3.1309 \(\int (a+b x)^2 (c+d x)^{10} \, dx\)

Optimal. Leaf size=65 \[ -\frac {b (c+d x)^{12} (b c-a d)}{6 d^3}+\frac {(c+d x)^{11} (b c-a d)^2}{11 d^3}+\frac {b^2 (c+d x)^{13}}{13 d^3} \]

[Out]

1/11*(-a*d+b*c)^2*(d*x+c)^11/d^3-1/6*b*(-a*d+b*c)*(d*x+c)^12/d^3+1/13*b^2*(d*x+c)^13/d^3

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Rubi [A]  time = 0.25, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \[ -\frac {b (c+d x)^{12} (b c-a d)}{6 d^3}+\frac {(c+d x)^{11} (b c-a d)^2}{11 d^3}+\frac {b^2 (c+d x)^{13}}{13 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*(c + d*x)^10,x]

[Out]

((b*c - a*d)^2*(c + d*x)^11)/(11*d^3) - (b*(b*c - a*d)*(c + d*x)^12)/(6*d^3) + (b^2*(c + d*x)^13)/(13*d^3)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int (a+b x)^2 (c+d x)^{10} \, dx &=\int \left (\frac {(-b c+a d)^2 (c+d x)^{10}}{d^2}-\frac {2 b (b c-a d) (c+d x)^{11}}{d^2}+\frac {b^2 (c+d x)^{12}}{d^2}\right ) \, dx\\ &=\frac {(b c-a d)^2 (c+d x)^{11}}{11 d^3}-\frac {b (b c-a d) (c+d x)^{12}}{6 d^3}+\frac {b^2 (c+d x)^{13}}{13 d^3}\\ \end {align*}

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Mathematica [B]  time = 0.05, size = 358, normalized size = 5.51 \[ \frac {1}{11} d^8 x^{11} \left (a^2 d^2+20 a b c d+45 b^2 c^2\right )+c d^7 x^{10} \left (a^2 d^2+9 a b c d+12 b^2 c^2\right )+\frac {5}{3} c^2 d^6 x^9 \left (3 a^2 d^2+16 a b c d+14 b^2 c^2\right )+\frac {1}{3} c^8 x^3 \left (45 a^2 d^2+20 a b c d+b^2 c^2\right )+\frac {5}{2} c^7 d x^4 \left (12 a^2 d^2+9 a b c d+b^2 c^2\right )+3 c^6 d^2 x^5 \left (14 a^2 d^2+16 a b c d+3 b^2 c^2\right )+2 c^5 d^3 x^6 \left (21 a^2 d^2+35 a b c d+10 b^2 c^2\right )+6 c^4 d^4 x^7 \left (5 a^2 d^2+12 a b c d+5 b^2 c^2\right )+\frac {3}{2} c^3 d^5 x^8 \left (10 a^2 d^2+35 a b c d+21 b^2 c^2\right )+a^2 c^{10} x+a c^9 x^2 (5 a d+b c)+\frac {1}{6} b d^9 x^{12} (a d+5 b c)+\frac {1}{13} b^2 d^{10} x^{13} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2*(c + d*x)^10,x]

[Out]

a^2*c^10*x + a*c^9*(b*c + 5*a*d)*x^2 + (c^8*(b^2*c^2 + 20*a*b*c*d + 45*a^2*d^2)*x^3)/3 + (5*c^7*d*(b^2*c^2 + 9
*a*b*c*d + 12*a^2*d^2)*x^4)/2 + 3*c^6*d^2*(3*b^2*c^2 + 16*a*b*c*d + 14*a^2*d^2)*x^5 + 2*c^5*d^3*(10*b^2*c^2 +
35*a*b*c*d + 21*a^2*d^2)*x^6 + 6*c^4*d^4*(5*b^2*c^2 + 12*a*b*c*d + 5*a^2*d^2)*x^7 + (3*c^3*d^5*(21*b^2*c^2 + 3
5*a*b*c*d + 10*a^2*d^2)*x^8)/2 + (5*c^2*d^6*(14*b^2*c^2 + 16*a*b*c*d + 3*a^2*d^2)*x^9)/3 + c*d^7*(12*b^2*c^2 +
9*a*b*c*d + a^2*d^2)*x^10 + (d^8*(45*b^2*c^2 + 20*a*b*c*d + a^2*d^2)*x^11)/11 + (b*d^9*(5*b*c + a*d)*x^12)/6
+ (b^2*d^10*x^13)/13

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fricas [B]  time = 0.39, size = 417, normalized size = 6.42 \[ \frac {1}{13} x^{13} d^{10} b^{2} + \frac {5}{6} x^{12} d^{9} c b^{2} + \frac {1}{6} x^{12} d^{10} b a + \frac {45}{11} x^{11} d^{8} c^{2} b^{2} + \frac {20}{11} x^{11} d^{9} c b a + \frac {1}{11} x^{11} d^{10} a^{2} + 12 x^{10} d^{7} c^{3} b^{2} + 9 x^{10} d^{8} c^{2} b a + x^{10} d^{9} c a^{2} + \frac {70}{3} x^{9} d^{6} c^{4} b^{2} + \frac {80}{3} x^{9} d^{7} c^{3} b a + 5 x^{9} d^{8} c^{2} a^{2} + \frac {63}{2} x^{8} d^{5} c^{5} b^{2} + \frac {105}{2} x^{8} d^{6} c^{4} b a + 15 x^{8} d^{7} c^{3} a^{2} + 30 x^{7} d^{4} c^{6} b^{2} + 72 x^{7} d^{5} c^{5} b a + 30 x^{7} d^{6} c^{4} a^{2} + 20 x^{6} d^{3} c^{7} b^{2} + 70 x^{6} d^{4} c^{6} b a + 42 x^{6} d^{5} c^{5} a^{2} + 9 x^{5} d^{2} c^{8} b^{2} + 48 x^{5} d^{3} c^{7} b a + 42 x^{5} d^{4} c^{6} a^{2} + \frac {5}{2} x^{4} d c^{9} b^{2} + \frac {45}{2} x^{4} d^{2} c^{8} b a + 30 x^{4} d^{3} c^{7} a^{2} + \frac {1}{3} x^{3} c^{10} b^{2} + \frac {20}{3} x^{3} d c^{9} b a + 15 x^{3} d^{2} c^{8} a^{2} + x^{2} c^{10} b a + 5 x^{2} d c^{9} a^{2} + x c^{10} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(d*x+c)^10,x, algorithm="fricas")

[Out]

1/13*x^13*d^10*b^2 + 5/6*x^12*d^9*c*b^2 + 1/6*x^12*d^10*b*a + 45/11*x^11*d^8*c^2*b^2 + 20/11*x^11*d^9*c*b*a +
1/11*x^11*d^10*a^2 + 12*x^10*d^7*c^3*b^2 + 9*x^10*d^8*c^2*b*a + x^10*d^9*c*a^2 + 70/3*x^9*d^6*c^4*b^2 + 80/3*x
^9*d^7*c^3*b*a + 5*x^9*d^8*c^2*a^2 + 63/2*x^8*d^5*c^5*b^2 + 105/2*x^8*d^6*c^4*b*a + 15*x^8*d^7*c^3*a^2 + 30*x^
7*d^4*c^6*b^2 + 72*x^7*d^5*c^5*b*a + 30*x^7*d^6*c^4*a^2 + 20*x^6*d^3*c^7*b^2 + 70*x^6*d^4*c^6*b*a + 42*x^6*d^5
*c^5*a^2 + 9*x^5*d^2*c^8*b^2 + 48*x^5*d^3*c^7*b*a + 42*x^5*d^4*c^6*a^2 + 5/2*x^4*d*c^9*b^2 + 45/2*x^4*d^2*c^8*
b*a + 30*x^4*d^3*c^7*a^2 + 1/3*x^3*c^10*b^2 + 20/3*x^3*d*c^9*b*a + 15*x^3*d^2*c^8*a^2 + x^2*c^10*b*a + 5*x^2*d
*c^9*a^2 + x*c^10*a^2

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giac [B]  time = 1.26, size = 417, normalized size = 6.42 \[ \frac {1}{13} \, b^{2} d^{10} x^{13} + \frac {5}{6} \, b^{2} c d^{9} x^{12} + \frac {1}{6} \, a b d^{10} x^{12} + \frac {45}{11} \, b^{2} c^{2} d^{8} x^{11} + \frac {20}{11} \, a b c d^{9} x^{11} + \frac {1}{11} \, a^{2} d^{10} x^{11} + 12 \, b^{2} c^{3} d^{7} x^{10} + 9 \, a b c^{2} d^{8} x^{10} + a^{2} c d^{9} x^{10} + \frac {70}{3} \, b^{2} c^{4} d^{6} x^{9} + \frac {80}{3} \, a b c^{3} d^{7} x^{9} + 5 \, a^{2} c^{2} d^{8} x^{9} + \frac {63}{2} \, b^{2} c^{5} d^{5} x^{8} + \frac {105}{2} \, a b c^{4} d^{6} x^{8} + 15 \, a^{2} c^{3} d^{7} x^{8} + 30 \, b^{2} c^{6} d^{4} x^{7} + 72 \, a b c^{5} d^{5} x^{7} + 30 \, a^{2} c^{4} d^{6} x^{7} + 20 \, b^{2} c^{7} d^{3} x^{6} + 70 \, a b c^{6} d^{4} x^{6} + 42 \, a^{2} c^{5} d^{5} x^{6} + 9 \, b^{2} c^{8} d^{2} x^{5} + 48 \, a b c^{7} d^{3} x^{5} + 42 \, a^{2} c^{6} d^{4} x^{5} + \frac {5}{2} \, b^{2} c^{9} d x^{4} + \frac {45}{2} \, a b c^{8} d^{2} x^{4} + 30 \, a^{2} c^{7} d^{3} x^{4} + \frac {1}{3} \, b^{2} c^{10} x^{3} + \frac {20}{3} \, a b c^{9} d x^{3} + 15 \, a^{2} c^{8} d^{2} x^{3} + a b c^{10} x^{2} + 5 \, a^{2} c^{9} d x^{2} + a^{2} c^{10} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(d*x+c)^10,x, algorithm="giac")

[Out]

1/13*b^2*d^10*x^13 + 5/6*b^2*c*d^9*x^12 + 1/6*a*b*d^10*x^12 + 45/11*b^2*c^2*d^8*x^11 + 20/11*a*b*c*d^9*x^11 +
1/11*a^2*d^10*x^11 + 12*b^2*c^3*d^7*x^10 + 9*a*b*c^2*d^8*x^10 + a^2*c*d^9*x^10 + 70/3*b^2*c^4*d^6*x^9 + 80/3*a
*b*c^3*d^7*x^9 + 5*a^2*c^2*d^8*x^9 + 63/2*b^2*c^5*d^5*x^8 + 105/2*a*b*c^4*d^6*x^8 + 15*a^2*c^3*d^7*x^8 + 30*b^
2*c^6*d^4*x^7 + 72*a*b*c^5*d^5*x^7 + 30*a^2*c^4*d^6*x^7 + 20*b^2*c^7*d^3*x^6 + 70*a*b*c^6*d^4*x^6 + 42*a^2*c^5
*d^5*x^6 + 9*b^2*c^8*d^2*x^5 + 48*a*b*c^7*d^3*x^5 + 42*a^2*c^6*d^4*x^5 + 5/2*b^2*c^9*d*x^4 + 45/2*a*b*c^8*d^2*
x^4 + 30*a^2*c^7*d^3*x^4 + 1/3*b^2*c^10*x^3 + 20/3*a*b*c^9*d*x^3 + 15*a^2*c^8*d^2*x^3 + a*b*c^10*x^2 + 5*a^2*c
^9*d*x^2 + a^2*c^10*x

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maple [B]  time = 0.00, size = 391, normalized size = 6.02 \[ \frac {b^{2} d^{10} x^{13}}{13}+a^{2} c^{10} x +\frac {\left (2 a b \,d^{10}+10 b^{2} c \,d^{9}\right ) x^{12}}{12}+\frac {\left (a^{2} d^{10}+20 a b c \,d^{9}+45 b^{2} c^{2} d^{8}\right ) x^{11}}{11}+\frac {\left (10 a^{2} c \,d^{9}+90 a b \,c^{2} d^{8}+120 b^{2} c^{3} d^{7}\right ) x^{10}}{10}+\frac {\left (45 a^{2} c^{2} d^{8}+240 a b \,c^{3} d^{7}+210 b^{2} c^{4} d^{6}\right ) x^{9}}{9}+\frac {\left (120 a^{2} c^{3} d^{7}+420 a b \,c^{4} d^{6}+252 b^{2} c^{5} d^{5}\right ) x^{8}}{8}+\frac {\left (210 a^{2} c^{4} d^{6}+504 a b \,c^{5} d^{5}+210 b^{2} c^{6} d^{4}\right ) x^{7}}{7}+\frac {\left (252 a^{2} c^{5} d^{5}+420 a b \,c^{6} d^{4}+120 b^{2} c^{7} d^{3}\right ) x^{6}}{6}+\frac {\left (210 a^{2} c^{6} d^{4}+240 a b \,c^{7} d^{3}+45 b^{2} c^{8} d^{2}\right ) x^{5}}{5}+\frac {\left (120 a^{2} c^{7} d^{3}+90 a b \,c^{8} d^{2}+10 b^{2} c^{9} d \right ) x^{4}}{4}+\frac {\left (45 a^{2} c^{8} d^{2}+20 a b \,c^{9} d +b^{2} c^{10}\right ) x^{3}}{3}+\frac {\left (10 a^{2} c^{9} d +2 a b \,c^{10}\right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(d*x+c)^10,x)

[Out]

1/13*b^2*d^10*x^13+1/12*(2*a*b*d^10+10*b^2*c*d^9)*x^12+1/11*(a^2*d^10+20*a*b*c*d^9+45*b^2*c^2*d^8)*x^11+1/10*(
10*a^2*c*d^9+90*a*b*c^2*d^8+120*b^2*c^3*d^7)*x^10+1/9*(45*a^2*c^2*d^8+240*a*b*c^3*d^7+210*b^2*c^4*d^6)*x^9+1/8
*(120*a^2*c^3*d^7+420*a*b*c^4*d^6+252*b^2*c^5*d^5)*x^8+1/7*(210*a^2*c^4*d^6+504*a*b*c^5*d^5+210*b^2*c^6*d^4)*x
^7+1/6*(252*a^2*c^5*d^5+420*a*b*c^6*d^4+120*b^2*c^7*d^3)*x^6+1/5*(210*a^2*c^6*d^4+240*a*b*c^7*d^3+45*b^2*c^8*d
^2)*x^5+1/4*(120*a^2*c^7*d^3+90*a*b*c^8*d^2+10*b^2*c^9*d)*x^4+1/3*(45*a^2*c^8*d^2+20*a*b*c^9*d+b^2*c^10)*x^3+1
/2*(10*a^2*c^9*d+2*a*b*c^10)*x^2+a^2*c^10*x

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maxima [B]  time = 1.33, size = 384, normalized size = 5.91 \[ \frac {1}{13} \, b^{2} d^{10} x^{13} + a^{2} c^{10} x + \frac {1}{6} \, {\left (5 \, b^{2} c d^{9} + a b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (45 \, b^{2} c^{2} d^{8} + 20 \, a b c d^{9} + a^{2} d^{10}\right )} x^{11} + {\left (12 \, b^{2} c^{3} d^{7} + 9 \, a b c^{2} d^{8} + a^{2} c d^{9}\right )} x^{10} + \frac {5}{3} \, {\left (14 \, b^{2} c^{4} d^{6} + 16 \, a b c^{3} d^{7} + 3 \, a^{2} c^{2} d^{8}\right )} x^{9} + \frac {3}{2} \, {\left (21 \, b^{2} c^{5} d^{5} + 35 \, a b c^{4} d^{6} + 10 \, a^{2} c^{3} d^{7}\right )} x^{8} + 6 \, {\left (5 \, b^{2} c^{6} d^{4} + 12 \, a b c^{5} d^{5} + 5 \, a^{2} c^{4} d^{6}\right )} x^{7} + 2 \, {\left (10 \, b^{2} c^{7} d^{3} + 35 \, a b c^{6} d^{4} + 21 \, a^{2} c^{5} d^{5}\right )} x^{6} + 3 \, {\left (3 \, b^{2} c^{8} d^{2} + 16 \, a b c^{7} d^{3} + 14 \, a^{2} c^{6} d^{4}\right )} x^{5} + \frac {5}{2} \, {\left (b^{2} c^{9} d + 9 \, a b c^{8} d^{2} + 12 \, a^{2} c^{7} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} c^{10} + 20 \, a b c^{9} d + 45 \, a^{2} c^{8} d^{2}\right )} x^{3} + {\left (a b c^{10} + 5 \, a^{2} c^{9} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(d*x+c)^10,x, algorithm="maxima")

[Out]

1/13*b^2*d^10*x^13 + a^2*c^10*x + 1/6*(5*b^2*c*d^9 + a*b*d^10)*x^12 + 1/11*(45*b^2*c^2*d^8 + 20*a*b*c*d^9 + a^
2*d^10)*x^11 + (12*b^2*c^3*d^7 + 9*a*b*c^2*d^8 + a^2*c*d^9)*x^10 + 5/3*(14*b^2*c^4*d^6 + 16*a*b*c^3*d^7 + 3*a^
2*c^2*d^8)*x^9 + 3/2*(21*b^2*c^5*d^5 + 35*a*b*c^4*d^6 + 10*a^2*c^3*d^7)*x^8 + 6*(5*b^2*c^6*d^4 + 12*a*b*c^5*d^
5 + 5*a^2*c^4*d^6)*x^7 + 2*(10*b^2*c^7*d^3 + 35*a*b*c^6*d^4 + 21*a^2*c^5*d^5)*x^6 + 3*(3*b^2*c^8*d^2 + 16*a*b*
c^7*d^3 + 14*a^2*c^6*d^4)*x^5 + 5/2*(b^2*c^9*d + 9*a*b*c^8*d^2 + 12*a^2*c^7*d^3)*x^4 + 1/3*(b^2*c^10 + 20*a*b*
c^9*d + 45*a^2*c^8*d^2)*x^3 + (a*b*c^10 + 5*a^2*c^9*d)*x^2

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mupad [B]  time = 0.32, size = 348, normalized size = 5.35 \[ x^3\,\left (15\,a^2\,c^8\,d^2+\frac {20\,a\,b\,c^9\,d}{3}+\frac {b^2\,c^{10}}{3}\right )+x^{11}\,\left (\frac {a^2\,d^{10}}{11}+\frac {20\,a\,b\,c\,d^9}{11}+\frac {45\,b^2\,c^2\,d^8}{11}\right )+a^2\,c^{10}\,x+\frac {b^2\,d^{10}\,x^{13}}{13}+a\,c^9\,x^2\,\left (5\,a\,d+b\,c\right )+\frac {b\,d^9\,x^{12}\,\left (a\,d+5\,b\,c\right )}{6}+\frac {5\,c^7\,d\,x^4\,\left (12\,a^2\,d^2+9\,a\,b\,c\,d+b^2\,c^2\right )}{2}+c\,d^7\,x^{10}\,\left (a^2\,d^2+9\,a\,b\,c\,d+12\,b^2\,c^2\right )+6\,c^4\,d^4\,x^7\,\left (5\,a^2\,d^2+12\,a\,b\,c\,d+5\,b^2\,c^2\right )+3\,c^6\,d^2\,x^5\,\left (14\,a^2\,d^2+16\,a\,b\,c\,d+3\,b^2\,c^2\right )+\frac {5\,c^2\,d^6\,x^9\,\left (3\,a^2\,d^2+16\,a\,b\,c\,d+14\,b^2\,c^2\right )}{3}+2\,c^5\,d^3\,x^6\,\left (21\,a^2\,d^2+35\,a\,b\,c\,d+10\,b^2\,c^2\right )+\frac {3\,c^3\,d^5\,x^8\,\left (10\,a^2\,d^2+35\,a\,b\,c\,d+21\,b^2\,c^2\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2*(c + d*x)^10,x)

[Out]

x^3*((b^2*c^10)/3 + 15*a^2*c^8*d^2 + (20*a*b*c^9*d)/3) + x^11*((a^2*d^10)/11 + (45*b^2*c^2*d^8)/11 + (20*a*b*c
*d^9)/11) + a^2*c^10*x + (b^2*d^10*x^13)/13 + a*c^9*x^2*(5*a*d + b*c) + (b*d^9*x^12*(a*d + 5*b*c))/6 + (5*c^7*
d*x^4*(12*a^2*d^2 + b^2*c^2 + 9*a*b*c*d))/2 + c*d^7*x^10*(a^2*d^2 + 12*b^2*c^2 + 9*a*b*c*d) + 6*c^4*d^4*x^7*(5
*a^2*d^2 + 5*b^2*c^2 + 12*a*b*c*d) + 3*c^6*d^2*x^5*(14*a^2*d^2 + 3*b^2*c^2 + 16*a*b*c*d) + (5*c^2*d^6*x^9*(3*a
^2*d^2 + 14*b^2*c^2 + 16*a*b*c*d))/3 + 2*c^5*d^3*x^6*(21*a^2*d^2 + 10*b^2*c^2 + 35*a*b*c*d) + (3*c^3*d^5*x^8*(
10*a^2*d^2 + 21*b^2*c^2 + 35*a*b*c*d))/2

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sympy [B]  time = 0.14, size = 415, normalized size = 6.38 \[ a^{2} c^{10} x + \frac {b^{2} d^{10} x^{13}}{13} + x^{12} \left (\frac {a b d^{10}}{6} + \frac {5 b^{2} c d^{9}}{6}\right ) + x^{11} \left (\frac {a^{2} d^{10}}{11} + \frac {20 a b c d^{9}}{11} + \frac {45 b^{2} c^{2} d^{8}}{11}\right ) + x^{10} \left (a^{2} c d^{9} + 9 a b c^{2} d^{8} + 12 b^{2} c^{3} d^{7}\right ) + x^{9} \left (5 a^{2} c^{2} d^{8} + \frac {80 a b c^{3} d^{7}}{3} + \frac {70 b^{2} c^{4} d^{6}}{3}\right ) + x^{8} \left (15 a^{2} c^{3} d^{7} + \frac {105 a b c^{4} d^{6}}{2} + \frac {63 b^{2} c^{5} d^{5}}{2}\right ) + x^{7} \left (30 a^{2} c^{4} d^{6} + 72 a b c^{5} d^{5} + 30 b^{2} c^{6} d^{4}\right ) + x^{6} \left (42 a^{2} c^{5} d^{5} + 70 a b c^{6} d^{4} + 20 b^{2} c^{7} d^{3}\right ) + x^{5} \left (42 a^{2} c^{6} d^{4} + 48 a b c^{7} d^{3} + 9 b^{2} c^{8} d^{2}\right ) + x^{4} \left (30 a^{2} c^{7} d^{3} + \frac {45 a b c^{8} d^{2}}{2} + \frac {5 b^{2} c^{9} d}{2}\right ) + x^{3} \left (15 a^{2} c^{8} d^{2} + \frac {20 a b c^{9} d}{3} + \frac {b^{2} c^{10}}{3}\right ) + x^{2} \left (5 a^{2} c^{9} d + a b c^{10}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(d*x+c)**10,x)

[Out]

a**2*c**10*x + b**2*d**10*x**13/13 + x**12*(a*b*d**10/6 + 5*b**2*c*d**9/6) + x**11*(a**2*d**10/11 + 20*a*b*c*d
**9/11 + 45*b**2*c**2*d**8/11) + x**10*(a**2*c*d**9 + 9*a*b*c**2*d**8 + 12*b**2*c**3*d**7) + x**9*(5*a**2*c**2
*d**8 + 80*a*b*c**3*d**7/3 + 70*b**2*c**4*d**6/3) + x**8*(15*a**2*c**3*d**7 + 105*a*b*c**4*d**6/2 + 63*b**2*c*
*5*d**5/2) + x**7*(30*a**2*c**4*d**6 + 72*a*b*c**5*d**5 + 30*b**2*c**6*d**4) + x**6*(42*a**2*c**5*d**5 + 70*a*
b*c**6*d**4 + 20*b**2*c**7*d**3) + x**5*(42*a**2*c**6*d**4 + 48*a*b*c**7*d**3 + 9*b**2*c**8*d**2) + x**4*(30*a
**2*c**7*d**3 + 45*a*b*c**8*d**2/2 + 5*b**2*c**9*d/2) + x**3*(15*a**2*c**8*d**2 + 20*a*b*c**9*d/3 + b**2*c**10
/3) + x**2*(5*a**2*c**9*d + a*b*c**10)

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