\(\int (a+b x)^{12} (c+d x)^{10} \, dx\) [1299]
Optimal result
Integrand size = 15, antiderivative size = 275 \[
\int (a+b x)^{12} (c+d x)^{10} \, dx=\frac {(b c-a d)^{10} (a+b x)^{13}}{13 b^{11}}+\frac {5 d (b c-a d)^9 (a+b x)^{14}}{7 b^{11}}+\frac {3 d^2 (b c-a d)^8 (a+b x)^{15}}{b^{11}}+\frac {15 d^3 (b c-a d)^7 (a+b x)^{16}}{2 b^{11}}+\frac {210 d^4 (b c-a d)^6 (a+b x)^{17}}{17 b^{11}}+\frac {14 d^5 (b c-a d)^5 (a+b x)^{18}}{b^{11}}+\frac {210 d^6 (b c-a d)^4 (a+b x)^{19}}{19 b^{11}}+\frac {6 d^7 (b c-a d)^3 (a+b x)^{20}}{b^{11}}+\frac {15 d^8 (b c-a d)^2 (a+b x)^{21}}{7 b^{11}}+\frac {5 d^9 (b c-a d) (a+b x)^{22}}{11 b^{11}}+\frac {d^{10} (a+b x)^{23}}{23 b^{11}}
\]
[Out]
1/13*(-a*d+b*c)^10*(b*x+a)^13/b^11+5/7*d*(-a*d+b*c)^9*(b*x+a)^14/b^11+3*d^2*(-a*d+b*c)^8*(b*x+a)^15/b^11+15/2*
d^3*(-a*d+b*c)^7*(b*x+a)^16/b^11+210/17*d^4*(-a*d+b*c)^6*(b*x+a)^17/b^11+14*d^5*(-a*d+b*c)^5*(b*x+a)^18/b^11+2
10/19*d^6*(-a*d+b*c)^4*(b*x+a)^19/b^11+6*d^7*(-a*d+b*c)^3*(b*x+a)^20/b^11+15/7*d^8*(-a*d+b*c)^2*(b*x+a)^21/b^1
1+5/11*d^9*(-a*d+b*c)*(b*x+a)^22/b^11+1/23*d^10*(b*x+a)^23/b^11
Rubi [A] (verified)
Time = 0.99 (sec) , antiderivative size = 275, 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}
\[
\int (a+b x)^{12} (c+d x)^{10} \, dx=\frac {5 d^9 (a+b x)^{22} (b c-a d)}{11 b^{11}}+\frac {15 d^8 (a+b x)^{21} (b c-a d)^2}{7 b^{11}}+\frac {6 d^7 (a+b x)^{20} (b c-a d)^3}{b^{11}}+\frac {210 d^6 (a+b x)^{19} (b c-a d)^4}{19 b^{11}}+\frac {14 d^5 (a+b x)^{18} (b c-a d)^5}{b^{11}}+\frac {210 d^4 (a+b x)^{17} (b c-a d)^6}{17 b^{11}}+\frac {15 d^3 (a+b x)^{16} (b c-a d)^7}{2 b^{11}}+\frac {3 d^2 (a+b x)^{15} (b c-a d)^8}{b^{11}}+\frac {5 d (a+b x)^{14} (b c-a d)^9}{7 b^{11}}+\frac {(a+b x)^{13} (b c-a d)^{10}}{13 b^{11}}+\frac {d^{10} (a+b x)^{23}}{23 b^{11}}
\]
[In]
Int[(a + b*x)^12*(c + d*x)^10,x]
[Out]
((b*c - a*d)^10*(a + b*x)^13)/(13*b^11) + (5*d*(b*c - a*d)^9*(a + b*x)^14)/(7*b^11) + (3*d^2*(b*c - a*d)^8*(a
+ b*x)^15)/b^11 + (15*d^3*(b*c - a*d)^7*(a + b*x)^16)/(2*b^11) + (210*d^4*(b*c - a*d)^6*(a + b*x)^17)/(17*b^11
) + (14*d^5*(b*c - a*d)^5*(a + b*x)^18)/b^11 + (210*d^6*(b*c - a*d)^4*(a + b*x)^19)/(19*b^11) + (6*d^7*(b*c -
a*d)^3*(a + b*x)^20)/b^11 + (15*d^8*(b*c - a*d)^2*(a + b*x)^21)/(7*b^11) + (5*d^9*(b*c - a*d)*(a + b*x)^22)/(1
1*b^11) + (d^10*(a + b*x)^23)/(23*b^11)
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)^{10} (a+b x)^{12}}{b^{10}}+\frac {10 d (b c-a d)^9 (a+b x)^{13}}{b^{10}}+\frac {45 d^2 (b c-a d)^8 (a+b x)^{14}}{b^{10}}+\frac {120 d^3 (b c-a d)^7 (a+b x)^{15}}{b^{10}}+\frac {210 d^4 (b c-a d)^6 (a+b x)^{16}}{b^{10}}+\frac {252 d^5 (b c-a d)^5 (a+b x)^{17}}{b^{10}}+\frac {210 d^6 (b c-a d)^4 (a+b x)^{18}}{b^{10}}+\frac {120 d^7 (b c-a d)^3 (a+b x)^{19}}{b^{10}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^{20}}{b^{10}}+\frac {10 d^9 (b c-a d) (a+b x)^{21}}{b^{10}}+\frac {d^{10} (a+b x)^{22}}{b^{10}}\right ) \, dx \\ & = \frac {(b c-a d)^{10} (a+b x)^{13}}{13 b^{11}}+\frac {5 d (b c-a d)^9 (a+b x)^{14}}{7 b^{11}}+\frac {3 d^2 (b c-a d)^8 (a+b x)^{15}}{b^{11}}+\frac {15 d^3 (b c-a d)^7 (a+b x)^{16}}{2 b^{11}}+\frac {210 d^4 (b c-a d)^6 (a+b x)^{17}}{17 b^{11}}+\frac {14 d^5 (b c-a d)^5 (a+b x)^{18}}{b^{11}}+\frac {210 d^6 (b c-a d)^4 (a+b x)^{19}}{19 b^{11}}+\frac {6 d^7 (b c-a d)^3 (a+b x)^{20}}{b^{11}}+\frac {15 d^8 (b c-a d)^2 (a+b x)^{21}}{7 b^{11}}+\frac {5 d^9 (b c-a d) (a+b x)^{22}}{11 b^{11}}+\frac {d^{10} (a+b x)^{23}}{23 b^{11}} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(1817\) vs. \(2(275)=550\).
Time = 0.17 (sec) , antiderivative size = 1817, normalized size of antiderivative = 6.61
\[
\int (a+b x)^{12} (c+d x)^{10} \, dx=a^{12} c^{10} x+a^{11} c^9 (6 b c+5 a d) x^2+a^{10} c^8 \left (22 b^2 c^2+40 a b c d+15 a^2 d^2\right ) x^3+5 a^9 c^7 \left (11 b^3 c^3+33 a b^2 c^2 d+27 a^2 b c d^2+6 a^3 d^3\right ) x^4+a^8 c^6 \left (99 b^4 c^4+440 a b^3 c^3 d+594 a^2 b^2 c^2 d^2+288 a^3 b c d^3+42 a^4 d^4\right ) x^5+3 a^7 c^5 \left (44 b^5 c^5+275 a b^4 c^4 d+550 a^2 b^3 c^3 d^2+440 a^3 b^2 c^2 d^3+140 a^4 b c d^4+14 a^5 d^5\right ) x^6+\frac {3}{7} a^6 c^4 \left (308 b^6 c^6+2640 a b^5 c^5 d+7425 a^2 b^4 c^4 d^2+8800 a^3 b^3 c^3 d^3+4620 a^4 b^2 c^2 d^4+1008 a^5 b c d^5+70 a^6 d^6\right ) x^7+3 a^5 c^3 \left (33 b^7 c^7+385 a b^6 c^6 d+1485 a^2 b^5 c^5 d^2+2475 a^3 b^4 c^4 d^3+1925 a^4 b^3 c^3 d^4+693 a^5 b^2 c^2 d^5+105 a^6 b c d^6+5 a^7 d^7\right ) x^8+5 a^4 c^2 \left (11 b^8 c^8+176 a b^7 c^7 d+924 a^2 b^6 c^6 d^2+2112 a^3 b^5 c^5 d^3+2310 a^4 b^4 c^4 d^4+1232 a^5 b^3 c^3 d^5+308 a^6 b^2 c^2 d^6+32 a^7 b c d^7+a^8 d^8\right ) x^9+a^3 c \left (22 b^9 c^9+495 a b^8 c^8 d+3564 a^2 b^7 c^7 d^2+11088 a^3 b^6 c^6 d^3+16632 a^4 b^5 c^5 d^4+12474 a^5 b^4 c^4 d^5+4620 a^6 b^3 c^3 d^6+792 a^7 b^2 c^2 d^7+54 a^8 b c d^8+a^9 d^9\right ) x^{10}+\frac {1}{11} a^2 \left (66 b^{10} c^{10}+2200 a b^9 c^9 d+22275 a^2 b^8 c^8 d^2+95040 a^3 b^7 c^7 d^3+194040 a^4 b^6 c^6 d^4+199584 a^5 b^5 c^5 d^5+103950 a^6 b^4 c^4 d^6+26400 a^7 b^3 c^3 d^7+2970 a^8 b^2 c^2 d^8+120 a^9 b c d^9+a^{10} d^{10}\right ) x^{11}+a b \left (b^{10} c^{10}+55 a b^9 c^9 d+825 a^2 b^8 c^8 d^2+4950 a^3 b^7 c^7 d^3+13860 a^4 b^6 c^6 d^4+19404 a^5 b^5 c^5 d^5+13860 a^6 b^4 c^4 d^6+4950 a^7 b^3 c^3 d^7+825 a^8 b^2 c^2 d^8+55 a^9 b c d^9+a^{10} d^{10}\right ) x^{12}+\frac {1}{13} b^2 \left (b^{10} c^{10}+120 a b^9 c^9 d+2970 a^2 b^8 c^8 d^2+26400 a^3 b^7 c^7 d^3+103950 a^4 b^6 c^6 d^4+199584 a^5 b^5 c^5 d^5+194040 a^6 b^4 c^4 d^6+95040 a^7 b^3 c^3 d^7+22275 a^8 b^2 c^2 d^8+2200 a^9 b c d^9+66 a^{10} d^{10}\right ) x^{13}+\frac {5}{7} b^3 d \left (b^9 c^9+54 a b^8 c^8 d+792 a^2 b^7 c^7 d^2+4620 a^3 b^6 c^6 d^3+12474 a^4 b^5 c^5 d^4+16632 a^5 b^4 c^4 d^5+11088 a^6 b^3 c^3 d^6+3564 a^7 b^2 c^2 d^7+495 a^8 b c d^8+22 a^9 d^9\right ) x^{14}+3 b^4 d^2 \left (b^8 c^8+32 a b^7 c^7 d+308 a^2 b^6 c^6 d^2+1232 a^3 b^5 c^5 d^3+2310 a^4 b^4 c^4 d^4+2112 a^5 b^3 c^3 d^5+924 a^6 b^2 c^2 d^6+176 a^7 b c d^7+11 a^8 d^8\right ) x^{15}+\frac {3}{2} b^5 d^3 \left (5 b^7 c^7+105 a b^6 c^6 d+693 a^2 b^5 c^5 d^2+1925 a^3 b^4 c^4 d^3+2475 a^4 b^3 c^3 d^4+1485 a^5 b^2 c^2 d^5+385 a^6 b c d^6+33 a^7 d^7\right ) x^{16}+\frac {3}{17} b^6 d^4 \left (70 b^6 c^6+1008 a b^5 c^5 d+4620 a^2 b^4 c^4 d^2+8800 a^3 b^3 c^3 d^3+7425 a^4 b^2 c^2 d^4+2640 a^5 b c d^5+308 a^6 d^6\right ) x^{17}+b^7 d^5 \left (14 b^5 c^5+140 a b^4 c^4 d+440 a^2 b^3 c^3 d^2+550 a^3 b^2 c^2 d^3+275 a^4 b c d^4+44 a^5 d^5\right ) x^{18}+\frac {5}{19} b^8 d^6 \left (42 b^4 c^4+288 a b^3 c^3 d+594 a^2 b^2 c^2 d^2+440 a^3 b c d^3+99 a^4 d^4\right ) x^{19}+b^9 d^7 \left (6 b^3 c^3+27 a b^2 c^2 d+33 a^2 b c d^2+11 a^3 d^3\right ) x^{20}+\frac {1}{7} b^{10} d^8 \left (15 b^2 c^2+40 a b c d+22 a^2 d^2\right ) x^{21}+\frac {1}{11} b^{11} d^9 (5 b c+6 a d) x^{22}+\frac {1}{23} b^{12} d^{10} x^{23}
\]
[In]
Integrate[(a + b*x)^12*(c + d*x)^10,x]
[Out]
a^12*c^10*x + a^11*c^9*(6*b*c + 5*a*d)*x^2 + a^10*c^8*(22*b^2*c^2 + 40*a*b*c*d + 15*a^2*d^2)*x^3 + 5*a^9*c^7*(
11*b^3*c^3 + 33*a*b^2*c^2*d + 27*a^2*b*c*d^2 + 6*a^3*d^3)*x^4 + a^8*c^6*(99*b^4*c^4 + 440*a*b^3*c^3*d + 594*a^
2*b^2*c^2*d^2 + 288*a^3*b*c*d^3 + 42*a^4*d^4)*x^5 + 3*a^7*c^5*(44*b^5*c^5 + 275*a*b^4*c^4*d + 550*a^2*b^3*c^3*
d^2 + 440*a^3*b^2*c^2*d^3 + 140*a^4*b*c*d^4 + 14*a^5*d^5)*x^6 + (3*a^6*c^4*(308*b^6*c^6 + 2640*a*b^5*c^5*d + 7
425*a^2*b^4*c^4*d^2 + 8800*a^3*b^3*c^3*d^3 + 4620*a^4*b^2*c^2*d^4 + 1008*a^5*b*c*d^5 + 70*a^6*d^6)*x^7)/7 + 3*
a^5*c^3*(33*b^7*c^7 + 385*a*b^6*c^6*d + 1485*a^2*b^5*c^5*d^2 + 2475*a^3*b^4*c^4*d^3 + 1925*a^4*b^3*c^3*d^4 + 6
93*a^5*b^2*c^2*d^5 + 105*a^6*b*c*d^6 + 5*a^7*d^7)*x^8 + 5*a^4*c^2*(11*b^8*c^8 + 176*a*b^7*c^7*d + 924*a^2*b^6*
c^6*d^2 + 2112*a^3*b^5*c^5*d^3 + 2310*a^4*b^4*c^4*d^4 + 1232*a^5*b^3*c^3*d^5 + 308*a^6*b^2*c^2*d^6 + 32*a^7*b*
c*d^7 + a^8*d^8)*x^9 + a^3*c*(22*b^9*c^9 + 495*a*b^8*c^8*d + 3564*a^2*b^7*c^7*d^2 + 11088*a^3*b^6*c^6*d^3 + 16
632*a^4*b^5*c^5*d^4 + 12474*a^5*b^4*c^4*d^5 + 4620*a^6*b^3*c^3*d^6 + 792*a^7*b^2*c^2*d^7 + 54*a^8*b*c*d^8 + a^
9*d^9)*x^10 + (a^2*(66*b^10*c^10 + 2200*a*b^9*c^9*d + 22275*a^2*b^8*c^8*d^2 + 95040*a^3*b^7*c^7*d^3 + 194040*a
^4*b^6*c^6*d^4 + 199584*a^5*b^5*c^5*d^5 + 103950*a^6*b^4*c^4*d^6 + 26400*a^7*b^3*c^3*d^7 + 2970*a^8*b^2*c^2*d^
8 + 120*a^9*b*c*d^9 + a^10*d^10)*x^11)/11 + a*b*(b^10*c^10 + 55*a*b^9*c^9*d + 825*a^2*b^8*c^8*d^2 + 4950*a^3*b
^7*c^7*d^3 + 13860*a^4*b^6*c^6*d^4 + 19404*a^5*b^5*c^5*d^5 + 13860*a^6*b^4*c^4*d^6 + 4950*a^7*b^3*c^3*d^7 + 82
5*a^8*b^2*c^2*d^8 + 55*a^9*b*c*d^9 + a^10*d^10)*x^12 + (b^2*(b^10*c^10 + 120*a*b^9*c^9*d + 2970*a^2*b^8*c^8*d^
2 + 26400*a^3*b^7*c^7*d^3 + 103950*a^4*b^6*c^6*d^4 + 199584*a^5*b^5*c^5*d^5 + 194040*a^6*b^4*c^4*d^6 + 95040*a
^7*b^3*c^3*d^7 + 22275*a^8*b^2*c^2*d^8 + 2200*a^9*b*c*d^9 + 66*a^10*d^10)*x^13)/13 + (5*b^3*d*(b^9*c^9 + 54*a*
b^8*c^8*d + 792*a^2*b^7*c^7*d^2 + 4620*a^3*b^6*c^6*d^3 + 12474*a^4*b^5*c^5*d^4 + 16632*a^5*b^4*c^4*d^5 + 11088
*a^6*b^3*c^3*d^6 + 3564*a^7*b^2*c^2*d^7 + 495*a^8*b*c*d^8 + 22*a^9*d^9)*x^14)/7 + 3*b^4*d^2*(b^8*c^8 + 32*a*b^
7*c^7*d + 308*a^2*b^6*c^6*d^2 + 1232*a^3*b^5*c^5*d^3 + 2310*a^4*b^4*c^4*d^4 + 2112*a^5*b^3*c^3*d^5 + 924*a^6*b
^2*c^2*d^6 + 176*a^7*b*c*d^7 + 11*a^8*d^8)*x^15 + (3*b^5*d^3*(5*b^7*c^7 + 105*a*b^6*c^6*d + 693*a^2*b^5*c^5*d^
2 + 1925*a^3*b^4*c^4*d^3 + 2475*a^4*b^3*c^3*d^4 + 1485*a^5*b^2*c^2*d^5 + 385*a^6*b*c*d^6 + 33*a^7*d^7)*x^16)/2
+ (3*b^6*d^4*(70*b^6*c^6 + 1008*a*b^5*c^5*d + 4620*a^2*b^4*c^4*d^2 + 8800*a^3*b^3*c^3*d^3 + 7425*a^4*b^2*c^2*
d^4 + 2640*a^5*b*c*d^5 + 308*a^6*d^6)*x^17)/17 + b^7*d^5*(14*b^5*c^5 + 140*a*b^4*c^4*d + 440*a^2*b^3*c^3*d^2 +
550*a^3*b^2*c^2*d^3 + 275*a^4*b*c*d^4 + 44*a^5*d^5)*x^18 + (5*b^8*d^6*(42*b^4*c^4 + 288*a*b^3*c^3*d + 594*a^2
*b^2*c^2*d^2 + 440*a^3*b*c*d^3 + 99*a^4*d^4)*x^19)/19 + b^9*d^7*(6*b^3*c^3 + 27*a*b^2*c^2*d + 33*a^2*b*c*d^2 +
11*a^3*d^3)*x^20 + (b^10*d^8*(15*b^2*c^2 + 40*a*b*c*d + 22*a^2*d^2)*x^21)/7 + (b^11*d^9*(5*b*c + 6*a*d)*x^22)
/11 + (b^12*d^10*x^23)/23
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(1868\) vs. \(2(259)=518\).
Time = 0.21 (sec) , antiderivative size = 1869, normalized size of antiderivative =
6.80
| | |
| method | result | size |
| | |
| norman |
\(\text {Expression too large to display}\) |
\(1869\) |
| default |
\(\text {Expression too large to display}\) |
\(1891\) |
| gosper |
\(\text {Expression too large to display}\) |
\(2187\) |
| risch |
\(\text {Expression too large to display}\) |
\(2187\) |
| parallelrisch |
\(\text {Expression too large to display}\) |
\(2187\) |
| | |
|
|
|
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[In]
int((b*x+a)^12*(d*x+c)^10,x,method=_RETURNVERBOSE)
[Out]
a^12*c^10*x+(5*a^12*c^9*d+6*a^11*b*c^10)*x^2+(15*a^12*c^8*d^2+40*a^11*b*c^9*d+22*a^10*b^2*c^10)*x^3+(30*a^12*c
^7*d^3+135*a^11*b*c^8*d^2+165*a^10*b^2*c^9*d+55*a^9*b^3*c^10)*x^4+(42*a^12*c^6*d^4+288*a^11*b*c^7*d^3+594*a^10
*b^2*c^8*d^2+440*a^9*b^3*c^9*d+99*a^8*b^4*c^10)*x^5+(42*a^12*c^5*d^5+420*a^11*b*c^6*d^4+1320*a^10*b^2*c^7*d^3+
1650*a^9*b^3*c^8*d^2+825*a^8*b^4*c^9*d+132*a^7*b^5*c^10)*x^6+(30*a^12*c^4*d^6+432*a^11*b*c^5*d^5+1980*a^10*b^2
*c^6*d^4+26400/7*a^9*b^3*c^7*d^3+22275/7*a^8*b^4*c^8*d^2+7920/7*a^7*b^5*c^9*d+132*a^6*b^6*c^10)*x^7+(15*a^12*c
^3*d^7+315*a^11*b*c^4*d^6+2079*a^10*b^2*c^5*d^5+5775*a^9*b^3*c^6*d^4+7425*a^8*b^4*c^7*d^3+4455*a^7*b^5*c^8*d^2
+1155*a^6*b^6*c^9*d+99*a^5*b^7*c^10)*x^8+(5*a^12*c^2*d^8+160*a^11*b*c^3*d^7+1540*a^10*b^2*c^4*d^6+6160*a^9*b^3
*c^5*d^5+11550*a^8*b^4*c^6*d^4+10560*a^7*b^5*c^7*d^3+4620*a^6*b^6*c^8*d^2+880*a^5*b^7*c^9*d+55*a^4*b^8*c^10)*x
^9+(a^12*c*d^9+54*a^11*b*c^2*d^8+792*a^10*b^2*c^3*d^7+4620*a^9*b^3*c^4*d^6+12474*a^8*b^4*c^5*d^5+16632*a^7*b^5
*c^6*d^4+11088*a^6*b^6*c^7*d^3+3564*a^5*b^7*c^8*d^2+495*a^4*b^8*c^9*d+22*a^3*b^9*c^10)*x^10+(1/11*a^12*d^10+12
0/11*a^11*b*c*d^9+270*a^10*b^2*c^2*d^8+2400*a^9*b^3*c^3*d^7+9450*a^8*b^4*c^4*d^6+18144*a^7*b^5*c^5*d^5+17640*a
^6*b^6*c^6*d^4+8640*a^5*b^7*c^7*d^3+2025*a^4*b^8*c^8*d^2+200*a^3*b^9*c^9*d+6*a^2*b^10*c^10)*x^11+(a^11*b*d^10+
55*a^10*b^2*c*d^9+825*a^9*b^3*c^2*d^8+4950*a^8*b^4*c^3*d^7+13860*a^7*b^5*c^4*d^6+19404*a^6*b^6*c^5*d^5+13860*a
^5*b^7*c^6*d^4+4950*a^4*b^8*c^7*d^3+825*a^3*b^9*c^8*d^2+55*a^2*b^10*c^9*d+a*b^11*c^10)*x^12+(66/13*a^10*b^2*d^
10+2200/13*a^9*b^3*c*d^9+22275/13*a^8*b^4*c^2*d^8+95040/13*a^7*b^5*c^3*d^7+194040/13*a^6*b^6*c^4*d^6+199584/13
*a^5*b^7*c^5*d^5+103950/13*a^4*b^8*c^6*d^4+26400/13*a^3*b^9*c^7*d^3+2970/13*a^2*b^10*c^8*d^2+120/13*a*b^11*c^9
*d+1/13*b^12*c^10)*x^13+(110/7*a^9*b^3*d^10+2475/7*a^8*b^4*c*d^9+17820/7*a^7*b^5*c^2*d^8+7920*a^6*b^6*c^3*d^7+
11880*a^5*b^7*c^4*d^6+8910*a^4*b^8*c^5*d^5+3300*a^3*b^9*c^6*d^4+3960/7*a^2*b^10*c^7*d^3+270/7*a*b^11*c^8*d^2+5
/7*b^12*c^9*d)*x^14+(33*a^8*b^4*d^10+528*a^7*b^5*c*d^9+2772*a^6*b^6*c^2*d^8+6336*a^5*b^7*c^3*d^7+6930*a^4*b^8*
c^4*d^6+3696*a^3*b^9*c^5*d^5+924*a^2*b^10*c^6*d^4+96*a*b^11*c^7*d^3+3*b^12*c^8*d^2)*x^15+(99/2*a^7*b^5*d^10+11
55/2*a^6*b^6*c*d^9+4455/2*a^5*b^7*c^2*d^8+7425/2*a^4*b^8*c^3*d^7+5775/2*a^3*b^9*c^4*d^6+2079/2*a^2*b^10*c^5*d^
5+315/2*a*b^11*c^6*d^4+15/2*b^12*c^7*d^3)*x^16+(924/17*a^6*b^6*d^10+7920/17*a^5*b^7*c*d^9+22275/17*a^4*b^8*c^2
*d^8+26400/17*a^3*b^9*c^3*d^7+13860/17*a^2*b^10*c^4*d^6+3024/17*a*b^11*c^5*d^5+210/17*b^12*c^6*d^4)*x^17+(44*a
^5*b^7*d^10+275*a^4*b^8*c*d^9+550*a^3*b^9*c^2*d^8+440*a^2*b^10*c^3*d^7+140*a*b^11*c^4*d^6+14*b^12*c^5*d^5)*x^1
8+(495/19*a^4*b^8*d^10+2200/19*a^3*b^9*c*d^9+2970/19*a^2*b^10*c^2*d^8+1440/19*a*b^11*c^3*d^7+210/19*b^12*c^4*d
^6)*x^19+(11*a^3*b^9*d^10+33*a^2*b^10*c*d^9+27*a*b^11*c^2*d^8+6*b^12*c^3*d^7)*x^20+(22/7*a^2*b^10*d^10+40/7*a*
b^11*c*d^9+15/7*b^12*c^2*d^8)*x^21+(6/11*a*b^11*d^10+5/11*b^12*c*d^9)*x^22+1/23*b^12*d^10*x^23
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1877 vs. \(2 (259) = 518\).
Time = 0.24 (sec) , antiderivative size = 1877, normalized size of antiderivative = 6.83
\[
\int (a+b x)^{12} (c+d x)^{10} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^12*(d*x+c)^10,x, algorithm="fricas")
[Out]
1/23*b^12*d^10*x^23 + a^12*c^10*x + 1/11*(5*b^12*c*d^9 + 6*a*b^11*d^10)*x^22 + 1/7*(15*b^12*c^2*d^8 + 40*a*b^1
1*c*d^9 + 22*a^2*b^10*d^10)*x^21 + (6*b^12*c^3*d^7 + 27*a*b^11*c^2*d^8 + 33*a^2*b^10*c*d^9 + 11*a^3*b^9*d^10)*
x^20 + 5/19*(42*b^12*c^4*d^6 + 288*a*b^11*c^3*d^7 + 594*a^2*b^10*c^2*d^8 + 440*a^3*b^9*c*d^9 + 99*a^4*b^8*d^10
)*x^19 + (14*b^12*c^5*d^5 + 140*a*b^11*c^4*d^6 + 440*a^2*b^10*c^3*d^7 + 550*a^3*b^9*c^2*d^8 + 275*a^4*b^8*c*d^
9 + 44*a^5*b^7*d^10)*x^18 + 3/17*(70*b^12*c^6*d^4 + 1008*a*b^11*c^5*d^5 + 4620*a^2*b^10*c^4*d^6 + 8800*a^3*b^9
*c^3*d^7 + 7425*a^4*b^8*c^2*d^8 + 2640*a^5*b^7*c*d^9 + 308*a^6*b^6*d^10)*x^17 + 3/2*(5*b^12*c^7*d^3 + 105*a*b^
11*c^6*d^4 + 693*a^2*b^10*c^5*d^5 + 1925*a^3*b^9*c^4*d^6 + 2475*a^4*b^8*c^3*d^7 + 1485*a^5*b^7*c^2*d^8 + 385*a
^6*b^6*c*d^9 + 33*a^7*b^5*d^10)*x^16 + 3*(b^12*c^8*d^2 + 32*a*b^11*c^7*d^3 + 308*a^2*b^10*c^6*d^4 + 1232*a^3*b
^9*c^5*d^5 + 2310*a^4*b^8*c^4*d^6 + 2112*a^5*b^7*c^3*d^7 + 924*a^6*b^6*c^2*d^8 + 176*a^7*b^5*c*d^9 + 11*a^8*b^
4*d^10)*x^15 + 5/7*(b^12*c^9*d + 54*a*b^11*c^8*d^2 + 792*a^2*b^10*c^7*d^3 + 4620*a^3*b^9*c^6*d^4 + 12474*a^4*b
^8*c^5*d^5 + 16632*a^5*b^7*c^4*d^6 + 11088*a^6*b^6*c^3*d^7 + 3564*a^7*b^5*c^2*d^8 + 495*a^8*b^4*c*d^9 + 22*a^9
*b^3*d^10)*x^14 + 1/13*(b^12*c^10 + 120*a*b^11*c^9*d + 2970*a^2*b^10*c^8*d^2 + 26400*a^3*b^9*c^7*d^3 + 103950*
a^4*b^8*c^6*d^4 + 199584*a^5*b^7*c^5*d^5 + 194040*a^6*b^6*c^4*d^6 + 95040*a^7*b^5*c^3*d^7 + 22275*a^8*b^4*c^2*
d^8 + 2200*a^9*b^3*c*d^9 + 66*a^10*b^2*d^10)*x^13 + (a*b^11*c^10 + 55*a^2*b^10*c^9*d + 825*a^3*b^9*c^8*d^2 + 4
950*a^4*b^8*c^7*d^3 + 13860*a^5*b^7*c^6*d^4 + 19404*a^6*b^6*c^5*d^5 + 13860*a^7*b^5*c^4*d^6 + 4950*a^8*b^4*c^3
*d^7 + 825*a^9*b^3*c^2*d^8 + 55*a^10*b^2*c*d^9 + a^11*b*d^10)*x^12 + 1/11*(66*a^2*b^10*c^10 + 2200*a^3*b^9*c^9
*d + 22275*a^4*b^8*c^8*d^2 + 95040*a^5*b^7*c^7*d^3 + 194040*a^6*b^6*c^6*d^4 + 199584*a^7*b^5*c^5*d^5 + 103950*
a^8*b^4*c^4*d^6 + 26400*a^9*b^3*c^3*d^7 + 2970*a^10*b^2*c^2*d^8 + 120*a^11*b*c*d^9 + a^12*d^10)*x^11 + (22*a^3
*b^9*c^10 + 495*a^4*b^8*c^9*d + 3564*a^5*b^7*c^8*d^2 + 11088*a^6*b^6*c^7*d^3 + 16632*a^7*b^5*c^6*d^4 + 12474*a
^8*b^4*c^5*d^5 + 4620*a^9*b^3*c^4*d^6 + 792*a^10*b^2*c^3*d^7 + 54*a^11*b*c^2*d^8 + a^12*c*d^9)*x^10 + 5*(11*a^
4*b^8*c^10 + 176*a^5*b^7*c^9*d + 924*a^6*b^6*c^8*d^2 + 2112*a^7*b^5*c^7*d^3 + 2310*a^8*b^4*c^6*d^4 + 1232*a^9*
b^3*c^5*d^5 + 308*a^10*b^2*c^4*d^6 + 32*a^11*b*c^3*d^7 + a^12*c^2*d^8)*x^9 + 3*(33*a^5*b^7*c^10 + 385*a^6*b^6*
c^9*d + 1485*a^7*b^5*c^8*d^2 + 2475*a^8*b^4*c^7*d^3 + 1925*a^9*b^3*c^6*d^4 + 693*a^10*b^2*c^5*d^5 + 105*a^11*b
*c^4*d^6 + 5*a^12*c^3*d^7)*x^8 + 3/7*(308*a^6*b^6*c^10 + 2640*a^7*b^5*c^9*d + 7425*a^8*b^4*c^8*d^2 + 8800*a^9*
b^3*c^7*d^3 + 4620*a^10*b^2*c^6*d^4 + 1008*a^11*b*c^5*d^5 + 70*a^12*c^4*d^6)*x^7 + 3*(44*a^7*b^5*c^10 + 275*a^
8*b^4*c^9*d + 550*a^9*b^3*c^8*d^2 + 440*a^10*b^2*c^7*d^3 + 140*a^11*b*c^6*d^4 + 14*a^12*c^5*d^5)*x^6 + (99*a^8
*b^4*c^10 + 440*a^9*b^3*c^9*d + 594*a^10*b^2*c^8*d^2 + 288*a^11*b*c^7*d^3 + 42*a^12*c^6*d^4)*x^5 + 5*(11*a^9*b
^3*c^10 + 33*a^10*b^2*c^9*d + 27*a^11*b*c^8*d^2 + 6*a^12*c^7*d^3)*x^4 + (22*a^10*b^2*c^10 + 40*a^11*b*c^9*d +
15*a^12*c^8*d^2)*x^3 + (6*a^11*b*c^10 + 5*a^12*c^9*d)*x^2
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2088 vs. \(2 (255) = 510\).
Time = 0.17 (sec) , antiderivative size = 2088, normalized size of antiderivative = 7.59
\[
\int (a+b x)^{12} (c+d x)^{10} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)**12*(d*x+c)**10,x)
[Out]
a**12*c**10*x + b**12*d**10*x**23/23 + x**22*(6*a*b**11*d**10/11 + 5*b**12*c*d**9/11) + x**21*(22*a**2*b**10*d
**10/7 + 40*a*b**11*c*d**9/7 + 15*b**12*c**2*d**8/7) + x**20*(11*a**3*b**9*d**10 + 33*a**2*b**10*c*d**9 + 27*a
*b**11*c**2*d**8 + 6*b**12*c**3*d**7) + x**19*(495*a**4*b**8*d**10/19 + 2200*a**3*b**9*c*d**9/19 + 2970*a**2*b
**10*c**2*d**8/19 + 1440*a*b**11*c**3*d**7/19 + 210*b**12*c**4*d**6/19) + x**18*(44*a**5*b**7*d**10 + 275*a**4
*b**8*c*d**9 + 550*a**3*b**9*c**2*d**8 + 440*a**2*b**10*c**3*d**7 + 140*a*b**11*c**4*d**6 + 14*b**12*c**5*d**5
) + x**17*(924*a**6*b**6*d**10/17 + 7920*a**5*b**7*c*d**9/17 + 22275*a**4*b**8*c**2*d**8/17 + 26400*a**3*b**9*
c**3*d**7/17 + 13860*a**2*b**10*c**4*d**6/17 + 3024*a*b**11*c**5*d**5/17 + 210*b**12*c**6*d**4/17) + x**16*(99
*a**7*b**5*d**10/2 + 1155*a**6*b**6*c*d**9/2 + 4455*a**5*b**7*c**2*d**8/2 + 7425*a**4*b**8*c**3*d**7/2 + 5775*
a**3*b**9*c**4*d**6/2 + 2079*a**2*b**10*c**5*d**5/2 + 315*a*b**11*c**6*d**4/2 + 15*b**12*c**7*d**3/2) + x**15*
(33*a**8*b**4*d**10 + 528*a**7*b**5*c*d**9 + 2772*a**6*b**6*c**2*d**8 + 6336*a**5*b**7*c**3*d**7 + 6930*a**4*b
**8*c**4*d**6 + 3696*a**3*b**9*c**5*d**5 + 924*a**2*b**10*c**6*d**4 + 96*a*b**11*c**7*d**3 + 3*b**12*c**8*d**2
) + x**14*(110*a**9*b**3*d**10/7 + 2475*a**8*b**4*c*d**9/7 + 17820*a**7*b**5*c**2*d**8/7 + 7920*a**6*b**6*c**3
*d**7 + 11880*a**5*b**7*c**4*d**6 + 8910*a**4*b**8*c**5*d**5 + 3300*a**3*b**9*c**6*d**4 + 3960*a**2*b**10*c**7
*d**3/7 + 270*a*b**11*c**8*d**2/7 + 5*b**12*c**9*d/7) + x**13*(66*a**10*b**2*d**10/13 + 2200*a**9*b**3*c*d**9/
13 + 22275*a**8*b**4*c**2*d**8/13 + 95040*a**7*b**5*c**3*d**7/13 + 194040*a**6*b**6*c**4*d**6/13 + 199584*a**5
*b**7*c**5*d**5/13 + 103950*a**4*b**8*c**6*d**4/13 + 26400*a**3*b**9*c**7*d**3/13 + 2970*a**2*b**10*c**8*d**2/
13 + 120*a*b**11*c**9*d/13 + b**12*c**10/13) + x**12*(a**11*b*d**10 + 55*a**10*b**2*c*d**9 + 825*a**9*b**3*c**
2*d**8 + 4950*a**8*b**4*c**3*d**7 + 13860*a**7*b**5*c**4*d**6 + 19404*a**6*b**6*c**5*d**5 + 13860*a**5*b**7*c*
*6*d**4 + 4950*a**4*b**8*c**7*d**3 + 825*a**3*b**9*c**8*d**2 + 55*a**2*b**10*c**9*d + a*b**11*c**10) + x**11*(
a**12*d**10/11 + 120*a**11*b*c*d**9/11 + 270*a**10*b**2*c**2*d**8 + 2400*a**9*b**3*c**3*d**7 + 9450*a**8*b**4*
c**4*d**6 + 18144*a**7*b**5*c**5*d**5 + 17640*a**6*b**6*c**6*d**4 + 8640*a**5*b**7*c**7*d**3 + 2025*a**4*b**8*
c**8*d**2 + 200*a**3*b**9*c**9*d + 6*a**2*b**10*c**10) + x**10*(a**12*c*d**9 + 54*a**11*b*c**2*d**8 + 792*a**1
0*b**2*c**3*d**7 + 4620*a**9*b**3*c**4*d**6 + 12474*a**8*b**4*c**5*d**5 + 16632*a**7*b**5*c**6*d**4 + 11088*a*
*6*b**6*c**7*d**3 + 3564*a**5*b**7*c**8*d**2 + 495*a**4*b**8*c**9*d + 22*a**3*b**9*c**10) + x**9*(5*a**12*c**2
*d**8 + 160*a**11*b*c**3*d**7 + 1540*a**10*b**2*c**4*d**6 + 6160*a**9*b**3*c**5*d**5 + 11550*a**8*b**4*c**6*d*
*4 + 10560*a**7*b**5*c**7*d**3 + 4620*a**6*b**6*c**8*d**2 + 880*a**5*b**7*c**9*d + 55*a**4*b**8*c**10) + x**8*
(15*a**12*c**3*d**7 + 315*a**11*b*c**4*d**6 + 2079*a**10*b**2*c**5*d**5 + 5775*a**9*b**3*c**6*d**4 + 7425*a**8
*b**4*c**7*d**3 + 4455*a**7*b**5*c**8*d**2 + 1155*a**6*b**6*c**9*d + 99*a**5*b**7*c**10) + x**7*(30*a**12*c**4
*d**6 + 432*a**11*b*c**5*d**5 + 1980*a**10*b**2*c**6*d**4 + 26400*a**9*b**3*c**7*d**3/7 + 22275*a**8*b**4*c**8
*d**2/7 + 7920*a**7*b**5*c**9*d/7 + 132*a**6*b**6*c**10) + x**6*(42*a**12*c**5*d**5 + 420*a**11*b*c**6*d**4 +
1320*a**10*b**2*c**7*d**3 + 1650*a**9*b**3*c**8*d**2 + 825*a**8*b**4*c**9*d + 132*a**7*b**5*c**10) + x**5*(42*
a**12*c**6*d**4 + 288*a**11*b*c**7*d**3 + 594*a**10*b**2*c**8*d**2 + 440*a**9*b**3*c**9*d + 99*a**8*b**4*c**10
) + x**4*(30*a**12*c**7*d**3 + 135*a**11*b*c**8*d**2 + 165*a**10*b**2*c**9*d + 55*a**9*b**3*c**10) + x**3*(15*
a**12*c**8*d**2 + 40*a**11*b*c**9*d + 22*a**10*b**2*c**10) + x**2*(5*a**12*c**9*d + 6*a**11*b*c**10)
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1877 vs. \(2 (259) = 518\).
Time = 0.22 (sec) , antiderivative size = 1877, normalized size of antiderivative = 6.83
\[
\int (a+b x)^{12} (c+d x)^{10} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^12*(d*x+c)^10,x, algorithm="maxima")
[Out]
1/23*b^12*d^10*x^23 + a^12*c^10*x + 1/11*(5*b^12*c*d^9 + 6*a*b^11*d^10)*x^22 + 1/7*(15*b^12*c^2*d^8 + 40*a*b^1
1*c*d^9 + 22*a^2*b^10*d^10)*x^21 + (6*b^12*c^3*d^7 + 27*a*b^11*c^2*d^8 + 33*a^2*b^10*c*d^9 + 11*a^3*b^9*d^10)*
x^20 + 5/19*(42*b^12*c^4*d^6 + 288*a*b^11*c^3*d^7 + 594*a^2*b^10*c^2*d^8 + 440*a^3*b^9*c*d^9 + 99*a^4*b^8*d^10
)*x^19 + (14*b^12*c^5*d^5 + 140*a*b^11*c^4*d^6 + 440*a^2*b^10*c^3*d^7 + 550*a^3*b^9*c^2*d^8 + 275*a^4*b^8*c*d^
9 + 44*a^5*b^7*d^10)*x^18 + 3/17*(70*b^12*c^6*d^4 + 1008*a*b^11*c^5*d^5 + 4620*a^2*b^10*c^4*d^6 + 8800*a^3*b^9
*c^3*d^7 + 7425*a^4*b^8*c^2*d^8 + 2640*a^5*b^7*c*d^9 + 308*a^6*b^6*d^10)*x^17 + 3/2*(5*b^12*c^7*d^3 + 105*a*b^
11*c^6*d^4 + 693*a^2*b^10*c^5*d^5 + 1925*a^3*b^9*c^4*d^6 + 2475*a^4*b^8*c^3*d^7 + 1485*a^5*b^7*c^2*d^8 + 385*a
^6*b^6*c*d^9 + 33*a^7*b^5*d^10)*x^16 + 3*(b^12*c^8*d^2 + 32*a*b^11*c^7*d^3 + 308*a^2*b^10*c^6*d^4 + 1232*a^3*b
^9*c^5*d^5 + 2310*a^4*b^8*c^4*d^6 + 2112*a^5*b^7*c^3*d^7 + 924*a^6*b^6*c^2*d^8 + 176*a^7*b^5*c*d^9 + 11*a^8*b^
4*d^10)*x^15 + 5/7*(b^12*c^9*d + 54*a*b^11*c^8*d^2 + 792*a^2*b^10*c^7*d^3 + 4620*a^3*b^9*c^6*d^4 + 12474*a^4*b
^8*c^5*d^5 + 16632*a^5*b^7*c^4*d^6 + 11088*a^6*b^6*c^3*d^7 + 3564*a^7*b^5*c^2*d^8 + 495*a^8*b^4*c*d^9 + 22*a^9
*b^3*d^10)*x^14 + 1/13*(b^12*c^10 + 120*a*b^11*c^9*d + 2970*a^2*b^10*c^8*d^2 + 26400*a^3*b^9*c^7*d^3 + 103950*
a^4*b^8*c^6*d^4 + 199584*a^5*b^7*c^5*d^5 + 194040*a^6*b^6*c^4*d^6 + 95040*a^7*b^5*c^3*d^7 + 22275*a^8*b^4*c^2*
d^8 + 2200*a^9*b^3*c*d^9 + 66*a^10*b^2*d^10)*x^13 + (a*b^11*c^10 + 55*a^2*b^10*c^9*d + 825*a^3*b^9*c^8*d^2 + 4
950*a^4*b^8*c^7*d^3 + 13860*a^5*b^7*c^6*d^4 + 19404*a^6*b^6*c^5*d^5 + 13860*a^7*b^5*c^4*d^6 + 4950*a^8*b^4*c^3
*d^7 + 825*a^9*b^3*c^2*d^8 + 55*a^10*b^2*c*d^9 + a^11*b*d^10)*x^12 + 1/11*(66*a^2*b^10*c^10 + 2200*a^3*b^9*c^9
*d + 22275*a^4*b^8*c^8*d^2 + 95040*a^5*b^7*c^7*d^3 + 194040*a^6*b^6*c^6*d^4 + 199584*a^7*b^5*c^5*d^5 + 103950*
a^8*b^4*c^4*d^6 + 26400*a^9*b^3*c^3*d^7 + 2970*a^10*b^2*c^2*d^8 + 120*a^11*b*c*d^9 + a^12*d^10)*x^11 + (22*a^3
*b^9*c^10 + 495*a^4*b^8*c^9*d + 3564*a^5*b^7*c^8*d^2 + 11088*a^6*b^6*c^7*d^3 + 16632*a^7*b^5*c^6*d^4 + 12474*a
^8*b^4*c^5*d^5 + 4620*a^9*b^3*c^4*d^6 + 792*a^10*b^2*c^3*d^7 + 54*a^11*b*c^2*d^8 + a^12*c*d^9)*x^10 + 5*(11*a^
4*b^8*c^10 + 176*a^5*b^7*c^9*d + 924*a^6*b^6*c^8*d^2 + 2112*a^7*b^5*c^7*d^3 + 2310*a^8*b^4*c^6*d^4 + 1232*a^9*
b^3*c^5*d^5 + 308*a^10*b^2*c^4*d^6 + 32*a^11*b*c^3*d^7 + a^12*c^2*d^8)*x^9 + 3*(33*a^5*b^7*c^10 + 385*a^6*b^6*
c^9*d + 1485*a^7*b^5*c^8*d^2 + 2475*a^8*b^4*c^7*d^3 + 1925*a^9*b^3*c^6*d^4 + 693*a^10*b^2*c^5*d^5 + 105*a^11*b
*c^4*d^6 + 5*a^12*c^3*d^7)*x^8 + 3/7*(308*a^6*b^6*c^10 + 2640*a^7*b^5*c^9*d + 7425*a^8*b^4*c^8*d^2 + 8800*a^9*
b^3*c^7*d^3 + 4620*a^10*b^2*c^6*d^4 + 1008*a^11*b*c^5*d^5 + 70*a^12*c^4*d^6)*x^7 + 3*(44*a^7*b^5*c^10 + 275*a^
8*b^4*c^9*d + 550*a^9*b^3*c^8*d^2 + 440*a^10*b^2*c^7*d^3 + 140*a^11*b*c^6*d^4 + 14*a^12*c^5*d^5)*x^6 + (99*a^8
*b^4*c^10 + 440*a^9*b^3*c^9*d + 594*a^10*b^2*c^8*d^2 + 288*a^11*b*c^7*d^3 + 42*a^12*c^6*d^4)*x^5 + 5*(11*a^9*b
^3*c^10 + 33*a^10*b^2*c^9*d + 27*a^11*b*c^8*d^2 + 6*a^12*c^7*d^3)*x^4 + (22*a^10*b^2*c^10 + 40*a^11*b*c^9*d +
15*a^12*c^8*d^2)*x^3 + (6*a^11*b*c^10 + 5*a^12*c^9*d)*x^2
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2186 vs. \(2 (259) = 518\).
Time = 0.30 (sec) , antiderivative size = 2186, normalized size of antiderivative = 7.95
\[
\int (a+b x)^{12} (c+d x)^{10} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^12*(d*x+c)^10,x, algorithm="giac")
[Out]
1/23*b^12*d^10*x^23 + 5/11*b^12*c*d^9*x^22 + 6/11*a*b^11*d^10*x^22 + 15/7*b^12*c^2*d^8*x^21 + 40/7*a*b^11*c*d^
9*x^21 + 22/7*a^2*b^10*d^10*x^21 + 6*b^12*c^3*d^7*x^20 + 27*a*b^11*c^2*d^8*x^20 + 33*a^2*b^10*c*d^9*x^20 + 11*
a^3*b^9*d^10*x^20 + 210/19*b^12*c^4*d^6*x^19 + 1440/19*a*b^11*c^3*d^7*x^19 + 2970/19*a^2*b^10*c^2*d^8*x^19 + 2
200/19*a^3*b^9*c*d^9*x^19 + 495/19*a^4*b^8*d^10*x^19 + 14*b^12*c^5*d^5*x^18 + 140*a*b^11*c^4*d^6*x^18 + 440*a^
2*b^10*c^3*d^7*x^18 + 550*a^3*b^9*c^2*d^8*x^18 + 275*a^4*b^8*c*d^9*x^18 + 44*a^5*b^7*d^10*x^18 + 210/17*b^12*c
^6*d^4*x^17 + 3024/17*a*b^11*c^5*d^5*x^17 + 13860/17*a^2*b^10*c^4*d^6*x^17 + 26400/17*a^3*b^9*c^3*d^7*x^17 + 2
2275/17*a^4*b^8*c^2*d^8*x^17 + 7920/17*a^5*b^7*c*d^9*x^17 + 924/17*a^6*b^6*d^10*x^17 + 15/2*b^12*c^7*d^3*x^16
+ 315/2*a*b^11*c^6*d^4*x^16 + 2079/2*a^2*b^10*c^5*d^5*x^16 + 5775/2*a^3*b^9*c^4*d^6*x^16 + 7425/2*a^4*b^8*c^3*
d^7*x^16 + 4455/2*a^5*b^7*c^2*d^8*x^16 + 1155/2*a^6*b^6*c*d^9*x^16 + 99/2*a^7*b^5*d^10*x^16 + 3*b^12*c^8*d^2*x
^15 + 96*a*b^11*c^7*d^3*x^15 + 924*a^2*b^10*c^6*d^4*x^15 + 3696*a^3*b^9*c^5*d^5*x^15 + 6930*a^4*b^8*c^4*d^6*x^
15 + 6336*a^5*b^7*c^3*d^7*x^15 + 2772*a^6*b^6*c^2*d^8*x^15 + 528*a^7*b^5*c*d^9*x^15 + 33*a^8*b^4*d^10*x^15 + 5
/7*b^12*c^9*d*x^14 + 270/7*a*b^11*c^8*d^2*x^14 + 3960/7*a^2*b^10*c^7*d^3*x^14 + 3300*a^3*b^9*c^6*d^4*x^14 + 89
10*a^4*b^8*c^5*d^5*x^14 + 11880*a^5*b^7*c^4*d^6*x^14 + 7920*a^6*b^6*c^3*d^7*x^14 + 17820/7*a^7*b^5*c^2*d^8*x^1
4 + 2475/7*a^8*b^4*c*d^9*x^14 + 110/7*a^9*b^3*d^10*x^14 + 1/13*b^12*c^10*x^13 + 120/13*a*b^11*c^9*d*x^13 + 297
0/13*a^2*b^10*c^8*d^2*x^13 + 26400/13*a^3*b^9*c^7*d^3*x^13 + 103950/13*a^4*b^8*c^6*d^4*x^13 + 199584/13*a^5*b^
7*c^5*d^5*x^13 + 194040/13*a^6*b^6*c^4*d^6*x^13 + 95040/13*a^7*b^5*c^3*d^7*x^13 + 22275/13*a^8*b^4*c^2*d^8*x^1
3 + 2200/13*a^9*b^3*c*d^9*x^13 + 66/13*a^10*b^2*d^10*x^13 + a*b^11*c^10*x^12 + 55*a^2*b^10*c^9*d*x^12 + 825*a^
3*b^9*c^8*d^2*x^12 + 4950*a^4*b^8*c^7*d^3*x^12 + 13860*a^5*b^7*c^6*d^4*x^12 + 19404*a^6*b^6*c^5*d^5*x^12 + 138
60*a^7*b^5*c^4*d^6*x^12 + 4950*a^8*b^4*c^3*d^7*x^12 + 825*a^9*b^3*c^2*d^8*x^12 + 55*a^10*b^2*c*d^9*x^12 + a^11
*b*d^10*x^12 + 6*a^2*b^10*c^10*x^11 + 200*a^3*b^9*c^9*d*x^11 + 2025*a^4*b^8*c^8*d^2*x^11 + 8640*a^5*b^7*c^7*d^
3*x^11 + 17640*a^6*b^6*c^6*d^4*x^11 + 18144*a^7*b^5*c^5*d^5*x^11 + 9450*a^8*b^4*c^4*d^6*x^11 + 2400*a^9*b^3*c^
3*d^7*x^11 + 270*a^10*b^2*c^2*d^8*x^11 + 120/11*a^11*b*c*d^9*x^11 + 1/11*a^12*d^10*x^11 + 22*a^3*b^9*c^10*x^10
+ 495*a^4*b^8*c^9*d*x^10 + 3564*a^5*b^7*c^8*d^2*x^10 + 11088*a^6*b^6*c^7*d^3*x^10 + 16632*a^7*b^5*c^6*d^4*x^1
0 + 12474*a^8*b^4*c^5*d^5*x^10 + 4620*a^9*b^3*c^4*d^6*x^10 + 792*a^10*b^2*c^3*d^7*x^10 + 54*a^11*b*c^2*d^8*x^1
0 + a^12*c*d^9*x^10 + 55*a^4*b^8*c^10*x^9 + 880*a^5*b^7*c^9*d*x^9 + 4620*a^6*b^6*c^8*d^2*x^9 + 10560*a^7*b^5*c
^7*d^3*x^9 + 11550*a^8*b^4*c^6*d^4*x^9 + 6160*a^9*b^3*c^5*d^5*x^9 + 1540*a^10*b^2*c^4*d^6*x^9 + 160*a^11*b*c^3
*d^7*x^9 + 5*a^12*c^2*d^8*x^9 + 99*a^5*b^7*c^10*x^8 + 1155*a^6*b^6*c^9*d*x^8 + 4455*a^7*b^5*c^8*d^2*x^8 + 7425
*a^8*b^4*c^7*d^3*x^8 + 5775*a^9*b^3*c^6*d^4*x^8 + 2079*a^10*b^2*c^5*d^5*x^8 + 315*a^11*b*c^4*d^6*x^8 + 15*a^12
*c^3*d^7*x^8 + 132*a^6*b^6*c^10*x^7 + 7920/7*a^7*b^5*c^9*d*x^7 + 22275/7*a^8*b^4*c^8*d^2*x^7 + 26400/7*a^9*b^3
*c^7*d^3*x^7 + 1980*a^10*b^2*c^6*d^4*x^7 + 432*a^11*b*c^5*d^5*x^7 + 30*a^12*c^4*d^6*x^7 + 132*a^7*b^5*c^10*x^6
+ 825*a^8*b^4*c^9*d*x^6 + 1650*a^9*b^3*c^8*d^2*x^6 + 1320*a^10*b^2*c^7*d^3*x^6 + 420*a^11*b*c^6*d^4*x^6 + 42*
a^12*c^5*d^5*x^6 + 99*a^8*b^4*c^10*x^5 + 440*a^9*b^3*c^9*d*x^5 + 594*a^10*b^2*c^8*d^2*x^5 + 288*a^11*b*c^7*d^3
*x^5 + 42*a^12*c^6*d^4*x^5 + 55*a^9*b^3*c^10*x^4 + 165*a^10*b^2*c^9*d*x^4 + 135*a^11*b*c^8*d^2*x^4 + 30*a^12*c
^7*d^3*x^4 + 22*a^10*b^2*c^10*x^3 + 40*a^11*b*c^9*d*x^3 + 15*a^12*c^8*d^2*x^3 + 6*a^11*b*c^10*x^2 + 5*a^12*c^9
*d*x^2 + a^12*c^10*x
Mupad [B] (verification not implemented)
Time = 1.30 (sec) , antiderivative size = 1847, normalized size of antiderivative = 6.72
\[
\int (a+b x)^{12} (c+d x)^{10} \, dx=\text {Too large to display}
\]
[In]
int((a + b*x)^12*(c + d*x)^10,x)
[Out]
x^12*(a*b^11*c^10 + a^11*b*d^10 + 55*a^2*b^10*c^9*d + 55*a^10*b^2*c*d^9 + 825*a^3*b^9*c^8*d^2 + 4950*a^4*b^8*c
^7*d^3 + 13860*a^5*b^7*c^6*d^4 + 19404*a^6*b^6*c^5*d^5 + 13860*a^7*b^5*c^4*d^6 + 4950*a^8*b^4*c^3*d^7 + 825*a^
9*b^3*c^2*d^8) + x^7*(132*a^6*b^6*c^10 + 30*a^12*c^4*d^6 + (7920*a^7*b^5*c^9*d)/7 + 432*a^11*b*c^5*d^5 + (2227
5*a^8*b^4*c^8*d^2)/7 + (26400*a^9*b^3*c^7*d^3)/7 + 1980*a^10*b^2*c^6*d^4) + x^17*((924*a^6*b^6*d^10)/17 + (210
*b^12*c^6*d^4)/17 + (3024*a*b^11*c^5*d^5)/17 + (7920*a^5*b^7*c*d^9)/17 + (13860*a^2*b^10*c^4*d^6)/17 + (26400*
a^3*b^9*c^3*d^7)/17 + (22275*a^4*b^8*c^2*d^8)/17) + x^5*(99*a^8*b^4*c^10 + 42*a^12*c^6*d^4 + 440*a^9*b^3*c^9*d
+ 288*a^11*b*c^7*d^3 + 594*a^10*b^2*c^8*d^2) + x^19*((495*a^4*b^8*d^10)/19 + (210*b^12*c^4*d^6)/19 + (1440*a*
b^11*c^3*d^7)/19 + (2200*a^3*b^9*c*d^9)/19 + (2970*a^2*b^10*c^2*d^8)/19) + x^8*(99*a^5*b^7*c^10 + 15*a^12*c^3*
d^7 + 1155*a^6*b^6*c^9*d + 315*a^11*b*c^4*d^6 + 4455*a^7*b^5*c^8*d^2 + 7425*a^8*b^4*c^7*d^3 + 5775*a^9*b^3*c^6
*d^4 + 2079*a^10*b^2*c^5*d^5) + x^16*((99*a^7*b^5*d^10)/2 + (15*b^12*c^7*d^3)/2 + (315*a*b^11*c^6*d^4)/2 + (11
55*a^6*b^6*c*d^9)/2 + (2079*a^2*b^10*c^5*d^5)/2 + (5775*a^3*b^9*c^4*d^6)/2 + (7425*a^4*b^8*c^3*d^7)/2 + (4455*
a^5*b^7*c^2*d^8)/2) + x^11*((a^12*d^10)/11 + 6*a^2*b^10*c^10 + 200*a^3*b^9*c^9*d + 2025*a^4*b^8*c^8*d^2 + 8640
*a^5*b^7*c^7*d^3 + 17640*a^6*b^6*c^6*d^4 + 18144*a^7*b^5*c^5*d^5 + 9450*a^8*b^4*c^4*d^6 + 2400*a^9*b^3*c^3*d^7
+ 270*a^10*b^2*c^2*d^8 + (120*a^11*b*c*d^9)/11) + x^13*((b^12*c^10)/13 + (66*a^10*b^2*d^10)/13 + (2200*a^9*b^
3*c*d^9)/13 + (2970*a^2*b^10*c^8*d^2)/13 + (26400*a^3*b^9*c^7*d^3)/13 + (103950*a^4*b^8*c^6*d^4)/13 + (199584*
a^5*b^7*c^5*d^5)/13 + (194040*a^6*b^6*c^4*d^6)/13 + (95040*a^7*b^5*c^3*d^7)/13 + (22275*a^8*b^4*c^2*d^8)/13 +
(120*a*b^11*c^9*d)/13) + x^6*(132*a^7*b^5*c^10 + 42*a^12*c^5*d^5 + 825*a^8*b^4*c^9*d + 420*a^11*b*c^6*d^4 + 16
50*a^9*b^3*c^8*d^2 + 1320*a^10*b^2*c^7*d^3) + x^18*(44*a^5*b^7*d^10 + 14*b^12*c^5*d^5 + 140*a*b^11*c^4*d^6 + 2
75*a^4*b^8*c*d^9 + 440*a^2*b^10*c^3*d^7 + 550*a^3*b^9*c^2*d^8) + x^9*(55*a^4*b^8*c^10 + 5*a^12*c^2*d^8 + 880*a
^5*b^7*c^9*d + 160*a^11*b*c^3*d^7 + 4620*a^6*b^6*c^8*d^2 + 10560*a^7*b^5*c^7*d^3 + 11550*a^8*b^4*c^6*d^4 + 616
0*a^9*b^3*c^5*d^5 + 1540*a^10*b^2*c^4*d^6) + x^15*(33*a^8*b^4*d^10 + 3*b^12*c^8*d^2 + 96*a*b^11*c^7*d^3 + 528*
a^7*b^5*c*d^9 + 924*a^2*b^10*c^6*d^4 + 3696*a^3*b^9*c^5*d^5 + 6930*a^4*b^8*c^4*d^6 + 6336*a^5*b^7*c^3*d^7 + 27
72*a^6*b^6*c^2*d^8) + x^10*(a^12*c*d^9 + 22*a^3*b^9*c^10 + 495*a^4*b^8*c^9*d + 54*a^11*b*c^2*d^8 + 3564*a^5*b^
7*c^8*d^2 + 11088*a^6*b^6*c^7*d^3 + 16632*a^7*b^5*c^6*d^4 + 12474*a^8*b^4*c^5*d^5 + 4620*a^9*b^3*c^4*d^6 + 792
*a^10*b^2*c^3*d^7) + x^14*((5*b^12*c^9*d)/7 + (110*a^9*b^3*d^10)/7 + (270*a*b^11*c^8*d^2)/7 + (2475*a^8*b^4*c*
d^9)/7 + (3960*a^2*b^10*c^7*d^3)/7 + 3300*a^3*b^9*c^6*d^4 + 8910*a^4*b^8*c^5*d^5 + 11880*a^5*b^7*c^4*d^6 + 792
0*a^6*b^6*c^3*d^7 + (17820*a^7*b^5*c^2*d^8)/7) + a^12*c^10*x + (b^12*d^10*x^23)/23 + 5*a^9*c^7*x^4*(6*a^3*d^3
+ 11*b^3*c^3 + 33*a*b^2*c^2*d + 27*a^2*b*c*d^2) + b^9*d^7*x^20*(11*a^3*d^3 + 6*b^3*c^3 + 27*a*b^2*c^2*d + 33*a
^2*b*c*d^2) + a^11*c^9*x^2*(5*a*d + 6*b*c) + (b^11*d^9*x^22*(6*a*d + 5*b*c))/11 + a^10*c^8*x^3*(15*a^2*d^2 + 2
2*b^2*c^2 + 40*a*b*c*d) + (b^10*d^8*x^21*(22*a^2*d^2 + 15*b^2*c^2 + 40*a*b*c*d))/7