3.2487 \(\int \frac{(A+B x) (d+e x)^6}{(a+b x+c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=1401 \[ \text{result too large to display} \]

[Out]

(2*(d + e*x)^5*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*c*d - a*B*e))*x))/
(5*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) + (2*(d + e*x)^3*(b^3*B*e*(3*c*d^2 - 7*a*e^2) - 8*a*c^2*e*(3*A*c*d
^2 + 12*a*B*d*e + 5*a*A*e^2) - 2*b^2*c*(4*B*c*d^3 + 9*A*c*d^2*e - a*A*e^3) + 4*b*c*(4*A*c*d*(c*d^2 + 3*a*e^2)
+ a*B*e*(9*c*d^2 + 11*a*e^2)) - (7*b^4*B*e^3 - 2*b^3*c*e^2*(3*B*d + A*e) - 12*b^2*c*e*(B*c*d^2 + A*c*d*e + 4*a
*B*e^2) + 8*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e + 9*a*B*d*e^2 + 3*a*A*e^3) - 16*c^2*(3*a*B*e*(c*d^2 - a*e^2) + A*c*
d*(2*c*d^2 + 3*a*e^2)))*x))/(15*c^2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) + (2*(d + e*x)*(7*b^5*B*e^3*(c*d^
2 - 5*a*e^2) - 2*b^4*c*e^3*(A*c*d^2 - 16*a*B*d*e - 5*a*A*e^2) - 8*b^3*c*e*(A*c*d*e*(21*c*d^2 - a*e^2) + 6*B*(2
*c^2*d^4 + a*c*d^2*e^2 - 7*a^2*e^4)) + 32*a*c^3*e*(6*a*B*d*e*(c*d^2 + 11*a*e^2) + A*(4*c^2*d^4 + 9*a*c*d^2*e^2
+ 15*a^2*e^4)) - 16*b*c^2*(2*A*c*d*(4*c^2*d^4 + 19*a*c*d^2*e^2 + 21*a^2*e^4) + a*B*e*(16*c^2*d^4 + 75*a*c*d^2
*e^2 + 57*a^2*e^4)) + 16*b^2*c^2*(6*A*e*(3*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4) + B*(4*c^2*d^5 + 37*a*c*d^3*e^2
- 21*a^2*d*e^4)) - (35*b^6*B*e^5 - 2*b^5*c*e^4*(23*B*d + 5*A*e) - 4*b^4*c*e^3*(5*B*c*d^2 + A*c*d*e + 91*a*B*e^
2) - 8*b^3*c^2*e^2*(5*B*c*d^3 + 7*A*c*d^2*e - 63*a*B*d*e^2 - 13*a*A*e^3) + 64*c^3*(6*a*B*e*(c^2*d^4 + 4*a*c*d^
2*e^2 - 2*a^2*e^4) + A*c*d*(4*c^2*d^4 + 11*a*c*d^2*e^2 + 12*a^2*e^4)) + 16*b^2*c^2*e*(A*c*d*e*(29*c*d^2 + 9*a*
e^2) + B*(14*c^2*d^4 + 21*a*c*d^2*e^2 + 72*a^2*e^4)) - 32*b*c^3*(A*e*(20*c^2*d^4 + 33*a*c*d^2*e^2 + 12*a^2*e^4
) + B*(4*c^2*d^5 + 35*a*c*d^3*e^2 + 60*a^2*d*e^4)))*x))/(15*c^3*(b^2 - 4*a*c)^3*Sqrt[a + b*x + c*x^2]) + (e*(1
05*b^6*B*e^5 - 10*b^5*c*e^4*(11*B*d + 3*A*e) - 16*b^3*c^2*e^2*(3*B*c*d^3 + A*c*d^2*e - 78*a*B*d*e^2 - 20*a*A*e
^3) - 4*b^4*c*e^3*(5*A*c*d*e + 8*B*(2*c*d^2 + 35*a*e^2)) + 16*b^2*c^2*e*(6*A*c*d*e*(9*c*d^2 + 2*a*e^2) + 7*B*(
4*c^2*d^4 + 6*a*c*d^2*e^2 + 33*a^2*e^4)) + 64*c^3*(6*a*B*e*(2*c^2*d^4 + 9*a*c*d^2*e^2 - 8*a^2*e^4) + A*c*d*(8*
c^2*d^4 + 26*a*c*d^2*e^2 + 33*a^2*e^4)) - 32*b*c^3*(A*e*(40*c^2*d^4 + 78*a*c*d^2*e^2 + 33*a^2*e^4) + B*(8*c^2*
d^5 + 74*a*c*d^3*e^2 + 141*a^2*d*e^4)))*Sqrt[a + b*x + c*x^2])/(15*c^4*(b^2 - 4*a*c)^3) + (e^5*(12*B*c*d - 7*b
*B*e + 2*A*c*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 2.19519, antiderivative size = 1401, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {818, 640, 621, 206} \[ \frac{(12 B c d-7 b B e+2 A c e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) e^5}{2 c^{9/2}}+\frac{\left (105 B e^5 b^6-10 c e^4 (11 B d+3 A e) b^5-4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right ) b^4-16 c^2 e^2 \left (3 B c d^3+A c e d^2-78 a B e^2 d-20 a A e^3\right ) b^3+16 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c e^2 d^2+33 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (40 c^2 d^4+78 a c e^2 d^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c e^2 d^3+141 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c e^2 d^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c e^2 d^2+33 a^2 e^4\right )\right )\right ) \sqrt{c x^2+b x+a} e}{15 c^4 \left (b^2-4 a c\right )^3}+\frac{2 (d+e x) \left (7 B e^3 \left (c d^2-5 a e^2\right ) b^5-2 c e^3 \left (A c d^2-16 a B e d-5 a A e^2\right ) b^4-8 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c e^2 d^2-7 a^2 e^4\right )\right ) b^3+16 c^2 \left (6 A e \left (3 c^2 d^4+6 a c e^2 d^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c e^2 d^3-21 a^2 e^4 d\right )\right ) b^2-16 c^2 \left (2 A c d \left (4 c^2 d^4+19 a c e^2 d^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c e^2 d^2+57 a^2 e^4\right )\right ) b+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c e^2 d^2+15 a^2 e^4\right )\right )-\left (35 B e^5 b^6-2 c e^4 (23 B d+5 A e) b^5-4 c e^3 \left (5 B c d^2+A c e d+91 a B e^2\right ) b^4-8 c^2 e^2 \left (5 B c d^3+7 A c e d^2-63 a B e^2 d-13 a A e^3\right ) b^3+16 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c e^2 d^2+72 a^2 e^4\right )\right ) b^2-32 c^3 \left (A e \left (20 c^2 d^4+33 a c e^2 d^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c e^2 d^3+60 a^2 e^4 d\right )\right ) b+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c e^2 d^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c e^2 d^2+12 a^2 e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt{c x^2+b x+a}}+\frac{2 (d+e x)^3 \left (B e \left (3 c d^2-7 a e^2\right ) b^3-2 c \left (4 B c d^3+9 A c e d^2-a A e^3\right ) b^2+4 c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right ) b-8 a c^2 e \left (3 A c d^2+12 a B e d+5 a A e^2\right )-\left (7 B e^3 b^4-2 c e^2 (3 B d+A e) b^3-12 c e \left (B c d^2+A c e d+4 a B e^2\right ) b^2+8 c^2 \left (2 B c d^3+6 A c e d^2+9 a B e^2 d+3 a A e^3\right ) b-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (c x^2+b x+a\right )^{3/2}}+\frac{2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^6)/(a + b*x + c*x^2)^(7/2),x]

[Out]

(2*(d + e*x)^5*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*c*d - a*B*e))*x))/
(5*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) + (2*(d + e*x)^3*(b^3*B*e*(3*c*d^2 - 7*a*e^2) - 8*a*c^2*e*(3*A*c*d
^2 + 12*a*B*d*e + 5*a*A*e^2) - 2*b^2*c*(4*B*c*d^3 + 9*A*c*d^2*e - a*A*e^3) + 4*b*c*(4*A*c*d*(c*d^2 + 3*a*e^2)
+ a*B*e*(9*c*d^2 + 11*a*e^2)) - (7*b^4*B*e^3 - 2*b^3*c*e^2*(3*B*d + A*e) - 12*b^2*c*e*(B*c*d^2 + A*c*d*e + 4*a
*B*e^2) + 8*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e + 9*a*B*d*e^2 + 3*a*A*e^3) - 16*c^2*(3*a*B*e*(c*d^2 - a*e^2) + A*c*
d*(2*c*d^2 + 3*a*e^2)))*x))/(15*c^2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) + (2*(d + e*x)*(7*b^5*B*e^3*(c*d^
2 - 5*a*e^2) - 2*b^4*c*e^3*(A*c*d^2 - 16*a*B*d*e - 5*a*A*e^2) - 8*b^3*c*e*(A*c*d*e*(21*c*d^2 - a*e^2) + 6*B*(2
*c^2*d^4 + a*c*d^2*e^2 - 7*a^2*e^4)) + 32*a*c^3*e*(6*a*B*d*e*(c*d^2 + 11*a*e^2) + A*(4*c^2*d^4 + 9*a*c*d^2*e^2
+ 15*a^2*e^4)) - 16*b*c^2*(2*A*c*d*(4*c^2*d^4 + 19*a*c*d^2*e^2 + 21*a^2*e^4) + a*B*e*(16*c^2*d^4 + 75*a*c*d^2
*e^2 + 57*a^2*e^4)) + 16*b^2*c^2*(6*A*e*(3*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4) + B*(4*c^2*d^5 + 37*a*c*d^3*e^2
- 21*a^2*d*e^4)) - (35*b^6*B*e^5 - 2*b^5*c*e^4*(23*B*d + 5*A*e) - 4*b^4*c*e^3*(5*B*c*d^2 + A*c*d*e + 91*a*B*e^
2) - 8*b^3*c^2*e^2*(5*B*c*d^3 + 7*A*c*d^2*e - 63*a*B*d*e^2 - 13*a*A*e^3) + 64*c^3*(6*a*B*e*(c^2*d^4 + 4*a*c*d^
2*e^2 - 2*a^2*e^4) + A*c*d*(4*c^2*d^4 + 11*a*c*d^2*e^2 + 12*a^2*e^4)) + 16*b^2*c^2*e*(A*c*d*e*(29*c*d^2 + 9*a*
e^2) + B*(14*c^2*d^4 + 21*a*c*d^2*e^2 + 72*a^2*e^4)) - 32*b*c^3*(A*e*(20*c^2*d^4 + 33*a*c*d^2*e^2 + 12*a^2*e^4
) + B*(4*c^2*d^5 + 35*a*c*d^3*e^2 + 60*a^2*d*e^4)))*x))/(15*c^3*(b^2 - 4*a*c)^3*Sqrt[a + b*x + c*x^2]) + (e*(1
05*b^6*B*e^5 - 10*b^5*c*e^4*(11*B*d + 3*A*e) - 16*b^3*c^2*e^2*(3*B*c*d^3 + A*c*d^2*e - 78*a*B*d*e^2 - 20*a*A*e
^3) - 4*b^4*c*e^3*(5*A*c*d*e + 8*B*(2*c*d^2 + 35*a*e^2)) + 16*b^2*c^2*e*(6*A*c*d*e*(9*c*d^2 + 2*a*e^2) + 7*B*(
4*c^2*d^4 + 6*a*c*d^2*e^2 + 33*a^2*e^4)) + 64*c^3*(6*a*B*e*(2*c^2*d^4 + 9*a*c*d^2*e^2 - 8*a^2*e^4) + A*c*d*(8*
c^2*d^4 + 26*a*c*d^2*e^2 + 33*a^2*e^4)) - 32*b*c^3*(A*e*(40*c^2*d^4 + 78*a*c*d^2*e^2 + 33*a^2*e^4) + B*(8*c^2*
d^5 + 74*a*c*d^3*e^2 + 141*a^2*d*e^4)))*Sqrt[a + b*x + c*x^2])/(15*c^4*(b^2 - 4*a*c)^3) + (e^5*(12*B*c*d - 7*b
*B*e + 2*A*c*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(9/2))

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

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{(A+B x) (d+e x)^6}{\left (a+b x+c x^2\right )^{7/2}} \, dx &=\frac{2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{2 \int \frac{(d+e x)^4 \left (\frac{1}{2} \left (-16 A c^2 d^2-2 b B e \left (\frac{3 b d}{2}-5 a e\right )-4 a c e (6 B d+5 A e)+2 b c d (4 B d+9 A e)\right )+\frac{1}{2} e \left (7 b^2 B e-2 b c (B d+A e)+4 c (A c d-6 a B e)\right ) x\right )}{\left (a+b x+c x^2\right )^{5/2}} \, dx}{5 c \left (b^2-4 a c\right )}\\ &=\frac{2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{2 (d+e x)^3 \left (b^3 B e \left (3 c d^2-7 a e^2\right )-8 a c^2 e \left (3 A c d^2+12 a B d e+5 a A e^2\right )-2 b^2 c \left (4 B c d^3+9 A c d^2 e-a A e^3\right )+4 b c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right )-\left (7 b^4 B e^3-2 b^3 c e^2 (3 B d+A e)-12 b^2 c e \left (B c d^2+A c d e+4 a B e^2\right )+8 b c^2 \left (2 B c d^3+6 A c d^2 e+9 a B d e^2+3 a A e^3\right )-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{4 \int \frac{(d+e x)^2 \left (\frac{1}{4} \left (-7 b^4 B d e^3+2 b^3 e^3 (A c d+21 a B e)+12 b^2 c e \left (8 B c d^3+14 A c d^2 e+3 a B d e^2-a A e^3\right )+16 c^2 \left (6 a B d e \left (2 c d^2+7 a e^2\right )+A \left (8 c^2 d^4+18 a c d^2 e^2+15 a^2 e^4\right )\right )-8 b c \left (9 A c d e \left (4 c d^2+5 a e^2\right )+B \left (8 c^2 d^4+60 a c d^2 e^2+33 a^2 e^4\right )\right )\right )+\frac{1}{4} e \left (35 b^4 B e^3-2 b^3 c e^2 (9 B d+5 A e)-12 b^2 c e \left (2 B c d^2+A c d e+21 a B e^2\right )+8 b c^2 \left (4 B c d^3+12 A c d^2 e+21 a B d e^2+9 a A e^3\right )-16 c^2 \left (6 a B e \left (c d^2-4 a e^2\right )+A c d \left (4 c d^2+9 a e^2\right )\right )\right ) x\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{15 c^2 \left (b^2-4 a c\right )^2}\\ &=\frac{2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{2 (d+e x)^3 \left (b^3 B e \left (3 c d^2-7 a e^2\right )-8 a c^2 e \left (3 A c d^2+12 a B d e+5 a A e^2\right )-2 b^2 c \left (4 B c d^3+9 A c d^2 e-a A e^3\right )+4 b c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right )-\left (7 b^4 B e^3-2 b^3 c e^2 (3 B d+A e)-12 b^2 c e \left (B c d^2+A c d e+4 a B e^2\right )+8 b c^2 \left (2 B c d^3+6 A c d^2 e+9 a B d e^2+3 a A e^3\right )-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{2 (d+e x) \left (7 b^5 B e^3 \left (c d^2-5 a e^2\right )-2 b^4 c e^3 \left (A c d^2-16 a B d e-5 a A e^2\right )-8 b^3 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c d^2 e^2-7 a^2 e^4\right )\right )+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c d^2 e^2+15 a^2 e^4\right )\right )-16 b c^2 \left (2 A c d \left (4 c^2 d^4+19 a c d^2 e^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c d^2 e^2+57 a^2 e^4\right )\right )+16 b^2 c^2 \left (6 A e \left (3 c^2 d^4+6 a c d^2 e^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c d^3 e^2-21 a^2 d e^4\right )\right )-\left (35 b^6 B e^5-2 b^5 c e^4 (23 B d+5 A e)-4 b^4 c e^3 \left (5 B c d^2+A c d e+91 a B e^2\right )-8 b^3 c^2 e^2 \left (5 B c d^3+7 A c d^2 e-63 a B d e^2-13 a A e^3\right )+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c d^2 e^2+12 a^2 e^4\right )\right )+16 b^2 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c d^2 e^2+72 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (20 c^2 d^4+33 a c d^2 e^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c d^3 e^2+60 a^2 d e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}+\frac{8 \int \frac{\frac{1}{8} e \left (35 b^6 B d e^4-960 a^3 c^3 e^4 (6 B d+A e)-2 b^5 e^3 \left (16 B c d^2+5 A c d e-35 a B e^2\right )-4 b^4 c e^2 \left (6 B c d^3+2 A c d^2 e+114 a B d e^2+5 a A e^3\right )+16 b^3 c e \left (3 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (2 c^2 d^4+3 a c d^2 e^2-6 a^2 e^4\right )\right )+32 b c^2 \left (3 a B e \left (4 c^2 d^4+18 a c d^2 e^2+19 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c d^2 e^2+33 a^2 e^4\right )\right )-16 b^2 c^2 \left (2 A e \left (20 c^2 d^4+39 a c d^2 e^2-6 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c d^3 e^2-129 a^2 d e^4\right )\right )\right )+\frac{1}{8} e \left (105 b^6 B e^5-10 b^5 c e^4 (11 B d+3 A e)-16 b^3 c^2 e^2 \left (3 B c d^3+A c d^2 e-78 a B d e^2-20 a A e^3\right )-4 b^4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right )+16 b^2 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c d^2 e^2+33 a^2 e^4\right )\right )+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c d^2 e^2+33 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (40 c^2 d^4+78 a c d^2 e^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c d^3 e^2+141 a^2 d e^4\right )\right )\right ) x}{\sqrt{a+b x+c x^2}} \, dx}{15 c^3 \left (b^2-4 a c\right )^3}\\ &=\frac{2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{2 (d+e x)^3 \left (b^3 B e \left (3 c d^2-7 a e^2\right )-8 a c^2 e \left (3 A c d^2+12 a B d e+5 a A e^2\right )-2 b^2 c \left (4 B c d^3+9 A c d^2 e-a A e^3\right )+4 b c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right )-\left (7 b^4 B e^3-2 b^3 c e^2 (3 B d+A e)-12 b^2 c e \left (B c d^2+A c d e+4 a B e^2\right )+8 b c^2 \left (2 B c d^3+6 A c d^2 e+9 a B d e^2+3 a A e^3\right )-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{2 (d+e x) \left (7 b^5 B e^3 \left (c d^2-5 a e^2\right )-2 b^4 c e^3 \left (A c d^2-16 a B d e-5 a A e^2\right )-8 b^3 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c d^2 e^2-7 a^2 e^4\right )\right )+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c d^2 e^2+15 a^2 e^4\right )\right )-16 b c^2 \left (2 A c d \left (4 c^2 d^4+19 a c d^2 e^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c d^2 e^2+57 a^2 e^4\right )\right )+16 b^2 c^2 \left (6 A e \left (3 c^2 d^4+6 a c d^2 e^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c d^3 e^2-21 a^2 d e^4\right )\right )-\left (35 b^6 B e^5-2 b^5 c e^4 (23 B d+5 A e)-4 b^4 c e^3 \left (5 B c d^2+A c d e+91 a B e^2\right )-8 b^3 c^2 e^2 \left (5 B c d^3+7 A c d^2 e-63 a B d e^2-13 a A e^3\right )+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c d^2 e^2+12 a^2 e^4\right )\right )+16 b^2 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c d^2 e^2+72 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (20 c^2 d^4+33 a c d^2 e^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c d^3 e^2+60 a^2 d e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}+\frac{e \left (105 b^6 B e^5-10 b^5 c e^4 (11 B d+3 A e)-16 b^3 c^2 e^2 \left (3 B c d^3+A c d^2 e-78 a B d e^2-20 a A e^3\right )-4 b^4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right )+16 b^2 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c d^2 e^2+33 a^2 e^4\right )\right )+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c d^2 e^2+33 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (40 c^2 d^4+78 a c d^2 e^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c d^3 e^2+141 a^2 d e^4\right )\right )\right ) \sqrt{a+b x+c x^2}}{15 c^4 \left (b^2-4 a c\right )^3}+\frac{\left (e^5 (12 B c d-7 b B e+2 A c e)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 c^4}\\ &=\frac{2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{2 (d+e x)^3 \left (b^3 B e \left (3 c d^2-7 a e^2\right )-8 a c^2 e \left (3 A c d^2+12 a B d e+5 a A e^2\right )-2 b^2 c \left (4 B c d^3+9 A c d^2 e-a A e^3\right )+4 b c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right )-\left (7 b^4 B e^3-2 b^3 c e^2 (3 B d+A e)-12 b^2 c e \left (B c d^2+A c d e+4 a B e^2\right )+8 b c^2 \left (2 B c d^3+6 A c d^2 e+9 a B d e^2+3 a A e^3\right )-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{2 (d+e x) \left (7 b^5 B e^3 \left (c d^2-5 a e^2\right )-2 b^4 c e^3 \left (A c d^2-16 a B d e-5 a A e^2\right )-8 b^3 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c d^2 e^2-7 a^2 e^4\right )\right )+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c d^2 e^2+15 a^2 e^4\right )\right )-16 b c^2 \left (2 A c d \left (4 c^2 d^4+19 a c d^2 e^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c d^2 e^2+57 a^2 e^4\right )\right )+16 b^2 c^2 \left (6 A e \left (3 c^2 d^4+6 a c d^2 e^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c d^3 e^2-21 a^2 d e^4\right )\right )-\left (35 b^6 B e^5-2 b^5 c e^4 (23 B d+5 A e)-4 b^4 c e^3 \left (5 B c d^2+A c d e+91 a B e^2\right )-8 b^3 c^2 e^2 \left (5 B c d^3+7 A c d^2 e-63 a B d e^2-13 a A e^3\right )+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c d^2 e^2+12 a^2 e^4\right )\right )+16 b^2 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c d^2 e^2+72 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (20 c^2 d^4+33 a c d^2 e^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c d^3 e^2+60 a^2 d e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}+\frac{e \left (105 b^6 B e^5-10 b^5 c e^4 (11 B d+3 A e)-16 b^3 c^2 e^2 \left (3 B c d^3+A c d^2 e-78 a B d e^2-20 a A e^3\right )-4 b^4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right )+16 b^2 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c d^2 e^2+33 a^2 e^4\right )\right )+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c d^2 e^2+33 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (40 c^2 d^4+78 a c d^2 e^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c d^3 e^2+141 a^2 d e^4\right )\right )\right ) \sqrt{a+b x+c x^2}}{15 c^4 \left (b^2-4 a c\right )^3}+\frac{\left (e^5 (12 B c d-7 b B e+2 A c e)\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 )}{c^4}\\ &=\frac{2 (d+e x)^5 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{2 (d+e x)^3 \left (b^3 B e \left (3 c d^2-7 a e^2\right )-8 a c^2 e \left (3 A c d^2+12 a B d e+5 a A e^2\right )-2 b^2 c \left (4 B c d^3+9 A c d^2 e-a A e^3\right )+4 b c \left (4 A c d \left (c d^2+3 a e^2\right )+a B e \left (9 c d^2+11 a e^2\right )\right )-\left (7 b^4 B e^3-2 b^3 c e^2 (3 B d+A e)-12 b^2 c e \left (B c d^2+A c d e+4 a B e^2\right )+8 b c^2 \left (2 B c d^3+6 A c d^2 e+9 a B d e^2+3 a A e^3\right )-16 c^2 \left (3 a B e \left (c d^2-a e^2\right )+A c d \left (2 c d^2+3 a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{2 (d+e x) \left (7 b^5 B e^3 \left (c d^2-5 a e^2\right )-2 b^4 c e^3 \left (A c d^2-16 a B d e-5 a A e^2\right )-8 b^3 c e \left (A c d e \left (21 c d^2-a e^2\right )+6 B \left (2 c^2 d^4+a c d^2 e^2-7 a^2 e^4\right )\right )+32 a c^3 e \left (6 a B d e \left (c d^2+11 a e^2\right )+A \left (4 c^2 d^4+9 a c d^2 e^2+15 a^2 e^4\right )\right )-16 b c^2 \left (2 A c d \left (4 c^2 d^4+19 a c d^2 e^2+21 a^2 e^4\right )+a B e \left (16 c^2 d^4+75 a c d^2 e^2+57 a^2 e^4\right )\right )+16 b^2 c^2 \left (6 A e \left (3 c^2 d^4+6 a c d^2 e^2-a^2 e^4\right )+B \left (4 c^2 d^5+37 a c d^3 e^2-21 a^2 d e^4\right )\right )-\left (35 b^6 B e^5-2 b^5 c e^4 (23 B d+5 A e)-4 b^4 c e^3 \left (5 B c d^2+A c d e+91 a B e^2\right )-8 b^3 c^2 e^2 \left (5 B c d^3+7 A c d^2 e-63 a B d e^2-13 a A e^3\right )+64 c^3 \left (6 a B e \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+A c d \left (4 c^2 d^4+11 a c d^2 e^2+12 a^2 e^4\right )\right )+16 b^2 c^2 e \left (A c d e \left (29 c d^2+9 a e^2\right )+B \left (14 c^2 d^4+21 a c d^2 e^2+72 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (20 c^2 d^4+33 a c d^2 e^2+12 a^2 e^4\right )+B \left (4 c^2 d^5+35 a c d^3 e^2+60 a^2 d e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}+\frac{e \left (105 b^6 B e^5-10 b^5 c e^4 (11 B d+3 A e)-16 b^3 c^2 e^2 \left (3 B c d^3+A c d^2 e-78 a B d e^2-20 a A e^3\right )-4 b^4 c e^3 \left (5 A c d e+8 B \left (2 c d^2+35 a e^2\right )\right )+16 b^2 c^2 e \left (6 A c d e \left (9 c d^2+2 a e^2\right )+7 B \left (4 c^2 d^4+6 a c d^2 e^2+33 a^2 e^4\right )\right )+64 c^3 \left (6 a B e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right )+A c d \left (8 c^2 d^4+26 a c d^2 e^2+33 a^2 e^4\right )\right )-32 b c^3 \left (A e \left (40 c^2 d^4+78 a c d^2 e^2+33 a^2 e^4\right )+B \left (8 c^2 d^5+74 a c d^3 e^2+141 a^2 d e^4\right )\right )\right ) \sqrt{a+b x+c x^2}}{15 c^4 \left (b^2-4 a c\right )^3}+\frac{e^5 (12 B c d-7 b B e+2 A c e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{9/2}}\\ \end{align*}

Mathematica [B]  time = 12.0471, size = 3199, normalized size = 2.28 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^6)/(a + b*x + c*x^2)^(7/2),x]

[Out]

((a + b*x + c*x^2)^4*((B*e^6)/c^4 + ((-2*(-(A*b*c^6*d^6) + 2*a*B*c^6*d^6 - 6*a*b*B*c^5*d^5*e + 12*a*A*c^6*d^5*
e + 15*a*b^2*B*c^4*d^4*e^2 - 15*a*A*b*c^5*d^4*e^2 - 30*a^2*B*c^5*d^4*e^2 - 20*a*b^3*B*c^3*d^3*e^3 + 20*a*A*b^2
*c^4*d^3*e^3 + 60*a^2*b*B*c^4*d^3*e^3 - 40*a^2*A*c^5*d^3*e^3 + 15*a*b^4*B*c^2*d^2*e^4 - 15*a*A*b^3*c^3*d^2*e^4
- 60*a^2*b^2*B*c^3*d^2*e^4 + 45*a^2*A*b*c^4*d^2*e^4 + 30*a^3*B*c^4*d^2*e^4 - 6*a*b^5*B*c*d*e^5 + 6*a*A*b^4*c^
2*d*e^5 + 30*a^2*b^3*B*c^2*d*e^5 - 24*a^2*A*b^2*c^3*d*e^5 - 30*a^3*b*B*c^3*d*e^5 + 12*a^3*A*c^4*d*e^5 + a*b^6*
B*e^6 - a*A*b^5*c*e^6 - 6*a^2*b^4*B*c*e^6 + 5*a^2*A*b^3*c^2*e^6 + 9*a^3*b^2*B*c^2*e^6 - 5*a^3*A*b*c^3*e^6 - 2*
a^4*B*c^3*e^6))/(5*c^6*(-b^2 + 4*a*c)) - (2*(b*B*c^6*d^6 - 2*A*c^7*d^6 - 6*b^2*B*c^5*d^5*e + 6*A*b*c^6*d^5*e +
12*a*B*c^6*d^5*e + 15*b^3*B*c^4*d^4*e^2 - 15*A*b^2*c^5*d^4*e^2 - 45*a*b*B*c^5*d^4*e^2 + 30*a*A*c^6*d^4*e^2 -
20*b^4*B*c^3*d^3*e^3 + 20*A*b^3*c^4*d^3*e^3 + 80*a*b^2*B*c^4*d^3*e^3 - 60*a*A*b*c^5*d^3*e^3 - 40*a^2*B*c^5*d^3
*e^3 + 15*b^5*B*c^2*d^2*e^4 - 15*A*b^4*c^3*d^2*e^4 - 75*a*b^3*B*c^3*d^2*e^4 + 60*a*A*b^2*c^4*d^2*e^4 + 75*a^2*
b*B*c^4*d^2*e^4 - 30*a^2*A*c^5*d^2*e^4 - 6*b^6*B*c*d*e^5 + 6*A*b^5*c^2*d*e^5 + 36*a*b^4*B*c^2*d*e^5 - 30*a*A*b
^3*c^3*d*e^5 - 54*a^2*b^2*B*c^3*d*e^5 + 30*a^2*A*b*c^4*d*e^5 + 12*a^3*B*c^4*d*e^5 + b^7*B*e^6 - A*b^6*c*e^6 -
7*a*b^5*B*c*e^6 + 6*a*A*b^4*c^2*e^6 + 14*a^2*b^3*B*c^2*e^6 - 9*a^2*A*b^2*c^3*e^6 - 7*a^3*b*B*c^3*e^6 + 2*a^3*A
*c^4*e^6)*x)/(5*c^6*(-b^2 + 4*a*c)))/(a + b*x + c*x^2)^3 + ((-2*(8*b^2*B*c^6*d^6 - 16*A*b*c^7*d^6 - 18*b^3*B*c
^5*d^5*e + 48*A*b^2*c^6*d^5*e - 24*a*b*B*c^6*d^5*e + 45*b^4*B*c^4*d^4*e^2 - 45*A*b^3*c^5*d^4*e^2 - 210*a*b^2*B
*c^5*d^4*e^2 - 60*a*A*b*c^6*d^4*e^2 + 600*a^2*B*c^6*d^4*e^2 - 60*b^5*B*c^3*d^3*e^3 + 60*A*b^4*c^4*d^3*e^3 + 44
0*a*b^3*B*c^4*d^3*e^3 - 280*a*A*b^2*c^5*d^3*e^3 - 1120*a^2*b*B*c^5*d^3*e^3 + 800*a^2*A*c^6*d^3*e^3 + 45*b^6*B*
c^2*d^2*e^4 - 45*A*b^5*c^3*d^2*e^4 - 450*a*b^4*B*c^3*d^2*e^4 + 330*a*A*b^3*c^4*d^2*e^4 + 1500*a^2*b^2*B*c^4*d^
2*e^4 - 840*a^2*A*b*c^5*d^2*e^4 - 1200*a^3*B*c^5*d^2*e^4 - 18*b^7*B*c*d*e^5 + 18*A*b^6*c^2*d*e^5 + 228*a*b^5*B
*c^2*d*e^5 - 180*a*A*b^4*c^3*d*e^5 - 942*a^2*b^3*B*c^3*d*e^5 + 600*a^2*A*b^2*c^4*d*e^5 + 1176*a^3*b*B*c^4*d*e^
5 - 480*a^3*A*c^5*d*e^5 + 3*b^8*B*e^6 - 3*A*b^7*c*e^6 - 46*a*b^6*B*c*e^6 + 38*a*A*b^5*c^2*e^6 + 227*a^2*b^4*B*
c^2*e^6 - 157*a^2*A*b^3*c^3*e^6 - 386*a^3*b^2*B*c^3*e^6 + 196*a^3*A*b*c^4*e^6 + 120*a^4*B*c^4*e^6))/(15*c^6*(-
b^2 + 4*a*c)^2) - (2*(16*b*B*c^6*d^6 - 32*A*c^7*d^6 - 36*b^2*B*c^5*d^5*e + 96*A*b*c^6*d^5*e - 48*a*B*c^6*d^5*e
+ 15*b^3*B*c^4*d^4*e^2 - 90*A*b^2*c^5*d^4*e^2 + 180*a*b*B*c^5*d^4*e^2 - 120*a*A*c^6*d^4*e^2 + 80*b^4*B*c^3*d^
3*e^3 + 20*A*b^3*c^4*d^3*e^3 - 720*a*b^2*B*c^4*d^3*e^3 + 240*a*A*b*c^5*d^3*e^3 + 960*a^2*B*c^5*d^3*e^3 - 135*b
^5*B*c^2*d^2*e^4 + 60*A*b^4*c^3*d^2*e^4 + 1050*a*b^3*B*c^3*d^2*e^4 - 540*a*A*b^2*c^4*d^2*e^4 - 1800*a^2*b*B*c^
4*d^2*e^4 + 720*a^2*A*c^5*d^2*e^4 + 84*b^6*B*c*d*e^5 - 54*A*b^5*c^2*d*e^5 - 684*a*b^4*B*c^2*d*e^5 + 420*a*A*b^
3*c^3*d*e^5 + 1476*a^2*b^2*B*c^3*d*e^5 - 720*a^2*A*b*c^4*d*e^5 - 528*a^3*B*c^4*d*e^5 - 19*b^7*B*e^6 + 14*A*b^6
*c*e^6 + 168*a*b^5*B*c*e^6 - 114*a*A*b^4*c^2*e^6 - 441*a^2*b^3*B*c^2*e^6 + 246*a^2*A*b^2*c^3*e^6 + 308*a^3*b*B
*c^3*e^6 - 88*a^3*A*c^4*e^6)*x)/(15*c^5*(-b^2 + 4*a*c)^2))/(a + b*x + c*x^2)^2 + ((-2*(64*b^2*B*c^6*d^6 - 128*
A*b*c^7*d^6 - 144*b^3*B*c^5*d^5*e + 384*A*b^2*c^6*d^5*e - 192*a*b*B*c^6*d^5*e + 60*b^4*B*c^4*d^4*e^2 - 360*A*b
^3*c^5*d^4*e^2 + 720*a*b^2*B*c^5*d^4*e^2 - 480*a*A*b*c^6*d^4*e^2 + 20*b^5*B*c^3*d^3*e^3 + 80*A*b^4*c^4*d^3*e^3
- 480*a*b^3*B*c^4*d^3*e^3 + 960*a*A*b^2*c^5*d^3*e^3 - 960*a^2*b*B*c^5*d^3*e^3 - 90*b^6*B*c^2*d^2*e^4 + 15*A*b
^5*c^3*d^2*e^4 + 1050*a*b^4*B*c^3*d^2*e^4 - 360*a*A*b^3*c^4*d^2*e^4 - 3600*a^2*b^2*B*c^4*d^2*e^4 - 720*a^2*A*b
*c^5*d^2*e^4 + 7200*a^3*B*c^5*d^2*e^4 + 66*b^7*B*c*d*e^5 - 36*A*b^6*c^2*d*e^5 - 846*a*b^5*B*c^2*d*e^5 + 420*a*
A*b^4*c^3*d*e^5 + 3744*a^2*b^3*B*c^3*d*e^5 - 1440*a^2*A*b^2*c^4*d*e^5 - 6432*a^3*b*B*c^4*d*e^5 + 2880*a^3*A*c^
5*d*e^5 - 16*b^8*B*e^6 + 11*A*b^7*c*e^6 + 237*a*b^6*B*c*e^6 - 141*a*A*b^5*c^2*e^6 - 1254*a^2*b^4*B*c^2*e^6 + 6
24*a^2*A*b^3*c^3*e^6 + 2672*a^3*b^2*B*c^3*e^6 - 1072*a^3*A*b*c^4*e^6 - 1440*a^4*B*c^4*e^6))/(15*c^5*(-b^2 + 4*
a*c)^3) - (2*(128*b*B*c^6*d^6 - 256*A*c^7*d^6 - 288*b^2*B*c^5*d^5*e + 768*A*b*c^6*d^5*e - 384*a*B*c^6*d^5*e +
120*b^3*B*c^4*d^4*e^2 - 720*A*b^2*c^5*d^4*e^2 + 1440*a*b*B*c^5*d^4*e^2 - 960*a*A*c^6*d^4*e^2 + 40*b^4*B*c^3*d^
3*e^3 + 160*A*b^3*c^4*d^3*e^3 - 960*a*b^2*B*c^4*d^3*e^3 + 1920*a*A*b*c^5*d^3*e^3 - 1920*a^2*B*c^5*d^3*e^3 + 45
*b^5*B*c^2*d^2*e^4 + 30*A*b^4*c^3*d^2*e^4 - 600*a*b^3*B*c^3*d^2*e^4 - 720*a*A*b^2*c^4*d^2*e^4 + 3600*a^2*b*B*c
^4*d^2*e^4 - 1440*a^2*A*c^5*d^2*e^4 - 138*b^6*B*c*d*e^5 + 18*A*b^5*c^2*d*e^5 + 1548*a*b^4*B*c^2*d*e^5 - 240*a*
A*b^3*c^3*d*e^5 - 5472*a^2*b^2*B*c^3*d*e^5 + 1440*a^2*A*b*c^4*d*e^5 + 4416*a^3*B*c^4*d*e^5 + 58*b^7*B*e^6 - 23
*A*b^6*c*e^6 - 651*a*b^5*B*c*e^6 + 258*a*A*b^4*c^2*e^6 + 2352*a^2*b^3*B*c^2*e^6 - 912*a^2*A*b^2*c^3*e^6 - 2576
*a^3*b*B*c^3*e^6 + 736*a^3*A*c^4*e^6)*x)/(15*c^4*(-b^2 + 4*a*c)^3))/(a + b*x + c*x^2)))/(a + x*(b + c*x))^(7/2
) + (e^5*(12*B*c*d - 7*b*B*e + 2*A*c*e)*(a + b*x + c*x^2)^(7/2)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + b*x + c*x^2
]])/(2*c^(9/2)*(a + x*(b + c*x))^(7/2))

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Maple [B]  time = 0.043, size = 11346, normalized size = 8.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^6/(c*x^2+b*x+a)^(7/2),x)

[Out]

result too large to display

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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((B*x+A)*(e*x+d)^6/(c*x^2+b*x+a)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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((B*x+A)*(e*x+d)^6/(c*x^2+b*x+a)^(7/2),x, algorithm="fricas")

[Out]

Timed out

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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((B*x+A)*(e*x+d)**6/(c*x**2+b*x+a)**(7/2),x)

[Out]

Timed out

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Giac [B]  time = 1.40246, size = 4525, normalized size = 3.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^6/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")

[Out]

1/15*((((((15*(B*b^6*c^3*e^6 - 12*B*a*b^4*c^4*e^6 + 48*B*a^2*b^2*c^5*e^6 - 64*B*a^3*c^6*e^6)*x/(b^6*c^4 - 12*a
*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7) + (256*B*b*c^8*d^6 - 512*A*c^9*d^6 - 576*B*b^2*c^7*d^5*e - 768*B*a*c^8
*d^5*e + 1536*A*b*c^8*d^5*e + 240*B*b^3*c^6*d^4*e^2 + 2880*B*a*b*c^7*d^4*e^2 - 1440*A*b^2*c^7*d^4*e^2 - 1920*A
*a*c^8*d^4*e^2 + 80*B*b^4*c^5*d^3*e^3 - 1920*B*a*b^2*c^6*d^3*e^3 + 320*A*b^3*c^6*d^3*e^3 - 3840*B*a^2*c^7*d^3*
e^3 + 3840*A*a*b*c^7*d^3*e^3 + 90*B*b^5*c^4*d^2*e^4 - 1200*B*a*b^3*c^5*d^2*e^4 + 60*A*b^4*c^5*d^2*e^4 + 7200*B
*a^2*b*c^6*d^2*e^4 - 1440*A*a*b^2*c^6*d^2*e^4 - 2880*A*a^2*c^7*d^2*e^4 - 276*B*b^6*c^3*d*e^5 + 3096*B*a*b^4*c^
4*d*e^5 + 36*A*b^5*c^4*d*e^5 - 10944*B*a^2*b^2*c^5*d*e^5 - 480*A*a*b^3*c^5*d*e^5 + 8832*B*a^3*c^6*d*e^5 + 2880
*A*a^2*b*c^6*d*e^5 + 161*B*b^7*c^2*e^6 - 1842*B*a*b^5*c^3*e^6 - 46*A*b^6*c^3*e^6 + 6864*B*a^2*b^3*c^4*e^6 + 51
6*A*a*b^4*c^4*e^6 - 8032*B*a^3*b*c^5*e^6 - 1824*A*a^2*b^2*c^5*e^6 + 1472*A*a^3*c^6*e^6)/(b^6*c^4 - 12*a*b^4*c^
5 + 48*a^2*b^2*c^6 - 64*a^3*c^7))*x + 5*(128*B*b^2*c^7*d^6 - 256*A*b*c^8*d^6 - 288*B*b^3*c^6*d^5*e - 384*B*a*b
*c^7*d^5*e + 768*A*b^2*c^7*d^5*e + 120*B*b^4*c^5*d^4*e^2 + 1440*B*a*b^2*c^6*d^4*e^2 - 720*A*b^3*c^6*d^4*e^2 -
960*A*a*b*c^7*d^4*e^2 + 40*B*b^5*c^4*d^3*e^3 - 960*B*a*b^3*c^5*d^3*e^3 + 160*A*b^4*c^5*d^3*e^3 - 1920*B*a^2*b*
c^6*d^3*e^3 + 1920*A*a*b^2*c^6*d^3*e^3 - 60*B*a*b^4*c^4*d^2*e^4 + 30*A*b^5*c^4*d^2*e^4 + 1440*B*a^2*b^2*c^5*d^
2*e^4 - 720*A*a*b^3*c^5*d^2*e^4 + 2880*B*a^3*c^6*d^2*e^4 - 1440*A*a^2*b*c^6*d^2*e^4 - 84*B*b^7*c^2*d*e^5 + 900
*B*a*b^5*c^3*d*e^5 - 2880*B*a^2*b^3*c^4*d*e^5 - 24*A*a*b^4*c^4*d*e^5 + 960*B*a^3*b*c^5*d*e^5 + 576*A*a^2*b^2*c
^5*d*e^5 + 1152*A*a^3*c^6*d*e^5 + 49*B*b^8*c*e^6 - 525*B*a*b^6*c^2*e^6 - 14*A*b^7*c^2*e^6 + 1704*B*a^2*b^4*c^3
*e^6 + 150*A*a*b^5*c^3*e^6 - 1136*B*a^3*b^2*c^4*e^6 - 480*A*a^2*b^3*c^4*e^6 - 1152*B*a^4*c^5*e^6 + 160*A*a^3*b
*c^5*e^6)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7))*x + 5*(96*B*b^3*c^6*d^6 + 128*B*a*b*c^7*d^6
- 192*A*b^2*c^7*d^6 - 256*A*a*c^8*d^6 - 216*B*b^4*c^5*d^5*e - 576*B*a*b^2*c^6*d^5*e + 576*A*b^3*c^6*d^5*e - 38
4*B*a^2*c^7*d^5*e + 768*A*a*b*c^7*d^5*e + 90*B*b^5*c^4*d^4*e^2 + 1200*B*a*b^3*c^5*d^4*e^2 - 540*A*b^4*c^5*d^4*
e^2 + 1440*B*a^2*b*c^6*d^4*e^2 - 1440*A*a*b^2*c^6*d^4*e^2 - 960*A*a^2*c^7*d^4*e^2 - 320*B*a*b^4*c^4*d^3*e^3 +
120*A*b^5*c^4*d^3*e^3 - 3840*B*a^2*b^2*c^5*d^3*e^3 + 1600*A*a*b^3*c^5*d^3*e^3 + 1920*A*a^2*b*c^6*d^3*e^3 + 480
*B*a^2*b^3*c^4*d^2*e^4 - 240*A*a*b^4*c^4*d^2*e^4 + 5760*B*a^3*b*c^5*d^2*e^4 - 2880*A*a^2*b^2*c^5*d^2*e^4 - 36*
B*b^8*c*d*e^5 + 240*B*a*b^6*c^2*d*e^5 + 360*B*a^2*b^4*c^3*d*e^5 - 5184*B*a^3*b^2*c^4*d*e^5 + 192*A*a^2*b^3*c^4
*d*e^5 + 2688*B*a^4*c^5*d*e^5 + 2304*A*a^3*b*c^5*d*e^5 + 21*B*b^9*e^6 - 140*B*a*b^7*c*e^6 - 6*A*b^8*c*e^6 - 21
0*B*a^2*b^5*c^2*e^6 + 40*A*a*b^6*c^2*e^6 + 2832*B*a^3*b^3*c^3*e^6 + 60*A*a^2*b^4*c^3*e^6 - 3872*B*a^4*b*c^4*e^
6 - 864*A*a^3*b^2*c^4*e^6 + 448*A*a^4*c^5*e^6)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7))*x + 5*(
16*B*b^4*c^5*d^6 + 192*B*a*b^2*c^6*d^6 - 32*A*b^3*c^6*d^6 - 384*A*a*b*c^7*d^6 - 36*B*b^5*c^4*d^5*e - 480*B*a*b
^3*c^5*d^5*e + 96*A*b^4*c^5*d^5*e - 576*B*a^2*b*c^6*d^5*e + 1152*A*a*b^2*c^6*d^5*e + 540*B*a*b^4*c^4*d^4*e^2 -
90*A*b^5*c^4*d^4*e^2 + 1440*B*a^2*b^2*c^5*d^4*e^2 - 1200*A*a*b^3*c^5*d^4*e^2 + 960*B*a^3*c^6*d^4*e^2 - 1440*A
*a^2*b*c^6*d^4*e^2 - 1920*B*a^2*b^3*c^4*d^3*e^3 + 720*A*a*b^4*c^4*d^3*e^3 - 2560*B*a^3*b*c^5*d^3*e^3 + 1920*A*
a^2*b^2*c^5*d^3*e^3 + 1280*A*a^3*c^6*d^3*e^3 + 2880*B*a^3*b^2*c^4*d^2*e^4 - 1440*A*a^2*b^3*c^4*d^2*e^4 + 3840*
B*a^4*c^5*d^2*e^4 - 1920*A*a^3*b*c^5*d^2*e^4 - 108*B*a*b^7*c*d*e^5 + 1116*B*a^2*b^5*c^2*d*e^5 - 3360*B*a^3*b^3
*c^3*d*e^5 - 576*B*a^4*b*c^4*d*e^5 + 1152*A*a^3*b^2*c^4*d*e^5 + 1536*A*a^4*c^5*d*e^5 + 63*B*a*b^8*e^6 - 651*B*
a^2*b^6*c*e^6 - 18*A*a*b^7*c*e^6 + 1960*B*a^3*b^4*c^2*e^6 + 186*A*a^2*b^5*c^2*e^6 - 816*B*a^4*b^2*c^3*e^6 - 56
0*A*a^3*b^3*c^3*e^6 - 1536*B*a^5*c^4*e^6 - 96*A*a^4*b*c^4*e^6)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a
^3*c^7))*x - 5*(2*B*b^5*c^4*d^6 - 48*B*a*b^3*c^5*d^6 - 4*A*b^4*c^5*d^6 - 96*B*a^2*b*c^6*d^6 + 96*A*a*b^2*c^6*d
^6 + 192*A*a^2*c^7*d^6 + 48*B*a*b^4*c^4*d^5*e + 12*A*b^5*c^4*d^5*e + 576*B*a^2*b^2*c^5*d^5*e - 288*A*a*b^3*c^5
*d^5*e - 576*A*a^2*b*c^6*d^5*e - 720*B*a^2*b^3*c^4*d^4*e^2 + 120*A*a*b^4*c^4*d^4*e^2 - 960*B*a^3*b*c^5*d^4*e^2
+ 1440*A*a^2*b^2*c^5*d^4*e^2 + 2560*B*a^3*b^2*c^4*d^3*e^3 - 960*A*a^2*b^3*c^4*d^3*e^3 - 1280*A*a^3*b*c^5*d^3*
e^3 - 3840*B*a^4*b*c^4*d^2*e^4 + 1920*A*a^3*b^2*c^4*d^2*e^4 + 108*B*a^2*b^6*c*d*e^5 - 1176*B*a^3*b^4*c^2*d*e^5
+ 4032*B*a^4*b^2*c^3*d*e^5 - 1152*B*a^5*c^4*d*e^5 - 1536*A*a^4*b*c^4*d*e^5 - 63*B*a^2*b^7*e^6 + 686*B*a^3*b^5
*c*e^6 + 18*A*a^2*b^6*c*e^6 - 2352*B*a^4*b^3*c^2*e^6 - 196*A*a^3*b^4*c^2*e^6 + 2208*B*a^5*b*c^3*e^6 + 672*A*a^
4*b^2*c^3*e^6 - 192*A*a^5*c^4*e^6)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7))*x - (4*B*a*b^4*c^4*
d^6 + 6*A*b^5*c^4*d^6 - 96*B*a^2*b^2*c^5*d^6 - 80*A*a*b^3*c^5*d^6 - 192*B*a^3*c^6*d^6 + 480*A*a^2*b*c^6*d^6 +
96*B*a^2*b^3*c^4*d^5*e + 24*A*a*b^4*c^4*d^5*e + 1152*B*a^3*b*c^5*d^5*e - 576*A*a^2*b^2*c^5*d^5*e - 1152*A*a^3*
c^6*d^5*e - 1440*B*a^3*b^2*c^4*d^4*e^2 + 240*A*a^2*b^3*c^4*d^4*e^2 - 1920*B*a^4*c^5*d^4*e^2 + 2880*A*a^3*b*c^5
*d^4*e^2 + 5120*B*a^4*b*c^4*d^3*e^3 - 1920*A*a^3*b^2*c^4*d^3*e^3 - 2560*A*a^4*c^5*d^3*e^3 - 7680*B*a^5*c^4*d^2
*e^4 + 3840*A*a^4*b*c^4*d^2*e^4 + 180*B*a^3*b^5*c*d*e^5 - 1920*B*a^4*b^3*c^2*d*e^5 + 6336*B*a^5*b*c^3*d*e^5 -
3072*A*a^5*c^4*d*e^5 - 105*B*a^3*b^6*e^6 + 1120*B*a^4*b^4*c*e^6 + 30*A*a^3*b^5*c*e^6 - 3696*B*a^5*b^2*c^2*e^6
- 320*A*a^4*b^3*c^2*e^6 + 3072*B*a^6*c^3*e^6 + 1056*A*a^5*b*c^3*e^6)/(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6
- 64*a^3*c^7))/(c*x^2 + b*x + a)^(5/2) - 1/2*(12*B*c*d*e^5 - 7*B*b*e^6 + 2*A*c*e^6)*log(abs(-2*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(9/2)