3.180 \(\int (a+b x)^n (c+d x^3)^2 \, dx\)
Optimal. Leaf size=203 \[ \frac {\left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^7 (n+1)}-\frac {3 a d \left (2 b^3 c-5 a^3 d\right ) (a+b x)^{n+3}}{b^7 (n+3)}+\frac {2 d \left (b^3 c-10 a^3 d\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac {15 a^2 d^2 (a+b x)^{n+5}}{b^7 (n+5)}+\frac {6 a^2 d \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^7 (n+2)}-\frac {6 a d^2 (a+b x)^{n+6}}{b^7 (n+6)}+\frac {d^2 (a+b x)^{n+7}}{b^7 (n+7)} \]
[Out]
(-a^3*d+b^3*c)^2*(b*x+a)^(1+n)/b^7/(1+n)+6*a^2*d*(-a^3*d+b^3*c)*(b*x+a)^(2+n)/b^7/(2+n)-3*a*d*(-5*a^3*d+2*b^3*
c)*(b*x+a)^(3+n)/b^7/(3+n)+2*d*(-10*a^3*d+b^3*c)*(b*x+a)^(4+n)/b^7/(4+n)+15*a^2*d^2*(b*x+a)^(5+n)/b^7/(5+n)-6*
a*d^2*(b*x+a)^(6+n)/b^7/(6+n)+d^2*(b*x+a)^(7+n)/b^7/(7+n)
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 203, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used =
{1850} \[ \frac {\left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^7 (n+1)}+\frac {6 a^2 d \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^7 (n+2)}-\frac {3 a d \left (2 b^3 c-5 a^3 d\right ) (a+b x)^{n+3}}{b^7 (n+3)}+\frac {2 d \left (b^3 c-10 a^3 d\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac {15 a^2 d^2 (a+b x)^{n+5}}{b^7 (n+5)}-\frac {6 a d^2 (a+b x)^{n+6}}{b^7 (n+6)}+\frac {d^2 (a+b x)^{n+7}}{b^7 (n+7)} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^n*(c + d*x^3)^2,x]
[Out]
((b^3*c - a^3*d)^2*(a + b*x)^(1 + n))/(b^7*(1 + n)) + (6*a^2*d*(b^3*c - a^3*d)*(a + b*x)^(2 + n))/(b^7*(2 + n)
) - (3*a*d*(2*b^3*c - 5*a^3*d)*(a + b*x)^(3 + n))/(b^7*(3 + n)) + (2*d*(b^3*c - 10*a^3*d)*(a + b*x)^(4 + n))/(
b^7*(4 + n)) + (15*a^2*d^2*(a + b*x)^(5 + n))/(b^7*(5 + n)) - (6*a*d^2*(a + b*x)^(6 + n))/(b^7*(6 + n)) + (d^2
*(a + b*x)^(7 + n))/(b^7*(7 + n))
Rule 1850
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])
Rubi steps
\begin {align*} \int (a+b x)^n \left (c+d x^3\right )^2 \, dx &=\int \left (\frac {\left (b^3 c-a^3 d\right )^2 (a+b x)^n}{b^6}-\frac {6 a^2 d \left (-b^3 c+a^3 d\right ) (a+b x)^{1+n}}{b^6}+\frac {3 a d \left (-2 b^3 c+5 a^3 d\right ) (a+b x)^{2+n}}{b^6}+\frac {2 d \left (b^3 c-10 a^3 d\right ) (a+b x)^{3+n}}{b^6}+\frac {15 a^2 d^2 (a+b x)^{4+n}}{b^6}-\frac {6 a d^2 (a+b x)^{5+n}}{b^6}+\frac {d^2 (a+b x)^{6+n}}{b^6}\right ) \, dx\\ &=\frac {\left (b^3 c-a^3 d\right )^2 (a+b x)^{1+n}}{b^7 (1+n)}+\frac {6 a^2 d \left (b^3 c-a^3 d\right ) (a+b x)^{2+n}}{b^7 (2+n)}-\frac {3 a d \left (2 b^3 c-5 a^3 d\right ) (a+b x)^{3+n}}{b^7 (3+n)}+\frac {2 d \left (b^3 c-10 a^3 d\right ) (a+b x)^{4+n}}{b^7 (4+n)}+\frac {15 a^2 d^2 (a+b x)^{5+n}}{b^7 (5+n)}-\frac {6 a d^2 (a+b x)^{6+n}}{b^7 (6+n)}+\frac {d^2 (a+b x)^{7+n}}{b^7 (7+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 172, normalized size = 0.85 \[ \frac {(a+b x)^{n+1} \left (\frac {2 d (a+b x)^3 \left (b^3 c-10 a^3 d\right )}{n+4}+\frac {3 a d (a+b x)^2 \left (5 a^3 d-2 b^3 c\right )}{n+3}+\frac {\left (b^3 c-a^3 d\right )^2}{n+1}+\frac {15 a^2 d^2 (a+b x)^4}{n+5}+\frac {6 a^2 d (a+b x) \left (b^3 c-a^3 d\right )}{n+2}+\frac {d^2 (a+b x)^6}{n+7}-\frac {6 a d^2 (a+b x)^5}{n+6}\right )}{b^7} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^n*(c + d*x^3)^2,x]
[Out]
((a + b*x)^(1 + n)*((b^3*c - a^3*d)^2/(1 + n) + (6*a^2*d*(b^3*c - a^3*d)*(a + b*x))/(2 + n) + (3*a*d*(-2*b^3*c
+ 5*a^3*d)*(a + b*x)^2)/(3 + n) + (2*d*(b^3*c - 10*a^3*d)*(a + b*x)^3)/(4 + n) + (15*a^2*d^2*(a + b*x)^4)/(5
+ n) - (6*a*d^2*(a + b*x)^5)/(6 + n) + (d^2*(a + b*x)^6)/(7 + n)))/b^7
________________________________________________________________________________________
fricas [B] time = 0.44, size = 893, normalized size = 4.40 \[ \frac {{\left (a b^{6} c^{2} n^{6} + 27 \, a b^{6} c^{2} n^{5} + 295 \, a b^{6} c^{2} n^{4} + 5040 \, a b^{6} c^{2} - 2520 \, a^{4} b^{3} c d + 720 \, a^{7} d^{2} + {\left (b^{7} d^{2} n^{6} + 21 \, b^{7} d^{2} n^{5} + 175 \, b^{7} d^{2} n^{4} + 735 \, b^{7} d^{2} n^{3} + 1624 \, b^{7} d^{2} n^{2} + 1764 \, b^{7} d^{2} n + 720 \, b^{7} d^{2}\right )} x^{7} + {\left (a b^{6} d^{2} n^{6} + 15 \, a b^{6} d^{2} n^{5} + 85 \, a b^{6} d^{2} n^{4} + 225 \, a b^{6} d^{2} n^{3} + 274 \, a b^{6} d^{2} n^{2} + 120 \, a b^{6} d^{2} n\right )} x^{6} - 6 \, {\left (a^{2} b^{5} d^{2} n^{5} + 10 \, a^{2} b^{5} d^{2} n^{4} + 35 \, a^{2} b^{5} d^{2} n^{3} + 50 \, a^{2} b^{5} d^{2} n^{2} + 24 \, a^{2} b^{5} d^{2} n\right )} x^{5} + 2 \, {\left (b^{7} c d n^{6} + 24 \, b^{7} c d n^{5} + 1260 \, b^{7} c d + {\left (226 \, b^{7} c d + 15 \, a^{3} b^{4} d^{2}\right )} n^{4} + 6 \, {\left (176 \, b^{7} c d + 15 \, a^{3} b^{4} d^{2}\right )} n^{3} + 5 \, {\left (509 \, b^{7} c d + 33 \, a^{3} b^{4} d^{2}\right )} n^{2} + 18 \, {\left (164 \, b^{7} c d + 5 \, a^{3} b^{4} d^{2}\right )} n\right )} x^{4} + 3 \, {\left (555 \, a b^{6} c^{2} - 4 \, a^{4} b^{3} c d\right )} n^{3} + 2 \, {\left (a b^{6} c d n^{6} + 21 \, a b^{6} c d n^{5} + 163 \, a b^{6} c d n^{4} + 3 \, {\left (189 \, a b^{6} c d - 20 \, a^{4} b^{3} d^{2}\right )} n^{3} + 4 \, {\left (211 \, a b^{6} c d - 45 \, a^{4} b^{3} d^{2}\right )} n^{2} + 60 \, {\left (7 \, a b^{6} c d - 2 \, a^{4} b^{3} d^{2}\right )} n\right )} x^{3} + 8 \, {\left (638 \, a b^{6} c^{2} - 27 \, a^{4} b^{3} c d\right )} n^{2} - 6 \, {\left (a^{2} b^{5} c d n^{5} + 19 \, a^{2} b^{5} c d n^{4} + 125 \, a^{2} b^{5} c d n^{3} + {\left (317 \, a^{2} b^{5} c d - 60 \, a^{5} b^{2} d^{2}\right )} n^{2} + 30 \, {\left (7 \, a^{2} b^{5} c d - 2 \, a^{5} b^{2} d^{2}\right )} n\right )} x^{2} + 12 \, {\left (669 \, a b^{6} c^{2} - 107 \, a^{4} b^{3} c d\right )} n + {\left (b^{7} c^{2} n^{6} + 27 \, b^{7} c^{2} n^{5} + 5040 \, b^{7} c^{2} + {\left (295 \, b^{7} c^{2} + 12 \, a^{3} b^{4} c d\right )} n^{4} + 9 \, {\left (185 \, b^{7} c^{2} + 24 \, a^{3} b^{4} c d\right )} n^{3} + 4 \, {\left (1276 \, b^{7} c^{2} + 321 \, a^{3} b^{4} c d\right )} n^{2} + 36 \, {\left (223 \, b^{7} c^{2} + 70 \, a^{3} b^{4} c d - 20 \, a^{6} b d^{2}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{7} n^{7} + 28 \, b^{7} n^{6} + 322 \, b^{7} n^{5} + 1960 \, b^{7} n^{4} + 6769 \, b^{7} n^{3} + 13132 \, b^{7} n^{2} + 13068 \, b^{7} n + 5040 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x^3+c)^2,x, algorithm="fricas")
[Out]
(a*b^6*c^2*n^6 + 27*a*b^6*c^2*n^5 + 295*a*b^6*c^2*n^4 + 5040*a*b^6*c^2 - 2520*a^4*b^3*c*d + 720*a^7*d^2 + (b^7
*d^2*n^6 + 21*b^7*d^2*n^5 + 175*b^7*d^2*n^4 + 735*b^7*d^2*n^3 + 1624*b^7*d^2*n^2 + 1764*b^7*d^2*n + 720*b^7*d^
2)*x^7 + (a*b^6*d^2*n^6 + 15*a*b^6*d^2*n^5 + 85*a*b^6*d^2*n^4 + 225*a*b^6*d^2*n^3 + 274*a*b^6*d^2*n^2 + 120*a*
b^6*d^2*n)*x^6 - 6*(a^2*b^5*d^2*n^5 + 10*a^2*b^5*d^2*n^4 + 35*a^2*b^5*d^2*n^3 + 50*a^2*b^5*d^2*n^2 + 24*a^2*b^
5*d^2*n)*x^5 + 2*(b^7*c*d*n^6 + 24*b^7*c*d*n^5 + 1260*b^7*c*d + (226*b^7*c*d + 15*a^3*b^4*d^2)*n^4 + 6*(176*b^
7*c*d + 15*a^3*b^4*d^2)*n^3 + 5*(509*b^7*c*d + 33*a^3*b^4*d^2)*n^2 + 18*(164*b^7*c*d + 5*a^3*b^4*d^2)*n)*x^4 +
3*(555*a*b^6*c^2 - 4*a^4*b^3*c*d)*n^3 + 2*(a*b^6*c*d*n^6 + 21*a*b^6*c*d*n^5 + 163*a*b^6*c*d*n^4 + 3*(189*a*b^
6*c*d - 20*a^4*b^3*d^2)*n^3 + 4*(211*a*b^6*c*d - 45*a^4*b^3*d^2)*n^2 + 60*(7*a*b^6*c*d - 2*a^4*b^3*d^2)*n)*x^3
+ 8*(638*a*b^6*c^2 - 27*a^4*b^3*c*d)*n^2 - 6*(a^2*b^5*c*d*n^5 + 19*a^2*b^5*c*d*n^4 + 125*a^2*b^5*c*d*n^3 + (3
17*a^2*b^5*c*d - 60*a^5*b^2*d^2)*n^2 + 30*(7*a^2*b^5*c*d - 2*a^5*b^2*d^2)*n)*x^2 + 12*(669*a*b^6*c^2 - 107*a^4
*b^3*c*d)*n + (b^7*c^2*n^6 + 27*b^7*c^2*n^5 + 5040*b^7*c^2 + (295*b^7*c^2 + 12*a^3*b^4*c*d)*n^4 + 9*(185*b^7*c
^2 + 24*a^3*b^4*c*d)*n^3 + 4*(1276*b^7*c^2 + 321*a^3*b^4*c*d)*n^2 + 36*(223*b^7*c^2 + 70*a^3*b^4*c*d - 20*a^6*
b*d^2)*n)*x)*(b*x + a)^n/(b^7*n^7 + 28*b^7*n^6 + 322*b^7*n^5 + 1960*b^7*n^4 + 6769*b^7*n^3 + 13132*b^7*n^2 + 1
3068*b^7*n + 5040*b^7)
________________________________________________________________________________________
giac [B] time = 0.50, size = 1477, normalized size = 7.28 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x^3+c)^2,x, algorithm="giac")
[Out]
((b*x + a)^n*b^7*d^2*n^6*x^7 + (b*x + a)^n*a*b^6*d^2*n^6*x^6 + 21*(b*x + a)^n*b^7*d^2*n^5*x^7 + 15*(b*x + a)^n
*a*b^6*d^2*n^5*x^6 + 175*(b*x + a)^n*b^7*d^2*n^4*x^7 + 2*(b*x + a)^n*b^7*c*d*n^6*x^4 - 6*(b*x + a)^n*a^2*b^5*d
^2*n^5*x^5 + 85*(b*x + a)^n*a*b^6*d^2*n^4*x^6 + 735*(b*x + a)^n*b^7*d^2*n^3*x^7 + 2*(b*x + a)^n*a*b^6*c*d*n^6*
x^3 + 48*(b*x + a)^n*b^7*c*d*n^5*x^4 - 60*(b*x + a)^n*a^2*b^5*d^2*n^4*x^5 + 225*(b*x + a)^n*a*b^6*d^2*n^3*x^6
+ 1624*(b*x + a)^n*b^7*d^2*n^2*x^7 + 42*(b*x + a)^n*a*b^6*c*d*n^5*x^3 + 452*(b*x + a)^n*b^7*c*d*n^4*x^4 + 30*(
b*x + a)^n*a^3*b^4*d^2*n^4*x^4 - 210*(b*x + a)^n*a^2*b^5*d^2*n^3*x^5 + 274*(b*x + a)^n*a*b^6*d^2*n^2*x^6 + 176
4*(b*x + a)^n*b^7*d^2*n*x^7 + (b*x + a)^n*b^7*c^2*n^6*x - 6*(b*x + a)^n*a^2*b^5*c*d*n^5*x^2 + 326*(b*x + a)^n*
a*b^6*c*d*n^4*x^3 + 2112*(b*x + a)^n*b^7*c*d*n^3*x^4 + 180*(b*x + a)^n*a^3*b^4*d^2*n^3*x^4 - 300*(b*x + a)^n*a
^2*b^5*d^2*n^2*x^5 + 120*(b*x + a)^n*a*b^6*d^2*n*x^6 + 720*(b*x + a)^n*b^7*d^2*x^7 + (b*x + a)^n*a*b^6*c^2*n^6
+ 27*(b*x + a)^n*b^7*c^2*n^5*x - 114*(b*x + a)^n*a^2*b^5*c*d*n^4*x^2 + 1134*(b*x + a)^n*a*b^6*c*d*n^3*x^3 - 1
20*(b*x + a)^n*a^4*b^3*d^2*n^3*x^3 + 5090*(b*x + a)^n*b^7*c*d*n^2*x^4 + 330*(b*x + a)^n*a^3*b^4*d^2*n^2*x^4 -
144*(b*x + a)^n*a^2*b^5*d^2*n*x^5 + 27*(b*x + a)^n*a*b^6*c^2*n^5 + 295*(b*x + a)^n*b^7*c^2*n^4*x + 12*(b*x + a
)^n*a^3*b^4*c*d*n^4*x - 750*(b*x + a)^n*a^2*b^5*c*d*n^3*x^2 + 1688*(b*x + a)^n*a*b^6*c*d*n^2*x^3 - 360*(b*x +
a)^n*a^4*b^3*d^2*n^2*x^3 + 5904*(b*x + a)^n*b^7*c*d*n*x^4 + 180*(b*x + a)^n*a^3*b^4*d^2*n*x^4 + 295*(b*x + a)^
n*a*b^6*c^2*n^4 + 1665*(b*x + a)^n*b^7*c^2*n^3*x + 216*(b*x + a)^n*a^3*b^4*c*d*n^3*x - 1902*(b*x + a)^n*a^2*b^
5*c*d*n^2*x^2 + 360*(b*x + a)^n*a^5*b^2*d^2*n^2*x^2 + 840*(b*x + a)^n*a*b^6*c*d*n*x^3 - 240*(b*x + a)^n*a^4*b^
3*d^2*n*x^3 + 2520*(b*x + a)^n*b^7*c*d*x^4 + 1665*(b*x + a)^n*a*b^6*c^2*n^3 - 12*(b*x + a)^n*a^4*b^3*c*d*n^3 +
5104*(b*x + a)^n*b^7*c^2*n^2*x + 1284*(b*x + a)^n*a^3*b^4*c*d*n^2*x - 1260*(b*x + a)^n*a^2*b^5*c*d*n*x^2 + 36
0*(b*x + a)^n*a^5*b^2*d^2*n*x^2 + 5104*(b*x + a)^n*a*b^6*c^2*n^2 - 216*(b*x + a)^n*a^4*b^3*c*d*n^2 + 8028*(b*x
+ a)^n*b^7*c^2*n*x + 2520*(b*x + a)^n*a^3*b^4*c*d*n*x - 720*(b*x + a)^n*a^6*b*d^2*n*x + 8028*(b*x + a)^n*a*b^
6*c^2*n - 1284*(b*x + a)^n*a^4*b^3*c*d*n + 5040*(b*x + a)^n*b^7*c^2*x + 5040*(b*x + a)^n*a*b^6*c^2 - 2520*(b*x
+ a)^n*a^4*b^3*c*d + 720*(b*x + a)^n*a^7*d^2)/(b^7*n^7 + 28*b^7*n^6 + 322*b^7*n^5 + 1960*b^7*n^4 + 6769*b^7*n
^3 + 13132*b^7*n^2 + 13068*b^7*n + 5040*b^7)
________________________________________________________________________________________
maple [B] time = 0.01, size = 793, normalized size = 3.91 \[ \frac {\left (b^{6} d^{2} n^{6} x^{6}+21 b^{6} d^{2} n^{5} x^{6}-6 a \,b^{5} d^{2} n^{5} x^{5}+175 b^{6} d^{2} n^{4} x^{6}-90 a \,b^{5} d^{2} n^{4} x^{5}+2 b^{6} c d \,n^{6} x^{3}+735 b^{6} d^{2} n^{3} x^{6}+30 a^{2} b^{4} d^{2} n^{4} x^{4}-510 a \,b^{5} d^{2} n^{3} x^{5}+48 b^{6} c d \,n^{5} x^{3}+1624 b^{6} d^{2} n^{2} x^{6}+300 a^{2} b^{4} d^{2} n^{3} x^{4}-6 a \,b^{5} c d \,n^{5} x^{2}-1350 a \,b^{5} d^{2} n^{2} x^{5}+452 b^{6} c d \,n^{4} x^{3}+1764 b^{6} d^{2} n \,x^{6}-120 a^{3} b^{3} d^{2} n^{3} x^{3}+1050 a^{2} b^{4} d^{2} n^{2} x^{4}-126 a \,b^{5} c d \,n^{4} x^{2}-1644 a \,b^{5} d^{2} n \,x^{5}+b^{6} c^{2} n^{6}+2112 b^{6} c d \,n^{3} x^{3}+720 d^{2} x^{6} b^{6}-720 a^{3} b^{3} d^{2} n^{2} x^{3}+12 a^{2} b^{4} c d \,n^{4} x +1500 a^{2} b^{4} d^{2} n \,x^{4}-978 a \,b^{5} c d \,n^{3} x^{2}-720 a \,d^{2} x^{5} b^{5}+27 b^{6} c^{2} n^{5}+5090 b^{6} c d \,n^{2} x^{3}+360 a^{4} b^{2} d^{2} n^{2} x^{2}-1320 a^{3} b^{3} d^{2} n \,x^{3}+228 a^{2} b^{4} c d \,n^{3} x +720 a^{2} d^{2} x^{4} b^{4}-3402 a \,b^{5} c d \,n^{2} x^{2}+295 b^{6} c^{2} n^{4}+5904 b^{6} c d n \,x^{3}+1080 a^{4} b^{2} d^{2} n \,x^{2}-12 a^{3} b^{3} c d \,n^{3}-720 a^{3} b^{3} d^{2} x^{3}+1500 a^{2} b^{4} c d \,n^{2} x -5064 a \,b^{5} c d n \,x^{2}+1665 b^{6} c^{2} n^{3}+2520 b^{6} c d \,x^{3}-720 a^{5} b \,d^{2} n x +720 a^{4} b^{2} d^{2} x^{2}-216 a^{3} b^{3} c d \,n^{2}+3804 a^{2} b^{4} c d n x -2520 a \,b^{5} c d \,x^{2}+5104 b^{6} c^{2} n^{2}-720 a^{5} b \,d^{2} x -1284 a^{3} b^{3} c d n +2520 a^{2} b^{4} c d x +8028 b^{6} c^{2} n +720 a^{6} d^{2}-2520 a^{3} b^{3} c d +5040 b^{6} c^{2}\right ) \left (b x +a \right )^{n +1}}{\left (n^{7}+28 n^{6}+322 n^{5}+1960 n^{4}+6769 n^{3}+13132 n^{2}+13068 n +5040\right ) b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^n*(d*x^3+c)^2,x)
[Out]
(b*x+a)^(n+1)*(b^6*d^2*n^6*x^6+21*b^6*d^2*n^5*x^6-6*a*b^5*d^2*n^5*x^5+175*b^6*d^2*n^4*x^6-90*a*b^5*d^2*n^4*x^5
+2*b^6*c*d*n^6*x^3+735*b^6*d^2*n^3*x^6+30*a^2*b^4*d^2*n^4*x^4-510*a*b^5*d^2*n^3*x^5+48*b^6*c*d*n^5*x^3+1624*b^
6*d^2*n^2*x^6+300*a^2*b^4*d^2*n^3*x^4-6*a*b^5*c*d*n^5*x^2-1350*a*b^5*d^2*n^2*x^5+452*b^6*c*d*n^4*x^3+1764*b^6*
d^2*n*x^6-120*a^3*b^3*d^2*n^3*x^3+1050*a^2*b^4*d^2*n^2*x^4-126*a*b^5*c*d*n^4*x^2-1644*a*b^5*d^2*n*x^5+b^6*c^2*
n^6+2112*b^6*c*d*n^3*x^3+720*b^6*d^2*x^6-720*a^3*b^3*d^2*n^2*x^3+12*a^2*b^4*c*d*n^4*x+1500*a^2*b^4*d^2*n*x^4-9
78*a*b^5*c*d*n^3*x^2-720*a*b^5*d^2*x^5+27*b^6*c^2*n^5+5090*b^6*c*d*n^2*x^3+360*a^4*b^2*d^2*n^2*x^2-1320*a^3*b^
3*d^2*n*x^3+228*a^2*b^4*c*d*n^3*x+720*a^2*b^4*d^2*x^4-3402*a*b^5*c*d*n^2*x^2+295*b^6*c^2*n^4+5904*b^6*c*d*n*x^
3+1080*a^4*b^2*d^2*n*x^2-12*a^3*b^3*c*d*n^3-720*a^3*b^3*d^2*x^3+1500*a^2*b^4*c*d*n^2*x-5064*a*b^5*c*d*n*x^2+16
65*b^6*c^2*n^3+2520*b^6*c*d*x^3-720*a^5*b*d^2*n*x+720*a^4*b^2*d^2*x^2-216*a^3*b^3*c*d*n^2+3804*a^2*b^4*c*d*n*x
-2520*a*b^5*c*d*x^2+5104*b^6*c^2*n^2-720*a^5*b*d^2*x-1284*a^3*b^3*c*d*n+2520*a^2*b^4*c*d*x+8028*b^6*c^2*n+720*
a^6*d^2-2520*a^3*b^3*c*d+5040*b^6*c^2)/b^7/(n^7+28*n^6+322*n^5+1960*n^4+6769*n^3+13132*n^2+13068*n+5040)
________________________________________________________________________________________
maxima [A] time = 0.66, size = 359, normalized size = 1.77 \[ \frac {{\left (b x + a\right )}^{n + 1} c^{2}}{b {\left (n + 1\right )}} + \frac {2 \, {\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} c d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} + \frac {{\left ({\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{7} x^{7} + {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a b^{6} x^{6} - 6 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{2} b^{5} x^{5} + 30 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{3} b^{4} x^{4} - 120 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{4} b^{3} x^{3} + 360 \, {\left (n^{2} + n\right )} a^{5} b^{2} x^{2} - 720 \, a^{6} b n x + 720 \, a^{7}\right )} {\left (b x + a\right )}^{n} d^{2}}{{\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x^3+c)^2,x, algorithm="maxima")
[Out]
(b*x + a)^(n + 1)*c^2/(b*(n + 1)) + 2*((n^3 + 6*n^2 + 11*n + 6)*b^4*x^4 + (n^3 + 3*n^2 + 2*n)*a*b^3*x^3 - 3*(n
^2 + n)*a^2*b^2*x^2 + 6*a^3*b*n*x - 6*a^4)*(b*x + a)^n*c*d/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^4) + ((n^6 +
21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*b^7*x^7 + (n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 +
120*n)*a*b^6*x^6 - 6*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^2*b^5*x^5 + 30*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^3
*b^4*x^4 - 120*(n^3 + 3*n^2 + 2*n)*a^4*b^3*x^3 + 360*(n^2 + n)*a^5*b^2*x^2 - 720*a^6*b*n*x + 720*a^7)*(b*x + a
)^n*d^2/((n^7 + 28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 5040)*b^7)
________________________________________________________________________________________
mupad [B] time = 3.19, size = 878, normalized size = 4.33 \[ \frac {a\,{\left (a+b\,x\right )}^n\,\left (720\,a^6\,d^2-12\,a^3\,b^3\,c\,d\,n^3-216\,a^3\,b^3\,c\,d\,n^2-1284\,a^3\,b^3\,c\,d\,n-2520\,a^3\,b^3\,c\,d+b^6\,c^2\,n^6+27\,b^6\,c^2\,n^5+295\,b^6\,c^2\,n^4+1665\,b^6\,c^2\,n^3+5104\,b^6\,c^2\,n^2+8028\,b^6\,c^2\,n+5040\,b^6\,c^2\right )}{b^7\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {d^2\,x^7\,{\left (a+b\,x\right )}^n\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}{n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040}+\frac {x\,{\left (a+b\,x\right )}^n\,\left (-720\,a^6\,b\,d^2\,n+12\,a^3\,b^4\,c\,d\,n^4+216\,a^3\,b^4\,c\,d\,n^3+1284\,a^3\,b^4\,c\,d\,n^2+2520\,a^3\,b^4\,c\,d\,n+b^7\,c^2\,n^6+27\,b^7\,c^2\,n^5+295\,b^7\,c^2\,n^4+1665\,b^7\,c^2\,n^3+5104\,b^7\,c^2\,n^2+8028\,b^7\,c^2\,n+5040\,b^7\,c^2\right )}{b^7\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {2\,d\,x^4\,{\left (a+b\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )\,\left (15\,d\,a^3\,n+c\,b^3\,n^3+18\,c\,b^3\,n^2+107\,c\,b^3\,n+210\,c\,b^3\right )}{b^3\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {a\,d^2\,n\,x^6\,{\left (a+b\,x\right )}^n\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{b\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}-\frac {6\,a^2\,d^2\,n\,x^5\,{\left (a+b\,x\right )}^n\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{b^2\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {2\,a\,d\,n\,x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (-60\,d\,a^3+c\,b^3\,n^3+18\,c\,b^3\,n^2+107\,c\,b^3\,n+210\,c\,b^3\right )}{b^4\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}-\frac {6\,a^2\,d\,n\,x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (-60\,d\,a^3+c\,b^3\,n^3+18\,c\,b^3\,n^2+107\,c\,b^3\,n+210\,c\,b^3\right )}{b^5\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x^3)^2*(a + b*x)^n,x)
[Out]
(a*(a + b*x)^n*(720*a^6*d^2 + 5040*b^6*c^2 + 8028*b^6*c^2*n + 5104*b^6*c^2*n^2 + 1665*b^6*c^2*n^3 + 295*b^6*c^
2*n^4 + 27*b^6*c^2*n^5 + b^6*c^2*n^6 - 2520*a^3*b^3*c*d - 1284*a^3*b^3*c*d*n - 216*a^3*b^3*c*d*n^2 - 12*a^3*b^
3*c*d*n^3))/(b^7*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (d^2*x^7*(a +
b*x)^n*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720))/(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^
4 + 322*n^5 + 28*n^6 + n^7 + 5040) + (x*(a + b*x)^n*(5040*b^7*c^2 + 8028*b^7*c^2*n + 5104*b^7*c^2*n^2 + 1665*b
^7*c^2*n^3 + 295*b^7*c^2*n^4 + 27*b^7*c^2*n^5 + b^7*c^2*n^6 - 720*a^6*b*d^2*n + 2520*a^3*b^4*c*d*n + 1284*a^3*
b^4*c*d*n^2 + 216*a^3*b^4*c*d*n^3 + 12*a^3*b^4*c*d*n^4))/(b^7*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322
*n^5 + 28*n^6 + n^7 + 5040)) + (2*d*x^4*(a + b*x)^n*(11*n + 6*n^2 + n^3 + 6)*(210*b^3*c + 18*b^3*c*n^2 + b^3*c
*n^3 + 15*a^3*d*n + 107*b^3*c*n))/(b^3*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5
040)) + (a*d^2*n*x^6*(a + b*x)^n*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120))/(b*(13068*n + 13132*n^2 + 67
69*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) - (6*a^2*d^2*n*x^5*(a + b*x)^n*(50*n + 35*n^2 + 10*n^3 + n
^4 + 24))/(b^2*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (2*a*d*n*x^3*(a
+ b*x)^n*(3*n + n^2 + 2)*(210*b^3*c - 60*a^3*d + 18*b^3*c*n^2 + b^3*c*n^3 + 107*b^3*c*n))/(b^4*(13068*n + 1313
2*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) - (6*a^2*d*n*x^2*(n + 1)*(a + b*x)^n*(210*b^3*c
- 60*a^3*d + 18*b^3*c*n^2 + b^3*c*n^3 + 107*b^3*c*n))/(b^5*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^
5 + 28*n^6 + n^7 + 5040))
________________________________________________________________________________________
sympy [A] time = 14.44, size = 11851, normalized size = 58.38 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**n*(d*x**3+c)**2,x)
[Out]
Piecewise((a**n*(c**2*x + c*d*x**4/2 + d**2*x**7/7), Eq(b, 0)), (60*a**6*d**2*log(a/b + x)/(60*a**6*b**7 + 360
*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x
**6) + 147*a**6*d**2/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b*
*11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 360*a**5*b*d**2*x*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x
+ 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 822*a*
*5*b*d**2*x/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4
+ 360*a*b**12*x**5 + 60*b**13*x**6) + 900*a**4*b**2*d**2*x**2*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 9
00*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 1875*a**4
*b**2*d**2*x**2/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x
**4 + 360*a*b**12*x**5 + 60*b**13*x**6) - 2*a**3*b**3*c*d/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2
+ 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 1200*a**3*b**3*d**2*x**3*l
og(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4
+ 360*a*b**12*x**5 + 60*b**13*x**6) + 2200*a**3*b**3*d**2*x**3/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9
*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) - 12*a**2*b**4*c*d*x/(6
0*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*
x**5 + 60*b**13*x**6) + 900*a**2*b**4*d**2*x**4*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x
**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 1350*a**2*b**4*d**2*x**
4/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b*
*12*x**5 + 60*b**13*x**6) - 30*a*b**5*c*d*x**2/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**
3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 360*a*b**5*d**2*x**5*log(a/b + x)/(60
*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x
**5 + 60*b**13*x**6) + 360*a*b**5*d**2*x**5/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b
**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) - 10*b**6*c**2/(60*a**6*b**7 + 360*a**5*b*
*8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) - 4
0*b**6*c*d*x**3/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x
**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 60*b**6*d**2*x**6*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 900
*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6), Eq(n, -7)),
(-60*a**6*d**2*log(a/b + x)/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b
**11*x**4 + 10*b**12*x**5) - 137*a**6*d**2/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**1
0*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 300*a**5*b*d**2*x*log(a/b + x)/(10*a**5*b**7 + 50*a**4*b**8*x + 10
0*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 625*a**5*b*d**2*x/(10*a**5*b**7 +
50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 600*a**4*b**2*d
**2*x**2*log(a/b + x)/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x
**4 + 10*b**12*x**5) - 1100*a**4*b**2*d**2*x**2/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2
*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - a**3*b**3*c*d/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*
x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 600*a**3*b**3*d**2*x**3*log(a/b + x)/(10*a**5*
b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 900*a**3
*b**3*d**2*x**3/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 +
10*b**12*x**5) - 5*a**2*b**4*c*d*x/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 +
50*a*b**11*x**4 + 10*b**12*x**5) - 300*a**2*b**4*d**2*x**4*log(a/b + x)/(10*a**5*b**7 + 50*a**4*b**8*x + 100*
a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 300*a**2*b**4*d**2*x**4/(10*a**5*b**
7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 10*a*b**5*c
*d*x**2/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12
*x**5) - 60*a*b**5*d**2*x**5*log(a/b + x)/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10
*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 2*b**6*c**2/(10*a**5*b**7 + 50*a**4*b**8*x + 100*a**3*b**9*x**2 + 1
00*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) - 10*b**6*c*d*x**3/(10*a**5*b**7 + 50*a**4*b**8*x + 100*
a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5) + 10*b**6*d**2*x**6/(10*a**5*b**7 + 50
*a**4*b**8*x + 100*a**3*b**9*x**2 + 100*a**2*b**10*x**3 + 50*a*b**11*x**4 + 10*b**12*x**5), Eq(n, -6)), (60*a*
*6*d**2*log(a/b + x)/(4*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b**11*x**4) + 125
*a**6*d**2/(4*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b**11*x**4) + 240*a**5*b*d*
*2*x*log(a/b + x)/(4*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b**11*x**4) + 440*a*
*5*b*d**2*x/(4*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b**11*x**4) + 360*a**4*b**
2*d**2*x**2*log(a/b + x)/(4*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b**11*x**4) +
540*a**4*b**2*d**2*x**2/(4*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b**11*x**4) -
2*a**3*b**3*c*d/(4*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b**11*x**4) + 240*a**
3*b**3*d**2*x**3*log(a/b + x)/(4*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b**11*x*
*4) + 240*a**3*b**3*d**2*x**3/(4*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b**11*x*
*4) - 8*a**2*b**4*c*d*x/(4*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b**11*x**4) +
60*a**2*b**4*d**2*x**4*log(a/b + x)/(4*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b*
*11*x**4) - 12*a*b**5*c*d*x**2/(4*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b**11*x
**4) - 12*a*b**5*d**2*x**5/(4*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b**11*x**4)
- b**6*c**2/(4*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b**11*x**4) - 8*b**6*c*d*
x**3/(4*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b**11*x**4) + 2*b**6*d**2*x**6/(4
*a**4*b**7 + 16*a**3*b**8*x + 24*a**2*b**9*x**2 + 16*a*b**10*x**3 + 4*b**11*x**4), Eq(n, -5)), (-60*a**6*d**2*
log(a/b + x)/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 110*a**6*d**2/(3*a**3*b**7 + 9*a**
2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 180*a**5*b*d**2*x*log(a/b + x)/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b
**9*x**2 + 3*b**10*x**3) - 270*a**5*b*d**2*x/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 18
0*a**4*b**2*d**2*x**2*log(a/b + x)/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - 180*a**4*b**
2*d**2*x**2/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) + 6*a**3*b**3*c*d*log(a/b + x)/(3*a**
3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) + 11*a**3*b**3*c*d/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b
**9*x**2 + 3*b**10*x**3) - 60*a**3*b**3*d**2*x**3*log(a/b + x)/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 +
3*b**10*x**3) + 18*a**2*b**4*c*d*x*log(a/b + x)/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) +
27*a**2*b**4*c*d*x/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) + 15*a**2*b**4*d**2*x**4/(3*a
**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) + 18*a*b**5*c*d*x**2*log(a/b + x)/(3*a**3*b**7 + 9*a*
*2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) + 18*a*b**5*c*d*x**2/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 +
3*b**10*x**3) - 3*a*b**5*d**2*x**5/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) - b**6*c**2/(3
*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) + 6*b**6*c*d*x**3*log(a/b + x)/(3*a**3*b**7 + 9*a**
2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) + b**6*d**2*x**6/(3*a**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**
10*x**3), Eq(n, -4)), (60*a**6*d**2*log(a/b + x)/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 90*a**6*d**2/(4*a*
*2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 120*a**5*b*d**2*x*log(a/b + x)/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2)
+ 120*a**5*b*d**2*x/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 60*a**4*b**2*d**2*x**2*log(a/b + x)/(4*a**2*b**
7 + 8*a*b**8*x + 4*b**9*x**2) - 24*a**3*b**3*c*d*log(a/b + x)/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) - 36*a*
*3*b**3*c*d/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) - 20*a**3*b**3*d**2*x**3/(4*a**2*b**7 + 8*a*b**8*x + 4*b*
*9*x**2) - 48*a**2*b**4*c*d*x*log(a/b + x)/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) - 48*a**2*b**4*c*d*x/(4*a*
*2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 5*a**2*b**4*d**2*x**4/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) - 24*a*b*
*5*c*d*x**2*log(a/b + x)/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) - 2*a*b**5*d**2*x**5/(4*a**2*b**7 + 8*a*b**8
*x + 4*b**9*x**2) - 2*b**6*c**2/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2) + 8*b**6*c*d*x**3/(4*a**2*b**7 + 8*a*
b**8*x + 4*b**9*x**2) + b**6*d**2*x**6/(4*a**2*b**7 + 8*a*b**8*x + 4*b**9*x**2), Eq(n, -3)), (-60*a**6*d**2*lo
g(a/b + x)/(10*a*b**7 + 10*b**8*x) - 60*a**6*d**2/(10*a*b**7 + 10*b**8*x) - 60*a**5*b*d**2*x*log(a/b + x)/(10*
a*b**7 + 10*b**8*x) + 30*a**4*b**2*d**2*x**2/(10*a*b**7 + 10*b**8*x) + 60*a**3*b**3*c*d*log(a/b + x)/(10*a*b**
7 + 10*b**8*x) + 60*a**3*b**3*c*d/(10*a*b**7 + 10*b**8*x) - 10*a**3*b**3*d**2*x**3/(10*a*b**7 + 10*b**8*x) + 6
0*a**2*b**4*c*d*x*log(a/b + x)/(10*a*b**7 + 10*b**8*x) + 5*a**2*b**4*d**2*x**4/(10*a*b**7 + 10*b**8*x) - 30*a*
b**5*c*d*x**2/(10*a*b**7 + 10*b**8*x) - 3*a*b**5*d**2*x**5/(10*a*b**7 + 10*b**8*x) - 10*b**6*c**2/(10*a*b**7 +
10*b**8*x) + 10*b**6*c*d*x**3/(10*a*b**7 + 10*b**8*x) + 2*b**6*d**2*x**6/(10*a*b**7 + 10*b**8*x), Eq(n, -2)),
(a**6*d**2*log(a/b + x)/b**7 - a**5*d**2*x/b**6 + a**4*d**2*x**2/(2*b**5) - 2*a**3*c*d*log(a/b + x)/b**4 - a*
*3*d**2*x**3/(3*b**4) + 2*a**2*c*d*x/b**3 + a**2*d**2*x**4/(4*b**3) - a*c*d*x**2/b**2 - a*d**2*x**5/(5*b**2) +
c**2*log(a/b + x)/b + 2*c*d*x**3/(3*b) + d**2*x**6/(6*b), Eq(n, -1)), (720*a**7*d**2*(a + b*x)**n/(b**7*n**7
+ 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7)
- 720*a**6*b*d**2*n*x*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**
3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 360*a**5*b**2*d**2*n**2*x**2*(a + b*x)**n/(b**7*n**7 + 28*b*
*7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 360*
a**5*b**2*d**2*n*x**2*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3
+ 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 12*a**4*b**3*c*d*n**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6
+ 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 216*a**4*b**
3*c*d*n**2*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b*
*7*n**2 + 13068*b**7*n + 5040*b**7) - 1284*a**4*b**3*c*d*n*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n
**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 2520*a**4*b**3*c*d*(a +
b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*
b**7*n + 5040*b**7) - 120*a**4*b**3*d**2*n**3*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 19
60*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 360*a**4*b**3*d**2*n**2*x**3*(a
+ b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 1306
8*b**7*n + 5040*b**7) - 240*a**4*b**3*d**2*n*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 196
0*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 12*a**3*b**4*c*d*n**4*x*(a + b*x)
**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7
*n + 5040*b**7) + 216*a**3*b**4*c*d*n**3*x*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*
n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 1284*a**3*b**4*c*d*n**2*x*(a + b*x)**n/(
b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n +
5040*b**7) + 2520*a**3*b**4*c*d*n*x*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 +
6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 30*a**3*b**4*d**2*n**4*x**4*(a + b*x)**n/(b**7*
n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*
b**7) + 180*a**3*b**4*d**2*n**3*x**4*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 +
6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 330*a**3*b**4*d**2*n**2*x**4*(a + b*x)**n/(b**
7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 504
0*b**7) + 180*a**3*b**4*d**2*n*x**4*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 +
6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 6*a**2*b**5*c*d*n**5*x**2*(a + b*x)**n/(b**7*n*
*7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b*
*7) - 114*a**2*b**5*c*d*n**4*x**2*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 67
69*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 750*a**2*b**5*c*d*n**3*x**2*(a + b*x)**n/(b**7*n*
*7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b*
*7) - 1902*a**2*b**5*c*d*n**2*x**2*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6
769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 1260*a**2*b**5*c*d*n*x**2*(a + b*x)**n/(b**7*n**
7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**
7) - 6*a**2*b**5*d**2*n**5*x**5*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769
*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 60*a**2*b**5*d**2*n**4*x**5*(a + b*x)**n/(b**7*n**7
+ 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7
) - 210*a**2*b**5*d**2*n**3*x**5*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 676
9*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) - 300*a**2*b**5*d**2*n**2*x**5*(a + b*x)**n/(b**7*n*
*7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b*
*7) - 144*a**2*b**5*d**2*n*x**5*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769
*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + a*b**6*c**2*n**6*(a + b*x)**n/(b**7*n**7 + 28*b**7*
n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 27*a*b*
*6*c**2*n**5*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*
b**7*n**2 + 13068*b**7*n + 5040*b**7) + 295*a*b**6*c**2*n**4*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7
*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 1665*a*b**6*c**2*n**3*
(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 1
3068*b**7*n + 5040*b**7) + 5104*a*b**6*c**2*n**2*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960
*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 8028*a*b**6*c**2*n*(a + b*x)**n/(b
**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5
040*b**7) + 5040*a*b**6*c**2*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b*
*7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 2*a*b**6*c*d*n**6*x**3*(a + b*x)**n/(b**7*n**7 + 28*b*
*7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 42*a
*b**6*c*d*n**5*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 +
13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 326*a*b**6*c*d*n**4*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6
+ 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 1134*a*b**6
*c*d*n**3*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 1313
2*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 1688*a*b**6*c*d*n**2*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 3
22*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 840*a*b**6*c*d*
n*x**3*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n
**2 + 13068*b**7*n + 5040*b**7) + a*b**6*d**2*n**6*x**6*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5
+ 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 15*a*b**6*d**2*n**5*x**6*(a
+ b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 130
68*b**7*n + 5040*b**7) + 85*a*b**6*d**2*n**4*x**6*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 196
0*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 225*a*b**6*d**2*n**3*x**6*(a + b*
x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b*
*7*n + 5040*b**7) + 274*a*b**6*d**2*n**2*x**6*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b*
*7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 120*a*b**6*d**2*n*x**6*(a + b*x)**n/(
b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n +
5040*b**7) + b**7*c**2*n**6*x*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b
**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 27*b**7*c**2*n**5*x*(a + b*x)**n/(b**7*n**7 + 28*b**7
*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 295*b*
*7*c**2*n**4*x*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 1313
2*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 1665*b**7*c**2*n**3*x*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b
**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 5104*b**7*c**2*n**2
*x*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2
+ 13068*b**7*n + 5040*b**7) + 8028*b**7*c**2*n*x*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960
*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 5040*b**7*c**2*x*(a + b*x)**n/(b**
7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 504
0*b**7) + 2*b**7*c*d*n**6*x**4*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*
b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 48*b**7*c*d*n**5*x**4*(a + b*x)**n/(b**7*n**7 + 28*b
**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 452
*b**7*c*d*n**4*x**4*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 +
13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 2112*b**7*c*d*n**3*x**4*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6
+ 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 5090*b**7*c*
d*n**2*x**4*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b
**7*n**2 + 13068*b**7*n + 5040*b**7) + 5904*b**7*c*d*n*x**4*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*
n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 2520*b**7*c*d*x**4*(a +
b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068
*b**7*n + 5040*b**7) + b**7*d**2*n**6*x**7*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*
n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 21*b**7*d**2*n**5*x**7*(a + b*x)**n/(b**
7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 504
0*b**7) + 175*b**7*d**2*n**4*x**7*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 67
69*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 735*b**7*d**2*n**3*x**7*(a + b*x)**n/(b**7*n**7 +
28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7)
+ 1624*b**7*d**2*n**2*x**7*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7
*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 1764*b**7*d**2*n*x**7*(a + b*x)**n/(b**7*n**7 + 28*b**7*
n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b**7*n**2 + 13068*b**7*n + 5040*b**7) + 720*b**
7*d**2*x**7*(a + b*x)**n/(b**7*n**7 + 28*b**7*n**6 + 322*b**7*n**5 + 1960*b**7*n**4 + 6769*b**7*n**3 + 13132*b
**7*n**2 + 13068*b**7*n + 5040*b**7), True))
________________________________________________________________________________________