\(\int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx\) [121]

Optimal result

Integrand size = 15, antiderivative size = 151 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx=-\frac {(c+d x)^{11}}{15 (b c-a d) (a+b x)^{15}}+\frac {2 d (c+d x)^{11}}{105 (b c-a d)^2 (a+b x)^{14}}-\frac {2 d^2 (c+d x)^{11}}{455 (b c-a d)^3 (a+b x)^{13}}+\frac {d^3 (c+d x)^{11}}{1365 (b c-a d)^4 (a+b x)^{12}}-\frac {d^4 (c+d x)^{11}}{15015 (b c-a d)^5 (a+b x)^{11}} \] Output:

-1/15*(d*x+c)^11/(-a*d+b*c)/(b*x+a)^15+2/105*d*(d*x+c)^11/(-a*d+b*c)^2/(b* 
x+a)^14-2/455*d^2*(d*x+c)^11/(-a*d+b*c)^3/(b*x+a)^13+1/1365*d^3*(d*x+c)^11 
/(-a*d+b*c)^4/(b*x+a)^12-1/15015*d^4*(d*x+c)^11/(-a*d+b*c)^5/(b*x+a)^11
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(690\) vs. \(2(151)=302\).

Time = 0.16 (sec) , antiderivative size = 690, normalized size of antiderivative = 4.57 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx=-\frac {a^{10} d^{10}+5 a^9 b d^9 (c+3 d x)+15 a^8 b^2 d^8 \left (c^2+5 c d x+7 d^2 x^2\right )+5 a^7 b^3 d^7 \left (7 c^3+45 c^2 d x+105 c d^2 x^2+91 d^3 x^3\right )+35 a^6 b^4 d^6 \left (2 c^4+15 c^3 d x+45 c^2 d^2 x^2+65 c d^3 x^3+39 d^4 x^4\right )+21 a^5 b^5 d^5 \left (6 c^5+50 c^4 d x+175 c^3 d^2 x^2+325 c^2 d^3 x^3+325 c d^4 x^4+143 d^5 x^5\right )+35 a^4 b^6 d^4 \left (6 c^6+54 c^5 d x+210 c^4 d^2 x^2+455 c^3 d^3 x^3+585 c^2 d^4 x^4+429 c d^5 x^5+143 d^6 x^6\right )+5 a^3 b^7 d^3 \left (66 c^7+630 c^6 d x+2646 c^5 d^2 x^2+6370 c^4 d^3 x^3+9555 c^3 d^4 x^4+9009 c^2 d^5 x^5+5005 c d^6 x^6+1287 d^7 x^7\right )+15 a^2 b^8 d^2 \left (33 c^8+330 c^7 d x+1470 c^6 d^2 x^2+3822 c^5 d^3 x^3+6370 c^4 d^4 x^4+7007 c^3 d^5 x^5+5005 c^2 d^6 x^6+2145 c d^7 x^7+429 d^8 x^8\right )+5 a b^9 d \left (143 c^9+1485 c^8 d x+6930 c^7 d^2 x^2+19110 c^6 d^3 x^3+34398 c^5 d^4 x^4+42042 c^4 d^5 x^5+35035 c^3 d^6 x^6+19305 c^2 d^7 x^7+6435 c d^8 x^8+1001 d^9 x^9\right )+b^{10} \left (1001 c^{10}+10725 c^9 d x+51975 c^8 d^2 x^2+150150 c^7 d^3 x^3+286650 c^6 d^4 x^4+378378 c^5 d^5 x^5+350350 c^4 d^6 x^6+225225 c^3 d^7 x^7+96525 c^2 d^8 x^8+25025 c d^9 x^9+3003 d^{10} x^{10}\right )}{15015 b^{11} (a+b x)^{15}} \] Input:

Integrate[(c + d*x)^10/(a + b*x)^16,x]
 

Output:

-1/15015*(a^10*d^10 + 5*a^9*b*d^9*(c + 3*d*x) + 15*a^8*b^2*d^8*(c^2 + 5*c* 
d*x + 7*d^2*x^2) + 5*a^7*b^3*d^7*(7*c^3 + 45*c^2*d*x + 105*c*d^2*x^2 + 91* 
d^3*x^3) + 35*a^6*b^4*d^6*(2*c^4 + 15*c^3*d*x + 45*c^2*d^2*x^2 + 65*c*d^3* 
x^3 + 39*d^4*x^4) + 21*a^5*b^5*d^5*(6*c^5 + 50*c^4*d*x + 175*c^3*d^2*x^2 + 
 325*c^2*d^3*x^3 + 325*c*d^4*x^4 + 143*d^5*x^5) + 35*a^4*b^6*d^4*(6*c^6 + 
54*c^5*d*x + 210*c^4*d^2*x^2 + 455*c^3*d^3*x^3 + 585*c^2*d^4*x^4 + 429*c*d 
^5*x^5 + 143*d^6*x^6) + 5*a^3*b^7*d^3*(66*c^7 + 630*c^6*d*x + 2646*c^5*d^2 
*x^2 + 6370*c^4*d^3*x^3 + 9555*c^3*d^4*x^4 + 9009*c^2*d^5*x^5 + 5005*c*d^6 
*x^6 + 1287*d^7*x^7) + 15*a^2*b^8*d^2*(33*c^8 + 330*c^7*d*x + 1470*c^6*d^2 
*x^2 + 3822*c^5*d^3*x^3 + 6370*c^4*d^4*x^4 + 7007*c^3*d^5*x^5 + 5005*c^2*d 
^6*x^6 + 2145*c*d^7*x^7 + 429*d^8*x^8) + 5*a*b^9*d*(143*c^9 + 1485*c^8*d*x 
 + 6930*c^7*d^2*x^2 + 19110*c^6*d^3*x^3 + 34398*c^5*d^4*x^4 + 42042*c^4*d^ 
5*x^5 + 35035*c^3*d^6*x^6 + 19305*c^2*d^7*x^7 + 6435*c*d^8*x^8 + 1001*d^9* 
x^9) + b^10*(1001*c^10 + 10725*c^9*d*x + 51975*c^8*d^2*x^2 + 150150*c^7*d^ 
3*x^3 + 286650*c^6*d^4*x^4 + 378378*c^5*d^5*x^5 + 350350*c^4*d^6*x^6 + 225 
225*c^3*d^7*x^7 + 96525*c^2*d^8*x^8 + 25025*c*d^9*x^9 + 3003*d^10*x^10))/( 
b^11*(a + b*x)^15)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.26, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {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.

\(\displaystyle \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {4 d \int \frac {(c+d x)^{10}}{(a+b x)^{15}}dx}{15 (b c-a d)}-\frac {(c+d x)^{11}}{15 (a+b x)^{15} (b c-a d)}\)

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 48

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

Input:

Int[(c + d*x)^10/(a + b*x)^16,x]
 

Output:

-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))
 

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. \(2(141)=282\).

Time = 0.20 (sec) , antiderivative size = 831, normalized size of antiderivative = 5.50

method	result	size
risch	\(\frac {-\frac {a^{10} d^{10}+5 a^{9} b c \,d^{9}+15 a^{8} b^{2} c^{2} d^{8}+35 a^{7} b^{3} c^{3} d^{7}+70 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}+330 a^{3} b^{7} c^{7} d^{3}+495 a^{2} b^{8} c^{8} d^{2}+715 a \,b^{9} c^{9} d +1001 b^{10} c^{10}}{15015 b^{11}}-\frac {d \left (a^{9} d^{9}+5 a^{8} b c \,d^{8}+15 a^{7} b^{2} c^{2} d^{7}+35 a^{6} b^{3} c^{3} d^{6}+70 a^{5} b^{4} c^{4} d^{5}+126 a^{4} b^{5} c^{5} d^{4}+210 a^{3} b^{6} c^{6} d^{3}+330 a^{2} b^{7} c^{7} d^{2}+495 a \,b^{8} c^{8} d +715 c^{9} b^{9}\right ) x}{1001 b^{10}}-\frac {d^{2} \left (a^{8} d^{8}+5 a^{7} b c \,d^{7}+15 a^{6} b^{2} c^{2} d^{6}+35 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}+126 a^{3} b^{5} c^{5} d^{3}+210 a^{2} b^{6} c^{6} d^{2}+330 a \,b^{7} c^{7} d +495 c^{8} b^{8}\right ) x^{2}}{143 b^{9}}-\frac {d^{3} \left (a^{7} d^{7}+5 a^{6} b c \,d^{6}+15 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}+70 a^{3} b^{4} c^{4} d^{3}+126 a^{2} b^{5} c^{5} d^{2}+210 a \,b^{6} c^{6} d +330 b^{7} c^{7}\right ) x^{3}}{33 b^{8}}-\frac {d^{4} \left (a^{6} d^{6}+5 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}+35 a^{3} b^{3} c^{3} d^{3}+70 a^{2} b^{4} c^{4} d^{2}+126 a \,b^{5} c^{5} d +210 c^{6} b^{6}\right ) x^{4}}{11 b^{7}}-\frac {d^{5} \left (a^{5} d^{5}+5 a^{4} b c \,d^{4}+15 a^{3} b^{2} c^{2} d^{3}+35 a^{2} b^{3} c^{3} d^{2}+70 a \,b^{4} c^{4} d +126 c^{5} b^{5}\right ) x^{5}}{5 b^{6}}-\frac {d^{6} \left (d^{4} a^{4}+5 a^{3} b c \,d^{3}+15 a^{2} b^{2} c^{2} d^{2}+35 a \,b^{3} c^{3} d +70 c^{4} b^{4}\right ) x^{6}}{3 b^{5}}-\frac {3 d^{7} \left (a^{3} d^{3}+5 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d +35 b^{3} c^{3}\right ) x^{7}}{7 b^{4}}-\frac {3 d^{8} \left (a^{2} d^{2}+5 a b c d +15 b^{2} c^{2}\right ) x^{8}}{7 b^{3}}-\frac {d^{9} \left (a d +5 b c \right ) x^{9}}{3 b^{2}}-\frac {d^{10} x^{10}}{5 b}}{\left (b x +a \right )^{15}}\)	\(831\)
default	\(-\frac {70 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 )}{3 b^{11} \left (b x +a \right )^{9}}-\frac {d^{10}}{5 b^{11} \left (b x +a \right )^{5}}-\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 )}{13 b^{11} \left (b x +a \right )^{13}}-\frac {45 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{7 b^{11} \left (b x +a \right )^{7}}+\frac {10 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 )^{12}}+\frac {15 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 )}{b^{11} \left (b x +a \right )^{8}}+\frac {126 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 )}{5 b^{11} \left (b x +a \right )^{10}}+\frac {5 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 )}{7 b^{11} \left (b x +a \right )^{14}}-\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}}{15 b^{11} \left (b x +a \right )^{15}}+\frac {5 d^{9} \left (a d -b c \right )}{3 b^{11} \left (b x +a \right )^{6}}-\frac {210 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 )}{11 b^{11} \left (b x +a \right )^{11}}\)	\(867\)
norman	\(\frac {\frac {-a^{10} b^{4} d^{10}-5 a^{9} b^{5} c \,d^{9}-15 a^{8} b^{6} c^{2} d^{8}-35 a^{7} b^{7} c^{3} d^{7}-70 a^{6} b^{8} c^{4} d^{6}-126 a^{5} b^{9} c^{5} d^{5}-210 a^{4} b^{10} c^{6} d^{4}-330 a^{3} c^{7} d^{3} b^{11}-495 a^{2} b^{12} c^{8} d^{2}-715 a \,b^{13} c^{9} d -1001 b^{14} c^{10}}{15015 b^{15}}+\frac {\left (-a^{9} b^{4} d^{10}-5 a^{8} b^{5} c \,d^{9}-15 a^{7} b^{6} c^{2} d^{8}-35 a^{6} b^{7} c^{3} d^{7}-70 a^{5} b^{8} c^{4} d^{6}-126 a^{4} b^{9} c^{5} d^{5}-210 a^{3} b^{10} c^{6} d^{4}-330 a^{2} c^{7} d^{3} b^{11}-495 a \,b^{12} c^{8} d^{2}-715 b^{13} c^{9} d \right ) x}{1001 b^{14}}+\frac {\left (-a^{8} b^{4} d^{10}-5 a^{7} b^{5} c \,d^{9}-15 a^{6} b^{6} c^{2} d^{8}-35 a^{5} b^{7} c^{3} d^{7}-70 a^{4} b^{8} c^{4} d^{6}-126 a^{3} b^{9} c^{5} d^{5}-210 a^{2} b^{10} c^{6} d^{4}-330 a \,c^{7} d^{3} b^{11}-495 b^{12} c^{8} d^{2}\right ) x^{2}}{143 b^{13}}+\frac {\left (-a^{7} b^{4} d^{10}-5 a^{6} b^{5} c \,d^{9}-15 a^{5} b^{6} c^{2} d^{8}-35 a^{4} b^{7} c^{3} d^{7}-70 a^{3} b^{8} c^{4} d^{6}-126 a^{2} b^{9} c^{5} d^{5}-210 a \,b^{10} c^{6} d^{4}-330 b^{11} c^{7} d^{3}\right ) x^{3}}{33 b^{12}}+\frac {\left (-a^{6} b^{4} d^{10}-5 a^{5} b^{5} c \,d^{9}-15 a^{4} b^{6} c^{2} d^{8}-35 a^{3} b^{7} c^{3} d^{7}-70 a^{2} b^{8} c^{4} d^{6}-126 a \,b^{9} c^{5} d^{5}-210 b^{10} c^{6} d^{4}\right ) x^{4}}{11 b^{11}}+\frac {\left (-a^{5} b^{4} d^{10}-5 a^{4} b^{5} c \,d^{9}-15 a^{3} b^{6} c^{2} d^{8}-35 a^{2} b^{7} c^{3} d^{7}-70 a \,b^{8} c^{4} d^{6}-126 b^{9} c^{5} d^{5}\right ) x^{5}}{5 b^{10}}+\frac {\left (-a^{4} b^{4} d^{10}-5 a^{3} b^{5} c \,d^{9}-15 a^{2} b^{6} c^{2} d^{8}-35 a \,b^{7} c^{3} d^{7}-70 b^{8} c^{4} d^{6}\right ) x^{6}}{3 b^{9}}+\frac {3 \left (-a^{3} b^{4} d^{10}-5 a^{2} b^{5} c \,d^{9}-15 a \,b^{6} c^{2} d^{8}-35 b^{7} c^{3} d^{7}\right ) x^{7}}{7 b^{8}}+\frac {3 \left (-a^{2} b^{4} d^{10}-5 a \,b^{5} c \,d^{9}-15 b^{6} c^{2} d^{8}\right ) x^{8}}{7 b^{7}}+\frac {\left (-a \,b^{4} d^{10}-5 b^{5} c \,d^{9}\right ) x^{9}}{3 b^{6}}-\frac {d^{10} x^{10}}{5 b}}{\left (b x +a \right )^{15}}\)	\(909\)
gosper	\(-\frac {3003 x^{10} d^{10} b^{10}+5005 x^{9} a \,b^{9} d^{10}+25025 x^{9} b^{10} c \,d^{9}+6435 x^{8} a^{2} b^{8} d^{10}+32175 x^{8} a \,b^{9} c \,d^{9}+96525 x^{8} b^{10} c^{2} d^{8}+6435 x^{7} a^{3} b^{7} d^{10}+32175 x^{7} a^{2} b^{8} c \,d^{9}+96525 x^{7} a \,b^{9} c^{2} d^{8}+225225 x^{7} b^{10} c^{3} d^{7}+5005 x^{6} a^{4} b^{6} d^{10}+25025 x^{6} a^{3} b^{7} c \,d^{9}+75075 x^{6} a^{2} b^{8} c^{2} d^{8}+175175 x^{6} a \,b^{9} c^{3} d^{7}+350350 x^{6} b^{10} c^{4} d^{6}+3003 x^{5} a^{5} b^{5} d^{10}+15015 x^{5} a^{4} b^{6} c \,d^{9}+45045 x^{5} a^{3} b^{7} c^{2} d^{8}+105105 x^{5} a^{2} b^{8} c^{3} d^{7}+210210 x^{5} a \,b^{9} c^{4} d^{6}+378378 x^{5} b^{10} c^{5} d^{5}+1365 x^{4} a^{6} b^{4} d^{10}+6825 x^{4} a^{5} b^{5} c \,d^{9}+20475 x^{4} a^{4} b^{6} c^{2} d^{8}+47775 x^{4} a^{3} b^{7} c^{3} d^{7}+95550 x^{4} a^{2} b^{8} c^{4} d^{6}+171990 x^{4} a \,b^{9} c^{5} d^{5}+286650 x^{4} b^{10} c^{6} d^{4}+455 x^{3} a^{7} b^{3} d^{10}+2275 x^{3} a^{6} b^{4} c \,d^{9}+6825 x^{3} a^{5} b^{5} c^{2} d^{8}+15925 x^{3} a^{4} b^{6} c^{3} d^{7}+31850 x^{3} a^{3} b^{7} c^{4} d^{6}+57330 x^{3} a^{2} b^{8} c^{5} d^{5}+95550 x^{3} a \,b^{9} c^{6} d^{4}+150150 x^{3} b^{10} c^{7} d^{3}+105 x^{2} a^{8} b^{2} d^{10}+525 x^{2} a^{7} b^{3} c \,d^{9}+1575 x^{2} a^{6} b^{4} c^{2} d^{8}+3675 x^{2} a^{5} b^{5} c^{3} d^{7}+7350 x^{2} a^{4} b^{6} c^{4} d^{6}+13230 x^{2} a^{3} b^{7} c^{5} d^{5}+22050 x^{2} a^{2} b^{8} c^{6} d^{4}+34650 x^{2} a \,b^{9} c^{7} d^{3}+51975 x^{2} b^{10} c^{8} d^{2}+15 x \,a^{9} b \,d^{10}+75 x \,a^{8} b^{2} c \,d^{9}+225 x \,a^{7} b^{3} c^{2} d^{8}+525 x \,a^{6} b^{4} c^{3} d^{7}+1050 x \,a^{5} b^{5} c^{4} d^{6}+1890 x \,a^{4} b^{6} c^{5} d^{5}+3150 x \,a^{3} b^{7} c^{6} d^{4}+4950 x \,a^{2} b^{8} c^{7} d^{3}+7425 x a \,b^{9} c^{8} d^{2}+10725 x \,b^{10} c^{9} d +a^{10} d^{10}+5 a^{9} b c \,d^{9}+15 a^{8} b^{2} c^{2} d^{8}+35 a^{7} b^{3} c^{3} d^{7}+70 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}+330 a^{3} b^{7} c^{7} d^{3}+495 a^{2} b^{8} c^{8} d^{2}+715 a \,b^{9} c^{9} d +1001 b^{10} c^{10}}{15015 b^{11} \left (b x +a \right )^{15}}\)	\(962\)
orering	\(-\frac {3003 x^{10} d^{10} b^{10}+5005 x^{9} a \,b^{9} d^{10}+25025 x^{9} b^{10} c \,d^{9}+6435 x^{8} a^{2} b^{8} d^{10}+32175 x^{8} a \,b^{9} c \,d^{9}+96525 x^{8} b^{10} c^{2} d^{8}+6435 x^{7} a^{3} b^{7} d^{10}+32175 x^{7} a^{2} b^{8} c \,d^{9}+96525 x^{7} a \,b^{9} c^{2} d^{8}+225225 x^{7} b^{10} c^{3} d^{7}+5005 x^{6} a^{4} b^{6} d^{10}+25025 x^{6} a^{3} b^{7} c \,d^{9}+75075 x^{6} a^{2} b^{8} c^{2} d^{8}+175175 x^{6} a \,b^{9} c^{3} d^{7}+350350 x^{6} b^{10} c^{4} d^{6}+3003 x^{5} a^{5} b^{5} d^{10}+15015 x^{5} a^{4} b^{6} c \,d^{9}+45045 x^{5} a^{3} b^{7} c^{2} d^{8}+105105 x^{5} a^{2} b^{8} c^{3} d^{7}+210210 x^{5} a \,b^{9} c^{4} d^{6}+378378 x^{5} b^{10} c^{5} d^{5}+1365 x^{4} a^{6} b^{4} d^{10}+6825 x^{4} a^{5} b^{5} c \,d^{9}+20475 x^{4} a^{4} b^{6} c^{2} d^{8}+47775 x^{4} a^{3} b^{7} c^{3} d^{7}+95550 x^{4} a^{2} b^{8} c^{4} d^{6}+171990 x^{4} a \,b^{9} c^{5} d^{5}+286650 x^{4} b^{10} c^{6} d^{4}+455 x^{3} a^{7} b^{3} d^{10}+2275 x^{3} a^{6} b^{4} c \,d^{9}+6825 x^{3} a^{5} b^{5} c^{2} d^{8}+15925 x^{3} a^{4} b^{6} c^{3} d^{7}+31850 x^{3} a^{3} b^{7} c^{4} d^{6}+57330 x^{3} a^{2} b^{8} c^{5} d^{5}+95550 x^{3} a \,b^{9} c^{6} d^{4}+150150 x^{3} b^{10} c^{7} d^{3}+105 x^{2} a^{8} b^{2} d^{10}+525 x^{2} a^{7} b^{3} c \,d^{9}+1575 x^{2} a^{6} b^{4} c^{2} d^{8}+3675 x^{2} a^{5} b^{5} c^{3} d^{7}+7350 x^{2} a^{4} b^{6} c^{4} d^{6}+13230 x^{2} a^{3} b^{7} c^{5} d^{5}+22050 x^{2} a^{2} b^{8} c^{6} d^{4}+34650 x^{2} a \,b^{9} c^{7} d^{3}+51975 x^{2} b^{10} c^{8} d^{2}+15 x \,a^{9} b \,d^{10}+75 x \,a^{8} b^{2} c \,d^{9}+225 x \,a^{7} b^{3} c^{2} d^{8}+525 x \,a^{6} b^{4} c^{3} d^{7}+1050 x \,a^{5} b^{5} c^{4} d^{6}+1890 x \,a^{4} b^{6} c^{5} d^{5}+3150 x \,a^{3} b^{7} c^{6} d^{4}+4950 x \,a^{2} b^{8} c^{7} d^{3}+7425 x a \,b^{9} c^{8} d^{2}+10725 x \,b^{10} c^{9} d +a^{10} d^{10}+5 a^{9} b c \,d^{9}+15 a^{8} b^{2} c^{2} d^{8}+35 a^{7} b^{3} c^{3} d^{7}+70 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}+330 a^{3} b^{7} c^{7} d^{3}+495 a^{2} b^{8} c^{8} d^{2}+715 a \,b^{9} c^{9} d +1001 b^{10} c^{10}}{15015 b^{11} \left (b x +a \right )^{15}}\)	\(962\)
parallelrisch	\(\frac {-3003 d^{10} x^{10} b^{14}-5005 a \,b^{13} d^{10} x^{9}-25025 b^{14} c \,d^{9} x^{9}-6435 a^{2} b^{12} d^{10} x^{8}-32175 a \,b^{13} c \,d^{9} x^{8}-96525 b^{14} c^{2} d^{8} x^{8}-6435 a^{3} b^{11} d^{10} x^{7}-32175 a^{2} b^{12} c \,d^{9} x^{7}-96525 a \,b^{13} c^{2} d^{8} x^{7}-225225 b^{14} c^{3} d^{7} x^{7}-5005 a^{4} b^{10} d^{10} x^{6}-25025 a^{3} b^{11} c \,d^{9} x^{6}-75075 a^{2} b^{12} c^{2} d^{8} x^{6}-175175 a \,b^{13} c^{3} d^{7} x^{6}-350350 b^{14} c^{4} d^{6} x^{6}-3003 a^{5} b^{9} d^{10} x^{5}-15015 a^{4} b^{10} c \,d^{9} x^{5}-45045 a^{3} b^{11} c^{2} d^{8} x^{5}-105105 a^{2} b^{12} c^{3} d^{7} x^{5}-210210 a \,b^{13} c^{4} d^{6} x^{5}-378378 b^{14} c^{5} d^{5} x^{5}-1365 a^{6} b^{8} d^{10} x^{4}-6825 a^{5} b^{9} c \,d^{9} x^{4}-20475 a^{4} b^{10} c^{2} d^{8} x^{4}-47775 a^{3} b^{11} c^{3} d^{7} x^{4}-95550 a^{2} b^{12} c^{4} d^{6} x^{4}-171990 a \,b^{13} c^{5} d^{5} x^{4}-286650 b^{14} c^{6} d^{4} x^{4}-455 a^{7} b^{7} d^{10} x^{3}-2275 a^{6} b^{8} c \,d^{9} x^{3}-6825 a^{5} b^{9} c^{2} d^{8} x^{3}-15925 a^{4} b^{10} c^{3} d^{7} x^{3}-31850 a^{3} b^{11} c^{4} d^{6} x^{3}-57330 a^{2} b^{12} c^{5} d^{5} x^{3}-95550 a \,b^{13} c^{6} d^{4} x^{3}-150150 b^{14} c^{7} d^{3} x^{3}-105 a^{8} b^{6} d^{10} x^{2}-525 a^{7} b^{7} c \,d^{9} x^{2}-1575 a^{6} b^{8} c^{2} d^{8} x^{2}-3675 a^{5} b^{9} c^{3} d^{7} x^{2}-7350 a^{4} b^{10} c^{4} d^{6} x^{2}-13230 a^{3} b^{11} c^{5} d^{5} x^{2}-22050 a^{2} b^{12} c^{6} d^{4} x^{2}-34650 a \,b^{13} c^{7} d^{3} x^{2}-51975 b^{14} c^{8} d^{2} x^{2}-15 a^{9} b^{5} d^{10} x -75 a^{8} b^{6} c \,d^{9} x -225 a^{7} b^{7} c^{2} d^{8} x -525 a^{6} b^{8} c^{3} d^{7} x -1050 a^{5} b^{9} c^{4} d^{6} x -1890 a^{4} b^{10} c^{5} d^{5} x -3150 a^{3} b^{11} c^{6} d^{4} x -4950 a^{2} b^{12} c^{7} d^{3} x -7425 a \,b^{13} c^{8} d^{2} x -10725 b^{14} c^{9} d x -a^{10} b^{4} d^{10}-5 a^{9} b^{5} c \,d^{9}-15 a^{8} b^{6} c^{2} d^{8}-35 a^{7} b^{7} c^{3} d^{7}-70 a^{6} b^{8} c^{4} d^{6}-126 a^{5} b^{9} c^{5} d^{5}-210 a^{4} b^{10} c^{6} d^{4}-330 a^{3} c^{7} d^{3} b^{11}-495 a^{2} b^{12} c^{8} d^{2}-715 a \,b^{13} c^{9} d -1001 b^{14} c^{10}}{15015 b^{15} \left (b x +a \right )^{15}}\)	\(970\)

Input:

int((d*x+c)^10/(b*x+a)^16,x,method=_RETURNVERBOSE)
 

Output:

(-1/15015/b^11*(a^10*d^10+5*a^9*b*c*d^9+15*a^8*b^2*c^2*d^8+35*a^7*b^3*c^3* 
d^7+70*a^6*b^4*c^4*d^6+126*a^5*b^5*c^5*d^5+210*a^4*b^6*c^6*d^4+330*a^3*b^7 
*c^7*d^3+495*a^2*b^8*c^8*d^2+715*a*b^9*c^9*d+1001*b^10*c^10)-1/1001/b^10*d 
*(a^9*d^9+5*a^8*b*c*d^8+15*a^7*b^2*c^2*d^7+35*a^6*b^3*c^3*d^6+70*a^5*b^4*c 
^4*d^5+126*a^4*b^5*c^5*d^4+210*a^3*b^6*c^6*d^3+330*a^2*b^7*c^7*d^2+495*a*b 
^8*c^8*d+715*b^9*c^9)*x-1/143/b^9*d^2*(a^8*d^8+5*a^7*b*c*d^7+15*a^6*b^2*c^ 
2*d^6+35*a^5*b^3*c^3*d^5+70*a^4*b^4*c^4*d^4+126*a^3*b^5*c^5*d^3+210*a^2*b^ 
6*c^6*d^2+330*a*b^7*c^7*d+495*b^8*c^8)*x^2-1/33/b^8*d^3*(a^7*d^7+5*a^6*b*c 
*d^6+15*a^5*b^2*c^2*d^5+35*a^4*b^3*c^3*d^4+70*a^3*b^4*c^4*d^3+126*a^2*b^5* 
c^5*d^2+210*a*b^6*c^6*d+330*b^7*c^7)*x^3-1/11/b^7*d^4*(a^6*d^6+5*a^5*b*c*d 
^5+15*a^4*b^2*c^2*d^4+35*a^3*b^3*c^3*d^3+70*a^2*b^4*c^4*d^2+126*a*b^5*c^5* 
d+210*b^6*c^6)*x^4-1/5/b^6*d^5*(a^5*d^5+5*a^4*b*c*d^4+15*a^3*b^2*c^2*d^3+3 
5*a^2*b^3*c^3*d^2+70*a*b^4*c^4*d+126*b^5*c^5)*x^5-1/3/b^5*d^6*(a^4*d^4+5*a 
^3*b*c*d^3+15*a^2*b^2*c^2*d^2+35*a*b^3*c^3*d+70*b^4*c^4)*x^6-3/7/b^4*d^7*( 
a^3*d^3+5*a^2*b*c*d^2+15*a*b^2*c^2*d+35*b^3*c^3)*x^7-3/7/b^3*d^8*(a^2*d^2+ 
5*a*b*c*d+15*b^2*c^2)*x^8-1/3/b^2*d^9*(a*d+5*b*c)*x^9-1/5/b*d^10*x^10)/(b* 
x+a)^15
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1019 vs. \(2 (141) = 282\).

Time = 0.10 (sec) , antiderivative size = 1019, normalized size of antiderivative = 6.75 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^16,x, algorithm="fricas")
 

Output:

-1/15015*(3003*b^10*d^10*x^10 + 1001*b^10*c^10 + 715*a*b^9*c^9*d + 495*a^2 
*b^8*c^8*d^2 + 330*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5 
*d^5 + 70*a^6*b^4*c^4*d^6 + 35*a^7*b^3*c^3*d^7 + 15*a^8*b^2*c^2*d^8 + 5*a^ 
9*b*c*d^9 + a^10*d^10 + 5005*(5*b^10*c*d^9 + a*b^9*d^10)*x^9 + 6435*(15*b^ 
10*c^2*d^8 + 5*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 6435*(35*b^10*c^3*d^7 + 1 
5*a*b^9*c^2*d^8 + 5*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 5005*(70*b^10*c^4* 
d^6 + 35*a*b^9*c^3*d^7 + 15*a^2*b^8*c^2*d^8 + 5*a^3*b^7*c*d^9 + a^4*b^6*d^ 
10)*x^6 + 3003*(126*b^10*c^5*d^5 + 70*a*b^9*c^4*d^6 + 35*a^2*b^8*c^3*d^7 + 
 15*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 1365*(210*b^10 
*c^6*d^4 + 126*a*b^9*c^5*d^5 + 70*a^2*b^8*c^4*d^6 + 35*a^3*b^7*c^3*d^7 + 1 
5*a^4*b^6*c^2*d^8 + 5*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 455*(330*b^10*c^ 
7*d^3 + 210*a*b^9*c^6*d^4 + 126*a^2*b^8*c^5*d^5 + 70*a^3*b^7*c^4*d^6 + 35* 
a^4*b^6*c^3*d^7 + 15*a^5*b^5*c^2*d^8 + 5*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 
 + 105*(495*b^10*c^8*d^2 + 330*a*b^9*c^7*d^3 + 210*a^2*b^8*c^6*d^4 + 126*a 
^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 35*a^5*b^5*c^3*d^7 + 15*a^6*b^4*c^2* 
d^8 + 5*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 15*(715*b^10*c^9*d + 495*a*b^9 
*c^8*d^2 + 330*a^2*b^8*c^7*d^3 + 210*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 
 + 70*a^5*b^5*c^4*d^6 + 35*a^6*b^4*c^3*d^7 + 15*a^7*b^3*c^2*d^8 + 5*a^8*b^ 
2*c*d^9 + a^9*b*d^10)*x)/(b^26*x^15 + 15*a*b^25*x^14 + 105*a^2*b^24*x^13 + 
 455*a^3*b^23*x^12 + 1365*a^4*b^22*x^11 + 3003*a^5*b^21*x^10 + 5005*a^6...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**10/(b*x+a)**16,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1019 vs. \(2 (141) = 282\).

Time = 0.09 (sec) , antiderivative size = 1019, normalized size of antiderivative = 6.75 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^16,x, algorithm="maxima")
 

Output:

-1/15015*(3003*b^10*d^10*x^10 + 1001*b^10*c^10 + 715*a*b^9*c^9*d + 495*a^2 
*b^8*c^8*d^2 + 330*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5 
*d^5 + 70*a^6*b^4*c^4*d^6 + 35*a^7*b^3*c^3*d^7 + 15*a^8*b^2*c^2*d^8 + 5*a^ 
9*b*c*d^9 + a^10*d^10 + 5005*(5*b^10*c*d^9 + a*b^9*d^10)*x^9 + 6435*(15*b^ 
10*c^2*d^8 + 5*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 6435*(35*b^10*c^3*d^7 + 1 
5*a*b^9*c^2*d^8 + 5*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 5005*(70*b^10*c^4* 
d^6 + 35*a*b^9*c^3*d^7 + 15*a^2*b^8*c^2*d^8 + 5*a^3*b^7*c*d^9 + a^4*b^6*d^ 
10)*x^6 + 3003*(126*b^10*c^5*d^5 + 70*a*b^9*c^4*d^6 + 35*a^2*b^8*c^3*d^7 + 
 15*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 1365*(210*b^10 
*c^6*d^4 + 126*a*b^9*c^5*d^5 + 70*a^2*b^8*c^4*d^6 + 35*a^3*b^7*c^3*d^7 + 1 
5*a^4*b^6*c^2*d^8 + 5*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 455*(330*b^10*c^ 
7*d^3 + 210*a*b^9*c^6*d^4 + 126*a^2*b^8*c^5*d^5 + 70*a^3*b^7*c^4*d^6 + 35* 
a^4*b^6*c^3*d^7 + 15*a^5*b^5*c^2*d^8 + 5*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 
 + 105*(495*b^10*c^8*d^2 + 330*a*b^9*c^7*d^3 + 210*a^2*b^8*c^6*d^4 + 126*a 
^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 35*a^5*b^5*c^3*d^7 + 15*a^6*b^4*c^2* 
d^8 + 5*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 15*(715*b^10*c^9*d + 495*a*b^9 
*c^8*d^2 + 330*a^2*b^8*c^7*d^3 + 210*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 
 + 70*a^5*b^5*c^4*d^6 + 35*a^6*b^4*c^3*d^7 + 15*a^7*b^3*c^2*d^8 + 5*a^8*b^ 
2*c*d^9 + a^9*b*d^10)*x)/(b^26*x^15 + 15*a*b^25*x^14 + 105*a^2*b^24*x^13 + 
 455*a^3*b^23*x^12 + 1365*a^4*b^22*x^11 + 3003*a^5*b^21*x^10 + 5005*a^6...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (141) = 282\).

Time = 0.12 (sec) , antiderivative size = 961, normalized size of antiderivative = 6.36 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^16,x, algorithm="giac")
 

Output:

-1/15015*(3003*b^10*d^10*x^10 + 25025*b^10*c*d^9*x^9 + 5005*a*b^9*d^10*x^9 
 + 96525*b^10*c^2*d^8*x^8 + 32175*a*b^9*c*d^9*x^8 + 6435*a^2*b^8*d^10*x^8 
+ 225225*b^10*c^3*d^7*x^7 + 96525*a*b^9*c^2*d^8*x^7 + 32175*a^2*b^8*c*d^9* 
x^7 + 6435*a^3*b^7*d^10*x^7 + 350350*b^10*c^4*d^6*x^6 + 175175*a*b^9*c^3*d 
^7*x^6 + 75075*a^2*b^8*c^2*d^8*x^6 + 25025*a^3*b^7*c*d^9*x^6 + 5005*a^4*b^ 
6*d^10*x^6 + 378378*b^10*c^5*d^5*x^5 + 210210*a*b^9*c^4*d^6*x^5 + 105105*a 
^2*b^8*c^3*d^7*x^5 + 45045*a^3*b^7*c^2*d^8*x^5 + 15015*a^4*b^6*c*d^9*x^5 + 
 3003*a^5*b^5*d^10*x^5 + 286650*b^10*c^6*d^4*x^4 + 171990*a*b^9*c^5*d^5*x^ 
4 + 95550*a^2*b^8*c^4*d^6*x^4 + 47775*a^3*b^7*c^3*d^7*x^4 + 20475*a^4*b^6* 
c^2*d^8*x^4 + 6825*a^5*b^5*c*d^9*x^4 + 1365*a^6*b^4*d^10*x^4 + 150150*b^10 
*c^7*d^3*x^3 + 95550*a*b^9*c^6*d^4*x^3 + 57330*a^2*b^8*c^5*d^5*x^3 + 31850 
*a^3*b^7*c^4*d^6*x^3 + 15925*a^4*b^6*c^3*d^7*x^3 + 6825*a^5*b^5*c^2*d^8*x^ 
3 + 2275*a^6*b^4*c*d^9*x^3 + 455*a^7*b^3*d^10*x^3 + 51975*b^10*c^8*d^2*x^2 
 + 34650*a*b^9*c^7*d^3*x^2 + 22050*a^2*b^8*c^6*d^4*x^2 + 13230*a^3*b^7*c^5 
*d^5*x^2 + 7350*a^4*b^6*c^4*d^6*x^2 + 3675*a^5*b^5*c^3*d^7*x^2 + 1575*a^6* 
b^4*c^2*d^8*x^2 + 525*a^7*b^3*c*d^9*x^2 + 105*a^8*b^2*d^10*x^2 + 10725*b^1 
0*c^9*d*x + 7425*a*b^9*c^8*d^2*x + 4950*a^2*b^8*c^7*d^3*x + 3150*a^3*b^7*c 
^6*d^4*x + 1890*a^4*b^6*c^5*d^5*x + 1050*a^5*b^5*c^4*d^6*x + 525*a^6*b^4*c 
^3*d^7*x + 225*a^7*b^3*c^2*d^8*x + 75*a^8*b^2*c*d^9*x + 15*a^9*b*d^10*x + 
1001*b^10*c^10 + 715*a*b^9*c^9*d + 495*a^2*b^8*c^8*d^2 + 330*a^3*b^7*c^...
 

Mupad [B] (verification not implemented)

Time = 1.41 (sec) , antiderivative size = 1120, normalized size of antiderivative = 7.42 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx =\text {Too large to display} \] Input:

int((c + d*x)^10/(a + b*x)^16,x)
 

Output:

-(a^10*d^10 + 1001*b^10*c^10 + 3003*b^10*d^10*x^10 + 5005*a*b^9*d^10*x^9 + 
 25025*b^10*c*d^9*x^9 + 495*a^2*b^8*c^8*d^2 + 330*a^3*b^7*c^7*d^3 + 210*a^ 
4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 70*a^6*b^4*c^4*d^6 + 35*a^7*b^3*c^3* 
d^7 + 15*a^8*b^2*c^2*d^8 + 105*a^8*b^2*d^10*x^2 + 455*a^7*b^3*d^10*x^3 + 1 
365*a^6*b^4*d^10*x^4 + 3003*a^5*b^5*d^10*x^5 + 5005*a^4*b^6*d^10*x^6 + 643 
5*a^3*b^7*d^10*x^7 + 6435*a^2*b^8*d^10*x^8 + 51975*b^10*c^8*d^2*x^2 + 1501 
50*b^10*c^7*d^3*x^3 + 286650*b^10*c^6*d^4*x^4 + 378378*b^10*c^5*d^5*x^5 + 
350350*b^10*c^4*d^6*x^6 + 225225*b^10*c^3*d^7*x^7 + 96525*b^10*c^2*d^8*x^8 
 + 715*a*b^9*c^9*d + 5*a^9*b*c*d^9 + 15*a^9*b*d^10*x + 10725*b^10*c^9*d*x 
+ 22050*a^2*b^8*c^6*d^4*x^2 + 13230*a^3*b^7*c^5*d^5*x^2 + 7350*a^4*b^6*c^4 
*d^6*x^2 + 3675*a^5*b^5*c^3*d^7*x^2 + 1575*a^6*b^4*c^2*d^8*x^2 + 57330*a^2 
*b^8*c^5*d^5*x^3 + 31850*a^3*b^7*c^4*d^6*x^3 + 15925*a^4*b^6*c^3*d^7*x^3 + 
 6825*a^5*b^5*c^2*d^8*x^3 + 95550*a^2*b^8*c^4*d^6*x^4 + 47775*a^3*b^7*c^3* 
d^7*x^4 + 20475*a^4*b^6*c^2*d^8*x^4 + 105105*a^2*b^8*c^3*d^7*x^5 + 45045*a 
^3*b^7*c^2*d^8*x^5 + 75075*a^2*b^8*c^2*d^8*x^6 + 7425*a*b^9*c^8*d^2*x + 75 
*a^8*b^2*c*d^9*x + 32175*a*b^9*c*d^9*x^8 + 4950*a^2*b^8*c^7*d^3*x + 3150*a 
^3*b^7*c^6*d^4*x + 1890*a^4*b^6*c^5*d^5*x + 1050*a^5*b^5*c^4*d^6*x + 525*a 
^6*b^4*c^3*d^7*x + 225*a^7*b^3*c^2*d^8*x + 34650*a*b^9*c^7*d^3*x^2 + 525*a 
^7*b^3*c*d^9*x^2 + 95550*a*b^9*c^6*d^4*x^3 + 2275*a^6*b^4*c*d^9*x^3 + 1719 
90*a*b^9*c^5*d^5*x^4 + 6825*a^5*b^5*c*d^9*x^4 + 210210*a*b^9*c^4*d^6*x^...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1116, normalized size of antiderivative = 7.39 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^10/(b*x+a)^16,x)
 

Output:

( - a**10*d**10 - 5*a**9*b*c*d**9 - 15*a**9*b*d**10*x - 15*a**8*b**2*c**2* 
d**8 - 75*a**8*b**2*c*d**9*x - 105*a**8*b**2*d**10*x**2 - 35*a**7*b**3*c** 
3*d**7 - 225*a**7*b**3*c**2*d**8*x - 525*a**7*b**3*c*d**9*x**2 - 455*a**7* 
b**3*d**10*x**3 - 70*a**6*b**4*c**4*d**6 - 525*a**6*b**4*c**3*d**7*x - 157 
5*a**6*b**4*c**2*d**8*x**2 - 2275*a**6*b**4*c*d**9*x**3 - 1365*a**6*b**4*d 
**10*x**4 - 126*a**5*b**5*c**5*d**5 - 1050*a**5*b**5*c**4*d**6*x - 3675*a* 
*5*b**5*c**3*d**7*x**2 - 6825*a**5*b**5*c**2*d**8*x**3 - 6825*a**5*b**5*c* 
d**9*x**4 - 3003*a**5*b**5*d**10*x**5 - 210*a**4*b**6*c**6*d**4 - 1890*a** 
4*b**6*c**5*d**5*x - 7350*a**4*b**6*c**4*d**6*x**2 - 15925*a**4*b**6*c**3* 
d**7*x**3 - 20475*a**4*b**6*c**2*d**8*x**4 - 15015*a**4*b**6*c*d**9*x**5 - 
 5005*a**4*b**6*d**10*x**6 - 330*a**3*b**7*c**7*d**3 - 3150*a**3*b**7*c**6 
*d**4*x - 13230*a**3*b**7*c**5*d**5*x**2 - 31850*a**3*b**7*c**4*d**6*x**3 
- 47775*a**3*b**7*c**3*d**7*x**4 - 45045*a**3*b**7*c**2*d**8*x**5 - 25025* 
a**3*b**7*c*d**9*x**6 - 6435*a**3*b**7*d**10*x**7 - 495*a**2*b**8*c**8*d** 
2 - 4950*a**2*b**8*c**7*d**3*x - 22050*a**2*b**8*c**6*d**4*x**2 - 57330*a* 
*2*b**8*c**5*d**5*x**3 - 95550*a**2*b**8*c**4*d**6*x**4 - 105105*a**2*b**8 
*c**3*d**7*x**5 - 75075*a**2*b**8*c**2*d**8*x**6 - 32175*a**2*b**8*c*d**9* 
x**7 - 6435*a**2*b**8*d**10*x**8 - 715*a*b**9*c**9*d - 7425*a*b**9*c**8*d* 
*2*x - 34650*a*b**9*c**7*d**3*x**2 - 95550*a*b**9*c**6*d**4*x**3 - 171990* 
a*b**9*c**5*d**5*x**4 - 210210*a*b**9*c**4*d**6*x**5 - 175175*a*b**9*c*...