3.356 \(\int x (a+b x)^n \left (c+d x^2\right )^2 \, dx\)
Optimal. Leaf size=185 \[ -\frac{a \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+1}}{b^6 (n+1)}+\frac{\left (a^2 d+b^2 c\right ) \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^6 (n+2)}-\frac{2 a d \left (5 a^2 d+3 b^2 c\right ) (a+b x)^{n+3}}{b^6 (n+3)}+\frac{2 d \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+4}}{b^6 (n+4)}-\frac{5 a d^2 (a+b x)^{n+5}}{b^6 (n+5)}+\frac{d^2 (a+b x)^{n+6}}{b^6 (n+6)} \]
[Out]
-((a*(b^2*c + a^2*d)^2*(a + b*x)^(1 + n))/(b^6*(1 + n))) + ((b^2*c + a^2*d)*(b^2
*c + 5*a^2*d)*(a + b*x)^(2 + n))/(b^6*(2 + n)) - (2*a*d*(3*b^2*c + 5*a^2*d)*(a +
b*x)^(3 + n))/(b^6*(3 + n)) + (2*d*(b^2*c + 5*a^2*d)*(a + b*x)^(4 + n))/(b^6*(4
+ n)) - (5*a*d^2*(a + b*x)^(5 + n))/(b^6*(5 + n)) + (d^2*(a + b*x)^(6 + n))/(b^
6*(6 + n))
_______________________________________________________________________________________
Rubi [A] time = 0.230561, antiderivative size = 185, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056
\[ -\frac{a \left (a^2 d+b^2 c\right )^2 (a+b x)^{n+1}}{b^6 (n+1)}+\frac{\left (a^2 d+b^2 c\right ) \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^6 (n+2)}-\frac{2 a d \left (5 a^2 d+3 b^2 c\right ) (a+b x)^{n+3}}{b^6 (n+3)}+\frac{2 d \left (5 a^2 d+b^2 c\right ) (a+b x)^{n+4}}{b^6 (n+4)}-\frac{5 a d^2 (a+b x)^{n+5}}{b^6 (n+5)}+\frac{d^2 (a+b x)^{n+6}}{b^6 (n+6)} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x)^n*(c + d*x^2)^2,x]
[Out]
-((a*(b^2*c + a^2*d)^2*(a + b*x)^(1 + n))/(b^6*(1 + n))) + ((b^2*c + a^2*d)*(b^2
*c + 5*a^2*d)*(a + b*x)^(2 + n))/(b^6*(2 + n)) - (2*a*d*(3*b^2*c + 5*a^2*d)*(a +
b*x)^(3 + n))/(b^6*(3 + n)) + (2*d*(b^2*c + 5*a^2*d)*(a + b*x)^(4 + n))/(b^6*(4
+ n)) - (5*a*d^2*(a + b*x)^(5 + n))/(b^6*(5 + n)) + (d^2*(a + b*x)^(6 + n))/(b^
6*(6 + n))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 47.599, size = 170, normalized size = 0.92 \[ - \frac{5 a d^{2} \left (a + b x\right )^{n + 5}}{b^{6} \left (n + 5\right )} - \frac{2 a d \left (a + b x\right )^{n + 3} \left (5 a^{2} d + 3 b^{2} c\right )}{b^{6} \left (n + 3\right )} - \frac{a \left (a + b x\right )^{n + 1} \left (a^{2} d + b^{2} c\right )^{2}}{b^{6} \left (n + 1\right )} + \frac{d^{2} \left (a + b x\right )^{n + 6}}{b^{6} \left (n + 6\right )} + \frac{2 d \left (a + b x\right )^{n + 4} \left (5 a^{2} d + b^{2} c\right )}{b^{6} \left (n + 4\right )} + \frac{\left (a + b x\right )^{n + 2} \left (a^{2} d + b^{2} c\right ) \left (5 a^{2} d + b^{2} c\right )}{b^{6} \left (n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x+a)**n*(d*x**2+c)**2,x)
[Out]
-5*a*d**2*(a + b*x)**(n + 5)/(b**6*(n + 5)) - 2*a*d*(a + b*x)**(n + 3)*(5*a**2*d
+ 3*b**2*c)/(b**6*(n + 3)) - a*(a + b*x)**(n + 1)*(a**2*d + b**2*c)**2/(b**6*(n
+ 1)) + d**2*(a + b*x)**(n + 6)/(b**6*(n + 6)) + 2*d*(a + b*x)**(n + 4)*(5*a**2
*d + b**2*c)/(b**6*(n + 4)) + (a + b*x)**(n + 2)*(a**2*d + b**2*c)*(5*a**2*d + b
**2*c)/(b**6*(n + 2))
_______________________________________________________________________________________
Mathematica [A] time = 0.288424, size = 279, normalized size = 1.51 \[ \frac{(a+b x)^{n+1} \left (-120 a^5 d^2+120 a^4 b d^2 (n+1) x-12 a^3 b^2 d \left (c \left (n^2+11 n+30\right )+5 d \left (n^2+3 n+2\right ) x^2\right )+4 a^2 b^3 d (n+1) x \left (3 c \left (n^2+11 n+30\right )+5 d \left (n^2+5 n+6\right ) x^2\right )-a b^4 \left (c^2 \left (n^4+18 n^3+119 n^2+342 n+360\right )+6 c d \left (n^4+14 n^3+65 n^2+112 n+60\right ) x^2+5 d^2 \left (n^4+10 n^3+35 n^2+50 n+24\right ) x^4\right )+b^5 \left (n^3+9 n^2+23 n+15\right ) x \left (c^2 \left (n^2+10 n+24\right )+2 c d \left (n^2+8 n+12\right ) x^2+d^2 \left (n^2+6 n+8\right ) x^4\right )\right )}{b^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6)} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x)^n*(c + d*x^2)^2,x]
[Out]
((a + b*x)^(1 + n)*(-120*a^5*d^2 + 120*a^4*b*d^2*(1 + n)*x - 12*a^3*b^2*d*(c*(30
+ 11*n + n^2) + 5*d*(2 + 3*n + n^2)*x^2) + 4*a^2*b^3*d*(1 + n)*x*(3*c*(30 + 11*
n + n^2) + 5*d*(6 + 5*n + n^2)*x^2) + b^5*(15 + 23*n + 9*n^2 + n^3)*x*(c^2*(24 +
10*n + n^2) + 2*c*d*(12 + 8*n + n^2)*x^2 + d^2*(8 + 6*n + n^2)*x^4) - a*b^4*(c^
2*(360 + 342*n + 119*n^2 + 18*n^3 + n^4) + 6*c*d*(60 + 112*n + 65*n^2 + 14*n^3 +
n^4)*x^2 + 5*d^2*(24 + 50*n + 35*n^2 + 10*n^3 + n^4)*x^4)))/(b^6*(1 + n)*(2 + n
)*(3 + n)*(4 + n)*(5 + n)*(6 + n))
_______________________________________________________________________________________
Maple [B] time = 0.015, size = 677, normalized size = 3.7 \[ -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -{b}^{5}{d}^{2}{n}^{5}{x}^{5}-15\,{b}^{5}{d}^{2}{n}^{4}{x}^{5}+5\,a{b}^{4}{d}^{2}{n}^{4}{x}^{4}-2\,{b}^{5}cd{n}^{5}{x}^{3}-85\,{b}^{5}{d}^{2}{n}^{3}{x}^{5}+50\,a{b}^{4}{d}^{2}{n}^{3}{x}^{4}-34\,{b}^{5}cd{n}^{4}{x}^{3}-225\,{b}^{5}{d}^{2}{n}^{2}{x}^{5}-20\,{a}^{2}{b}^{3}{d}^{2}{n}^{3}{x}^{3}+6\,a{b}^{4}cd{n}^{4}{x}^{2}+175\,a{b}^{4}{d}^{2}{n}^{2}{x}^{4}-{b}^{5}{c}^{2}{n}^{5}x-214\,{b}^{5}cd{n}^{3}{x}^{3}-274\,{b}^{5}{d}^{2}n{x}^{5}-120\,{a}^{2}{b}^{3}{d}^{2}{n}^{2}{x}^{3}+84\,a{b}^{4}cd{n}^{3}{x}^{2}+250\,a{b}^{4}{d}^{2}n{x}^{4}-19\,{b}^{5}{c}^{2}{n}^{4}x-614\,{b}^{5}cd{n}^{2}{x}^{3}-120\,{d}^{2}{x}^{5}{b}^{5}+60\,{a}^{3}{b}^{2}{d}^{2}{n}^{2}{x}^{2}-12\,{a}^{2}{b}^{3}cd{n}^{3}x-220\,{a}^{2}{b}^{3}{d}^{2}n{x}^{3}+a{b}^{4}{c}^{2}{n}^{4}+390\,a{b}^{4}cd{n}^{2}{x}^{2}+120\,a{d}^{2}{x}^{4}{b}^{4}-137\,{b}^{5}{c}^{2}{n}^{3}x-792\,{b}^{5}cdn{x}^{3}+180\,{a}^{3}{b}^{2}{d}^{2}n{x}^{2}-144\,{a}^{2}{b}^{3}cd{n}^{2}x-120\,{a}^{2}{b}^{3}{d}^{2}{x}^{3}+18\,a{b}^{4}{c}^{2}{n}^{3}+672\,a{b}^{4}cdn{x}^{2}-461\,{b}^{5}{c}^{2}{n}^{2}x-360\,{b}^{5}cd{x}^{3}-120\,{a}^{4}b{d}^{2}nx+12\,{a}^{3}{b}^{2}cd{n}^{2}+120\,{a}^{3}{b}^{2}{d}^{2}{x}^{2}-492\,{a}^{2}{b}^{3}cdnx+119\,a{b}^{4}{c}^{2}{n}^{2}+360\,a{b}^{4}cd{x}^{2}-702\,{b}^{5}{c}^{2}nx-120\,{a}^{4}b{d}^{2}x+132\,{a}^{3}{b}^{2}cdn-360\,{a}^{2}{b}^{3}cdx+342\,a{b}^{4}{c}^{2}n-360\,{b}^{5}{c}^{2}x+120\,{a}^{5}{d}^{2}+360\,{a}^{3}{b}^{2}cd+360\,a{b}^{4}{c}^{2} \right ) }{{b}^{6} \left ({n}^{6}+21\,{n}^{5}+175\,{n}^{4}+735\,{n}^{3}+1624\,{n}^{2}+1764\,n+720 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x+a)^n*(d*x^2+c)^2,x)
[Out]
-(b*x+a)^(1+n)*(-b^5*d^2*n^5*x^5-15*b^5*d^2*n^4*x^5+5*a*b^4*d^2*n^4*x^4-2*b^5*c*
d*n^5*x^3-85*b^5*d^2*n^3*x^5+50*a*b^4*d^2*n^3*x^4-34*b^5*c*d*n^4*x^3-225*b^5*d^2
*n^2*x^5-20*a^2*b^3*d^2*n^3*x^3+6*a*b^4*c*d*n^4*x^2+175*a*b^4*d^2*n^2*x^4-b^5*c^
2*n^5*x-214*b^5*c*d*n^3*x^3-274*b^5*d^2*n*x^5-120*a^2*b^3*d^2*n^2*x^3+84*a*b^4*c
*d*n^3*x^2+250*a*b^4*d^2*n*x^4-19*b^5*c^2*n^4*x-614*b^5*c*d*n^2*x^3-120*b^5*d^2*
x^5+60*a^3*b^2*d^2*n^2*x^2-12*a^2*b^3*c*d*n^3*x-220*a^2*b^3*d^2*n*x^3+a*b^4*c^2*
n^4+390*a*b^4*c*d*n^2*x^2+120*a*b^4*d^2*x^4-137*b^5*c^2*n^3*x-792*b^5*c*d*n*x^3+
180*a^3*b^2*d^2*n*x^2-144*a^2*b^3*c*d*n^2*x-120*a^2*b^3*d^2*x^3+18*a*b^4*c^2*n^3
+672*a*b^4*c*d*n*x^2-461*b^5*c^2*n^2*x-360*b^5*c*d*x^3-120*a^4*b*d^2*n*x+12*a^3*
b^2*c*d*n^2+120*a^3*b^2*d^2*x^2-492*a^2*b^3*c*d*n*x+119*a*b^4*c^2*n^2+360*a*b^4*
c*d*x^2-702*b^5*c^2*n*x-120*a^4*b*d^2*x+132*a^3*b^2*c*d*n-360*a^2*b^3*c*d*x+342*
a*b^4*c^2*n-360*b^5*c^2*x+120*a^5*d^2+360*a^3*b^2*c*d+360*a*b^4*c^2)/b^6/(n^6+21
*n^5+175*n^4+735*n^3+1624*n^2+1764*n+720)
_______________________________________________________________________________________
Maxima [A] time = 0.722806, size = 452, normalized size = 2.44 \[ \frac{{\left (b^{2}{\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )}{\left (b x + a\right )}^{n} c^{2}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac{2 \,{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} +{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \,{\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )}{\left (b x + a\right )}^{n} c d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} + \frac{{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} x^{6} +{\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} x^{5} - 5 \,{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} x^{4} + 20 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} x^{3} - 60 \,{\left (n^{2} + n\right )} a^{4} b^{2} x^{2} + 120 \, a^{5} b n x - 120 \, a^{6}\right )}{\left (b x + a\right )}^{n} d^{2}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^2*(b*x + a)^n*x,x, algorithm="maxima")
[Out]
(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^2/((n^2 + 3*n + 2)*b^2) + 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^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^6*x^6 + (n^5 + 10*
n^4 + 35*n^3 + 50*n^2 + 24*n)*a*b^5*x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^2*b^4
*x^4 + 20*(n^3 + 3*n^2 + 2*n)*a^3*b^3*x^3 - 60*(n^2 + n)*a^4*b^2*x^2 + 120*a^5*b
*n*x - 120*a^6)*(b*x + a)^n*d^2/((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 +
1764*n + 720)*b^6)
_______________________________________________________________________________________
Fricas [A] time = 0.282535, size = 1022, normalized size = 5.52 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^2*(b*x + a)^n*x,x, algorithm="fricas")
[Out]
-(a^2*b^4*c^2*n^4 + 18*a^2*b^4*c^2*n^3 + 360*a^2*b^4*c^2 + 360*a^4*b^2*c*d + 120
*a^6*d^2 - (b^6*d^2*n^5 + 15*b^6*d^2*n^4 + 85*b^6*d^2*n^3 + 225*b^6*d^2*n^2 + 27
4*b^6*d^2*n + 120*b^6*d^2)*x^6 - (a*b^5*d^2*n^5 + 10*a*b^5*d^2*n^4 + 35*a*b^5*d^
2*n^3 + 50*a*b^5*d^2*n^2 + 24*a*b^5*d^2*n)*x^5 - (2*b^6*c*d*n^5 + 360*b^6*c*d +
(34*b^6*c*d - 5*a^2*b^4*d^2)*n^4 + 2*(107*b^6*c*d - 15*a^2*b^4*d^2)*n^3 + (614*b
^6*c*d - 55*a^2*b^4*d^2)*n^2 + 6*(132*b^6*c*d - 5*a^2*b^4*d^2)*n)*x^4 - 2*(a*b^5
*c*d*n^5 + 14*a*b^5*c*d*n^4 + 5*(13*a*b^5*c*d + 2*a^3*b^3*d^2)*n^3 + 2*(56*a*b^5
*c*d + 15*a^3*b^3*d^2)*n^2 + 20*(3*a*b^5*c*d + a^3*b^3*d^2)*n)*x^3 + (119*a^2*b^
4*c^2 + 12*a^4*b^2*c*d)*n^2 - (b^6*c^2*n^5 + 360*b^6*c^2 + (19*b^6*c^2 - 6*a^2*b
^4*c*d)*n^4 + (137*b^6*c^2 - 72*a^2*b^4*c*d)*n^3 + (461*b^6*c^2 - 246*a^2*b^4*c*
d - 60*a^4*b^2*d^2)*n^2 + 6*(117*b^6*c^2 - 30*a^2*b^4*c*d - 10*a^4*b^2*d^2)*n)*x
^2 + 6*(57*a^2*b^4*c^2 + 22*a^4*b^2*c*d)*n - (a*b^5*c^2*n^5 + 18*a*b^5*c^2*n^4 +
(119*a*b^5*c^2 + 12*a^3*b^3*c*d)*n^3 + 6*(57*a*b^5*c^2 + 22*a^3*b^3*c*d)*n^2 +
120*(3*a*b^5*c^2 + 3*a^3*b^3*c*d + a^5*b*d^2)*n)*x)*(b*x + a)^n/(b^6*n^6 + 21*b^
6*n^5 + 175*b^6*n^4 + 735*b^6*n^3 + 1624*b^6*n^2 + 1764*b^6*n + 720*b^6)
_______________________________________________________________________________________
Sympy [A] time = 25.6743, size = 9083, normalized size = 49.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x+a)**n*(d*x**2+c)**2,x)
[Out]
Piecewise((a**n*(c**2*x**2/2 + c*d*x**4/2 + d**2*x**6/6), Eq(b, 0)), (60*a**5*d*
*2*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*
b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 137*a**5*d**2/(60*a**5*b**6 + 30
0*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*
b**11*x**5) + 300*a**4*b*d**2*x*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 6
00*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 625
*a**4*b*d**2*x/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b
**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) - 6*a**3*b**2*c*d/(60*a**5*b**6 + 3
00*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60
*b**11*x**5) + 600*a**3*b**2*d**2*x**2*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**
7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5
) + 1100*a**3*b**2*d**2*x**2/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**
2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) - 30*a**2*b**3*c*d*x/
(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*
a*b**10*x**4 + 60*b**11*x**5) + 600*a**2*b**3*d**2*x**3*log(a/b + x)/(60*a**5*b*
*6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**
4 + 60*b**11*x**5) + 900*a**2*b**3*d**2*x**3/(60*a**5*b**6 + 300*a**4*b**7*x + 6
00*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) - 3*a
*b**4*c**2/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*
x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) - 60*a*b**4*c*d*x**2/(60*a**5*b**6 + 30
0*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*
b**11*x**5) + 300*a*b**4*d**2*x**4*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x
+ 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) +
300*a*b**4*d**2*x**4/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*
a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) - 15*b**5*c**2*x/(60*a**5*b**
6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4
+ 60*b**11*x**5) - 60*b**5*c*d*x**3/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*
b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 60*b**5*d**
2*x**5*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a
**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5), Eq(n, -6)), (-60*a**5*d**2*lo
g(a/b + x)/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 +
12*b**10*x**4) - 125*a**5*d**2/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x*
*2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 240*a**4*b*d**2*x*log(a/b + x)/(12*a**4*b
**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 440
*a**4*b*d**2*x/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x*
*3 + 12*b**10*x**4) - 6*a**3*b**2*c*d/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b
**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 360*a**3*b**2*d**2*x**2*log(a/b + x
)/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10
*x**4) - 540*a**3*b**2*d**2*x**2/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x
**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 24*a**2*b**3*c*d*x/(12*a**4*b**6 + 48*a*
*3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 240*a**2*b**3*
d**2*x**3*log(a/b + x)/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a
*b**9*x**3 + 12*b**10*x**4) - 240*a**2*b**3*d**2*x**3/(12*a**4*b**6 + 48*a**3*b*
*7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - a*b**4*c**2/(12*a**
4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) -
36*a*b**4*c*d*x**2/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**
9*x**3 + 12*b**10*x**4) - 60*a*b**4*d**2*x**4*log(a/b + x)/(12*a**4*b**6 + 48*a*
*3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 4*b**5*c**2*x/
(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x
**4) - 24*b**5*c*d*x**3/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*
a*b**9*x**3 + 12*b**10*x**4) + 12*b**5*d**2*x**5/(12*a**4*b**6 + 48*a**3*b**7*x
+ 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4), Eq(n, -5)), (60*a**7*d**2
*log(a/b + x)/(6*a**5*b**6 + 18*a**4*b**7*x + 18*a**3*b**8*x**2 + 6*a**2*b**9*x*
*3) + 20*a**7*d**2/(6*a**5*b**6 + 18*a**4*b**7*x + 18*a**3*b**8*x**2 + 6*a**2*b*
*9*x**3) + 180*a**6*b*d**2*x*log(a/b + x)/(6*a**5*b**6 + 18*a**4*b**7*x + 18*a**
3*b**8*x**2 + 6*a**2*b**9*x**3) + 12*a**5*b**2*c*d*log(a/b + x)/(6*a**5*b**6 + 1
8*a**4*b**7*x + 18*a**3*b**8*x**2 + 6*a**2*b**9*x**3) + 4*a**5*b**2*c*d/(6*a**5*
b**6 + 18*a**4*b**7*x + 18*a**3*b**8*x**2 + 6*a**2*b**9*x**3) + 180*a**5*b**2*d*
*2*x**2*log(a/b + x)/(6*a**5*b**6 + 18*a**4*b**7*x + 18*a**3*b**8*x**2 + 6*a**2*
b**9*x**3) - 90*a**5*b**2*d**2*x**2/(6*a**5*b**6 + 18*a**4*b**7*x + 18*a**3*b**8
*x**2 + 6*a**2*b**9*x**3) + 36*a**4*b**3*c*d*x*log(a/b + x)/(6*a**5*b**6 + 18*a*
*4*b**7*x + 18*a**3*b**8*x**2 + 6*a**2*b**9*x**3) + 60*a**4*b**3*d**2*x**3*log(a
/b + x)/(6*a**5*b**6 + 18*a**4*b**7*x + 18*a**3*b**8*x**2 + 6*a**2*b**9*x**3) -
90*a**4*b**3*d**2*x**3/(6*a**5*b**6 + 18*a**4*b**7*x + 18*a**3*b**8*x**2 + 6*a**
2*b**9*x**3) + 36*a**3*b**4*c*d*x**2*log(a/b + x)/(6*a**5*b**6 + 18*a**4*b**7*x
+ 18*a**3*b**8*x**2 + 6*a**2*b**9*x**3) - 18*a**3*b**4*c*d*x**2/(6*a**5*b**6 + 1
8*a**4*b**7*x + 18*a**3*b**8*x**2 + 6*a**2*b**9*x**3) - 15*a**3*b**4*d**2*x**4/(
6*a**5*b**6 + 18*a**4*b**7*x + 18*a**3*b**8*x**2 + 6*a**2*b**9*x**3) + 12*a**2*b
**5*c*d*x**3*log(a/b + x)/(6*a**5*b**6 + 18*a**4*b**7*x + 18*a**3*b**8*x**2 + 6*
a**2*b**9*x**3) - 18*a**2*b**5*c*d*x**3/(6*a**5*b**6 + 18*a**4*b**7*x + 18*a**3*
b**8*x**2 + 6*a**2*b**9*x**3) + 3*a**2*b**5*d**2*x**5/(6*a**5*b**6 + 18*a**4*b**
7*x + 18*a**3*b**8*x**2 + 6*a**2*b**9*x**3) + 3*a*b**6*c**2*x**2/(6*a**5*b**6 +
18*a**4*b**7*x + 18*a**3*b**8*x**2 + 6*a**2*b**9*x**3) + b**7*c**2*x**3/(6*a**5*
b**6 + 18*a**4*b**7*x + 18*a**3*b**8*x**2 + 6*a**2*b**9*x**3), Eq(n, -4)), (-60*
a**6*d**2*log(a/b + x)/(6*a**3*b**6 + 12*a**2*b**7*x + 6*a*b**8*x**2) - 30*a**6*
d**2/(6*a**3*b**6 + 12*a**2*b**7*x + 6*a*b**8*x**2) - 120*a**5*b*d**2*x*log(a/b
+ x)/(6*a**3*b**6 + 12*a**2*b**7*x + 6*a*b**8*x**2) - 36*a**4*b**2*c*d*log(a/b +
x)/(6*a**3*b**6 + 12*a**2*b**7*x + 6*a*b**8*x**2) - 18*a**4*b**2*c*d/(6*a**3*b*
*6 + 12*a**2*b**7*x + 6*a*b**8*x**2) - 60*a**4*b**2*d**2*x**2*log(a/b + x)/(6*a*
*3*b**6 + 12*a**2*b**7*x + 6*a*b**8*x**2) + 60*a**4*b**2*d**2*x**2/(6*a**3*b**6
+ 12*a**2*b**7*x + 6*a*b**8*x**2) - 72*a**3*b**3*c*d*x*log(a/b + x)/(6*a**3*b**6
+ 12*a**2*b**7*x + 6*a*b**8*x**2) + 20*a**3*b**3*d**2*x**3/(6*a**3*b**6 + 12*a*
*2*b**7*x + 6*a*b**8*x**2) - 36*a**2*b**4*c*d*x**2*log(a/b + x)/(6*a**3*b**6 + 1
2*a**2*b**7*x + 6*a*b**8*x**2) + 36*a**2*b**4*c*d*x**2/(6*a**3*b**6 + 12*a**2*b*
*7*x + 6*a*b**8*x**2) - 5*a**2*b**4*d**2*x**4/(6*a**3*b**6 + 12*a**2*b**7*x + 6*
a*b**8*x**2) + 12*a*b**5*c*d*x**3/(6*a**3*b**6 + 12*a**2*b**7*x + 6*a*b**8*x**2)
+ 2*a*b**5*d**2*x**5/(6*a**3*b**6 + 12*a**2*b**7*x + 6*a*b**8*x**2) + 3*b**6*c*
*2*x**2/(6*a**3*b**6 + 12*a**2*b**7*x + 6*a*b**8*x**2), Eq(n, -3)), (60*a**5*d**
2*log(a/b + x)/(12*a*b**6 + 12*b**7*x) + 60*a**5*d**2/(12*a*b**6 + 12*b**7*x) +
60*a**4*b*d**2*x*log(a/b + x)/(12*a*b**6 + 12*b**7*x) + 72*a**3*b**2*c*d*log(a/b
+ x)/(12*a*b**6 + 12*b**7*x) + 72*a**3*b**2*c*d/(12*a*b**6 + 12*b**7*x) - 30*a*
*3*b**2*d**2*x**2/(12*a*b**6 + 12*b**7*x) + 72*a**2*b**3*c*d*x*log(a/b + x)/(12*
a*b**6 + 12*b**7*x) + 10*a**2*b**3*d**2*x**3/(12*a*b**6 + 12*b**7*x) + 12*a*b**4
*c**2*log(a/b + x)/(12*a*b**6 + 12*b**7*x) + 12*a*b**4*c**2/(12*a*b**6 + 12*b**7
*x) - 36*a*b**4*c*d*x**2/(12*a*b**6 + 12*b**7*x) - 5*a*b**4*d**2*x**4/(12*a*b**6
+ 12*b**7*x) + 12*b**5*c**2*x*log(a/b + x)/(12*a*b**6 + 12*b**7*x) + 12*b**5*c*
d*x**3/(12*a*b**6 + 12*b**7*x) + 3*b**5*d**2*x**5/(12*a*b**6 + 12*b**7*x), Eq(n,
-2)), (-a**5*d**2*log(a/b + x)/b**6 + a**4*d**2*x/b**5 - 2*a**3*c*d*log(a/b + x
)/b**4 - a**3*d**2*x**2/(2*b**4) + 2*a**2*c*d*x/b**3 + a**2*d**2*x**3/(3*b**3) -
a*c**2*log(a/b + x)/b**2 - a*c*d*x**2/b**2 - a*d**2*x**4/(4*b**2) + c**2*x/b +
2*c*d*x**3/(3*b) + d**2*x**5/(5*b), Eq(n, -1)), (-120*a**6*d**2*(a + b*x)**n/(b*
*6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b
**6*n + 720*b**6) + 120*a**5*b*d**2*n*x*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 +
175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 12*a
**4*b**2*c*d*n**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b
**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 132*a**4*b**2*c*d*n*(a + b
*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**
2 + 1764*b**6*n + 720*b**6) - 360*a**4*b**2*c*d*(a + b*x)**n/(b**6*n**6 + 21*b**
6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6
) - 60*a**4*b**2*d**2*n**2*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**
6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 60*a**4*b**2
*d**2*n*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n
**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 12*a**3*b**3*c*d*n**3*x*(a + b*
x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2
+ 1764*b**6*n + 720*b**6) + 132*a**3*b**3*c*d*n**2*x*(a + b*x)**n/(b**6*n**6 +
21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 72
0*b**6) + 360*a**3*b**3*c*d*n*x*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**
6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 20*a**3*b**3
*d**2*n**3*x**3*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**
6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 60*a**3*b**3*d**2*n**2*x**3*
(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b*
*6*n**2 + 1764*b**6*n + 720*b**6) + 40*a**3*b**3*d**2*n*x**3*(a + b*x)**n/(b**6*
n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6
*n + 720*b**6) - a**2*b**4*c**2*n**4*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 17
5*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 18*a**2
*b**4*c**2*n**3*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**
6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 119*a**2*b**4*c**2*n**2*(a +
b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n
**2 + 1764*b**6*n + 720*b**6) - 342*a**2*b**4*c**2*n*(a + b*x)**n/(b**6*n**6 + 2
1*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720
*b**6) - 360*a**2*b**4*c**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n*
*4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 6*a**2*b**4*c*d*
n**4*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3
+ 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 72*a**2*b**4*c*d*n**3*x**2*(a + b*
x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2
+ 1764*b**6*n + 720*b**6) - 246*a**2*b**4*c*d*n**2*x**2*(a + b*x)**n/(b**6*n**6
+ 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n +
720*b**6) - 180*a**2*b**4*c*d*n*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 1
75*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 5*a**2
*b**4*d**2*n**4*x**4*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 73
5*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 30*a**2*b**4*d**2*n**3*
x**4*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 16
24*b**6*n**2 + 1764*b**6*n + 720*b**6) - 55*a**2*b**4*d**2*n**2*x**4*(a + b*x)**
n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1
764*b**6*n + 720*b**6) - 30*a**2*b**4*d**2*n*x**4*(a + b*x)**n/(b**6*n**6 + 21*b
**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b*
*6) + a*b**5*c**2*n**5*x*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4
+ 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 18*a*b**5*c**2*n**4
*x*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624
*b**6*n**2 + 1764*b**6*n + 720*b**6) + 119*a*b**5*c**2*n**3*x*(a + b*x)**n/(b**6
*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**
6*n + 720*b**6) + 342*a*b**5*c**2*n**2*x*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5
+ 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 360
*a*b**5*c**2*n*x*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b*
*6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 2*a*b**5*c*d*n**5*x**3*(a +
b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n
**2 + 1764*b**6*n + 720*b**6) + 28*a*b**5*c*d*n**4*x**3*(a + b*x)**n/(b**6*n**6
+ 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n +
720*b**6) + 130*a*b**5*c*d*n**3*x**3*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 17
5*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 224*a*b
**5*c*d*n**2*x**3*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b
**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 120*a*b**5*c*d*n*x**3*(a +
b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n
**2 + 1764*b**6*n + 720*b**6) + a*b**5*d**2*n**5*x**5*(a + b*x)**n/(b**6*n**6 +
21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 72
0*b**6) + 10*a*b**5*d**2*n**4*x**5*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*
b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 35*a*b**5
*d**2*n**3*x**5*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**
6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 50*a*b**5*d**2*n**2*x**5*(a
+ b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*
n**2 + 1764*b**6*n + 720*b**6) + 24*a*b**5*d**2*n*x**5*(a + b*x)**n/(b**6*n**6 +
21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 7
20*b**6) + b**6*c**2*n**5*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6
*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 19*b**6*c**2*
n**4*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3
+ 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 137*b**6*c**2*n**3*x**2*(a + b*x)*
*n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 +
1764*b**6*n + 720*b**6) + 461*b**6*c**2*n**2*x**2*(a + b*x)**n/(b**6*n**6 + 21*b
**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b*
*6) + 702*b**6*c**2*n*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**
4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 360*b**6*c**2*x**
2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*
b**6*n**2 + 1764*b**6*n + 720*b**6) + 2*b**6*c*d*n**5*x**4*(a + b*x)**n/(b**6*n*
*6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n
+ 720*b**6) + 34*b**6*c*d*n**4*x**4*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 17
5*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 214*b**
6*c*d*n**3*x**4*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**
6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 614*b**6*c*d*n**2*x**4*(a +
b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n*
*2 + 1764*b**6*n + 720*b**6) + 792*b**6*c*d*n*x**4*(a + b*x)**n/(b**6*n**6 + 21*
b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b
**6) + 360*b**6*c*d*x**4*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4
+ 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + b**6*d**2*n**5*x**6
*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b
**6*n**2 + 1764*b**6*n + 720*b**6) + 15*b**6*d**2*n**4*x**6*(a + b*x)**n/(b**6*n
**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*
n + 720*b**6) + 85*b**6*d**2*n**3*x**6*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 +
175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 225*b
**6*d**2*n**2*x**6*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*
b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 274*b**6*d**2*n*x**6*(a +
b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n
**2 + 1764*b**6*n + 720*b**6) + 120*b**6*d**2*x**6*(a + b*x)**n/(b**6*n**6 + 21*
b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b
**6), True))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.294322, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^2*(b*x + a)^n*x,x, algorithm="giac")
[Out]
Done