\(\int (a+b x)^9 (c+d x)^7 \, dx\) [1273]

Optimal result

Integrand size = 15, antiderivative size = 200 \[ \int (a+b x)^9 (c+d x)^7 \, dx=\frac {(b c-a d)^7 (a+b x)^{10}}{10 b^8}+\frac {7 d (b c-a d)^6 (a+b x)^{11}}{11 b^8}+\frac {7 d^2 (b c-a d)^5 (a+b x)^{12}}{4 b^8}+\frac {35 d^3 (b c-a d)^4 (a+b x)^{13}}{13 b^8}+\frac {5 d^4 (b c-a d)^3 (a+b x)^{14}}{2 b^8}+\frac {7 d^5 (b c-a d)^2 (a+b x)^{15}}{5 b^8}+\frac {7 d^6 (b c-a d) (a+b x)^{16}}{16 b^8}+\frac {d^7 (a+b x)^{17}}{17 b^8} \]

[Out]

1/10*(-a*d+b*c)^7*(b*x+a)^10/b^8+7/11*d*(-a*d+b*c)^6*(b*x+a)^11/b^8+7/4*d^2*(-a*d+b*c)^5*(b*x+a)^12/b^8+35/13*
d^3*(-a*d+b*c)^4*(b*x+a)^13/b^8+5/2*d^4*(-a*d+b*c)^3*(b*x+a)^14/b^8+7/5*d^5*(-a*d+b*c)^2*(b*x+a)^15/b^8+7/16*d
^6*(-a*d+b*c)*(b*x+a)^16/b^8+1/17*d^7*(b*x+a)^17/b^8

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int (a+b x)^9 (c+d x)^7 \, dx=\frac {7 d^6 (a+b x)^{16} (b c-a d)}{16 b^8}+\frac {7 d^5 (a+b x)^{15} (b c-a d)^2}{5 b^8}+\frac {5 d^4 (a+b x)^{14} (b c-a d)^3}{2 b^8}+\frac {35 d^3 (a+b x)^{13} (b c-a d)^4}{13 b^8}+\frac {7 d^2 (a+b x)^{12} (b c-a d)^5}{4 b^8}+\frac {7 d (a+b x)^{11} (b c-a d)^6}{11 b^8}+\frac {(a+b x)^{10} (b c-a d)^7}{10 b^8}+\frac {d^7 (a+b x)^{17}}{17 b^8} \]

[In]

Int[(a + b*x)^9*(c + d*x)^7,x]

[Out]

((b*c - a*d)^7*(a + b*x)^10)/(10*b^8) + (7*d*(b*c - a*d)^6*(a + b*x)^11)/(11*b^8) + (7*d^2*(b*c - a*d)^5*(a +
b*x)^12)/(4*b^8) + (35*d^3*(b*c - a*d)^4*(a + b*x)^13)/(13*b^8) + (5*d^4*(b*c - a*d)^3*(a + b*x)^14)/(2*b^8) +
(7*d^5*(b*c - a*d)^2*(a + b*x)^15)/(5*b^8) + (7*d^6*(b*c - a*d)*(a + b*x)^16)/(16*b^8) + (d^7*(a + b*x)^17)/(
17*b^8)

Rule 45

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*} \text {integral}& = \int \left (\frac {(b c-a d)^7 (a+b x)^9}{b^7}+\frac {7 d (b c-a d)^6 (a+b x)^{10}}{b^7}+\frac {21 d^2 (b c-a d)^5 (a+b x)^{11}}{b^7}+\frac {35 d^3 (b c-a d)^4 (a+b x)^{12}}{b^7}+\frac {35 d^4 (b c-a d)^3 (a+b x)^{13}}{b^7}+\frac {21 d^5 (b c-a d)^2 (a+b x)^{14}}{b^7}+\frac {7 d^6 (b c-a d) (a+b x)^{15}}{b^7}+\frac {d^7 (a+b x)^{16}}{b^7}\right ) \, dx \\ & = \frac {(b c-a d)^7 (a+b x)^{10}}{10 b^8}+\frac {7 d (b c-a d)^6 (a+b x)^{11}}{11 b^8}+\frac {7 d^2 (b c-a d)^5 (a+b x)^{12}}{4 b^8}+\frac {35 d^3 (b c-a d)^4 (a+b x)^{13}}{13 b^8}+\frac {5 d^4 (b c-a d)^3 (a+b x)^{14}}{2 b^8}+\frac {7 d^5 (b c-a d)^2 (a+b x)^{15}}{5 b^8}+\frac {7 d^6 (b c-a d) (a+b x)^{16}}{16 b^8}+\frac {d^7 (a+b x)^{17}}{17 b^8} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(993\) vs. \(2(200)=400\).

Time = 0.08 (sec) , antiderivative size = 993, normalized size of antiderivative = 4.96 \[ \int (a+b x)^9 (c+d x)^7 \, dx=a^9 c^7 x+\frac {1}{2} a^8 c^6 (9 b c+7 a d) x^2+a^7 c^5 \left (12 b^2 c^2+21 a b c d+7 a^2 d^2\right ) x^3+\frac {7}{4} a^6 c^4 \left (12 b^3 c^3+36 a b^2 c^2 d+27 a^2 b c d^2+5 a^3 d^3\right ) x^4+\frac {7}{5} a^5 c^3 \left (18 b^4 c^4+84 a b^3 c^3 d+108 a^2 b^2 c^2 d^2+45 a^3 b c d^3+5 a^4 d^4\right ) x^5+\frac {7}{2} a^4 c^2 \left (6 b^5 c^5+42 a b^4 c^4 d+84 a^2 b^3 c^3 d^2+60 a^3 b^2 c^2 d^3+15 a^4 b c d^4+a^5 d^5\right ) x^6+a^3 c \left (12 b^6 c^6+126 a b^5 c^5 d+378 a^2 b^4 c^4 d^2+420 a^3 b^3 c^3 d^3+180 a^4 b^2 c^2 d^4+27 a^5 b c d^5+a^6 d^6\right ) x^7+\frac {1}{8} a^2 \left (36 b^7 c^7+588 a b^6 c^6 d+2646 a^2 b^5 c^5 d^2+4410 a^3 b^4 c^4 d^3+2940 a^4 b^3 c^3 d^4+756 a^5 b^2 c^2 d^5+63 a^6 b c d^6+a^7 d^7\right ) x^8+a b \left (b^7 c^7+28 a b^6 c^6 d+196 a^2 b^5 c^5 d^2+490 a^3 b^4 c^4 d^3+490 a^4 b^3 c^3 d^4+196 a^5 b^2 c^2 d^5+28 a^6 b c d^6+a^7 d^7\right ) x^9+\frac {1}{10} b^2 \left (b^7 c^7+63 a b^6 c^6 d+756 a^2 b^5 c^5 d^2+2940 a^3 b^4 c^4 d^3+4410 a^4 b^3 c^3 d^4+2646 a^5 b^2 c^2 d^5+588 a^6 b c d^6+36 a^7 d^7\right ) x^{10}+\frac {7}{11} b^3 d \left (b^6 c^6+27 a b^5 c^5 d+180 a^2 b^4 c^4 d^2+420 a^3 b^3 c^3 d^3+378 a^4 b^2 c^2 d^4+126 a^5 b c d^5+12 a^6 d^6\right ) x^{11}+\frac {7}{4} b^4 d^2 \left (b^5 c^5+15 a b^4 c^4 d+60 a^2 b^3 c^3 d^2+84 a^3 b^2 c^2 d^3+42 a^4 b c d^4+6 a^5 d^5\right ) x^{12}+\frac {7}{13} b^5 d^3 \left (5 b^4 c^4+45 a b^3 c^3 d+108 a^2 b^2 c^2 d^2+84 a^3 b c d^3+18 a^4 d^4\right ) x^{13}+\frac {1}{2} b^6 d^4 \left (5 b^3 c^3+27 a b^2 c^2 d+36 a^2 b c d^2+12 a^3 d^3\right ) x^{14}+\frac {1}{5} b^7 d^5 \left (7 b^2 c^2+21 a b c d+12 a^2 d^2\right ) x^{15}+\frac {1}{16} b^8 d^6 (7 b c+9 a d) x^{16}+\frac {1}{17} b^9 d^7 x^{17} \]

[In]

Integrate[(a + b*x)^9*(c + d*x)^7,x]

[Out]

a^9*c^7*x + (a^8*c^6*(9*b*c + 7*a*d)*x^2)/2 + a^7*c^5*(12*b^2*c^2 + 21*a*b*c*d + 7*a^2*d^2)*x^3 + (7*a^6*c^4*(
12*b^3*c^3 + 36*a*b^2*c^2*d + 27*a^2*b*c*d^2 + 5*a^3*d^3)*x^4)/4 + (7*a^5*c^3*(18*b^4*c^4 + 84*a*b^3*c^3*d + 1
08*a^2*b^2*c^2*d^2 + 45*a^3*b*c*d^3 + 5*a^4*d^4)*x^5)/5 + (7*a^4*c^2*(6*b^5*c^5 + 42*a*b^4*c^4*d + 84*a^2*b^3*
c^3*d^2 + 60*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 + a^5*d^5)*x^6)/2 + a^3*c*(12*b^6*c^6 + 126*a*b^5*c^5*d + 378*a^
2*b^4*c^4*d^2 + 420*a^3*b^3*c^3*d^3 + 180*a^4*b^2*c^2*d^4 + 27*a^5*b*c*d^5 + a^6*d^6)*x^7 + (a^2*(36*b^7*c^7 +
588*a*b^6*c^6*d + 2646*a^2*b^5*c^5*d^2 + 4410*a^3*b^4*c^4*d^3 + 2940*a^4*b^3*c^3*d^4 + 756*a^5*b^2*c^2*d^5 +
63*a^6*b*c*d^6 + a^7*d^7)*x^8)/8 + a*b*(b^7*c^7 + 28*a*b^6*c^6*d + 196*a^2*b^5*c^5*d^2 + 490*a^3*b^4*c^4*d^3 +
490*a^4*b^3*c^3*d^4 + 196*a^5*b^2*c^2*d^5 + 28*a^6*b*c*d^6 + a^7*d^7)*x^9 + (b^2*(b^7*c^7 + 63*a*b^6*c^6*d +
756*a^2*b^5*c^5*d^2 + 2940*a^3*b^4*c^4*d^3 + 4410*a^4*b^3*c^3*d^4 + 2646*a^5*b^2*c^2*d^5 + 588*a^6*b*c*d^6 + 3
6*a^7*d^7)*x^10)/10 + (7*b^3*d*(b^6*c^6 + 27*a*b^5*c^5*d + 180*a^2*b^4*c^4*d^2 + 420*a^3*b^3*c^3*d^3 + 378*a^4
*b^2*c^2*d^4 + 126*a^5*b*c*d^5 + 12*a^6*d^6)*x^11)/11 + (7*b^4*d^2*(b^5*c^5 + 15*a*b^4*c^4*d + 60*a^2*b^3*c^3*
d^2 + 84*a^3*b^2*c^2*d^3 + 42*a^4*b*c*d^4 + 6*a^5*d^5)*x^12)/4 + (7*b^5*d^3*(5*b^4*c^4 + 45*a*b^3*c^3*d + 108*
a^2*b^2*c^2*d^2 + 84*a^3*b*c*d^3 + 18*a^4*d^4)*x^13)/13 + (b^6*d^4*(5*b^3*c^3 + 27*a*b^2*c^2*d + 36*a^2*b*c*d^
2 + 12*a^3*d^3)*x^14)/2 + (b^7*d^5*(7*b^2*c^2 + 21*a*b*c*d + 12*a^2*d^2)*x^15)/5 + (b^8*d^6*(7*b*c + 9*a*d)*x^
16)/16 + (b^9*d^7*x^17)/17

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1016\) vs. \(2(184)=368\).

Time = 0.20 (sec) , antiderivative size = 1017, normalized size of antiderivative = 5.08

method	result	size
norman	\(\text {Expression too large to display}\)	\(1017\)
default	\(\text {Expression too large to display}\)	\(1033\)
gosper	\(\text {Expression too large to display}\)	\(1176\)
risch	\(\text {Expression too large to display}\)	\(1176\)
parallelrisch	\(\text {Expression too large to display}\)	\(1176\)

[In]

int((b*x+a)^9*(d*x+c)^7,x,method=_RETURNVERBOSE)

[Out]

a^9*c^7*x+(7/2*a^9*c^6*d+9/2*a^8*b*c^7)*x^2+(7*a^9*c^5*d^2+21*a^8*b*c^6*d+12*a^7*b^2*c^7)*x^3+(35/4*a^9*c^4*d^
3+189/4*a^8*b*c^5*d^2+63*a^7*b^2*c^6*d+21*a^6*b^3*c^7)*x^4+(7*a^9*c^3*d^4+63*a^8*b*c^4*d^3+756/5*a^7*b^2*c^5*d
^2+588/5*a^6*b^3*c^6*d+126/5*a^5*b^4*c^7)*x^5+(7/2*a^9*c^2*d^5+105/2*a^8*b*c^3*d^4+210*a^7*b^2*c^4*d^3+294*a^6
*b^3*c^5*d^2+147*a^5*b^4*c^6*d+21*a^4*b^5*c^7)*x^6+(a^9*c*d^6+27*a^8*b*c^2*d^5+180*a^7*b^2*c^3*d^4+420*a^6*b^3
*c^4*d^3+378*a^5*b^4*c^5*d^2+126*a^4*b^5*c^6*d+12*a^3*b^6*c^7)*x^7+(1/8*a^9*d^7+63/8*a^8*b*c*d^6+189/2*a^7*b^2
*c^2*d^5+735/2*a^6*b^3*c^3*d^4+2205/4*a^5*b^4*c^4*d^3+1323/4*a^4*b^5*c^5*d^2+147/2*a^3*b^6*c^6*d+9/2*a^2*b^7*c
^7)*x^8+(a^8*b*d^7+28*a^7*b^2*c*d^6+196*a^6*b^3*c^2*d^5+490*a^5*b^4*c^3*d^4+490*a^4*b^5*c^4*d^3+196*a^3*b^6*c^
5*d^2+28*a^2*b^7*c^6*d+a*b^8*c^7)*x^9+(18/5*a^7*b^2*d^7+294/5*a^6*b^3*c*d^6+1323/5*a^5*b^4*c^2*d^5+441*a^4*b^5
*c^3*d^4+294*a^3*b^6*c^4*d^3+378/5*a^2*b^7*c^5*d^2+63/10*a*b^8*c^6*d+1/10*b^9*c^7)*x^10+(84/11*a^6*b^3*d^7+882
/11*a^5*b^4*c*d^6+2646/11*a^4*b^5*c^2*d^5+2940/11*a^3*b^6*c^3*d^4+1260/11*a^2*b^7*c^4*d^3+189/11*a*b^8*c^5*d^2
+7/11*b^9*c^6*d)*x^11+(21/2*a^5*b^4*d^7+147/2*a^4*b^5*c*d^6+147*a^3*b^6*c^2*d^5+105*a^2*b^7*c^3*d^4+105/4*a*b^
8*c^4*d^3+7/4*b^9*c^5*d^2)*x^12+(126/13*a^4*b^5*d^7+588/13*a^3*b^6*c*d^6+756/13*a^2*b^7*c^2*d^5+315/13*a*b^8*c
^3*d^4+35/13*b^9*c^4*d^3)*x^13+(6*a^3*b^6*d^7+18*a^2*b^7*c*d^6+27/2*a*b^8*c^2*d^5+5/2*b^9*c^3*d^4)*x^14+(12/5*
a^2*b^7*d^7+21/5*a*b^8*c*d^6+7/5*b^9*c^2*d^5)*x^15+(9/16*a*b^8*d^7+7/16*b^9*c*d^6)*x^16+1/17*b^9*d^7*x^17

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1023 vs. \(2 (184) = 368\).

Time = 0.23 (sec) , antiderivative size = 1023, normalized size of antiderivative = 5.12 \[ \int (a+b x)^9 (c+d x)^7 \, dx=\frac {1}{17} \, b^{9} d^{7} x^{17} + a^{9} c^{7} x + \frac {1}{16} \, {\left (7 \, b^{9} c d^{6} + 9 \, a b^{8} d^{7}\right )} x^{16} + \frac {1}{5} \, {\left (7 \, b^{9} c^{2} d^{5} + 21 \, a b^{8} c d^{6} + 12 \, a^{2} b^{7} d^{7}\right )} x^{15} + \frac {1}{2} \, {\left (5 \, b^{9} c^{3} d^{4} + 27 \, a b^{8} c^{2} d^{5} + 36 \, a^{2} b^{7} c d^{6} + 12 \, a^{3} b^{6} d^{7}\right )} x^{14} + \frac {7}{13} \, {\left (5 \, b^{9} c^{4} d^{3} + 45 \, a b^{8} c^{3} d^{4} + 108 \, a^{2} b^{7} c^{2} d^{5} + 84 \, a^{3} b^{6} c d^{6} + 18 \, a^{4} b^{5} d^{7}\right )} x^{13} + \frac {7}{4} \, {\left (b^{9} c^{5} d^{2} + 15 \, a b^{8} c^{4} d^{3} + 60 \, a^{2} b^{7} c^{3} d^{4} + 84 \, a^{3} b^{6} c^{2} d^{5} + 42 \, a^{4} b^{5} c d^{6} + 6 \, a^{5} b^{4} d^{7}\right )} x^{12} + \frac {7}{11} \, {\left (b^{9} c^{6} d + 27 \, a b^{8} c^{5} d^{2} + 180 \, a^{2} b^{7} c^{4} d^{3} + 420 \, a^{3} b^{6} c^{3} d^{4} + 378 \, a^{4} b^{5} c^{2} d^{5} + 126 \, a^{5} b^{4} c d^{6} + 12 \, a^{6} b^{3} d^{7}\right )} x^{11} + \frac {1}{10} \, {\left (b^{9} c^{7} + 63 \, a b^{8} c^{6} d + 756 \, a^{2} b^{7} c^{5} d^{2} + 2940 \, a^{3} b^{6} c^{4} d^{3} + 4410 \, a^{4} b^{5} c^{3} d^{4} + 2646 \, a^{5} b^{4} c^{2} d^{5} + 588 \, a^{6} b^{3} c d^{6} + 36 \, a^{7} b^{2} d^{7}\right )} x^{10} + {\left (a b^{8} c^{7} + 28 \, a^{2} b^{7} c^{6} d + 196 \, a^{3} b^{6} c^{5} d^{2} + 490 \, a^{4} b^{5} c^{4} d^{3} + 490 \, a^{5} b^{4} c^{3} d^{4} + 196 \, a^{6} b^{3} c^{2} d^{5} + 28 \, a^{7} b^{2} c d^{6} + a^{8} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (36 \, a^{2} b^{7} c^{7} + 588 \, a^{3} b^{6} c^{6} d + 2646 \, a^{4} b^{5} c^{5} d^{2} + 4410 \, a^{5} b^{4} c^{4} d^{3} + 2940 \, a^{6} b^{3} c^{3} d^{4} + 756 \, a^{7} b^{2} c^{2} d^{5} + 63 \, a^{8} b c d^{6} + a^{9} d^{7}\right )} x^{8} + {\left (12 \, a^{3} b^{6} c^{7} + 126 \, a^{4} b^{5} c^{6} d + 378 \, a^{5} b^{4} c^{5} d^{2} + 420 \, a^{6} b^{3} c^{4} d^{3} + 180 \, a^{7} b^{2} c^{3} d^{4} + 27 \, a^{8} b c^{2} d^{5} + a^{9} c d^{6}\right )} x^{7} + \frac {7}{2} \, {\left (6 \, a^{4} b^{5} c^{7} + 42 \, a^{5} b^{4} c^{6} d + 84 \, a^{6} b^{3} c^{5} d^{2} + 60 \, a^{7} b^{2} c^{4} d^{3} + 15 \, a^{8} b c^{3} d^{4} + a^{9} c^{2} d^{5}\right )} x^{6} + \frac {7}{5} \, {\left (18 \, a^{5} b^{4} c^{7} + 84 \, a^{6} b^{3} c^{6} d + 108 \, a^{7} b^{2} c^{5} d^{2} + 45 \, a^{8} b c^{4} d^{3} + 5 \, a^{9} c^{3} d^{4}\right )} x^{5} + \frac {7}{4} \, {\left (12 \, a^{6} b^{3} c^{7} + 36 \, a^{7} b^{2} c^{6} d + 27 \, a^{8} b c^{5} d^{2} + 5 \, a^{9} c^{4} d^{3}\right )} x^{4} + {\left (12 \, a^{7} b^{2} c^{7} + 21 \, a^{8} b c^{6} d + 7 \, a^{9} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (9 \, a^{8} b c^{7} + 7 \, a^{9} c^{6} d\right )} x^{2} \]

[In]

integrate((b*x+a)^9*(d*x+c)^7,x, algorithm="fricas")

[Out]

1/17*b^9*d^7*x^17 + a^9*c^7*x + 1/16*(7*b^9*c*d^6 + 9*a*b^8*d^7)*x^16 + 1/5*(7*b^9*c^2*d^5 + 21*a*b^8*c*d^6 +
12*a^2*b^7*d^7)*x^15 + 1/2*(5*b^9*c^3*d^4 + 27*a*b^8*c^2*d^5 + 36*a^2*b^7*c*d^6 + 12*a^3*b^6*d^7)*x^14 + 7/13*
(5*b^9*c^4*d^3 + 45*a*b^8*c^3*d^4 + 108*a^2*b^7*c^2*d^5 + 84*a^3*b^6*c*d^6 + 18*a^4*b^5*d^7)*x^13 + 7/4*(b^9*c
^5*d^2 + 15*a*b^8*c^4*d^3 + 60*a^2*b^7*c^3*d^4 + 84*a^3*b^6*c^2*d^5 + 42*a^4*b^5*c*d^6 + 6*a^5*b^4*d^7)*x^12 +
7/11*(b^9*c^6*d + 27*a*b^8*c^5*d^2 + 180*a^2*b^7*c^4*d^3 + 420*a^3*b^6*c^3*d^4 + 378*a^4*b^5*c^2*d^5 + 126*a^
5*b^4*c*d^6 + 12*a^6*b^3*d^7)*x^11 + 1/10*(b^9*c^7 + 63*a*b^8*c^6*d + 756*a^2*b^7*c^5*d^2 + 2940*a^3*b^6*c^4*d
^3 + 4410*a^4*b^5*c^3*d^4 + 2646*a^5*b^4*c^2*d^5 + 588*a^6*b^3*c*d^6 + 36*a^7*b^2*d^7)*x^10 + (a*b^8*c^7 + 28*
a^2*b^7*c^6*d + 196*a^3*b^6*c^5*d^2 + 490*a^4*b^5*c^4*d^3 + 490*a^5*b^4*c^3*d^4 + 196*a^6*b^3*c^2*d^5 + 28*a^7
*b^2*c*d^6 + a^8*b*d^7)*x^9 + 1/8*(36*a^2*b^7*c^7 + 588*a^3*b^6*c^6*d + 2646*a^4*b^5*c^5*d^2 + 4410*a^5*b^4*c^
4*d^3 + 2940*a^6*b^3*c^3*d^4 + 756*a^7*b^2*c^2*d^5 + 63*a^8*b*c*d^6 + a^9*d^7)*x^8 + (12*a^3*b^6*c^7 + 126*a^4
*b^5*c^6*d + 378*a^5*b^4*c^5*d^2 + 420*a^6*b^3*c^4*d^3 + 180*a^7*b^2*c^3*d^4 + 27*a^8*b*c^2*d^5 + a^9*c*d^6)*x
^7 + 7/2*(6*a^4*b^5*c^7 + 42*a^5*b^4*c^6*d + 84*a^6*b^3*c^5*d^2 + 60*a^7*b^2*c^4*d^3 + 15*a^8*b*c^3*d^4 + a^9*
c^2*d^5)*x^6 + 7/5*(18*a^5*b^4*c^7 + 84*a^6*b^3*c^6*d + 108*a^7*b^2*c^5*d^2 + 45*a^8*b*c^4*d^3 + 5*a^9*c^3*d^4
)*x^5 + 7/4*(12*a^6*b^3*c^7 + 36*a^7*b^2*c^6*d + 27*a^8*b*c^5*d^2 + 5*a^9*c^4*d^3)*x^4 + (12*a^7*b^2*c^7 + 21*
a^8*b*c^6*d + 7*a^9*c^5*d^2)*x^3 + 1/2*(9*a^8*b*c^7 + 7*a^9*c^6*d)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1163 vs. \(2 (184) = 368\).

Time = 0.10 (sec) , antiderivative size = 1163, normalized size of antiderivative = 5.82 \[ \int (a+b x)^9 (c+d x)^7 \, dx=a^{9} c^{7} x + \frac {b^{9} d^{7} x^{17}}{17} + x^{16} \cdot \left (\frac {9 a b^{8} d^{7}}{16} + \frac {7 b^{9} c d^{6}}{16}\right ) + x^{15} \cdot \left (\frac {12 a^{2} b^{7} d^{7}}{5} + \frac {21 a b^{8} c d^{6}}{5} + \frac {7 b^{9} c^{2} d^{5}}{5}\right ) + x^{14} \cdot \left (6 a^{3} b^{6} d^{7} + 18 a^{2} b^{7} c d^{6} + \frac {27 a b^{8} c^{2} d^{5}}{2} + \frac {5 b^{9} c^{3} d^{4}}{2}\right ) + x^{13} \cdot \left (\frac {126 a^{4} b^{5} d^{7}}{13} + \frac {588 a^{3} b^{6} c d^{6}}{13} + \frac {756 a^{2} b^{7} c^{2} d^{5}}{13} + \frac {315 a b^{8} c^{3} d^{4}}{13} + \frac {35 b^{9} c^{4} d^{3}}{13}\right ) + x^{12} \cdot \left (\frac {21 a^{5} b^{4} d^{7}}{2} + \frac {147 a^{4} b^{5} c d^{6}}{2} + 147 a^{3} b^{6} c^{2} d^{5} + 105 a^{2} b^{7} c^{3} d^{4} + \frac {105 a b^{8} c^{4} d^{3}}{4} + \frac {7 b^{9} c^{5} d^{2}}{4}\right ) + x^{11} \cdot \left (\frac {84 a^{6} b^{3} d^{7}}{11} + \frac {882 a^{5} b^{4} c d^{6}}{11} + \frac {2646 a^{4} b^{5} c^{2} d^{5}}{11} + \frac {2940 a^{3} b^{6} c^{3} d^{4}}{11} + \frac {1260 a^{2} b^{7} c^{4} d^{3}}{11} + \frac {189 a b^{8} c^{5} d^{2}}{11} + \frac {7 b^{9} c^{6} d}{11}\right ) + x^{10} \cdot \left (\frac {18 a^{7} b^{2} d^{7}}{5} + \frac {294 a^{6} b^{3} c d^{6}}{5} + \frac {1323 a^{5} b^{4} c^{2} d^{5}}{5} + 441 a^{4} b^{5} c^{3} d^{4} + 294 a^{3} b^{6} c^{4} d^{3} + \frac {378 a^{2} b^{7} c^{5} d^{2}}{5} + \frac {63 a b^{8} c^{6} d}{10} + \frac {b^{9} c^{7}}{10}\right ) + x^{9} \left (a^{8} b d^{7} + 28 a^{7} b^{2} c d^{6} + 196 a^{6} b^{3} c^{2} d^{5} + 490 a^{5} b^{4} c^{3} d^{4} + 490 a^{4} b^{5} c^{4} d^{3} + 196 a^{3} b^{6} c^{5} d^{2} + 28 a^{2} b^{7} c^{6} d + a b^{8} c^{7}\right ) + x^{8} \left (\frac {a^{9} d^{7}}{8} + \frac {63 a^{8} b c d^{6}}{8} + \frac {189 a^{7} b^{2} c^{2} d^{5}}{2} + \frac {735 a^{6} b^{3} c^{3} d^{4}}{2} + \frac {2205 a^{5} b^{4} c^{4} d^{3}}{4} + \frac {1323 a^{4} b^{5} c^{5} d^{2}}{4} + \frac {147 a^{3} b^{6} c^{6} d}{2} + \frac {9 a^{2} b^{7} c^{7}}{2}\right ) + x^{7} \left (a^{9} c d^{6} + 27 a^{8} b c^{2} d^{5} + 180 a^{7} b^{2} c^{3} d^{4} + 420 a^{6} b^{3} c^{4} d^{3} + 378 a^{5} b^{4} c^{5} d^{2} + 126 a^{4} b^{5} c^{6} d + 12 a^{3} b^{6} c^{7}\right ) + x^{6} \cdot \left (\frac {7 a^{9} c^{2} d^{5}}{2} + \frac {105 a^{8} b c^{3} d^{4}}{2} + 210 a^{7} b^{2} c^{4} d^{3} + 294 a^{6} b^{3} c^{5} d^{2} + 147 a^{5} b^{4} c^{6} d + 21 a^{4} b^{5} c^{7}\right ) + x^{5} \cdot \left (7 a^{9} c^{3} d^{4} + 63 a^{8} b c^{4} d^{3} + \frac {756 a^{7} b^{2} c^{5} d^{2}}{5} + \frac {588 a^{6} b^{3} c^{6} d}{5} + \frac {126 a^{5} b^{4} c^{7}}{5}\right ) + x^{4} \cdot \left (\frac {35 a^{9} c^{4} d^{3}}{4} + \frac {189 a^{8} b c^{5} d^{2}}{4} + 63 a^{7} b^{2} c^{6} d + 21 a^{6} b^{3} c^{7}\right ) + x^{3} \cdot \left (7 a^{9} c^{5} d^{2} + 21 a^{8} b c^{6} d + 12 a^{7} b^{2} c^{7}\right ) + x^{2} \cdot \left (\frac {7 a^{9} c^{6} d}{2} + \frac {9 a^{8} b c^{7}}{2}\right ) \]

[In]

integrate((b*x+a)**9*(d*x+c)**7,x)

[Out]

a**9*c**7*x + b**9*d**7*x**17/17 + x**16*(9*a*b**8*d**7/16 + 7*b**9*c*d**6/16) + x**15*(12*a**2*b**7*d**7/5 +
21*a*b**8*c*d**6/5 + 7*b**9*c**2*d**5/5) + x**14*(6*a**3*b**6*d**7 + 18*a**2*b**7*c*d**6 + 27*a*b**8*c**2*d**5
/2 + 5*b**9*c**3*d**4/2) + x**13*(126*a**4*b**5*d**7/13 + 588*a**3*b**6*c*d**6/13 + 756*a**2*b**7*c**2*d**5/13
+ 315*a*b**8*c**3*d**4/13 + 35*b**9*c**4*d**3/13) + x**12*(21*a**5*b**4*d**7/2 + 147*a**4*b**5*c*d**6/2 + 147
*a**3*b**6*c**2*d**5 + 105*a**2*b**7*c**3*d**4 + 105*a*b**8*c**4*d**3/4 + 7*b**9*c**5*d**2/4) + x**11*(84*a**6
*b**3*d**7/11 + 882*a**5*b**4*c*d**6/11 + 2646*a**4*b**5*c**2*d**5/11 + 2940*a**3*b**6*c**3*d**4/11 + 1260*a**
2*b**7*c**4*d**3/11 + 189*a*b**8*c**5*d**2/11 + 7*b**9*c**6*d/11) + x**10*(18*a**7*b**2*d**7/5 + 294*a**6*b**3
*c*d**6/5 + 1323*a**5*b**4*c**2*d**5/5 + 441*a**4*b**5*c**3*d**4 + 294*a**3*b**6*c**4*d**3 + 378*a**2*b**7*c**
5*d**2/5 + 63*a*b**8*c**6*d/10 + b**9*c**7/10) + x**9*(a**8*b*d**7 + 28*a**7*b**2*c*d**6 + 196*a**6*b**3*c**2*
d**5 + 490*a**5*b**4*c**3*d**4 + 490*a**4*b**5*c**4*d**3 + 196*a**3*b**6*c**5*d**2 + 28*a**2*b**7*c**6*d + a*b
**8*c**7) + x**8*(a**9*d**7/8 + 63*a**8*b*c*d**6/8 + 189*a**7*b**2*c**2*d**5/2 + 735*a**6*b**3*c**3*d**4/2 + 2
205*a**5*b**4*c**4*d**3/4 + 1323*a**4*b**5*c**5*d**2/4 + 147*a**3*b**6*c**6*d/2 + 9*a**2*b**7*c**7/2) + x**7*(
a**9*c*d**6 + 27*a**8*b*c**2*d**5 + 180*a**7*b**2*c**3*d**4 + 420*a**6*b**3*c**4*d**3 + 378*a**5*b**4*c**5*d**
2 + 126*a**4*b**5*c**6*d + 12*a**3*b**6*c**7) + x**6*(7*a**9*c**2*d**5/2 + 105*a**8*b*c**3*d**4/2 + 210*a**7*b
**2*c**4*d**3 + 294*a**6*b**3*c**5*d**2 + 147*a**5*b**4*c**6*d + 21*a**4*b**5*c**7) + x**5*(7*a**9*c**3*d**4 +
63*a**8*b*c**4*d**3 + 756*a**7*b**2*c**5*d**2/5 + 588*a**6*b**3*c**6*d/5 + 126*a**5*b**4*c**7/5) + x**4*(35*a
**9*c**4*d**3/4 + 189*a**8*b*c**5*d**2/4 + 63*a**7*b**2*c**6*d + 21*a**6*b**3*c**7) + x**3*(7*a**9*c**5*d**2 +
21*a**8*b*c**6*d + 12*a**7*b**2*c**7) + x**2*(7*a**9*c**6*d/2 + 9*a**8*b*c**7/2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1023 vs. \(2 (184) = 368\).

Time = 0.21 (sec) , antiderivative size = 1023, normalized size of antiderivative = 5.12 \[ \int (a+b x)^9 (c+d x)^7 \, dx=\frac {1}{17} \, b^{9} d^{7} x^{17} + a^{9} c^{7} x + \frac {1}{16} \, {\left (7 \, b^{9} c d^{6} + 9 \, a b^{8} d^{7}\right )} x^{16} + \frac {1}{5} \, {\left (7 \, b^{9} c^{2} d^{5} + 21 \, a b^{8} c d^{6} + 12 \, a^{2} b^{7} d^{7}\right )} x^{15} + \frac {1}{2} \, {\left (5 \, b^{9} c^{3} d^{4} + 27 \, a b^{8} c^{2} d^{5} + 36 \, a^{2} b^{7} c d^{6} + 12 \, a^{3} b^{6} d^{7}\right )} x^{14} + \frac {7}{13} \, {\left (5 \, b^{9} c^{4} d^{3} + 45 \, a b^{8} c^{3} d^{4} + 108 \, a^{2} b^{7} c^{2} d^{5} + 84 \, a^{3} b^{6} c d^{6} + 18 \, a^{4} b^{5} d^{7}\right )} x^{13} + \frac {7}{4} \, {\left (b^{9} c^{5} d^{2} + 15 \, a b^{8} c^{4} d^{3} + 60 \, a^{2} b^{7} c^{3} d^{4} + 84 \, a^{3} b^{6} c^{2} d^{5} + 42 \, a^{4} b^{5} c d^{6} + 6 \, a^{5} b^{4} d^{7}\right )} x^{12} + \frac {7}{11} \, {\left (b^{9} c^{6} d + 27 \, a b^{8} c^{5} d^{2} + 180 \, a^{2} b^{7} c^{4} d^{3} + 420 \, a^{3} b^{6} c^{3} d^{4} + 378 \, a^{4} b^{5} c^{2} d^{5} + 126 \, a^{5} b^{4} c d^{6} + 12 \, a^{6} b^{3} d^{7}\right )} x^{11} + \frac {1}{10} \, {\left (b^{9} c^{7} + 63 \, a b^{8} c^{6} d + 756 \, a^{2} b^{7} c^{5} d^{2} + 2940 \, a^{3} b^{6} c^{4} d^{3} + 4410 \, a^{4} b^{5} c^{3} d^{4} + 2646 \, a^{5} b^{4} c^{2} d^{5} + 588 \, a^{6} b^{3} c d^{6} + 36 \, a^{7} b^{2} d^{7}\right )} x^{10} + {\left (a b^{8} c^{7} + 28 \, a^{2} b^{7} c^{6} d + 196 \, a^{3} b^{6} c^{5} d^{2} + 490 \, a^{4} b^{5} c^{4} d^{3} + 490 \, a^{5} b^{4} c^{3} d^{4} + 196 \, a^{6} b^{3} c^{2} d^{5} + 28 \, a^{7} b^{2} c d^{6} + a^{8} b d^{7}\right )} x^{9} + \frac {1}{8} \, {\left (36 \, a^{2} b^{7} c^{7} + 588 \, a^{3} b^{6} c^{6} d + 2646 \, a^{4} b^{5} c^{5} d^{2} + 4410 \, a^{5} b^{4} c^{4} d^{3} + 2940 \, a^{6} b^{3} c^{3} d^{4} + 756 \, a^{7} b^{2} c^{2} d^{5} + 63 \, a^{8} b c d^{6} + a^{9} d^{7}\right )} x^{8} + {\left (12 \, a^{3} b^{6} c^{7} + 126 \, a^{4} b^{5} c^{6} d + 378 \, a^{5} b^{4} c^{5} d^{2} + 420 \, a^{6} b^{3} c^{4} d^{3} + 180 \, a^{7} b^{2} c^{3} d^{4} + 27 \, a^{8} b c^{2} d^{5} + a^{9} c d^{6}\right )} x^{7} + \frac {7}{2} \, {\left (6 \, a^{4} b^{5} c^{7} + 42 \, a^{5} b^{4} c^{6} d + 84 \, a^{6} b^{3} c^{5} d^{2} + 60 \, a^{7} b^{2} c^{4} d^{3} + 15 \, a^{8} b c^{3} d^{4} + a^{9} c^{2} d^{5}\right )} x^{6} + \frac {7}{5} \, {\left (18 \, a^{5} b^{4} c^{7} + 84 \, a^{6} b^{3} c^{6} d + 108 \, a^{7} b^{2} c^{5} d^{2} + 45 \, a^{8} b c^{4} d^{3} + 5 \, a^{9} c^{3} d^{4}\right )} x^{5} + \frac {7}{4} \, {\left (12 \, a^{6} b^{3} c^{7} + 36 \, a^{7} b^{2} c^{6} d + 27 \, a^{8} b c^{5} d^{2} + 5 \, a^{9} c^{4} d^{3}\right )} x^{4} + {\left (12 \, a^{7} b^{2} c^{7} + 21 \, a^{8} b c^{6} d + 7 \, a^{9} c^{5} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (9 \, a^{8} b c^{7} + 7 \, a^{9} c^{6} d\right )} x^{2} \]

[In]

integrate((b*x+a)^9*(d*x+c)^7,x, algorithm="maxima")

[Out]

1/17*b^9*d^7*x^17 + a^9*c^7*x + 1/16*(7*b^9*c*d^6 + 9*a*b^8*d^7)*x^16 + 1/5*(7*b^9*c^2*d^5 + 21*a*b^8*c*d^6 +
12*a^2*b^7*d^7)*x^15 + 1/2*(5*b^9*c^3*d^4 + 27*a*b^8*c^2*d^5 + 36*a^2*b^7*c*d^6 + 12*a^3*b^6*d^7)*x^14 + 7/13*
(5*b^9*c^4*d^3 + 45*a*b^8*c^3*d^4 + 108*a^2*b^7*c^2*d^5 + 84*a^3*b^6*c*d^6 + 18*a^4*b^5*d^7)*x^13 + 7/4*(b^9*c
^5*d^2 + 15*a*b^8*c^4*d^3 + 60*a^2*b^7*c^3*d^4 + 84*a^3*b^6*c^2*d^5 + 42*a^4*b^5*c*d^6 + 6*a^5*b^4*d^7)*x^12 +
7/11*(b^9*c^6*d + 27*a*b^8*c^5*d^2 + 180*a^2*b^7*c^4*d^3 + 420*a^3*b^6*c^3*d^4 + 378*a^4*b^5*c^2*d^5 + 126*a^
5*b^4*c*d^6 + 12*a^6*b^3*d^7)*x^11 + 1/10*(b^9*c^7 + 63*a*b^8*c^6*d + 756*a^2*b^7*c^5*d^2 + 2940*a^3*b^6*c^4*d
^3 + 4410*a^4*b^5*c^3*d^4 + 2646*a^5*b^4*c^2*d^5 + 588*a^6*b^3*c*d^6 + 36*a^7*b^2*d^7)*x^10 + (a*b^8*c^7 + 28*
a^2*b^7*c^6*d + 196*a^3*b^6*c^5*d^2 + 490*a^4*b^5*c^4*d^3 + 490*a^5*b^4*c^3*d^4 + 196*a^6*b^3*c^2*d^5 + 28*a^7
*b^2*c*d^6 + a^8*b*d^7)*x^9 + 1/8*(36*a^2*b^7*c^7 + 588*a^3*b^6*c^6*d + 2646*a^4*b^5*c^5*d^2 + 4410*a^5*b^4*c^
4*d^3 + 2940*a^6*b^3*c^3*d^4 + 756*a^7*b^2*c^2*d^5 + 63*a^8*b*c*d^6 + a^9*d^7)*x^8 + (12*a^3*b^6*c^7 + 126*a^4
*b^5*c^6*d + 378*a^5*b^4*c^5*d^2 + 420*a^6*b^3*c^4*d^3 + 180*a^7*b^2*c^3*d^4 + 27*a^8*b*c^2*d^5 + a^9*c*d^6)*x
^7 + 7/2*(6*a^4*b^5*c^7 + 42*a^5*b^4*c^6*d + 84*a^6*b^3*c^5*d^2 + 60*a^7*b^2*c^4*d^3 + 15*a^8*b*c^3*d^4 + a^9*
c^2*d^5)*x^6 + 7/5*(18*a^5*b^4*c^7 + 84*a^6*b^3*c^6*d + 108*a^7*b^2*c^5*d^2 + 45*a^8*b*c^4*d^3 + 5*a^9*c^3*d^4
)*x^5 + 7/4*(12*a^6*b^3*c^7 + 36*a^7*b^2*c^6*d + 27*a^8*b*c^5*d^2 + 5*a^9*c^4*d^3)*x^4 + (12*a^7*b^2*c^7 + 21*
a^8*b*c^6*d + 7*a^9*c^5*d^2)*x^3 + 1/2*(9*a^8*b*c^7 + 7*a^9*c^6*d)*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1175 vs. \(2 (184) = 368\).

Time = 0.32 (sec) , antiderivative size = 1175, normalized size of antiderivative = 5.88 \[ \int (a+b x)^9 (c+d x)^7 \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^9*(d*x+c)^7,x, algorithm="giac")

[Out]

1/17*b^9*d^7*x^17 + 7/16*b^9*c*d^6*x^16 + 9/16*a*b^8*d^7*x^16 + 7/5*b^9*c^2*d^5*x^15 + 21/5*a*b^8*c*d^6*x^15 +
12/5*a^2*b^7*d^7*x^15 + 5/2*b^9*c^3*d^4*x^14 + 27/2*a*b^8*c^2*d^5*x^14 + 18*a^2*b^7*c*d^6*x^14 + 6*a^3*b^6*d^
7*x^14 + 35/13*b^9*c^4*d^3*x^13 + 315/13*a*b^8*c^3*d^4*x^13 + 756/13*a^2*b^7*c^2*d^5*x^13 + 588/13*a^3*b^6*c*d
^6*x^13 + 126/13*a^4*b^5*d^7*x^13 + 7/4*b^9*c^5*d^2*x^12 + 105/4*a*b^8*c^4*d^3*x^12 + 105*a^2*b^7*c^3*d^4*x^12
+ 147*a^3*b^6*c^2*d^5*x^12 + 147/2*a^4*b^5*c*d^6*x^12 + 21/2*a^5*b^4*d^7*x^12 + 7/11*b^9*c^6*d*x^11 + 189/11*
a*b^8*c^5*d^2*x^11 + 1260/11*a^2*b^7*c^4*d^3*x^11 + 2940/11*a^3*b^6*c^3*d^4*x^11 + 2646/11*a^4*b^5*c^2*d^5*x^1
1 + 882/11*a^5*b^4*c*d^6*x^11 + 84/11*a^6*b^3*d^7*x^11 + 1/10*b^9*c^7*x^10 + 63/10*a*b^8*c^6*d*x^10 + 378/5*a^
2*b^7*c^5*d^2*x^10 + 294*a^3*b^6*c^4*d^3*x^10 + 441*a^4*b^5*c^3*d^4*x^10 + 1323/5*a^5*b^4*c^2*d^5*x^10 + 294/5
*a^6*b^3*c*d^6*x^10 + 18/5*a^7*b^2*d^7*x^10 + a*b^8*c^7*x^9 + 28*a^2*b^7*c^6*d*x^9 + 196*a^3*b^6*c^5*d^2*x^9 +
490*a^4*b^5*c^4*d^3*x^9 + 490*a^5*b^4*c^3*d^4*x^9 + 196*a^6*b^3*c^2*d^5*x^9 + 28*a^7*b^2*c*d^6*x^9 + a^8*b*d^
7*x^9 + 9/2*a^2*b^7*c^7*x^8 + 147/2*a^3*b^6*c^6*d*x^8 + 1323/4*a^4*b^5*c^5*d^2*x^8 + 2205/4*a^5*b^4*c^4*d^3*x^
8 + 735/2*a^6*b^3*c^3*d^4*x^8 + 189/2*a^7*b^2*c^2*d^5*x^8 + 63/8*a^8*b*c*d^6*x^8 + 1/8*a^9*d^7*x^8 + 12*a^3*b^
6*c^7*x^7 + 126*a^4*b^5*c^6*d*x^7 + 378*a^5*b^4*c^5*d^2*x^7 + 420*a^6*b^3*c^4*d^3*x^7 + 180*a^7*b^2*c^3*d^4*x^
7 + 27*a^8*b*c^2*d^5*x^7 + a^9*c*d^6*x^7 + 21*a^4*b^5*c^7*x^6 + 147*a^5*b^4*c^6*d*x^6 + 294*a^6*b^3*c^5*d^2*x^
6 + 210*a^7*b^2*c^4*d^3*x^6 + 105/2*a^8*b*c^3*d^4*x^6 + 7/2*a^9*c^2*d^5*x^6 + 126/5*a^5*b^4*c^7*x^5 + 588/5*a^
6*b^3*c^6*d*x^5 + 756/5*a^7*b^2*c^5*d^2*x^5 + 63*a^8*b*c^4*d^3*x^5 + 7*a^9*c^3*d^4*x^5 + 21*a^6*b^3*c^7*x^4 +
63*a^7*b^2*c^6*d*x^4 + 189/4*a^8*b*c^5*d^2*x^4 + 35/4*a^9*c^4*d^3*x^4 + 12*a^7*b^2*c^7*x^3 + 21*a^8*b*c^6*d*x^
3 + 7*a^9*c^5*d^2*x^3 + 9/2*a^8*b*c^7*x^2 + 7/2*a^9*c^6*d*x^2 + a^9*c^7*x

Mupad [B] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 997, normalized size of antiderivative = 4.98 \[ \int (a+b x)^9 (c+d x)^7 \, dx=x^5\,\left (7\,a^9\,c^3\,d^4+63\,a^8\,b\,c^4\,d^3+\frac {756\,a^7\,b^2\,c^5\,d^2}{5}+\frac {588\,a^6\,b^3\,c^6\,d}{5}+\frac {126\,a^5\,b^4\,c^7}{5}\right )+x^{13}\,\left (\frac {126\,a^4\,b^5\,d^7}{13}+\frac {588\,a^3\,b^6\,c\,d^6}{13}+\frac {756\,a^2\,b^7\,c^2\,d^5}{13}+\frac {315\,a\,b^8\,c^3\,d^4}{13}+\frac {35\,b^9\,c^4\,d^3}{13}\right )+x^8\,\left (\frac {a^9\,d^7}{8}+\frac {63\,a^8\,b\,c\,d^6}{8}+\frac {189\,a^7\,b^2\,c^2\,d^5}{2}+\frac {735\,a^6\,b^3\,c^3\,d^4}{2}+\frac {2205\,a^5\,b^4\,c^4\,d^3}{4}+\frac {1323\,a^4\,b^5\,c^5\,d^2}{4}+\frac {147\,a^3\,b^6\,c^6\,d}{2}+\frac {9\,a^2\,b^7\,c^7}{2}\right )+x^{10}\,\left (\frac {18\,a^7\,b^2\,d^7}{5}+\frac {294\,a^6\,b^3\,c\,d^6}{5}+\frac {1323\,a^5\,b^4\,c^2\,d^5}{5}+441\,a^4\,b^5\,c^3\,d^4+294\,a^3\,b^6\,c^4\,d^3+\frac {378\,a^2\,b^7\,c^5\,d^2}{5}+\frac {63\,a\,b^8\,c^6\,d}{10}+\frac {b^9\,c^7}{10}\right )+x^6\,\left (\frac {7\,a^9\,c^2\,d^5}{2}+\frac {105\,a^8\,b\,c^3\,d^4}{2}+210\,a^7\,b^2\,c^4\,d^3+294\,a^6\,b^3\,c^5\,d^2+147\,a^5\,b^4\,c^6\,d+21\,a^4\,b^5\,c^7\right )+x^{12}\,\left (\frac {21\,a^5\,b^4\,d^7}{2}+\frac {147\,a^4\,b^5\,c\,d^6}{2}+147\,a^3\,b^6\,c^2\,d^5+105\,a^2\,b^7\,c^3\,d^4+\frac {105\,a\,b^8\,c^4\,d^3}{4}+\frac {7\,b^9\,c^5\,d^2}{4}\right )+x^7\,\left (a^9\,c\,d^6+27\,a^8\,b\,c^2\,d^5+180\,a^7\,b^2\,c^3\,d^4+420\,a^6\,b^3\,c^4\,d^3+378\,a^5\,b^4\,c^5\,d^2+126\,a^4\,b^5\,c^6\,d+12\,a^3\,b^6\,c^7\right )+x^{11}\,\left (\frac {84\,a^6\,b^3\,d^7}{11}+\frac {882\,a^5\,b^4\,c\,d^6}{11}+\frac {2646\,a^4\,b^5\,c^2\,d^5}{11}+\frac {2940\,a^3\,b^6\,c^3\,d^4}{11}+\frac {1260\,a^2\,b^7\,c^4\,d^3}{11}+\frac {189\,a\,b^8\,c^5\,d^2}{11}+\frac {7\,b^9\,c^6\,d}{11}\right )+x^9\,\left (a^8\,b\,d^7+28\,a^7\,b^2\,c\,d^6+196\,a^6\,b^3\,c^2\,d^5+490\,a^5\,b^4\,c^3\,d^4+490\,a^4\,b^5\,c^4\,d^3+196\,a^3\,b^6\,c^5\,d^2+28\,a^2\,b^7\,c^6\,d+a\,b^8\,c^7\right )+a^9\,c^7\,x+\frac {b^9\,d^7\,x^{17}}{17}+\frac {7\,a^6\,c^4\,x^4\,\left (5\,a^3\,d^3+27\,a^2\,b\,c\,d^2+36\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )}{4}+\frac {b^6\,d^4\,x^{14}\,\left (12\,a^3\,d^3+36\,a^2\,b\,c\,d^2+27\,a\,b^2\,c^2\,d+5\,b^3\,c^3\right )}{2}+\frac {a^8\,c^6\,x^2\,\left (7\,a\,d+9\,b\,c\right )}{2}+\frac {b^8\,d^6\,x^{16}\,\left (9\,a\,d+7\,b\,c\right )}{16}+a^7\,c^5\,x^3\,\left (7\,a^2\,d^2+21\,a\,b\,c\,d+12\,b^2\,c^2\right )+\frac {b^7\,d^5\,x^{15}\,\left (12\,a^2\,d^2+21\,a\,b\,c\,d+7\,b^2\,c^2\right )}{5} \]

[In]

int((a + b*x)^9*(c + d*x)^7,x)

[Out]

x^5*((126*a^5*b^4*c^7)/5 + 7*a^9*c^3*d^4 + (588*a^6*b^3*c^6*d)/5 + 63*a^8*b*c^4*d^3 + (756*a^7*b^2*c^5*d^2)/5)
+ x^13*((126*a^4*b^5*d^7)/13 + (35*b^9*c^4*d^3)/13 + (315*a*b^8*c^3*d^4)/13 + (588*a^3*b^6*c*d^6)/13 + (756*a
^2*b^7*c^2*d^5)/13) + x^8*((a^9*d^7)/8 + (9*a^2*b^7*c^7)/2 + (147*a^3*b^6*c^6*d)/2 + (1323*a^4*b^5*c^5*d^2)/4
+ (2205*a^5*b^4*c^4*d^3)/4 + (735*a^6*b^3*c^3*d^4)/2 + (189*a^7*b^2*c^2*d^5)/2 + (63*a^8*b*c*d^6)/8) + x^10*((
b^9*c^7)/10 + (18*a^7*b^2*d^7)/5 + (294*a^6*b^3*c*d^6)/5 + (378*a^2*b^7*c^5*d^2)/5 + 294*a^3*b^6*c^4*d^3 + 441
*a^4*b^5*c^3*d^4 + (1323*a^5*b^4*c^2*d^5)/5 + (63*a*b^8*c^6*d)/10) + x^6*(21*a^4*b^5*c^7 + (7*a^9*c^2*d^5)/2 +
147*a^5*b^4*c^6*d + (105*a^8*b*c^3*d^4)/2 + 294*a^6*b^3*c^5*d^2 + 210*a^7*b^2*c^4*d^3) + x^12*((21*a^5*b^4*d^
7)/2 + (7*b^9*c^5*d^2)/4 + (105*a*b^8*c^4*d^3)/4 + (147*a^4*b^5*c*d^6)/2 + 105*a^2*b^7*c^3*d^4 + 147*a^3*b^6*c
^2*d^5) + x^7*(a^9*c*d^6 + 12*a^3*b^6*c^7 + 126*a^4*b^5*c^6*d + 27*a^8*b*c^2*d^5 + 378*a^5*b^4*c^5*d^2 + 420*a
^6*b^3*c^4*d^3 + 180*a^7*b^2*c^3*d^4) + x^11*((7*b^9*c^6*d)/11 + (84*a^6*b^3*d^7)/11 + (189*a*b^8*c^5*d^2)/11
+ (882*a^5*b^4*c*d^6)/11 + (1260*a^2*b^7*c^4*d^3)/11 + (2940*a^3*b^6*c^3*d^4)/11 + (2646*a^4*b^5*c^2*d^5)/11)
+ x^9*(a*b^8*c^7 + a^8*b*d^7 + 28*a^2*b^7*c^6*d + 28*a^7*b^2*c*d^6 + 196*a^3*b^6*c^5*d^2 + 490*a^4*b^5*c^4*d^3
+ 490*a^5*b^4*c^3*d^4 + 196*a^6*b^3*c^2*d^5) + a^9*c^7*x + (b^9*d^7*x^17)/17 + (7*a^6*c^4*x^4*(5*a^3*d^3 + 12
*b^3*c^3 + 36*a*b^2*c^2*d + 27*a^2*b*c*d^2))/4 + (b^6*d^4*x^14*(12*a^3*d^3 + 5*b^3*c^3 + 27*a*b^2*c^2*d + 36*a
^2*b*c*d^2))/2 + (a^8*c^6*x^2*(7*a*d + 9*b*c))/2 + (b^8*d^6*x^16*(9*a*d + 7*b*c))/16 + a^7*c^5*x^3*(7*a^2*d^2
+ 12*b^2*c^2 + 21*a*b*c*d) + (b^7*d^5*x^15*(12*a^2*d^2 + 7*b^2*c^2 + 21*a*b*c*d))/5