\(\int \frac {(c+d x)^{10}}{(a+b x)^{19}} \, dx\) [124]

Optimal result

Integrand size = 15, antiderivative size = 244 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{19}} \, dx=-\frac {(c+d x)^{11}}{18 (b c-a d) (a+b x)^{18}}+\frac {7 d (c+d x)^{11}}{306 (b c-a d)^2 (a+b x)^{17}}-\frac {7 d^2 (c+d x)^{11}}{816 (b c-a d)^3 (a+b x)^{16}}+\frac {7 d^3 (c+d x)^{11}}{2448 (b c-a d)^4 (a+b x)^{15}}-\frac {d^4 (c+d x)^{11}}{1224 (b c-a d)^5 (a+b x)^{14}}+\frac {d^5 (c+d x)^{11}}{5304 (b c-a d)^6 (a+b x)^{13}}-\frac {d^6 (c+d x)^{11}}{31824 (b c-a d)^7 (a+b x)^{12}}+\frac {d^7 (c+d x)^{11}}{350064 (b c-a d)^8 (a+b x)^{11}} \] Output:

-1/18*(d*x+c)^11/(-a*d+b*c)/(b*x+a)^18+7/306*d*(d*x+c)^11/(-a*d+b*c)^2/(b* 
x+a)^17-7/816*d^2*(d*x+c)^11/(-a*d+b*c)^3/(b*x+a)^16+7/2448*d^3*(d*x+c)^11 
/(-a*d+b*c)^4/(b*x+a)^15-1/1224*d^4*(d*x+c)^11/(-a*d+b*c)^5/(b*x+a)^14+1/5 
304*d^5*(d*x+c)^11/(-a*d+b*c)^6/(b*x+a)^13-1/31824*d^6*(d*x+c)^11/(-a*d+b* 
c)^7/(b*x+a)^12+1/350064*d^7*(d*x+c)^11/(-a*d+b*c)^8/(b*x+a)^11
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(694\) vs. \(2(244)=488\).

Time = 0.17 (sec) , antiderivative size = 694, normalized size of antiderivative = 2.84 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{19}} \, dx=-\frac {a^{10} d^{10}+2 a^9 b d^9 (4 c+9 d x)+9 a^8 b^2 d^8 \left (4 c^2+16 c d x+17 d^2 x^2\right )+24 a^7 b^3 d^7 \left (5 c^3+27 c^2 d x+51 c d^2 x^2+34 d^3 x^3\right )+6 a^6 b^4 d^6 \left (55 c^4+360 c^3 d x+918 c^2 d^2 x^2+1088 c d^3 x^3+510 d^4 x^4\right )+36 a^5 b^5 d^5 \left (22 c^5+165 c^4 d x+510 c^3 d^2 x^2+816 c^2 d^3 x^3+680 c d^4 x^4+238 d^5 x^5\right )+6 a^4 b^6 d^4 \left (286 c^6+2376 c^5 d x+8415 c^4 d^2 x^2+16320 c^3 d^3 x^3+18360 c^2 d^4 x^4+11424 c d^5 x^5+3094 d^6 x^6\right )+24 a^3 b^7 d^3 \left (143 c^7+1287 c^6 d x+5049 c^5 d^2 x^2+11220 c^4 d^3 x^3+15300 c^3 d^4 x^4+12852 c^2 d^5 x^5+6188 c d^6 x^6+1326 d^7 x^7\right )+9 a^2 b^8 d^2 \left (715 c^8+6864 c^7 d x+29172 c^6 d^2 x^2+71808 c^5 d^3 x^3+112200 c^4 d^4 x^4+114240 c^3 d^5 x^5+74256 c^2 d^6 x^6+28288 c d^7 x^7+4862 d^8 x^8\right )+2 a b^9 d \left (5720 c^9+57915 c^8 d x+262548 c^7 d^2 x^2+700128 c^6 d^3 x^3+1211760 c^5 d^4 x^4+1413720 c^4 d^5 x^5+1113840 c^3 d^6 x^6+572832 c^2 d^7 x^7+175032 c d^8 x^8+24310 d^9 x^9\right )+b^{10} \left (19448 c^{10}+205920 c^9 d x+984555 c^8 d^2 x^2+2800512 c^7 d^3 x^3+5250960 c^6 d^4 x^4+6785856 c^5 d^5 x^5+6126120 c^4 d^6 x^6+3818880 c^3 d^7 x^7+1575288 c^2 d^8 x^8+388960 c d^9 x^9+43758 d^{10} x^{10}\right )}{350064 b^{11} (a+b x)^{18}} \] Input:

Integrate[(c + d*x)^10/(a + b*x)^19,x]
 

Output:

-1/350064*(a^10*d^10 + 2*a^9*b*d^9*(4*c + 9*d*x) + 9*a^8*b^2*d^8*(4*c^2 + 
16*c*d*x + 17*d^2*x^2) + 24*a^7*b^3*d^7*(5*c^3 + 27*c^2*d*x + 51*c*d^2*x^2 
 + 34*d^3*x^3) + 6*a^6*b^4*d^6*(55*c^4 + 360*c^3*d*x + 918*c^2*d^2*x^2 + 1 
088*c*d^3*x^3 + 510*d^4*x^4) + 36*a^5*b^5*d^5*(22*c^5 + 165*c^4*d*x + 510* 
c^3*d^2*x^2 + 816*c^2*d^3*x^3 + 680*c*d^4*x^4 + 238*d^5*x^5) + 6*a^4*b^6*d 
^4*(286*c^6 + 2376*c^5*d*x + 8415*c^4*d^2*x^2 + 16320*c^3*d^3*x^3 + 18360* 
c^2*d^4*x^4 + 11424*c*d^5*x^5 + 3094*d^6*x^6) + 24*a^3*b^7*d^3*(143*c^7 + 
1287*c^6*d*x + 5049*c^5*d^2*x^2 + 11220*c^4*d^3*x^3 + 15300*c^3*d^4*x^4 + 
12852*c^2*d^5*x^5 + 6188*c*d^6*x^6 + 1326*d^7*x^7) + 9*a^2*b^8*d^2*(715*c^ 
8 + 6864*c^7*d*x + 29172*c^6*d^2*x^2 + 71808*c^5*d^3*x^3 + 112200*c^4*d^4* 
x^4 + 114240*c^3*d^5*x^5 + 74256*c^2*d^6*x^6 + 28288*c*d^7*x^7 + 4862*d^8* 
x^8) + 2*a*b^9*d*(5720*c^9 + 57915*c^8*d*x + 262548*c^7*d^2*x^2 + 700128*c 
^6*d^3*x^3 + 1211760*c^5*d^4*x^4 + 1413720*c^4*d^5*x^5 + 1113840*c^3*d^6*x 
^6 + 572832*c^2*d^7*x^7 + 175032*c*d^8*x^8 + 24310*d^9*x^9) + b^10*(19448* 
c^10 + 205920*c^9*d*x + 984555*c^8*d^2*x^2 + 2800512*c^7*d^3*x^3 + 5250960 
*c^6*d^4*x^4 + 6785856*c^5*d^5*x^5 + 6126120*c^4*d^6*x^6 + 3818880*c^3*d^7 
*x^7 + 1575288*c^2*d^8*x^8 + 388960*c*d^9*x^9 + 43758*d^10*x^10))/(b^11*(a 
 + b*x)^18)
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.32, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {55, 55, 55, 55, 55, 55, 55, 48}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(c+d x)^{10}}{(a+b x)^{19}} \, dx\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {7 d \int \frac {(c+d x)^{10}}{(a+b x)^{18}}dx}{18 (b c-a d)}-\frac {(c+d x)^{11}}{18 (a+b x)^{18} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {7 d \left (-\frac {6 d \int \frac {(c+d x)^{10}}{(a+b x)^{17}}dx}{17 (b c-a d)}-\frac {(c+d x)^{11}}{17 (a+b x)^{17} (b c-a d)}\right )}{18 (b c-a d)}-\frac {(c+d x)^{11}}{18 (a+b x)^{18} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {7 d \left (-\frac {6 d \left (-\frac {5 d \int \frac {(c+d x)^{10}}{(a+b x)^{16}}dx}{16 (b c-a d)}-\frac {(c+d x)^{11}}{16 (a+b x)^{16} (b c-a d)}\right )}{17 (b c-a d)}-\frac {(c+d x)^{11}}{17 (a+b x)^{17} (b c-a d)}\right )}{18 (b c-a d)}-\frac {(c+d x)^{11}}{18 (a+b x)^{18} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {7 d \left (-\frac {6 d \left (-\frac {5 d \left (-\frac {4 d \int \frac {(c+d x)^{10}}{(a+b x)^{15}}dx}{15 (b c-a d)}-\frac {(c+d x)^{11}}{15 (a+b x)^{15} (b c-a d)}\right )}{16 (b c-a d)}-\frac {(c+d x)^{11}}{16 (a+b x)^{16} (b c-a d)}\right )}{17 (b c-a d)}-\frac {(c+d x)^{11}}{17 (a+b x)^{17} (b c-a d)}\right )}{18 (b c-a d)}-\frac {(c+d x)^{11}}{18 (a+b x)^{18} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {7 d \left (-\frac {6 d \left (-\frac {5 d \left (-\frac {4 d \left (-\frac {3 d \int \frac {(c+d x)^{10}}{(a+b x)^{14}}dx}{14 (b c-a d)}-\frac {(c+d x)^{11}}{14 (a+b x)^{14} (b c-a d)}\right )}{15 (b c-a d)}-\frac {(c+d x)^{11}}{15 (a+b x)^{15} (b c-a d)}\right )}{16 (b c-a d)}-\frac {(c+d x)^{11}}{16 (a+b x)^{16} (b c-a d)}\right )}{17 (b c-a d)}-\frac {(c+d x)^{11}}{17 (a+b x)^{17} (b c-a d)}\right )}{18 (b c-a d)}-\frac {(c+d x)^{11}}{18 (a+b x)^{18} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {7 d \left (-\frac {6 d \left (-\frac {5 d \left (-\frac {4 d \left (-\frac {3 d \left (-\frac {2 d \int \frac {(c+d x)^{10}}{(a+b x)^{13}}dx}{13 (b c-a d)}-\frac {(c+d x)^{11}}{13 (a+b x)^{13} (b c-a d)}\right )}{14 (b c-a d)}-\frac {(c+d x)^{11}}{14 (a+b x)^{14} (b c-a d)}\right )}{15 (b c-a d)}-\frac {(c+d x)^{11}}{15 (a+b x)^{15} (b c-a d)}\right )}{16 (b c-a d)}-\frac {(c+d x)^{11}}{16 (a+b x)^{16} (b c-a d)}\right )}{17 (b c-a d)}-\frac {(c+d x)^{11}}{17 (a+b x)^{17} (b c-a d)}\right )}{18 (b c-a d)}-\frac {(c+d x)^{11}}{18 (a+b x)^{18} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {7 d \left (-\frac {6 d \left (-\frac {5 d \left (-\frac {4 d \left (-\frac {3 d \left (-\frac {2 d \left (-\frac {d \int \frac {(c+d x)^{10}}{(a+b x)^{12}}dx}{12 (b c-a d)}-\frac {(c+d x)^{11}}{12 (a+b x)^{12} (b c-a d)}\right )}{13 (b c-a d)}-\frac {(c+d x)^{11}}{13 (a+b x)^{13} (b c-a d)}\right )}{14 (b c-a d)}-\frac {(c+d x)^{11}}{14 (a+b x)^{14} (b c-a d)}\right )}{15 (b c-a d)}-\frac {(c+d x)^{11}}{15 (a+b x)^{15} (b c-a d)}\right )}{16 (b c-a d)}-\frac {(c+d x)^{11}}{16 (a+b x)^{16} (b c-a d)}\right )}{17 (b c-a d)}-\frac {(c+d x)^{11}}{17 (a+b x)^{17} (b c-a d)}\right )}{18 (b c-a d)}-\frac {(c+d x)^{11}}{18 (a+b x)^{18} (b c-a d)}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {(c+d x)^{11}}{18 (a+b x)^{18} (b c-a d)}-\frac {7 d \left (-\frac {(c+d x)^{11}}{17 (a+b x)^{17} (b c-a d)}-\frac {6 d \left (-\frac {(c+d x)^{11}}{16 (a+b x)^{16} (b c-a d)}-\frac {5 d \left (-\frac {(c+d x)^{11}}{15 (a+b x)^{15} (b c-a d)}-\frac {4 d \left (-\frac {(c+d x)^{11}}{14 (a+b x)^{14} (b c-a d)}-\frac {3 d \left (-\frac {(c+d x)^{11}}{13 (a+b x)^{13} (b c-a d)}-\frac {2 d \left (\frac {d (c+d x)^{11}}{132 (a+b x)^{11} (b c-a d)^2}-\frac {(c+d x)^{11}}{12 (a+b x)^{12} (b c-a d)}\right )}{13 (b c-a d)}\right )}{14 (b c-a d)}\right )}{15 (b c-a d)}\right )}{16 (b c-a d)}\right )}{17 (b c-a d)}\right )}{18 (b c-a d)}\)

Input:

Int[(c + d*x)^10/(a + b*x)^19,x]
 

Output:

-1/18*(c + d*x)^11/((b*c - a*d)*(a + b*x)^18) - (7*d*(-1/17*(c + d*x)^11/( 
(b*c - a*d)*(a + b*x)^17) - (6*d*(-1/16*(c + d*x)^11/((b*c - a*d)*(a + b*x 
)^16) - (5*d*(-1/15*(c + d*x)^11/((b*c - a*d)*(a + b*x)^15) - (4*d*(-1/14* 
(c + d*x)^11/((b*c - a*d)*(a + b*x)^14) - (3*d*(-1/13*(c + d*x)^11/((b*c - 
 a*d)*(a + b*x)^13) - (2*d*(-1/12*(c + d*x)^11/((b*c - a*d)*(a + b*x)^12) 
+ (d*(c + d*x)^11)/(132*(b*c - a*d)^2*(a + b*x)^11)))/(13*(b*c - a*d))))/( 
14*(b*c - a*d))))/(15*(b*c - a*d))))/(16*(b*c - a*d))))/(17*(b*c - a*d)))) 
/(18*(b*c - a*d))
 

Defintions of rubi rules used

rule 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(830\) vs. \(2(228)=456\).

Time = 0.24 (sec) , antiderivative size = 831, normalized size of antiderivative = 3.41

method	result	size
risch	\(\frac {-\frac {a^{10} d^{10}+8 a^{9} b c \,d^{9}+36 a^{8} b^{2} c^{2} d^{8}+120 a^{7} b^{3} c^{3} d^{7}+330 a^{6} b^{4} c^{4} d^{6}+792 a^{5} b^{5} c^{5} d^{5}+1716 a^{4} b^{6} c^{6} d^{4}+3432 a^{3} b^{7} c^{7} d^{3}+6435 a^{2} b^{8} c^{8} d^{2}+11440 a \,b^{9} c^{9} d +19448 b^{10} c^{10}}{350064 b^{11}}-\frac {d \left (a^{9} d^{9}+8 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}+120 a^{6} b^{3} c^{3} d^{6}+330 a^{5} b^{4} c^{4} d^{5}+792 a^{4} b^{5} c^{5} d^{4}+1716 a^{3} b^{6} c^{6} d^{3}+3432 a^{2} b^{7} c^{7} d^{2}+6435 a \,b^{8} c^{8} d +11440 c^{9} b^{9}\right ) x}{19448 b^{10}}-\frac {d^{2} \left (a^{8} d^{8}+8 a^{7} b c \,d^{7}+36 a^{6} b^{2} c^{2} d^{6}+120 a^{5} b^{3} c^{3} d^{5}+330 a^{4} b^{4} c^{4} d^{4}+792 a^{3} b^{5} c^{5} d^{3}+1716 a^{2} b^{6} c^{6} d^{2}+3432 a \,b^{7} c^{7} d +6435 c^{8} b^{8}\right ) x^{2}}{2288 b^{9}}-\frac {d^{3} \left (a^{7} d^{7}+8 a^{6} b c \,d^{6}+36 a^{5} b^{2} c^{2} d^{5}+120 a^{4} b^{3} c^{3} d^{4}+330 a^{3} b^{4} c^{4} d^{3}+792 a^{2} b^{5} c^{5} d^{2}+1716 a \,b^{6} c^{6} d +3432 b^{7} c^{7}\right ) x^{3}}{429 b^{8}}-\frac {5 d^{4} \left (a^{6} d^{6}+8 a^{5} b c \,d^{5}+36 a^{4} b^{2} c^{2} d^{4}+120 a^{3} b^{3} c^{3} d^{3}+330 a^{2} b^{4} c^{4} d^{2}+792 a \,b^{5} c^{5} d +1716 c^{6} b^{6}\right ) x^{4}}{572 b^{7}}-\frac {7 d^{5} \left (a^{5} d^{5}+8 a^{4} b c \,d^{4}+36 a^{3} b^{2} c^{2} d^{3}+120 a^{2} b^{3} c^{3} d^{2}+330 a \,b^{4} c^{4} d +792 c^{5} b^{5}\right ) x^{5}}{286 b^{6}}-\frac {7 d^{6} \left (d^{4} a^{4}+8 a^{3} b c \,d^{3}+36 a^{2} b^{2} c^{2} d^{2}+120 a \,b^{3} c^{3} d +330 c^{4} b^{4}\right ) x^{6}}{132 b^{5}}-\frac {d^{7} \left (a^{3} d^{3}+8 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d +120 b^{3} c^{3}\right ) x^{7}}{11 b^{4}}-\frac {d^{8} \left (a^{2} d^{2}+8 a b c d +36 b^{2} c^{2}\right ) x^{8}}{8 b^{3}}-\frac {5 d^{9} \left (a d +8 b c \right ) x^{9}}{36 b^{2}}-\frac {d^{10} x^{10}}{8 b}}{\left (b x +a \right )^{18}}\)	\(831\)
default	\(\frac {10 d^{9} \left (a d -b c \right )}{9 b^{11} \left (b x +a \right )^{9}}+\frac {10 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -c^{9} b^{9}\right )}{17 b^{11} \left (b x +a \right )^{17}}+\frac {252 d^{5} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}{13 b^{11} \left (b x +a \right )^{13}}-\frac {35 d^{6} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}{2 b^{11} \left (b x +a \right )^{12}}-\frac {d^{10}}{8 b^{11} \left (b x +a \right )^{8}}-\frac {9 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{11} \left (b x +a \right )^{10}}-\frac {15 d^{4} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +c^{6} b^{6}\right )}{b^{11} \left (b x +a \right )^{14}}+\frac {8 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{b^{11} \left (b x +a \right )^{15}}-\frac {45 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +c^{8} b^{8}\right )}{16 b^{11} \left (b x +a \right )^{16}}-\frac {a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}}{18 b^{11} \left (b x +a \right )^{18}}+\frac {120 d^{7} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{11 b^{11} \left (b x +a \right )^{11}}\)	\(867\)
norman	\(\frac {\frac {-a^{10} b^{7} d^{10}-8 a^{9} b^{8} c \,d^{9}-36 a^{8} b^{9} c^{2} d^{8}-120 a^{7} b^{10} c^{3} d^{7}-330 a^{6} b^{11} c^{4} d^{6}-792 a^{5} b^{12} c^{5} d^{5}-1716 a^{4} b^{13} c^{6} d^{4}-3432 a^{3} b^{14} c^{7} d^{3}-6435 a^{2} b^{15} c^{8} d^{2}-11440 a \,c^{9} d \,b^{16}-19448 b^{17} c^{10}}{350064 b^{18}}+\frac {\left (-a^{9} b^{7} d^{10}-8 a^{8} b^{8} c \,d^{9}-36 a^{7} b^{9} c^{2} d^{8}-120 a^{6} b^{10} c^{3} d^{7}-330 a^{5} b^{11} c^{4} d^{6}-792 a^{4} b^{12} c^{5} d^{5}-1716 a^{3} b^{13} c^{6} d^{4}-3432 a^{2} b^{14} c^{7} d^{3}-6435 a \,b^{15} c^{8} d^{2}-11440 c^{9} d \,b^{16}\right ) x}{19448 b^{17}}+\frac {\left (-a^{8} b^{7} d^{10}-8 a^{7} b^{8} c \,d^{9}-36 a^{6} b^{9} c^{2} d^{8}-120 a^{5} b^{10} c^{3} d^{7}-330 a^{4} b^{11} c^{4} d^{6}-792 a^{3} b^{12} c^{5} d^{5}-1716 a^{2} b^{13} c^{6} d^{4}-3432 a \,b^{14} c^{7} d^{3}-6435 b^{15} c^{8} d^{2}\right ) x^{2}}{2288 b^{16}}+\frac {\left (-a^{7} b^{7} d^{10}-8 a^{6} b^{8} c \,d^{9}-36 a^{5} b^{9} c^{2} d^{8}-120 a^{4} b^{10} c^{3} d^{7}-330 a^{3} b^{11} c^{4} d^{6}-792 a^{2} b^{12} c^{5} d^{5}-1716 a \,b^{13} c^{6} d^{4}-3432 b^{14} c^{7} d^{3}\right ) x^{3}}{429 b^{15}}+\frac {5 \left (-a^{6} b^{7} d^{10}-8 a^{5} b^{8} c \,d^{9}-36 a^{4} b^{9} c^{2} d^{8}-120 a^{3} b^{10} c^{3} d^{7}-330 a^{2} b^{11} c^{4} d^{6}-792 a \,c^{5} d^{5} b^{12}-1716 b^{13} c^{6} d^{4}\right ) x^{4}}{572 b^{14}}+\frac {7 \left (-a^{5} b^{7} d^{10}-8 a^{4} b^{8} c \,d^{9}-36 a^{3} b^{9} c^{2} d^{8}-120 a^{2} b^{10} c^{3} d^{7}-330 a \,b^{11} c^{4} d^{6}-792 c^{5} d^{5} b^{12}\right ) x^{5}}{286 b^{13}}+\frac {7 \left (-a^{4} b^{7} d^{10}-8 a^{3} b^{8} c \,d^{9}-36 a^{2} b^{9} c^{2} d^{8}-120 a \,b^{10} c^{3} d^{7}-330 b^{11} c^{4} d^{6}\right ) x^{6}}{132 b^{12}}+\frac {\left (-a^{3} b^{7} d^{10}-8 a^{2} b^{8} c \,d^{9}-36 a \,b^{9} c^{2} d^{8}-120 b^{10} c^{3} d^{7}\right ) x^{7}}{11 b^{11}}+\frac {\left (-a^{2} b^{7} d^{10}-8 a \,b^{8} c \,d^{9}-36 b^{9} c^{2} d^{8}\right ) x^{8}}{8 b^{10}}+\frac {5 \left (-a \,b^{7} d^{10}-8 b^{8} c \,d^{9}\right ) x^{9}}{36 b^{9}}-\frac {d^{10} x^{10}}{8 b}}{\left (b x +a \right )^{18}}\)	\(909\)
gosper	\(-\frac {43758 x^{10} d^{10} b^{10}+48620 x^{9} a \,b^{9} d^{10}+388960 x^{9} b^{10} c \,d^{9}+43758 x^{8} a^{2} b^{8} d^{10}+350064 x^{8} a \,b^{9} c \,d^{9}+1575288 x^{8} b^{10} c^{2} d^{8}+31824 x^{7} a^{3} b^{7} d^{10}+254592 x^{7} a^{2} b^{8} c \,d^{9}+1145664 x^{7} a \,b^{9} c^{2} d^{8}+3818880 x^{7} b^{10} c^{3} d^{7}+18564 x^{6} a^{4} b^{6} d^{10}+148512 x^{6} a^{3} b^{7} c \,d^{9}+668304 x^{6} a^{2} b^{8} c^{2} d^{8}+2227680 x^{6} a \,b^{9} c^{3} d^{7}+6126120 x^{6} b^{10} c^{4} d^{6}+8568 x^{5} a^{5} b^{5} d^{10}+68544 x^{5} a^{4} b^{6} c \,d^{9}+308448 x^{5} a^{3} b^{7} c^{2} d^{8}+1028160 x^{5} a^{2} b^{8} c^{3} d^{7}+2827440 x^{5} a \,b^{9} c^{4} d^{6}+6785856 x^{5} b^{10} c^{5} d^{5}+3060 x^{4} a^{6} b^{4} d^{10}+24480 x^{4} a^{5} b^{5} c \,d^{9}+110160 x^{4} a^{4} b^{6} c^{2} d^{8}+367200 x^{4} a^{3} b^{7} c^{3} d^{7}+1009800 x^{4} a^{2} b^{8} c^{4} d^{6}+2423520 x^{4} a \,b^{9} c^{5} d^{5}+5250960 x^{4} b^{10} c^{6} d^{4}+816 x^{3} a^{7} b^{3} d^{10}+6528 x^{3} a^{6} b^{4} c \,d^{9}+29376 x^{3} a^{5} b^{5} c^{2} d^{8}+97920 x^{3} a^{4} b^{6} c^{3} d^{7}+269280 x^{3} a^{3} b^{7} c^{4} d^{6}+646272 x^{3} a^{2} b^{8} c^{5} d^{5}+1400256 x^{3} a \,b^{9} c^{6} d^{4}+2800512 x^{3} b^{10} c^{7} d^{3}+153 x^{2} a^{8} b^{2} d^{10}+1224 x^{2} a^{7} b^{3} c \,d^{9}+5508 x^{2} a^{6} b^{4} c^{2} d^{8}+18360 x^{2} a^{5} b^{5} c^{3} d^{7}+50490 x^{2} a^{4} b^{6} c^{4} d^{6}+121176 x^{2} a^{3} b^{7} c^{5} d^{5}+262548 x^{2} a^{2} b^{8} c^{6} d^{4}+525096 x^{2} a \,b^{9} c^{7} d^{3}+984555 x^{2} b^{10} c^{8} d^{2}+18 x \,a^{9} b \,d^{10}+144 x \,a^{8} b^{2} c \,d^{9}+648 x \,a^{7} b^{3} c^{2} d^{8}+2160 x \,a^{6} b^{4} c^{3} d^{7}+5940 x \,a^{5} b^{5} c^{4} d^{6}+14256 x \,a^{4} b^{6} c^{5} d^{5}+30888 x \,a^{3} b^{7} c^{6} d^{4}+61776 x \,a^{2} b^{8} c^{7} d^{3}+115830 x a \,b^{9} c^{8} d^{2}+205920 x \,b^{10} c^{9} d +a^{10} d^{10}+8 a^{9} b c \,d^{9}+36 a^{8} b^{2} c^{2} d^{8}+120 a^{7} b^{3} c^{3} d^{7}+330 a^{6} b^{4} c^{4} d^{6}+792 a^{5} b^{5} c^{5} d^{5}+1716 a^{4} b^{6} c^{6} d^{4}+3432 a^{3} b^{7} c^{7} d^{3}+6435 a^{2} b^{8} c^{8} d^{2}+11440 a \,b^{9} c^{9} d +19448 b^{10} c^{10}}{350064 b^{11} \left (b x +a \right )^{18}}\)	\(962\)
orering	\(-\frac {43758 x^{10} d^{10} b^{10}+48620 x^{9} a \,b^{9} d^{10}+388960 x^{9} b^{10} c \,d^{9}+43758 x^{8} a^{2} b^{8} d^{10}+350064 x^{8} a \,b^{9} c \,d^{9}+1575288 x^{8} b^{10} c^{2} d^{8}+31824 x^{7} a^{3} b^{7} d^{10}+254592 x^{7} a^{2} b^{8} c \,d^{9}+1145664 x^{7} a \,b^{9} c^{2} d^{8}+3818880 x^{7} b^{10} c^{3} d^{7}+18564 x^{6} a^{4} b^{6} d^{10}+148512 x^{6} a^{3} b^{7} c \,d^{9}+668304 x^{6} a^{2} b^{8} c^{2} d^{8}+2227680 x^{6} a \,b^{9} c^{3} d^{7}+6126120 x^{6} b^{10} c^{4} d^{6}+8568 x^{5} a^{5} b^{5} d^{10}+68544 x^{5} a^{4} b^{6} c \,d^{9}+308448 x^{5} a^{3} b^{7} c^{2} d^{8}+1028160 x^{5} a^{2} b^{8} c^{3} d^{7}+2827440 x^{5} a \,b^{9} c^{4} d^{6}+6785856 x^{5} b^{10} c^{5} d^{5}+3060 x^{4} a^{6} b^{4} d^{10}+24480 x^{4} a^{5} b^{5} c \,d^{9}+110160 x^{4} a^{4} b^{6} c^{2} d^{8}+367200 x^{4} a^{3} b^{7} c^{3} d^{7}+1009800 x^{4} a^{2} b^{8} c^{4} d^{6}+2423520 x^{4} a \,b^{9} c^{5} d^{5}+5250960 x^{4} b^{10} c^{6} d^{4}+816 x^{3} a^{7} b^{3} d^{10}+6528 x^{3} a^{6} b^{4} c \,d^{9}+29376 x^{3} a^{5} b^{5} c^{2} d^{8}+97920 x^{3} a^{4} b^{6} c^{3} d^{7}+269280 x^{3} a^{3} b^{7} c^{4} d^{6}+646272 x^{3} a^{2} b^{8} c^{5} d^{5}+1400256 x^{3} a \,b^{9} c^{6} d^{4}+2800512 x^{3} b^{10} c^{7} d^{3}+153 x^{2} a^{8} b^{2} d^{10}+1224 x^{2} a^{7} b^{3} c \,d^{9}+5508 x^{2} a^{6} b^{4} c^{2} d^{8}+18360 x^{2} a^{5} b^{5} c^{3} d^{7}+50490 x^{2} a^{4} b^{6} c^{4} d^{6}+121176 x^{2} a^{3} b^{7} c^{5} d^{5}+262548 x^{2} a^{2} b^{8} c^{6} d^{4}+525096 x^{2} a \,b^{9} c^{7} d^{3}+984555 x^{2} b^{10} c^{8} d^{2}+18 x \,a^{9} b \,d^{10}+144 x \,a^{8} b^{2} c \,d^{9}+648 x \,a^{7} b^{3} c^{2} d^{8}+2160 x \,a^{6} b^{4} c^{3} d^{7}+5940 x \,a^{5} b^{5} c^{4} d^{6}+14256 x \,a^{4} b^{6} c^{5} d^{5}+30888 x \,a^{3} b^{7} c^{6} d^{4}+61776 x \,a^{2} b^{8} c^{7} d^{3}+115830 x a \,b^{9} c^{8} d^{2}+205920 x \,b^{10} c^{9} d +a^{10} d^{10}+8 a^{9} b c \,d^{9}+36 a^{8} b^{2} c^{2} d^{8}+120 a^{7} b^{3} c^{3} d^{7}+330 a^{6} b^{4} c^{4} d^{6}+792 a^{5} b^{5} c^{5} d^{5}+1716 a^{4} b^{6} c^{6} d^{4}+3432 a^{3} b^{7} c^{7} d^{3}+6435 a^{2} b^{8} c^{8} d^{2}+11440 a \,b^{9} c^{9} d +19448 b^{10} c^{10}}{350064 b^{11} \left (b x +a \right )^{18}}\)	\(962\)
parallelrisch	\(\frac {-43758 d^{10} x^{10} b^{17}-48620 a \,b^{16} d^{10} x^{9}-388960 b^{17} c \,d^{9} x^{9}-43758 a^{2} b^{15} d^{10} x^{8}-350064 a \,b^{16} c \,d^{9} x^{8}-1575288 b^{17} c^{2} d^{8} x^{8}-31824 a^{3} b^{14} d^{10} x^{7}-254592 a^{2} b^{15} c \,d^{9} x^{7}-1145664 a \,b^{16} c^{2} d^{8} x^{7}-3818880 b^{17} c^{3} d^{7} x^{7}-18564 a^{4} b^{13} d^{10} x^{6}-148512 a^{3} b^{14} c \,d^{9} x^{6}-668304 a^{2} b^{15} c^{2} d^{8} x^{6}-2227680 a \,b^{16} c^{3} d^{7} x^{6}-6126120 b^{17} c^{4} d^{6} x^{6}-8568 a^{5} b^{12} d^{10} x^{5}-68544 a^{4} b^{13} c \,d^{9} x^{5}-308448 a^{3} b^{14} c^{2} d^{8} x^{5}-1028160 a^{2} b^{15} c^{3} d^{7} x^{5}-2827440 a \,b^{16} c^{4} d^{6} x^{5}-6785856 b^{17} c^{5} d^{5} x^{5}-3060 a^{6} b^{11} d^{10} x^{4}-24480 a^{5} b^{12} c \,d^{9} x^{4}-110160 a^{4} b^{13} c^{2} d^{8} x^{4}-367200 a^{3} b^{14} c^{3} d^{7} x^{4}-1009800 a^{2} b^{15} c^{4} d^{6} x^{4}-2423520 a \,b^{16} c^{5} d^{5} x^{4}-5250960 b^{17} c^{6} d^{4} x^{4}-816 a^{7} b^{10} d^{10} x^{3}-6528 a^{6} b^{11} c \,d^{9} x^{3}-29376 a^{5} b^{12} c^{2} d^{8} x^{3}-97920 a^{4} b^{13} c^{3} d^{7} x^{3}-269280 a^{3} b^{14} c^{4} d^{6} x^{3}-646272 a^{2} b^{15} c^{5} d^{5} x^{3}-1400256 a \,b^{16} c^{6} d^{4} x^{3}-2800512 b^{17} c^{7} d^{3} x^{3}-153 a^{8} b^{9} d^{10} x^{2}-1224 a^{7} b^{10} c \,d^{9} x^{2}-5508 a^{6} b^{11} c^{2} d^{8} x^{2}-18360 a^{5} b^{12} c^{3} d^{7} x^{2}-50490 a^{4} b^{13} c^{4} d^{6} x^{2}-121176 a^{3} b^{14} c^{5} d^{5} x^{2}-262548 a^{2} b^{15} c^{6} d^{4} x^{2}-525096 a \,b^{16} c^{7} d^{3} x^{2}-984555 b^{17} c^{8} d^{2} x^{2}-18 a^{9} b^{8} d^{10} x -144 a^{8} b^{9} c \,d^{9} x -648 a^{7} b^{10} c^{2} d^{8} x -2160 a^{6} b^{11} c^{3} d^{7} x -5940 a^{5} b^{12} c^{4} d^{6} x -14256 a^{4} b^{13} c^{5} d^{5} x -30888 a^{3} b^{14} c^{6} d^{4} x -61776 a^{2} b^{15} c^{7} d^{3} x -115830 a \,b^{16} c^{8} d^{2} x -205920 b^{17} c^{9} d x -a^{10} b^{7} d^{10}-8 a^{9} b^{8} c \,d^{9}-36 a^{8} b^{9} c^{2} d^{8}-120 a^{7} b^{10} c^{3} d^{7}-330 a^{6} b^{11} c^{4} d^{6}-792 a^{5} b^{12} c^{5} d^{5}-1716 a^{4} b^{13} c^{6} d^{4}-3432 a^{3} b^{14} c^{7} d^{3}-6435 a^{2} b^{15} c^{8} d^{2}-11440 a \,c^{9} d \,b^{16}-19448 b^{17} c^{10}}{350064 b^{18} \left (b x +a \right )^{18}}\)	\(970\)

Input:

int((d*x+c)^10/(b*x+a)^19,x,method=_RETURNVERBOSE)
 

Output:

(-1/350064/b^11*(a^10*d^10+8*a^9*b*c*d^9+36*a^8*b^2*c^2*d^8+120*a^7*b^3*c^ 
3*d^7+330*a^6*b^4*c^4*d^6+792*a^5*b^5*c^5*d^5+1716*a^4*b^6*c^6*d^4+3432*a^ 
3*b^7*c^7*d^3+6435*a^2*b^8*c^8*d^2+11440*a*b^9*c^9*d+19448*b^10*c^10)-1/19 
448/b^10*d*(a^9*d^9+8*a^8*b*c*d^8+36*a^7*b^2*c^2*d^7+120*a^6*b^3*c^3*d^6+3 
30*a^5*b^4*c^4*d^5+792*a^4*b^5*c^5*d^4+1716*a^3*b^6*c^6*d^3+3432*a^2*b^7*c 
^7*d^2+6435*a*b^8*c^8*d+11440*b^9*c^9)*x-1/2288/b^9*d^2*(a^8*d^8+8*a^7*b*c 
*d^7+36*a^6*b^2*c^2*d^6+120*a^5*b^3*c^3*d^5+330*a^4*b^4*c^4*d^4+792*a^3*b^ 
5*c^5*d^3+1716*a^2*b^6*c^6*d^2+3432*a*b^7*c^7*d+6435*b^8*c^8)*x^2-1/429/b^ 
8*d^3*(a^7*d^7+8*a^6*b*c*d^6+36*a^5*b^2*c^2*d^5+120*a^4*b^3*c^3*d^4+330*a^ 
3*b^4*c^4*d^3+792*a^2*b^5*c^5*d^2+1716*a*b^6*c^6*d+3432*b^7*c^7)*x^3-5/572 
/b^7*d^4*(a^6*d^6+8*a^5*b*c*d^5+36*a^4*b^2*c^2*d^4+120*a^3*b^3*c^3*d^3+330 
*a^2*b^4*c^4*d^2+792*a*b^5*c^5*d+1716*b^6*c^6)*x^4-7/286/b^6*d^5*(a^5*d^5+ 
8*a^4*b*c*d^4+36*a^3*b^2*c^2*d^3+120*a^2*b^3*c^3*d^2+330*a*b^4*c^4*d+792*b 
^5*c^5)*x^5-7/132/b^5*d^6*(a^4*d^4+8*a^3*b*c*d^3+36*a^2*b^2*c^2*d^2+120*a* 
b^3*c^3*d+330*b^4*c^4)*x^6-1/11/b^4*d^7*(a^3*d^3+8*a^2*b*c*d^2+36*a*b^2*c^ 
2*d+120*b^3*c^3)*x^7-1/8/b^3*d^8*(a^2*d^2+8*a*b*c*d+36*b^2*c^2)*x^8-5/36/b 
^2*d^9*(a*d+8*b*c)*x^9-1/8/b*d^10*x^10)/(b*x+a)^18
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1052 vs. \(2 (228) = 456\).

Time = 0.09 (sec) , antiderivative size = 1052, normalized size of antiderivative = 4.31 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{19}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^19,x, algorithm="fricas")
 

Output:

-1/350064*(43758*b^10*d^10*x^10 + 19448*b^10*c^10 + 11440*a*b^9*c^9*d + 64 
35*a^2*b^8*c^8*d^2 + 3432*a^3*b^7*c^7*d^3 + 1716*a^4*b^6*c^6*d^4 + 792*a^5 
*b^5*c^5*d^5 + 330*a^6*b^4*c^4*d^6 + 120*a^7*b^3*c^3*d^7 + 36*a^8*b^2*c^2* 
d^8 + 8*a^9*b*c*d^9 + a^10*d^10 + 48620*(8*b^10*c*d^9 + a*b^9*d^10)*x^9 + 
43758*(36*b^10*c^2*d^8 + 8*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 31824*(120*b^ 
10*c^3*d^7 + 36*a*b^9*c^2*d^8 + 8*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 1856 
4*(330*b^10*c^4*d^6 + 120*a*b^9*c^3*d^7 + 36*a^2*b^8*c^2*d^8 + 8*a^3*b^7*c 
*d^9 + a^4*b^6*d^10)*x^6 + 8568*(792*b^10*c^5*d^5 + 330*a*b^9*c^4*d^6 + 12 
0*a^2*b^8*c^3*d^7 + 36*a^3*b^7*c^2*d^8 + 8*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x 
^5 + 3060*(1716*b^10*c^6*d^4 + 792*a*b^9*c^5*d^5 + 330*a^2*b^8*c^4*d^6 + 1 
20*a^3*b^7*c^3*d^7 + 36*a^4*b^6*c^2*d^8 + 8*a^5*b^5*c*d^9 + a^6*b^4*d^10)* 
x^4 + 816*(3432*b^10*c^7*d^3 + 1716*a*b^9*c^6*d^4 + 792*a^2*b^8*c^5*d^5 + 
330*a^3*b^7*c^4*d^6 + 120*a^4*b^6*c^3*d^7 + 36*a^5*b^5*c^2*d^8 + 8*a^6*b^4 
*c*d^9 + a^7*b^3*d^10)*x^3 + 153*(6435*b^10*c^8*d^2 + 3432*a*b^9*c^7*d^3 + 
 1716*a^2*b^8*c^6*d^4 + 792*a^3*b^7*c^5*d^5 + 330*a^4*b^6*c^4*d^6 + 120*a^ 
5*b^5*c^3*d^7 + 36*a^6*b^4*c^2*d^8 + 8*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 
 18*(11440*b^10*c^9*d + 6435*a*b^9*c^8*d^2 + 3432*a^2*b^8*c^7*d^3 + 1716*a 
^3*b^7*c^6*d^4 + 792*a^4*b^6*c^5*d^5 + 330*a^5*b^5*c^4*d^6 + 120*a^6*b^4*c 
^3*d^7 + 36*a^7*b^3*c^2*d^8 + 8*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^29*x^18 
+ 18*a*b^28*x^17 + 153*a^2*b^27*x^16 + 816*a^3*b^26*x^15 + 3060*a^4*b^2...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{19}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**10/(b*x+a)**19,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1052 vs. \(2 (228) = 456\).

Time = 0.09 (sec) , antiderivative size = 1052, normalized size of antiderivative = 4.31 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{19}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^19,x, algorithm="maxima")
 

Output:

-1/350064*(43758*b^10*d^10*x^10 + 19448*b^10*c^10 + 11440*a*b^9*c^9*d + 64 
35*a^2*b^8*c^8*d^2 + 3432*a^3*b^7*c^7*d^3 + 1716*a^4*b^6*c^6*d^4 + 792*a^5 
*b^5*c^5*d^5 + 330*a^6*b^4*c^4*d^6 + 120*a^7*b^3*c^3*d^7 + 36*a^8*b^2*c^2* 
d^8 + 8*a^9*b*c*d^9 + a^10*d^10 + 48620*(8*b^10*c*d^9 + a*b^9*d^10)*x^9 + 
43758*(36*b^10*c^2*d^8 + 8*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 31824*(120*b^ 
10*c^3*d^7 + 36*a*b^9*c^2*d^8 + 8*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 1856 
4*(330*b^10*c^4*d^6 + 120*a*b^9*c^3*d^7 + 36*a^2*b^8*c^2*d^8 + 8*a^3*b^7*c 
*d^9 + a^4*b^6*d^10)*x^6 + 8568*(792*b^10*c^5*d^5 + 330*a*b^9*c^4*d^6 + 12 
0*a^2*b^8*c^3*d^7 + 36*a^3*b^7*c^2*d^8 + 8*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x 
^5 + 3060*(1716*b^10*c^6*d^4 + 792*a*b^9*c^5*d^5 + 330*a^2*b^8*c^4*d^6 + 1 
20*a^3*b^7*c^3*d^7 + 36*a^4*b^6*c^2*d^8 + 8*a^5*b^5*c*d^9 + a^6*b^4*d^10)* 
x^4 + 816*(3432*b^10*c^7*d^3 + 1716*a*b^9*c^6*d^4 + 792*a^2*b^8*c^5*d^5 + 
330*a^3*b^7*c^4*d^6 + 120*a^4*b^6*c^3*d^7 + 36*a^5*b^5*c^2*d^8 + 8*a^6*b^4 
*c*d^9 + a^7*b^3*d^10)*x^3 + 153*(6435*b^10*c^8*d^2 + 3432*a*b^9*c^7*d^3 + 
 1716*a^2*b^8*c^6*d^4 + 792*a^3*b^7*c^5*d^5 + 330*a^4*b^6*c^4*d^6 + 120*a^ 
5*b^5*c^3*d^7 + 36*a^6*b^4*c^2*d^8 + 8*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 
 18*(11440*b^10*c^9*d + 6435*a*b^9*c^8*d^2 + 3432*a^2*b^8*c^7*d^3 + 1716*a 
^3*b^7*c^6*d^4 + 792*a^4*b^6*c^5*d^5 + 330*a^5*b^5*c^4*d^6 + 120*a^6*b^4*c 
^3*d^7 + 36*a^7*b^3*c^2*d^8 + 8*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^29*x^18 
+ 18*a*b^28*x^17 + 153*a^2*b^27*x^16 + 816*a^3*b^26*x^15 + 3060*a^4*b^2...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (228) = 456\).

Time = 0.13 (sec) , antiderivative size = 961, normalized size of antiderivative = 3.94 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{19}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^19,x, algorithm="giac")
 

Output:

-1/350064*(43758*b^10*d^10*x^10 + 388960*b^10*c*d^9*x^9 + 48620*a*b^9*d^10 
*x^9 + 1575288*b^10*c^2*d^8*x^8 + 350064*a*b^9*c*d^9*x^8 + 43758*a^2*b^8*d 
^10*x^8 + 3818880*b^10*c^3*d^7*x^7 + 1145664*a*b^9*c^2*d^8*x^7 + 254592*a^ 
2*b^8*c*d^9*x^7 + 31824*a^3*b^7*d^10*x^7 + 6126120*b^10*c^4*d^6*x^6 + 2227 
680*a*b^9*c^3*d^7*x^6 + 668304*a^2*b^8*c^2*d^8*x^6 + 148512*a^3*b^7*c*d^9* 
x^6 + 18564*a^4*b^6*d^10*x^6 + 6785856*b^10*c^5*d^5*x^5 + 2827440*a*b^9*c^ 
4*d^6*x^5 + 1028160*a^2*b^8*c^3*d^7*x^5 + 308448*a^3*b^7*c^2*d^8*x^5 + 685 
44*a^4*b^6*c*d^9*x^5 + 8568*a^5*b^5*d^10*x^5 + 5250960*b^10*c^6*d^4*x^4 + 
2423520*a*b^9*c^5*d^5*x^4 + 1009800*a^2*b^8*c^4*d^6*x^4 + 367200*a^3*b^7*c 
^3*d^7*x^4 + 110160*a^4*b^6*c^2*d^8*x^4 + 24480*a^5*b^5*c*d^9*x^4 + 3060*a 
^6*b^4*d^10*x^4 + 2800512*b^10*c^7*d^3*x^3 + 1400256*a*b^9*c^6*d^4*x^3 + 6 
46272*a^2*b^8*c^5*d^5*x^3 + 269280*a^3*b^7*c^4*d^6*x^3 + 97920*a^4*b^6*c^3 
*d^7*x^3 + 29376*a^5*b^5*c^2*d^8*x^3 + 6528*a^6*b^4*c*d^9*x^3 + 816*a^7*b^ 
3*d^10*x^3 + 984555*b^10*c^8*d^2*x^2 + 525096*a*b^9*c^7*d^3*x^2 + 262548*a 
^2*b^8*c^6*d^4*x^2 + 121176*a^3*b^7*c^5*d^5*x^2 + 50490*a^4*b^6*c^4*d^6*x^ 
2 + 18360*a^5*b^5*c^3*d^7*x^2 + 5508*a^6*b^4*c^2*d^8*x^2 + 1224*a^7*b^3*c* 
d^9*x^2 + 153*a^8*b^2*d^10*x^2 + 205920*b^10*c^9*d*x + 115830*a*b^9*c^8*d^ 
2*x + 61776*a^2*b^8*c^7*d^3*x + 30888*a^3*b^7*c^6*d^4*x + 14256*a^4*b^6*c^ 
5*d^5*x + 5940*a^5*b^5*c^4*d^6*x + 2160*a^6*b^4*c^3*d^7*x + 648*a^7*b^3*c^ 
2*d^8*x + 144*a^8*b^2*c*d^9*x + 18*a^9*b*d^10*x + 19448*b^10*c^10 + 114...
 

Mupad [B] (verification not implemented)

Time = 7.41 (sec) , antiderivative size = 1153, normalized size of antiderivative = 4.73 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{19}} \, dx =\text {Too large to display} \] Input:

int((c + d*x)^10/(a + b*x)^19,x)
 

Output:

-(a^10*d^10 + 19448*b^10*c^10 + 43758*b^10*d^10*x^10 + 48620*a*b^9*d^10*x^ 
9 + 388960*b^10*c*d^9*x^9 + 6435*a^2*b^8*c^8*d^2 + 3432*a^3*b^7*c^7*d^3 + 
1716*a^4*b^6*c^6*d^4 + 792*a^5*b^5*c^5*d^5 + 330*a^6*b^4*c^4*d^6 + 120*a^7 
*b^3*c^3*d^7 + 36*a^8*b^2*c^2*d^8 + 153*a^8*b^2*d^10*x^2 + 816*a^7*b^3*d^1 
0*x^3 + 3060*a^6*b^4*d^10*x^4 + 8568*a^5*b^5*d^10*x^5 + 18564*a^4*b^6*d^10 
*x^6 + 31824*a^3*b^7*d^10*x^7 + 43758*a^2*b^8*d^10*x^8 + 984555*b^10*c^8*d 
^2*x^2 + 2800512*b^10*c^7*d^3*x^3 + 5250960*b^10*c^6*d^4*x^4 + 6785856*b^1 
0*c^5*d^5*x^5 + 6126120*b^10*c^4*d^6*x^6 + 3818880*b^10*c^3*d^7*x^7 + 1575 
288*b^10*c^2*d^8*x^8 + 11440*a*b^9*c^9*d + 8*a^9*b*c*d^9 + 18*a^9*b*d^10*x 
 + 205920*b^10*c^9*d*x + 262548*a^2*b^8*c^6*d^4*x^2 + 121176*a^3*b^7*c^5*d 
^5*x^2 + 50490*a^4*b^6*c^4*d^6*x^2 + 18360*a^5*b^5*c^3*d^7*x^2 + 5508*a^6* 
b^4*c^2*d^8*x^2 + 646272*a^2*b^8*c^5*d^5*x^3 + 269280*a^3*b^7*c^4*d^6*x^3 
+ 97920*a^4*b^6*c^3*d^7*x^3 + 29376*a^5*b^5*c^2*d^8*x^3 + 1009800*a^2*b^8* 
c^4*d^6*x^4 + 367200*a^3*b^7*c^3*d^7*x^4 + 110160*a^4*b^6*c^2*d^8*x^4 + 10 
28160*a^2*b^8*c^3*d^7*x^5 + 308448*a^3*b^7*c^2*d^8*x^5 + 668304*a^2*b^8*c^ 
2*d^8*x^6 + 115830*a*b^9*c^8*d^2*x + 144*a^8*b^2*c*d^9*x + 350064*a*b^9*c* 
d^9*x^8 + 61776*a^2*b^8*c^7*d^3*x + 30888*a^3*b^7*c^6*d^4*x + 14256*a^4*b^ 
6*c^5*d^5*x + 5940*a^5*b^5*c^4*d^6*x + 2160*a^6*b^4*c^3*d^7*x + 648*a^7*b^ 
3*c^2*d^8*x + 525096*a*b^9*c^7*d^3*x^2 + 1224*a^7*b^3*c*d^9*x^2 + 1400256* 
a*b^9*c^6*d^4*x^3 + 6528*a^6*b^4*c*d^9*x^3 + 2423520*a*b^9*c^5*d^5*x^4 ...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1149, normalized size of antiderivative = 4.71 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{19}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^10/(b*x+a)^19,x)
 

Output:

( - a**10*d**10 - 8*a**9*b*c*d**9 - 18*a**9*b*d**10*x - 36*a**8*b**2*c**2* 
d**8 - 144*a**8*b**2*c*d**9*x - 153*a**8*b**2*d**10*x**2 - 120*a**7*b**3*c 
**3*d**7 - 648*a**7*b**3*c**2*d**8*x - 1224*a**7*b**3*c*d**9*x**2 - 816*a* 
*7*b**3*d**10*x**3 - 330*a**6*b**4*c**4*d**6 - 2160*a**6*b**4*c**3*d**7*x 
- 5508*a**6*b**4*c**2*d**8*x**2 - 6528*a**6*b**4*c*d**9*x**3 - 3060*a**6*b 
**4*d**10*x**4 - 792*a**5*b**5*c**5*d**5 - 5940*a**5*b**5*c**4*d**6*x - 18 
360*a**5*b**5*c**3*d**7*x**2 - 29376*a**5*b**5*c**2*d**8*x**3 - 24480*a**5 
*b**5*c*d**9*x**4 - 8568*a**5*b**5*d**10*x**5 - 1716*a**4*b**6*c**6*d**4 - 
 14256*a**4*b**6*c**5*d**5*x - 50490*a**4*b**6*c**4*d**6*x**2 - 97920*a**4 
*b**6*c**3*d**7*x**3 - 110160*a**4*b**6*c**2*d**8*x**4 - 68544*a**4*b**6*c 
*d**9*x**5 - 18564*a**4*b**6*d**10*x**6 - 3432*a**3*b**7*c**7*d**3 - 30888 
*a**3*b**7*c**6*d**4*x - 121176*a**3*b**7*c**5*d**5*x**2 - 269280*a**3*b** 
7*c**4*d**6*x**3 - 367200*a**3*b**7*c**3*d**7*x**4 - 308448*a**3*b**7*c**2 
*d**8*x**5 - 148512*a**3*b**7*c*d**9*x**6 - 31824*a**3*b**7*d**10*x**7 - 6 
435*a**2*b**8*c**8*d**2 - 61776*a**2*b**8*c**7*d**3*x - 262548*a**2*b**8*c 
**6*d**4*x**2 - 646272*a**2*b**8*c**5*d**5*x**3 - 1009800*a**2*b**8*c**4*d 
**6*x**4 - 1028160*a**2*b**8*c**3*d**7*x**5 - 668304*a**2*b**8*c**2*d**8*x 
**6 - 254592*a**2*b**8*c*d**9*x**7 - 43758*a**2*b**8*d**10*x**8 - 11440*a* 
b**9*c**9*d - 115830*a*b**9*c**8*d**2*x - 525096*a*b**9*c**7*d**3*x**2 - 1 
400256*a*b**9*c**6*d**4*x**3 - 2423520*a*b**9*c**5*d**5*x**4 - 2827440*...