3.2.14 \(\int \frac {(d+e x+f x^2)^3}{(a+b x+c x^2)^{3/2}} \, dx\) [114]

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

[Out]

3/128*(105*b^4*f^3-280*b^2*c*f^2*(a*f+b*e)+128*c^4*d*(d*f+e^2)+80*c^2*f*(6*a*b*e*f+a^2*f^2+3*b^2*(d*f+e^2))-64
*c^3*(3*a*f*(d*f+e^2)+b*(6*d*e*f+e^3)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(11/2)+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*(d*f+e^2))-b*c^2*(c^3*d^3+5*a^3*f^3+3*a*c^2*d*(d*f+e^2)-9*a^2*c*f*(d
*f+e^2))-a*b^2*c^2*e*(12*a*f^2-c*(6*d*f+e^2))+2*a*c^3*e*(3*c^2*d^2+3*a^2*f^2-a*c*(6*d*f+e^2))-(-2*a*c*f+b^2*f-
b*c*e+2*c^2*d)*(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^4*f^2-2*b^3*c*e*f+b^2*c^2*d*
f+b^2*c^2*e^2-b*c^3*d*e+c^4*d^2)*x)/c^5/(-4*a*c+b^2)/(c*x^2+b*x+a)^(1/2)-1/64*(187*b^3*f^3-4*b*c*f^2*(73*a*f+1
14*b*e)-64*c^3*(6*d*e*f+e^3)+16*c^2*f*(20*a*e*f+21*b*(d*f+e^2)))*(c*x^2+b*x+a)^(1/2)/c^5+1/32*f*(41*b^2*f^2-4*
c*f*(7*a*f+22*b*e)+48*c^2*(d*f+e^2))*x*(c*x^2+b*x+a)^(1/2)/c^4+1/8*f^2*(-5*b*f+8*c*e)*x^2*(c*x^2+b*x+a)^(1/2)/
c^3+1/4*f^3*x^3*(c*x^2+b*x+a)^(1/2)/c^2

________________________________________________________________________________________

Rubi [A]
time = 1.27, antiderivative size = 649, normalized size of antiderivative = 1.00, 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 = {1674, 1675, 654, 635, 212} \begin {gather*} \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 {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^4 f^2-2 b^3 c e f+b^2 c^2 d f+b^2 c^2 e^2-b c^3 d e+c^4 d^2\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 )-b c^2 \left (5 a^3 f^3-9 a^2 c f \left (d f+e^2\right )+3 a c^2 d \left (d f+e^2\right )+c^3 d^3\right )-a b^5 f^3+3 a b^4 c e f^2+a b^3 c f \left (5 a f^2-3 c \left (d f+e^2\right )\right )-a b^2 c^2 e \left (12 a f^2-c \left (6 d f+e^2\right )\right )\right )}{c^5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\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 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 {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} \end {gather*}

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

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 654

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 1674

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 1675

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]

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 ) \text {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*}

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Mathematica [A]
time = 4.24, size = 768, normalized size = 1.18 \begin {gather*} \frac {2 \sqrt {c} \left (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 \left (105 a f (4 e+9 f x)+c x \left (-360 e^2+140 e f x+3 f \left (-120 d+7 f x^2\right )\right )\right )+8 b^3 c \left (-210 a^2 f^3+a c f \left (90 e^2+530 e f x+f \left (90 d-77 f x^2\right )\right )+c^2 x \left (-24 e^3+30 e^2 f x+3 f^2 x \left (10 d+f x^2\right )+2 e f \left (-72 d+7 f x^2\right )\right )\right )-16 b^2 c^2 \left (-a^2 f^2 (230 e+169 f x)+a c \left (12 e^3+186 e^2 f x+2 e f \left (36 d-43 f x^2\right )+f^2 x \left (186 d-13 f x^2\right )\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 (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )\right )\right )+32 c^3 \left (8 c^3 d^3 x-a^3 f^2 (64 e+15 f x)+a^2 c \left (16 e^3+36 e^2 f x+f^2 x \left (36 d-5 f x^2\right )-32 e f \left (-3 d+f x^2\right )\right )+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 (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )\right )\right )+16 b c^2 \left (113 a^3 f^3+8 c^3 d^2 (d-3 e x)+a^2 c f \left (-156 e^2-244 e f x+f \left (-156 d+49 f x^2\right )\right )+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 (-20 e^3+30 e^2 f x+14 e f^2 x^2+3 f^3 x^3\right )\right )\right )\right )+3 \left (b^2-4 a c\right ) \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 ) \sqrt {a+x (b+c x)} \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{128 c^{11/2} \left (-b^2+4 a c\right ) \sqrt {a+x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c]*(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 + 140*e*f*x + 3*f*(-120*d + 7*f*x^2))) + 8*b^3*c*(-210*a^2*f^3 + a*c*f*(90*e^2 + 530*e*f*x + f*(90*d -
77*f*x^2)) + 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))) - 16*b^2*c^2*(-
(a^2*f^2*(230*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)))) + 3*(b^2 - 4*a*c)*(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)))*Sqrt[a + x*(b +
c*x)]*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(128*c^(11/2)*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2298\) vs. \(2(621)=1242\).
time = 0.16, size = 2299, normalized size = 3.54

method	result	size
default	\(\text {Expression too large to display}\)	\(2299\)
risch	\(\text {Expression too large to display}\)	\(2715\)

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,method=_RETURNVERBOSE)

[Out]

f^3*(1/4*x^5/c/(c*x^2+b*x+a)^(1/2)-9/8*b/c*(1/3*x^4/c/(c*x^2+b*x+a)^(1/2)-7/6*b/c*(1/2*x^3/c/(c*x^2+b*x+a)^(1/
2)-5/4*b/c*(x^2/c/(c*x^2+b*x+a)^(1/2)-3/2*b/c*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*
(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))-2*a/c*(-1/c/
(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)))-3/2*a/c*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*
(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x+a)^(1/2))))-4/3*a/c*(x^2/c/(c*x^2+b*x+a)^(1/2)-3/2*b/c*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b
*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2
)))-2*a/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))))-5/4*a/c*(1/2*x^3/c/(c*x^2
+b*x+a)^(1/2)-5/4*b/c*(x^2/c/(c*x^2+b*x+a)^(1/2)-3/2*b/c*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a)
^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))-2
*a/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)))-3/2*a/c*(-x/c/(c*x^2+b*x+a)^(1/
2)-1/2*b/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c
^(1/2)+(c*x^2+b*x+a)^(1/2)))))+3*e*f^2*(1/3*x^4/c/(c*x^2+b*x+a)^(1/2)-7/6*b/c*(1/2*x^3/c/(c*x^2+b*x+a)^(1/2)-5
/4*b/c*(x^2/c/(c*x^2+b*x+a)^(1/2)-3/2*b/c*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c
*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))-2*a/c*(-1/c/(c*x
^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)))-3/2*a/c*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/
c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))))-4/3*a/c*(x^2/c/(c*x^2+b*x+a)^(1/2)-3/2*b/c*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a
)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))-
2*a/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))))+(d*f^2+2*e^2*f+f*(2*d*f+e^2))
*(1/2*x^3/c/(c*x^2+b*x+a)^(1/2)-5/4*b/c*(x^2/c/(c*x^2+b*x+a)^(1/2)-3/2*b/c*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(
-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2)))-2*a/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)))-3/2*a/c*(-x/c
/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2
)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+(4*d*e*f+e*(2*d*f+e^2))*(x^2/c/(c*x^2+b*x+a)^(1/2)-3/2*b/c*(-x
/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3
/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))-2*a/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*
x^2+b*x+a)^(1/2)))+(d*(2*d*f+e^2)+2*d*e^2+f*d^2)*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b
/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+3*d^2*e*(
-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(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)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1553 vs. \(2 (619) = 1238\).
time = 4.19, size = 3109, normalized size = 4.79 \begin {gather*} \text {Too large to display} \end {gather*}

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^2*f + 48*(5*a*b^4*c^2 - 24*a^2*b^2*c^3 + 16*a^3*c^4)*d*f^2 + 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 + (128*(b^2*c^5 - 4*a*c^6)*d^2*f + 48*(5*b^4*c^3 - 24*
a*b^2*c^4 + 16*a^2*c^5)*d*f^2 + 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)*x^2 + (128*(b
^3*c^4 - 4*a*b*c^5)*d^2*f + 48*(5*b^5*c^2 - 24*a*b^3*c^3 + 16*a^2*b*c^4)*d*f^2 + 5*(21*b^7 - 140*a*b^5*c + 240
*a^2*b^3*c^2 - 64*a^3*b*c^3)*f^3)*x - 64*(a*b^3*c^3 - 4*a^2*b*c^4 + (b^3*c^4 - 4*a*b*c^5)*x^2 + (b^4*c^3 - 4*a
*b^2*c^4)*x)*e^3 + 16*((8*(b^2*c^5 - 4*a*c^6)*d + 3*(5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*c^5)*f)*x^2 + 8*(a*b^2*
c^4 - 4*a^2*c^5)*d + 3*(5*a*b^4*c^2 - 24*a^2*b^2*c^3 + 16*a^3*c^4)*f + (8*(b^3*c^4 - 4*a*b*c^5)*d + 3*(5*b^5*c
^2 - 24*a*b^3*c^3 + 16*a^2*b*c^4)*f)*x)*e^2 - 8*(48*(a*b^3*c^3 - 4*a^2*b*c^4)*d*f + 5*(7*a*b^5*c - 40*a^2*b^3*
c^2 + 48*a^3*b*c^3)*f^2 + (48*(b^3*c^4 - 4*a*b*c^5)*d*f + 5*(7*b^5*c^2 - 40*a*b^3*c^3 + 48*a^2*b*c^4)*f^2)*x^2
+ (48*(b^4*c^3 - 4*a*b^2*c^4)*d*f + 5*(7*b^6*c - 40*a*b^4*c^2 + 48*a^2*b^2*c^3)*f^2)*x)*e)*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*(128*b*c^6*d^3 + 384*a*b*c^5*d
^2*f - 16*(b^2*c^5 - 4*a*c^6)*f^3*x^5 + 24*(b^3*c^4 - 4*a*b*c^5)*f^3*x^4 + 48*(15*a*b^3*c^3 - 52*a^2*b*c^4)*d*
f^2 + (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)*d*f^2 + (21*b^4*c^3 -
104*a*b^2*c^4 + 80*a^2*c^5)*f^3)*x^3 + (240*(b^3*c^4 - 4*a*b*c^5)*d*f^2 + 7*(15*b^5*c^2 - 88*a*b^3*c^3 + 112*a
^2*b*c^4)*f^3)*x^2 + (256*c^7*d^3 + 384*(b^2*c^5 - 2*a*c^6)*d^2*f + 48*(15*b^4*c^3 - 62*a*b^2*c^4 + 24*a^2*c^5
)*d*f^2 + (315*b^6*c - 1890*a*b^4*c^2 + 2704*a^2*b^2*c^3 - 480*a^3*c^4)*f^3)*x - 64*(3*a*b^2*c^4 - 8*a^2*c^5 +
(b^2*c^5 - 4*a*c^6)*x^2 + (3*b^3*c^4 - 10*a*b*c^5)*x)*e^3 + 48*(8*a*b*c^5*d - 2*(b^2*c^5 - 4*a*c^6)*f*x^3 + 5
*(b^3*c^4 - 4*a*b*c^5)*f*x^2 + (15*a*b^3*c^3 - 52*a^2*b*c^4)*f + (8*(b^2*c^5 - 2*a*c^6)*d + (15*b^4*c^3 - 62*a
*b^2*c^4 + 24*a^2*c^5)*f)*x)*e^2 - 8*(96*a*c^6*d^2 + 8*(b^2*c^5 - 4*a*c^6)*f^2*x^4 - 14*(b^3*c^4 - 4*a*b*c^5)*
f^2*x^3 + 48*(3*a*b^2*c^4 - 8*a^2*c^5)*d*f + (105*a*b^4*c^2 - 460*a^2*b^2*c^3 + 256*a^3*c^4)*f^2 + (48*(b^2*c^
5 - 4*a*c^6)*d*f + (35*b^4*c^3 - 172*a*b^2*c^4 + 128*a^2*c^5)*f^2)*x^2 + (48*b*c^6*d^2 + 48*(3*b^3*c^4 - 10*a*
b*c^5)*d*f + (105*b^5*c^2 - 530*a*b^3*c^3 + 488*a^2*b*c^4)*f^2)*x)*e)*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^2*f + 48*
(5*a*b^4*c^2 - 24*a^2*b^2*c^3 + 16*a^3*c^4)*d*f^2 + 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 + (128*(b^2*c^5 - 4*a*c^6)*d^2*f + 48*(5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*c^5)*d*f^2 + 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)*x^2 + (128*(b^3*c^4 - 4*a*b*c^5)*d^2*f + 48*(5*b^5*c^2 - 24*a*b
^3*c^3 + 16*a^2*b*c^4)*d*f^2 + 5*(21*b^7 - 140*a*b^5*c + 240*a^2*b^3*c^2 - 64*a^3*b*c^3)*f^3)*x - 64*(a*b^3*c^
3 - 4*a^2*b*c^4 + (b^3*c^4 - 4*a*b*c^5)*x^2 + (b^4*c^3 - 4*a*b^2*c^4)*x)*e^3 + 16*((8*(b^2*c^5 - 4*a*c^6)*d +
3*(5*b^4*c^3 - 24*a*b^2*c^4 + 16*a^2*c^5)*f)*x^2 + 8*(a*b^2*c^4 - 4*a^2*c^5)*d + 3*(5*a*b^4*c^2 - 24*a^2*b^2*c
^3 + 16*a^3*c^4)*f + (8*(b^3*c^4 - 4*a*b*c^5)*d + 3*(5*b^5*c^2 - 24*a*b^3*c^3 + 16*a^2*b*c^4)*f)*x)*e^2 - 8*(4
8*(a*b^3*c^3 - 4*a^2*b*c^4)*d*f + 5*(7*a*b^5*c - 40*a^2*b^3*c^2 + 48*a^3*b*c^3)*f^2 + (48*(b^3*c^4 - 4*a*b*c^5
)*d*f + 5*(7*b^5*c^2 - 40*a*b^3*c^3 + 48*a^2*b*c^4)*f^2)*x^2 + (48*(b^4*c^3 - 4*a*b^2*c^4)*d*f + 5*(7*b^6*c -
40*a*b^4*c^2 + 48*a^2*b^2*c^3)*f^2)*x)*e)*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 + 384*a*b*c^5*d^2*f - 16*(b^2*c^5 - 4*a*c^6)*f^3*x^5 + 24*(b^3*c^4 - 4*
a*b*c^5)*f^3*x^4 + 48*(15*a*b^3*c^3 - 52*a^2*b*c^4)*d*f^2 + (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)*d*f^2 + (21*b^4*c^3 - 104*a*b^2*c^4 + 80*a^2*c^5)*f^3)*x^3 + (240*(b^3*c^4 - 4
*a*b*c^5)*d*f^2 + 7*(15*b^5*c^2 - 88*a*b^3*c^3 + 112*a^2*b*c^4)*f^3)*x^2 + (256*c^7*d^3 + 384*(b^2*c^5 - 2*a*c
^6)*d^2*f + 48*(15*b^4*c^3 - 62*a*b^2*c^4 + 24*a^2*c^5)*d*f^2 + (315*b^6*c - 1890*a*b^4*c^2 + 2704*a^2*b^2*c^3
- 480*a^3*c^4)*f^3)*x - 64*(3*a*b^2*c^4 - 8*a^2*c^5 + (b^2*c^5 - 4*a*c^6)*x^2 + (3*b^3*c^4 - 10*a*b*c^5)*x)*e
^3 + 48*(8*a*b*c^5*d - 2*(b^2*c^5 - 4*a*c^6)*f*x^3 + 5*(b^3*c^4 - 4*a*b*c^5)*f*x^2 + (15*a*b^3*c^3 - 52*a^2*b*
c^4)*f + (8*(b^2*c^5 - 2*a*c^6)*d + (15*b^4*c^3 - 62*a*b^2*c^4 + 24*a^2*c^5)*f)*x)*e^2 - 8*(96*a*c^6*d^2 + 8*(
b^2*c^5 - 4*a*c^6)*f^2*x^4 - 14*(b^3*c^4 - 4*a*b*c^5)*f^2*x^3 + 48*(3*a*b^2*c^4 - 8*a^2*c^5)*d*f + (105*a*b^4*
c^2 - 460*a^2*b^2*c^3 + 256*a^3*c^4)*f^2 + (48*(b^2*c^5 - 4*a*c^6)*d*f + (35*b^4*c^3 - 172*a*b^2*c^4 + 128*a^2
*c^5)*f^2)*x^2 + (48*b*c^6*d^2 + 48*(3*b^3*c^4 - 10*a*b*c^5)*d*f + (105*b^5*c^2 - 530*a*b^3*c^3 + 488*a^2*b*c^
4)*f^2)*x)*e)*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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x + f x^{2}\right )^{3}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

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]

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

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Giac [A]
time = 3.94, size = 1099, normalized size = 1.69 \begin {gather*} \frac {{\left ({\left (2 \, {\left (4 \, {\left (\frac {2 \, {\left (b^{2} c^{4} f^{3} - 4 \, a c^{5} f^{3}\right )} x}{b^{2} c^{5} - 4 \, a c^{6}} - \frac {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}}\right )} x + \frac {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}}\right )} x - \frac {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}}\right )} x - \frac {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}}\right )} x - \frac {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} + 315 \, 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}}}{64 \, \sqrt {c x^{2} + b x + a}} - \frac {3 \, {\left (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}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{128 \, c^{\frac {11}{2}}} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f\,x^2+e\,x+d\right )}^3}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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