^2+b*x+a)^(1/2)-3*x/c/(c*x^2+b*x+a)^(1/2)*f*d^2-3*x/c/(c*x^2+b*x+a)^(1/2)*d*e^2+3/2*b/c^2/
(c*x^2+b*x+a)^(1/2)*f*d^2+3/2*b/c^2/(c*x^2+b*x+a)^(1/2)*d*e^2+3/2*x^3/c/(c*x^2+b*x+a)^(1/2)*d*f^2+3/2*x^3/c/(c
*x^2+b*x+a)^(1/2)*e^2*f+45/16*b^3/c^4/(c*x^2+b*x+a)^(1/2)*d*f^2+45/16*b^3/c^4/(c*x^2+b*x+a)^(1/2)*e^2*f+45/8*b
^2/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*f^2+45/8*b^2/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*
x+a)^(1/2))*e^2*f-9/2*a/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*f^2-9/2*a/c^(5/2)*ln((1/2*b+c*x)
/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^2*f+e*f^2*x^4/c/(c*x^2+b*x+a)^(1/2)-39/4*b/c^3*a/(c*x^2+b*x+a)^(1/2)*d*f^2-39/
4*b/c^3*a/(c*x^2+b*x+a)^(1/2)*e^2*f+9/2*a/c^2*x/(c*x^2+b*x+a)^(1/2)*d*f^2+9/2*a/c^2*x/(c*x^2+b*x+a)^(1/2)*e^2*
f+115/8*e*f^2*b^2/c^4*a/(c*x^2+b*x+a)^(1/2)+45/4*e*f^2*b/c^(7/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
-4*e*f^2*a/c^2*x^2/(c*x^2+b*x+a)^(1/2)+3/2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*f*d^2+3/2*b^3/c^2/(4*a*c-b^
2)/(c*x^2+b*x+a)^(1/2)*d*e^2+45/16*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d*f^2+45/16*b^5/c^4/(4*a*c-b^2)/(c*
x^2+b*x+a)^(1/2)*e^2*f+x^2/c/(c*x^2+b*x+a)^(1/2)*e^3-3/4*b^2/c^3/(c*x^2+b*x+a)^(1/2)*e^3-3/2*b/c^(5/2)*ln((1/2
*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^3+2*a/c^2/(c*x^2+b*x+a)^(1/2)*e^3+315/256*f^3*b^5/c^6/(c*x^2+b*x+a)^(1/
2)+315/128*f^3*b^4/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/4*f^3*x^5/c/(c*x^2+b*x+a)^(1/2)+2*a/
c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*e^3+15/8*f^3*a^2/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3
/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f*d^2+3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))*d*e^2-3*d^2*e/c/(c*x^2+b*x+a)^(1/2)+2*d^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+105/16*f^3*b^2/c^4*a*x/
(c*x^2+b*x+a)^(1/2)+49/16*f^3*b/c^3*a*x^2/(c*x^2+b*x+a)^(1/2)+113/16*f^3*b^3/c^4*a^2/(4*a*c-b^2)/(c*x^2+b*x+a)
^(1/2)-15/4*b/c^2*x^2/(c*x^2+b*x+a)^(1/2)*d*f^2-15/4*b/c^2*x^2/(c*x^2+b*x+a)^(1/2)*e^2*f-45/8*b^2/c^3*x/(c*x^2
+b*x+a)^(1/2)*d*f^2-45/8*b^2/c^3*x/(c*x^2+b*x+a)^(1/2)*e^2*f-105/16*e*f^2*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1
/2)*x-105/8*f^3*b^4/c^4*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-7/4*e*f^2*b/c^2*x^3/(c*x^2+b*x+a)^(1/2)+35/8*e*f^2
*b^2/c^3*x^2/(c*x^2+b*x+a)^(1/2)+105/16*e*f^2*b^3/c^4*x/(c*x^2+b*x+a)^(1/2)-105/32*e*f^2*b^6/c^5/(4*a*c-b^2)/(
c*x^2+b*x+a)^(1/2)+315/128*f^3*b^6/c^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-105/16*f^3*b^5/c^5*a/(4*a*c-b^2)/(c*x
^2+b*x+a)^(1/2)-6*d^2*e*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-3*d^2*e*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)

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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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 15.911, size = 6743, normalized size = 10.39 \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)^(3/2),x, algorithm="fricas")

[Out]

[1/256*(3*(128*(a*b^2*c^4 - 4*a^2*c^5)*d*e^2 - 64*(a*b^3*c^3 - 4*a^2*b*c^4)*e^3 + 5*(21*a*b^6 - 140*a^2*b^4*c
+ 240*a^3*b^2*c^2 - 64*a^4*c^3)*f^3 + 8*(6*(5*a*b^4*c^2 - 24*a^2*b^2*c^3 + 16*a^3*c^4)*d - 5*(7*a*b^5*c - 40*a
^2*b^3*c^2 + 48*a^3*b*c^3)*e)*f^2 + (128*(b^2*c^5 - 4*a*c^6)*d*e^2 - 64*(b^3*c^4 - 4*a*b*c^5)*e^3 + 5*(21*b^6*
c - 140*a*b^4*c^2 + 240*a^2*b^2*c^3 - 64*a^3*c^4)*f^3 + 8*(6*(5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*c^5)*d - 5*(7*
b^5*c^2 - 40*a*b^3*c^3 + 48*a^2*b*c^4)*e)*f^2 + 16*(8*(b^2*c^5 - 4*a*c^6)*d^2 - 24*(b^3*c^4 - 4*a*b*c^5)*d*e +
3*(5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*c^5)*e^2)*f)*x^2 + 16*(8*(a*b^2*c^4 - 4*a^2*c^5)*d^2 - 24*(a*b^3*c^3 - 4
*a^2*b*c^4)*d*e + 3*(5*a*b^4*c^2 - 24*a^2*b^2*c^3 + 16*a^3*c^4)*e^2)*f + (128*(b^3*c^4 - 4*a*b*c^5)*d*e^2 - 64
*(b^4*c^3 - 4*a*b^2*c^4)*e^3 + 5*(21*b^7 - 140*a*b^5*c + 240*a^2*b^3*c^2 - 64*a^3*b*c^3)*f^3 + 8*(6*(5*b^5*c^2
- 24*a*b^3*c^3 + 16*a^2*b*c^4)*d - 5*(7*b^6*c - 40*a*b^4*c^2 + 48*a^2*b^2*c^3)*e)*f^2 + 16*(8*(b^3*c^4 - 4*a*
b*c^5)*d^2 - 24*(b^4*c^3 - 4*a*b^2*c^4)*d*e + 3*(5*b^5*c^2 - 24*a*b^3*c^3 + 16*a^2*b*c^4)*e^2)*f)*x)*sqrt(c)*l
og(-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*(128*b*c^6*d^3 - 768*
a*c^6*d^2*e + 384*a*b*c^5*d*e^2 - 16*(b^2*c^5 - 4*a*c^6)*f^3*x^5 - 8*(8*(b^2*c^5 - 4*a*c^6)*e*f^2 - 3*(b^3*c^4
- 4*a*b*c^5)*f^3)*x^4 - 64*(3*a*b^2*c^4 - 8*a^2*c^5)*e^3 + (315*a*b^5*c - 1680*a^2*b^3*c^2 + 1808*a^3*b*c^3)*
f^3 - 2*(48*(b^2*c^5 - 4*a*c^6)*e^2*f + (21*b^4*c^3 - 104*a*b^2*c^4 + 80*a^2*c^5)*f^3 + 8*(6*(b^2*c^5 - 4*a*c^
6)*d - 7*(b^3*c^4 - 4*a*b*c^5)*e)*f^2)*x^3 + 8*(6*(15*a*b^3*c^3 - 52*a^2*b*c^4)*d - (105*a*b^4*c^2 - 460*a^2*b
^2*c^3 + 256*a^3*c^4)*e)*f^2 - (64*(b^2*c^5 - 4*a*c^6)*e^3 - 7*(15*b^5*c^2 - 88*a*b^3*c^3 + 112*a^2*b*c^4)*f^3
- 8*(30*(b^3*c^4 - 4*a*b*c^5)*d - (35*b^4*c^3 - 172*a*b^2*c^4 + 128*a^2*c^5)*e)*f^2 + 48*(8*(b^2*c^5 - 4*a*c^
6)*d*e - 5*(b^3*c^4 - 4*a*b*c^5)*e^2)*f)*x^2 + 48*(8*a*b*c^5*d^2 - 8*(3*a*b^2*c^4 - 8*a^2*c^5)*d*e + (15*a*b^3
*c^3 - 52*a^2*b*c^4)*e^2)*f + (256*c^7*d^3 - 384*b*c^6*d^2*e + 384*(b^2*c^5 - 2*a*c^6)*d*e^2 - 64*(3*b^3*c^4 -
10*a*b*c^5)*e^3 + (315*b^6*c - 1890*a*b^4*c^2 + 2704*a^2*b^2*c^3 - 480*a^3*c^4)*f^3 + 8*(6*(15*b^4*c^3 - 62*a
*b^2*c^4 + 24*a^2*c^5)*d - (105*b^5*c^2 - 530*a*b^3*c^3 + 488*a^2*b*c^4)*e)*f^2 + 48*(8*(b^2*c^5 - 2*a*c^6)*d^
2 - 8*(3*b^3*c^4 - 10*a*b*c^5)*d*e + (15*b^4*c^3 - 62*a*b^2*c^4 + 24*a^2*c^5)*e^2)*f)*x)*sqrt(c*x^2 + b*x + a)
)/(a*b^2*c^6 - 4*a^2*c^7 + (b^2*c^7 - 4*a*c^8)*x^2 + (b^3*c^6 - 4*a*b*c^7)*x), -1/128*(3*(128*(a*b^2*c^4 - 4*a
^2*c^5)*d*e^2 - 64*(a*b^3*c^3 - 4*a^2*b*c^4)*e^3 + 5*(21*a*b^6 - 140*a^2*b^4*c + 240*a^3*b^2*c^2 - 64*a^4*c^3)
*f^3 + 8*(6*(5*a*b^4*c^2 - 24*a^2*b^2*c^3 + 16*a^3*c^4)*d - 5*(7*a*b^5*c - 40*a^2*b^3*c^2 + 48*a^3*b*c^3)*e)*f
^2 + (128*(b^2*c^5 - 4*a*c^6)*d*e^2 - 64*(b^3*c^4 - 4*a*b*c^5)*e^3 + 5*(21*b^6*c - 140*a*b^4*c^2 + 240*a^2*b^2
*c^3 - 64*a^3*c^4)*f^3 + 8*(6*(5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*c^5)*d - 5*(7*b^5*c^2 - 40*a*b^3*c^3 + 48*a^2
*b*c^4)*e)*f^2 + 16*(8*(b^2*c^5 - 4*a*c^6)*d^2 - 24*(b^3*c^4 - 4*a*b*c^5)*d*e + 3*(5*b^4*c^3 - 24*a*b^2*c^4 +
16*a^2*c^5)*e^2)*f)*x^2 + 16*(8*(a*b^2*c^4 - 4*a^2*c^5)*d^2 - 24*(a*b^3*c^3 - 4*a^2*b*c^4)*d*e + 3*(5*a*b^4*c^
2 - 24*a^2*b^2*c^3 + 16*a^3*c^4)*e^2)*f + (128*(b^3*c^4 - 4*a*b*c^5)*d*e^2 - 64*(b^4*c^3 - 4*a*b^2*c^4)*e^3 +
5*(21*b^7 - 140*a*b^5*c + 240*a^2*b^3*c^2 - 64*a^3*b*c^3)*f^3 + 8*(6*(5*b^5*c^2 - 24*a*b^3*c^3 + 16*a^2*b*c^4)
*d - 5*(7*b^6*c - 40*a*b^4*c^2 + 48*a^2*b^2*c^3)*e)*f^2 + 16*(8*(b^3*c^4 - 4*a*b*c^5)*d^2 - 24*(b^4*c^3 - 4*a*
b^2*c^4)*d*e + 3*(5*b^5*c^2 - 24*a*b^3*c^3 + 16*a^2*b*c^4)*e^2)*f)*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*(128*b*c^6*d^3 - 768*a*c^6*d^2*e + 384*a*b*c^5*d*e^2 - 16*
(b^2*c^5 - 4*a*c^6)*f^3*x^5 - 8*(8*(b^2*c^5 - 4*a*c^6)*e*f^2 - 3*(b^3*c^4 - 4*a*b*c^5)*f^3)*x^4 - 64*(3*a*b^2*
c^4 - 8*a^2*c^5)*e^3 + (315*a*b^5*c - 1680*a^2*b^3*c^2 + 1808*a^3*b*c^3)*f^3 - 2*(48*(b^2*c^5 - 4*a*c^6)*e^2*f
+ (21*b^4*c^3 - 104*a*b^2*c^4 + 80*a^2*c^5)*f^3 + 8*(6*(b^2*c^5 - 4*a*c^6)*d - 7*(b^3*c^4 - 4*a*b*c^5)*e)*f^2
)*x^3 + 8*(6*(15*a*b^3*c^3 - 52*a^2*b*c^4)*d - (105*a*b^4*c^2 - 460*a^2*b^2*c^3 + 256*a^3*c^4)*e)*f^2 - (64*(b
^2*c^5 - 4*a*c^6)*e^3 - 7*(15*b^5*c^2 - 88*a*b^3*c^3 + 112*a^2*b*c^4)*f^3 - 8*(30*(b^3*c^4 - 4*a*b*c^5)*d - (3
5*b^4*c^3 - 172*a*b^2*c^4 + 128*a^2*c^5)*e)*f^2 + 48*(8*(b^2*c^5 - 4*a*c^6)*d*e - 5*(b^3*c^4 - 4*a*b*c^5)*e^2)
*f)*x^2 + 48*(8*a*b*c^5*d^2 - 8*(3*a*b^2*c^4 - 8*a^2*c^5)*d*e + (15*a*b^3*c^3 - 52*a^2*b*c^4)*e^2)*f + (256*c^
7*d^3 - 384*b*c^6*d^2*e + 384*(b^2*c^5 - 2*a*c^6)*d*e^2 - 64*(3*b^3*c^4 - 10*a*b*c^5)*e^3 + (315*b^6*c - 1890*
a*b^4*c^2 + 2704*a^2*b^2*c^3 - 480*a^3*c^4)*f^3 + 8*(6*(15*b^4*c^3 - 62*a*b^2*c^4 + 24*a^2*c^5)*d - (105*b^5*c
^2 - 530*a*b^3*c^3 + 488*a^2*b*c^4)*e)*f^2 + 48*(8*(b^2*c^5 - 2*a*c^6)*d^2 - 8*(3*b^3*c^4 - 10*a*b*c^5)*d*e +
(15*b^4*c^3 - 62*a*b^2*c^4 + 24*a^2*c^5)*e^2)*f)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^6 - 4*a^2*c^7 + (b^2*c^7 -
4*a*c^8)*x^2 + (b^3*c^6 - 4*a*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)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.3817, size = 1484, normalized size = 2.29 \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)^(3/2),x, algorithm="giac")

[Out]

1/64*(((2*(4*(2*(b^2*c^4*f^3 - 4*a*c^5*f^3)*x/(b^2*c^5 - 4*a*c^6) - (3*b^3*c^3*f^3 - 12*a*b*c^4*f^3 - 8*b^2*c^
4*f^2*e + 32*a*c^5*f^2*e)/(b^2*c^5 - 4*a*c^6))*x + (48*b^2*c^4*d*f^2 - 192*a*c^5*d*f^2 + 21*b^4*c^2*f^3 - 104*
a*b^2*c^3*f^3 + 80*a^2*c^4*f^3 - 56*b^3*c^3*f^2*e + 224*a*b*c^4*f^2*e + 48*b^2*c^4*f*e^2 - 192*a*c^5*f*e^2)/(b
^2*c^5 - 4*a*c^6))*x - (240*b^3*c^3*d*f^2 - 960*a*b*c^4*d*f^2 + 105*b^5*c*f^3 - 616*a*b^3*c^2*f^3 + 784*a^2*b*
c^3*f^3 - 384*b^2*c^4*d*f*e + 1536*a*c^5*d*f*e - 280*b^4*c^2*f^2*e + 1376*a*b^2*c^3*f^2*e - 1024*a^2*c^4*f^2*e
+ 240*b^3*c^3*f*e^2 - 960*a*b*c^4*f*e^2 - 64*b^2*c^4*e^3 + 256*a*c^5*e^3)/(b^2*c^5 - 4*a*c^6))*x - (256*c^6*d
^3 + 384*b^2*c^4*d^2*f - 768*a*c^5*d^2*f + 720*b^4*c^2*d*f^2 - 2976*a*b^2*c^3*d*f^2 + 1152*a^2*c^4*d*f^2 + 315
*b^6*f^3 - 1890*a*b^4*c*f^3 + 2704*a^2*b^2*c^2*f^3 - 480*a^3*c^3*f^3 - 384*b*c^5*d^2*e - 1152*b^3*c^3*d*f*e +
3840*a*b*c^4*d*f*e - 840*b^5*c*f^2*e + 4240*a*b^3*c^2*f^2*e - 3904*a^2*b*c^3*f^2*e + 384*b^2*c^4*d*e^2 - 768*a
*c^5*d*e^2 + 720*b^4*c^2*f*e^2 - 2976*a*b^2*c^3*f*e^2 + 1152*a^2*c^4*f*e^2 - 192*b^3*c^3*e^3 + 640*a*b*c^4*e^3
)/(b^2*c^5 - 4*a*c^6))*x - (128*b*c^5*d^3 + 384*a*b*c^4*d^2*f + 720*a*b^3*c^2*d*f^2 - 2496*a^2*b*c^3*d*f^2 + 3
15*a*b^5*f^3 - 1680*a^2*b^3*c*f^3 + 1808*a^3*b*c^2*f^3 - 768*a*c^5*d^2*e - 1152*a*b^2*c^3*d*f*e + 3072*a^2*c^4
*d*f*e - 840*a*b^4*c*f^2*e + 3680*a^2*b^2*c^2*f^2*e - 2048*a^3*c^3*f^2*e + 384*a*b*c^4*d*e^2 + 720*a*b^3*c^2*f
*e^2 - 2496*a^2*b*c^3*f*e^2 - 192*a*b^2*c^3*e^3 + 512*a^2*c^4*e^3)/(b^2*c^5 - 4*a*c^6))/sqrt(c*x^2 + b*x + a)
- 3/128*(128*c^4*d^2*f + 240*b^2*c^2*d*f^2 - 192*a*c^3*d*f^2 + 105*b^4*f^3 - 280*a*b^2*c*f^3 + 80*a^2*c^2*f^3
- 384*b*c^3*d*f*e - 280*b^3*c*f^2*e + 480*a*b*c^2*f^2*e + 128*c^4*d*e^2 + 240*b^2*c^2*f*e^2 - 192*a*c^3*f*e^2
- 64*b*c^3*e^3)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)