3.2.18 \(\int \frac {(d+e x+f x^2)^3}{(a+b x+c x^2)^{5/2}} \, dx\) [118]

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

[Out]

2/3*(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)^(3/2)+1/8*f*(35*b^2*f^2-20*c*f*
(a*f+3*b*e)+24*c^2*(d*f+e^2))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(9/2)-2/3*(3*b^6*c*e*f^2-b^
7*f^3+3*b^5*c*f*(6*a*f^2-c*(d*f+e^2))-3*b^3*c^2*(29*a^2*f^3+c^2*d*(d*f+e^2)-10*a*c*f*(d*f+e^2))-4*b*c^3*(2*c^3
*d^3-29*a^3*f^3+3*a*c^2*d*(d*f+e^2)+24*a^2*c*f*(d*f+e^2))-24*a^2*c^4*e*(6*a*f^2-c*(6*d*f+e^2))-b^4*c^2*e*(42*a
*f^2-c*(6*d*f+e^2))+6*b^2*c^3*e*(2*c^2*d^2+28*a^2*f^2-a*c*(6*d*f+e^2))-c*(16*c^6*d^3-10*b^6*f^3+3*b^4*c*f^2*(2
6*a*f+7*b*e)-24*c^5*d*(b*d*e-a*(d*f+e^2))-6*b^2*c^2*f*(25*a*b*e*f+27*a^2*f^2+2*b^2*(d*f+e^2))+6*c^4*(b^2*d*(d*
f+e^2)-16*a^2*f*(d*f+e^2)-2*a*b*e*(6*d*f+e^2))+c^3*(240*a^2*b*e*f^2+56*a^3*f^3+84*a*b^2*f*(d*f+e^2)+b^3*(6*d*e
*f+e^3)))*x)/c^5/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(1/2)+1/4*f^2*(-11*b*f+12*c*e)*(c*x^2+b*x+a)^(1/2)/c^4+1/2*f^3*x
*(c*x^2+b*x+a)^(1/2)/c^3

________________________________________________________________________________________

Rubi [A]
time = 1.10, antiderivative size = 891, normalized size of antiderivative = 1.00, 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 = {1674, 1675, 654, 635, 212} \begin {gather*} \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}} \end {gather*}

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 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 )^{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 ) \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 )}{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*}

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Mathematica [A]
time = 10.90, size = 872, normalized size = 0.98 \begin {gather*} \frac {-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 \left (5 a^2 f (6 e+53 f x)-6 a c x \left (4 e^2+4 d f+30 e f x-31 f^2 x^2\right )+c^2 x^3 \left (-16 e^2-16 d f+6 e f x+f^2 x^2\right )\right )-3 b^5 f \left (35 a^2 f^2-10 a c f x (12 e+23 f x)+c^2 x^2 \left (24 e^2+24 d f-80 e f x+7 f^2 x^2\right )\right )-48 b c^2 \left (27 a^4 f^3-4 c^4 d^2 x^2 (d-e x)+a^2 c^2 \left (-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)\right )-2 a c^3 \left (d^3-e^3 x^3+3 d e x^2 (e-2 f x)+3 d^2 x (-e+f x)\right )-2 a^3 c f \left (5 e^2+39 e f x+f \left (5 d-14 f x^2\right )\right )\right )-8 b^3 c \left (-95 a^3 f^3+c^3 \left (d^3-e^3 x^3+9 d^2 x (e-f x)-3 d e x^2 (3 e+2 f x)\right )-3 a c^2 f x^2 \left (18 e^2-74 e f x+f \left (18 d+7 f x^2\right )\right )+3 a^2 c f \left (3 e^2+105 e f x+f \left (3 d+29 f x^2\right )\right )\right )+32 c^3 \left (4 c^4 d^3 x^3+3 a^4 f^2 (16 e+5 f x)+6 a c^3 d x \left (d^2+e^2 x^2+d f x^2\right )-2 a^3 c \left (2 e^3+9 e^2 f x+f^2 x \left (9 d-10 f x^2\right )+12 e f \left (d-3 f x^2\right )\right )-3 a^2 c^2 \left (2 d^2 e+4 d f x^2 (3 e+2 f x)+x^2 \left (2 e^3+8 e^2 f x-6 e f^2 x^2-f^3 x^3\right )\right )\right )-48 b^2 c^2 \left (a^3 f^2 (25 e+63 f x)-c^3 d x \left (d^2+e^2 x^2+d x (-6 e+f x)\right )+a^2 c f x \left (-21 e^2-12 e f x+7 f \left (-3 d+7 f x^2\right )\right )+a c^2 \left (d^2 (e-6 f x)-2 d x \left (3 e^2-3 e f x+7 f^2 x^2\right )+x^2 \left (e^3-14 e^2 f x+6 e f^2 x^2+f^3 x^3\right )\right )\right )}{12 c^4 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}+\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 ) \log \left (b+2 c x+2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{8 c^{9/2}} \end {gather*}

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))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3250\) vs. \(2(865)=1730\).
time = 0.20, size = 3251, normalized size = 3.65

method	result	size
default	\(\text {Expression too large to display}\)	\(3251\)
risch	\(\text {Expression too large to display}\)	\(19191\)

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,method=_RETURNVERBOSE)

[Out]

f^3*(1/2*x^5/c/(c*x^2+b*x+a)^(3/2)-7/4*b/c*(x^4/c/(c*x^2+b*x+a)^(3/2)-5/2*b/c*(-1/3*x^3/c/(c*x^2+b*x+a)^(3/2)-
1/2*b/c*(-x^2/c/(c*x^2+b*x+a)^(3/2)+1/2*b/c*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-
1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/
2*a/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+2*a/
c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c
*x+b)/(c*x^2+b*x+a)^(1/2))))+1/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*a/c*(-x^2/c/(c*x^2+b*x
+a)^(3/2)+1/2*b/c*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*
a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/c*(2/3*(2*c*x+b)/(4*a*
c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+2*a/c*(-1/3/c/(c*x^2+b*x+a)^(3
/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))
)))-5/2*a/c*(-1/3*x^3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(-x^2/c/(c*x^2+b*x+a)^(3/2)+1/2*b/c*(-1/2*x/c/(c*x^2+b*x+a
)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a
*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c
-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+2*a/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(
c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))))+1/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*(x^4/c/(c*x^2+b*x+a)^(3/2)-5/2*b/c*(-1/3*x^3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(-x^2/
c/(c*x^2+b*x+a)^(3/2)+1/2*b/c*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(
2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/c*(2/3*(2*
c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+2*a/c*(-1/3/c/(c*x
^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b
*x+a)^(1/2))))+1/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*a/c*(-x^2/c/(c*x^2+b*x+a)^(3/2)+1/2*
b/c*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^
2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+
b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+2*a/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2
/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))))+(d*f^2+2*e
^2*f+f*(2*d*f+e^2))*(-1/3*x^3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(-x^2/c/(c*x^2+b*x+a)^(3/2)+1/2*b/c*(-1/2*x/c/(c*x
^2+b*x+a)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/
3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*
c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+2*a/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*
c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))))+1/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)^(3/2)+1/2*b/c*(-1/2*x/c/(c*x^2+b*x+a
)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a
*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c
-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+2*a/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(
c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))))+(d*(2*d*f+e^2)+2*d*e^2+f*d^2)*(-1/2*x
/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3
/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2
)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+3*d^2*e*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x
+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+d^3*(2/3*(2*c*x+b)/(4
*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))

________________________________________________________________________________________

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)^(5/2),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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2001 vs. \(2 (855) = 1710\).
time = 9.65, size = 4005, normalized size = 4.49 \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)^(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)*d*f^2 + 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)*x^4 + 24*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d*f^2 + 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)*d*f^2 + 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)*x^3 + (24*(b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d*f^2 + 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)*x^2 + 2*(24*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*
c^4)*d*f^2 + 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)*x + 24*((b^4*c^4 - 8*a*b^2*c^5 +
16*a^2*c^6)*f*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*f*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*f*x
^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*f*x + (a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*f)*e^2 - 60
*((b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*f^2*x^4 + 2*(b^6*c^2 - 8*a*b^4*c^3 + 16*a^2*b^2*c^4)*f^2*x^3 + (b^7*c
- 6*a*b^5*c^2 + 32*a^3*b*c^4)*f^2*x^2 + 2*(a*b^6*c - 8*a^2*b^4*c^2 + 16*a^3*b^2*c^3)*f^2*x + (a^2*b^5*c - 8*a
^3*b^3*c^2 + 16*a^4*b*c^3)*f^2)*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*(192*a^2*b*c^5*d^2*f + 6*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*f^3*x^5 - 21*(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 - 24*(3*a^2*b^3*c^3 - 20*a^3*b*c^4)*d*f^2 -
(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 + 12*(b^2*c^6 + 4*a*c^7)*d^2*f - 24*(b^
4*c^4 - 7*a*b^2*c^5 + 8*a^2*c^6)*d*f^2 - (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)*x^3
+ 3*(64*b*c^7*d^3 + 24*(b^3*c^5 + 4*a*b*c^6)*d^2*f - 24*(b^5*c^3 - 6*a*b^3*c^4)*d*f^2 - (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)*x^2 + 6*(48*a*b^2*c^5*d^2*f + 8*(b^2*c^6 + 4*a*c^7)*d^3 - 24*(a*b
^4*c^3 - 7*a^2*b^2*c^4 + 4*a^3*c^5)*d*f^2 - (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)
*x - 8*(24*a^2*b*c^5*x + 16*a^3*c^5 - (b^3*c^5 - 12*a*b*c^6)*x^3 + 6*(a*b^2*c^5 + 4*a^2*c^6)*x^2)*e^3 + 24*(8*
a^2*b*c^5*d + 2*((b^2*c^6 + 4*a*c^7)*d - 2*(b^4*c^4 - 7*a*b^2*c^5 + 8*a^2*c^6)*f)*x^3 + 3*((b^3*c^5 + 4*a*b*c^
6)*d - (b^5*c^3 - 6*a*b^3*c^4)*f)*x^2 - (3*a^2*b^3*c^3 - 20*a^3*b*c^4)*f + 6*(2*a*b^2*c^5*d - (a*b^4*c^3 - 7*a
^2*b^2*c^4 + 4*a^3*c^5)*f)*x)*e^2 - 12*(64*a^3*c^5*d*f - 3*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*f^2*x^4 + 4*(4
*b*c^7*d^2 - (b^3*c^5 - 12*a*b*c^6)*d*f - (5*b^5*c^3 - 37*a*b^3*c^4 + 64*a^2*b*c^5)*f^2)*x^3 + 4*(a*b^2*c^5 +
4*a^2*c^6)*d^2 - (15*a^2*b^4*c^2 - 100*a^3*b^2*c^3 + 128*a^4*c^4)*f^2 + 3*(8*b^2*c^6*d^2 + 8*(a*b^2*c^5 + 4*a^
2*c^6)*d*f - (5*b^6*c^2 - 30*a*b^4*c^3 + 16*a^2*b^2*c^4 + 64*a^3*c^5)*f^2)*x^2 + 6*(16*a^2*b*c^5*d*f + (b^3*c^
5 + 4*a*b*c^6)*d^2 - (5*a*b^5*c^2 - 35*a^2*b^3*c^3 + 52*a^3*b*c^4)*f^2)*x)*e)*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)*d*f^2 + 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)*x^4 + 24*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d*f^2 + 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)*d*f^2 + 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)*x^3 + (24*(b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d*f^2 + 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)*x^2 + 2*(24*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4
)*d*f^2 + 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)*x + 24*((b^4*c^4 - 8*a*b^2*c^5 + 16
*a^2*c^6)*f*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*f*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*f*x^2
+ 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*f*x + (a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*f)*e^2 - 60*((
b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*f^2*x^4 + 2*(b^6*c^2 - 8*a*b^4*c^3 + 16*a^2*b^2*c^4)*f^2*x^3 + (b^7*c -
6*a*b^5*c^2 + 32*a^3*b*c^4)*f^2*x^2 + 2*(a*b^6*c - 8*a^2*b^4*c^2 + 16*a^3*b^2*c^3)*f^2*x + (a^2*b^5*c - 8*a^3*
b^3*c^2 + 16*a^4*b*c^3)*f^2)*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*(192*a^2*b*c^5*d^2*f + 6*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*f^3*x^5 - 21*(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 - 24*(3*a^2*b^3*c^3 - 20*a^3*b*c^4)*d*f^2 - (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 + 12*(b^2*c^6 + 4*a*c^7)*d^2*f - 24*(b^4*c^4 - 7
*a*b^2*c^5 + 8*a^2*c^6)*d*f^2 - (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)*x^3 + 3*(64*
b*c^7*d^3 + 24*(b^3*c^5 + 4*a*b*c^6)*d^2*f - 24*(b^5*c^3 - 6*a*b^3*c^4)*d*f^2 - (35*b^7*c - 230*a*b^5*c^2 + 23
2*a^2*b^3*c^3 + 448*a^3*b*c^4)*f^3)*x^2 + 6*(48*a*b^2*c^5*d^2*f + 8*(b^2*c^6 + 4*a*c^7)*d^3 - 24*(a*b^4*c^3 -
7*a^2*b^2*c^4 + 4*a^3*c^5)*d*f^2 - (35*a*b^6*c ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**(5/2),x)

[Out]

Timed out

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Giac [A]
time = 1.97, size = 1401, normalized size = 1.57 \begin {gather*} \frac {{\left ({\left ({\left (3 \, {\left (\frac {2 \, {\left (b^{4} c^{3} f^{3} - 8 \, a b^{2} c^{4} f^{3} + 16 \, a^{2} c^{5} f^{3}\right )} x}{b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}} - \frac {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}}\right )} x + \frac {4 \, {\left (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}\right )}}{b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}}\right )} x + \frac {3 \, {\left (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}\right )}}{b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}}\right )} x + \frac {6 \, {\left (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}\right )}}{b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}}\right )} x - \frac {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}}}{12 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} - \frac {{\left (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}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{8 \, c^{\frac {9}{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)^(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)

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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 )}^{5/2}} \,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)^(5/2),x)

[Out]

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

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