\(\int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx\) [120]

Optimal result

Integrand size = 15, antiderivative size = 120 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx=-\frac {(c+d x)^{11}}{14 (b c-a d) (a+b x)^{14}}+\frac {3 d (c+d x)^{11}}{182 (b c-a d)^2 (a+b x)^{13}}-\frac {d^2 (c+d x)^{11}}{364 (b c-a d)^3 (a+b x)^{12}}+\frac {d^3 (c+d x)^{11}}{4004 (b c-a d)^4 (a+b x)^{11}} \] Output:

-1/14*(d*x+c)^11/(-a*d+b*c)/(b*x+a)^14+3/182*d*(d*x+c)^11/(-a*d+b*c)^2/(b* 
x+a)^13-1/364*d^2*(d*x+c)^11/(-a*d+b*c)^3/(b*x+a)^12+1/4004*d^3*(d*x+c)^11 
/(-a*d+b*c)^4/(b*x+a)^11
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(692\) vs. \(2(120)=240\).

Time = 0.17 (sec) , antiderivative size = 692, normalized size of antiderivative = 5.77 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx=-\frac {a^{10} d^{10}+2 a^9 b d^9 (2 c+7 d x)+a^8 b^2 d^8 \left (10 c^2+56 c d x+91 d^2 x^2\right )+4 a^7 b^3 d^7 \left (5 c^3+35 c^2 d x+91 c d^2 x^2+91 d^3 x^3\right )+7 a^6 b^4 d^6 \left (5 c^4+40 c^3 d x+130 c^2 d^2 x^2+208 c d^3 x^3+143 d^4 x^4\right )+14 a^5 b^5 d^5 \left (4 c^5+35 c^4 d x+130 c^3 d^2 x^2+260 c^2 d^3 x^3+286 c d^4 x^4+143 d^5 x^5\right )+7 a^4 b^6 d^4 \left (12 c^6+112 c^5 d x+455 c^4 d^2 x^2+1040 c^3 d^3 x^3+1430 c^2 d^4 x^4+1144 c d^5 x^5+429 d^6 x^6\right )+4 a^3 b^7 d^3 \left (30 c^7+294 c^6 d x+1274 c^5 d^2 x^2+3185 c^4 d^3 x^3+5005 c^3 d^4 x^4+5005 c^2 d^5 x^5+3003 c d^6 x^6+858 d^7 x^7\right )+a^2 b^8 d^2 \left (165 c^8+1680 c^7 d x+7644 c^6 d^2 x^2+20384 c^5 d^3 x^3+35035 c^4 d^4 x^4+40040 c^3 d^5 x^5+30030 c^2 d^6 x^6+13728 c d^7 x^7+3003 d^8 x^8\right )+2 a b^9 d \left (110 c^9+1155 c^8 d x+5460 c^7 d^2 x^2+15288 c^6 d^3 x^3+28028 c^5 d^4 x^4+35035 c^4 d^5 x^5+30030 c^3 d^6 x^6+17160 c^2 d^7 x^7+6006 c d^8 x^8+1001 d^9 x^9\right )+b^{10} \left (286 c^{10}+3080 c^9 d x+15015 c^8 d^2 x^2+43680 c^7 d^3 x^3+84084 c^6 d^4 x^4+112112 c^5 d^5 x^5+105105 c^4 d^6 x^6+68640 c^3 d^7 x^7+30030 c^2 d^8 x^8+8008 c d^9 x^9+1001 d^{10} x^{10}\right )}{4004 b^{11} (a+b x)^{14}} \] Input:

Integrate[(c + d*x)^10/(a + b*x)^15,x]
 

Output:

-1/4004*(a^10*d^10 + 2*a^9*b*d^9*(2*c + 7*d*x) + a^8*b^2*d^8*(10*c^2 + 56* 
c*d*x + 91*d^2*x^2) + 4*a^7*b^3*d^7*(5*c^3 + 35*c^2*d*x + 91*c*d^2*x^2 + 9 
1*d^3*x^3) + 7*a^6*b^4*d^6*(5*c^4 + 40*c^3*d*x + 130*c^2*d^2*x^2 + 208*c*d 
^3*x^3 + 143*d^4*x^4) + 14*a^5*b^5*d^5*(4*c^5 + 35*c^4*d*x + 130*c^3*d^2*x 
^2 + 260*c^2*d^3*x^3 + 286*c*d^4*x^4 + 143*d^5*x^5) + 7*a^4*b^6*d^4*(12*c^ 
6 + 112*c^5*d*x + 455*c^4*d^2*x^2 + 1040*c^3*d^3*x^3 + 1430*c^2*d^4*x^4 + 
1144*c*d^5*x^5 + 429*d^6*x^6) + 4*a^3*b^7*d^3*(30*c^7 + 294*c^6*d*x + 1274 
*c^5*d^2*x^2 + 3185*c^4*d^3*x^3 + 5005*c^3*d^4*x^4 + 5005*c^2*d^5*x^5 + 30 
03*c*d^6*x^6 + 858*d^7*x^7) + a^2*b^8*d^2*(165*c^8 + 1680*c^7*d*x + 7644*c 
^6*d^2*x^2 + 20384*c^5*d^3*x^3 + 35035*c^4*d^4*x^4 + 40040*c^3*d^5*x^5 + 3 
0030*c^2*d^6*x^6 + 13728*c*d^7*x^7 + 3003*d^8*x^8) + 2*a*b^9*d*(110*c^9 + 
1155*c^8*d*x + 5460*c^7*d^2*x^2 + 15288*c^6*d^3*x^3 + 28028*c^5*d^4*x^4 + 
35035*c^4*d^5*x^5 + 30030*c^3*d^6*x^6 + 17160*c^2*d^7*x^7 + 6006*c*d^8*x^8 
 + 1001*d^9*x^9) + b^10*(286*c^10 + 3080*c^9*d*x + 15015*c^8*d^2*x^2 + 436 
80*c^7*d^3*x^3 + 84084*c^6*d^4*x^4 + 112112*c^5*d^5*x^5 + 105105*c^4*d^6*x 
^6 + 68640*c^3*d^7*x^7 + 30030*c^2*d^8*x^8 + 8008*c*d^9*x^9 + 1001*d^10*x^ 
10))/(b^11*(a + b*x)^14)
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.22, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {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.

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 48

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

Input:

Int[(c + d*x)^10/(a + b*x)^15,x]
 

Output:

-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))
 

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. \(2(112)=224\).

Time = 0.19 (sec) , antiderivative size = 831, normalized size of antiderivative = 6.92

method	result	size
risch	\(\frac {-\frac {d^{10} x^{10}}{4 b}-\frac {d^{9} \left (a d +4 b c \right ) x^{9}}{2 b^{2}}-\frac {3 d^{8} \left (a^{2} d^{2}+4 a b c d +10 b^{2} c^{2}\right ) x^{8}}{4 b^{3}}-\frac {6 d^{7} \left (a^{3} d^{3}+4 a^{2} b c \,d^{2}+10 a \,b^{2} c^{2} d +20 b^{3} c^{3}\right ) x^{7}}{7 b^{4}}-\frac {3 d^{6} \left (d^{4} a^{4}+4 a^{3} b c \,d^{3}+10 a^{2} b^{2} c^{2} d^{2}+20 a \,b^{3} c^{3} d +35 c^{4} b^{4}\right ) x^{6}}{4 b^{5}}-\frac {d^{5} \left (a^{5} d^{5}+4 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}+20 a^{2} b^{3} c^{3} d^{2}+35 a \,b^{4} c^{4} d +56 c^{5} b^{5}\right ) x^{5}}{2 b^{6}}-\frac {d^{4} \left (a^{6} d^{6}+4 a^{5} b c \,d^{5}+10 a^{4} b^{2} c^{2} d^{4}+20 a^{3} b^{3} c^{3} d^{3}+35 a^{2} b^{4} c^{4} d^{2}+56 a \,b^{5} c^{5} d +84 c^{6} b^{6}\right ) x^{4}}{4 b^{7}}-\frac {d^{3} \left (a^{7} d^{7}+4 a^{6} b c \,d^{6}+10 a^{5} b^{2} c^{2} d^{5}+20 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}+56 a^{2} b^{5} c^{5} d^{2}+84 a \,b^{6} c^{6} d +120 b^{7} c^{7}\right ) x^{3}}{11 b^{8}}-\frac {d^{2} \left (a^{8} d^{8}+4 a^{7} b c \,d^{7}+10 a^{6} b^{2} c^{2} d^{6}+20 a^{5} b^{3} c^{3} d^{5}+35 a^{4} b^{4} c^{4} d^{4}+56 a^{3} b^{5} c^{5} d^{3}+84 a^{2} b^{6} c^{6} d^{2}+120 a \,b^{7} c^{7} d +165 c^{8} b^{8}\right ) x^{2}}{44 b^{9}}-\frac {d \left (a^{9} d^{9}+4 a^{8} b c \,d^{8}+10 a^{7} b^{2} c^{2} d^{7}+20 a^{6} b^{3} c^{3} d^{6}+35 a^{5} b^{4} c^{4} d^{5}+56 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}+120 a^{2} b^{7} c^{7} d^{2}+165 a \,b^{8} c^{8} d +220 c^{9} b^{9}\right ) x}{286 b^{10}}-\frac {a^{10} d^{10}+4 a^{9} b c \,d^{9}+10 a^{8} b^{2} c^{2} d^{8}+20 a^{7} b^{3} c^{3} d^{7}+35 a^{6} b^{4} c^{4} d^{6}+56 a^{5} b^{5} c^{5} d^{5}+84 a^{4} b^{6} c^{6} d^{4}+120 a^{3} b^{7} c^{7} d^{3}+165 a^{2} b^{8} c^{8} d^{2}+220 a \,b^{9} c^{9} d +286 b^{10} c^{10}}{4004 b^{11}}}{\left (b x +a \right )^{14}}\)	\(831\)
default	\(\frac {28 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 )}{b^{11} \left (b x +a \right )^{9}}+\frac {2 d^{9} \left (a d -b c \right )}{b^{11} \left (b x +a \right )^{5}}-\frac {d^{10}}{4 b^{11} \left (b x +a \right )^{4}}+\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 )}{13 b^{11} \left (b x +a \right )^{13}}+\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 )}{7 b^{11} \left (b x +a \right )^{7}}-\frac {15 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 )}{4 b^{11} \left (b x +a \right )^{12}}-\frac {105 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 )}{4 b^{11} \left (b x +a \right )^{8}}-\frac {21 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 )^{10}}-\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}}{14 b^{11} \left (b x +a \right )^{14}}-\frac {15 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 )^{6}}+\frac {120 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 )}{11 b^{11} \left (b x +a \right )^{11}}\)	\(867\)
norman	\(\frac {\frac {-a^{10} b^{3} d^{10}-4 a^{9} b^{4} c \,d^{9}-10 a^{8} b^{5} c^{2} d^{8}-20 a^{7} b^{6} c^{3} d^{7}-35 a^{6} b^{7} c^{4} d^{6}-56 a^{5} b^{8} c^{5} d^{5}-84 a^{4} b^{9} c^{6} d^{4}-120 a^{3} b^{10} c^{7} d^{3}-165 a^{2} c^{8} d^{2} b^{11}-220 a \,b^{12} c^{9} d -286 b^{13} c^{10}}{4004 b^{14}}+\frac {\left (-a^{9} b^{3} d^{10}-4 a^{8} b^{4} c \,d^{9}-10 a^{7} b^{5} c^{2} d^{8}-20 a^{6} b^{6} c^{3} d^{7}-35 a^{5} b^{7} c^{4} d^{6}-56 a^{4} b^{8} c^{5} d^{5}-84 a^{3} b^{9} c^{6} d^{4}-120 a^{2} b^{10} c^{7} d^{3}-165 a \,b^{11} c^{8} d^{2}-220 b^{12} c^{9} d \right ) x}{286 b^{13}}+\frac {\left (-b^{3} a^{8} d^{10}-4 a^{7} b^{4} c \,d^{9}-10 a^{6} b^{5} c^{2} d^{8}-20 a^{5} b^{6} c^{3} d^{7}-35 a^{4} b^{7} c^{4} d^{6}-56 a^{3} b^{8} c^{5} d^{5}-84 a^{2} b^{9} c^{6} d^{4}-120 a \,b^{10} c^{7} d^{3}-165 b^{11} c^{8} d^{2}\right ) x^{2}}{44 b^{12}}+\frac {\left (-a^{7} b^{3} d^{10}-4 a^{6} b^{4} c \,d^{9}-10 a^{5} b^{5} c^{2} d^{8}-20 a^{4} b^{6} c^{3} d^{7}-35 a^{3} b^{7} c^{4} d^{6}-56 a^{2} b^{8} c^{5} d^{5}-84 a \,b^{9} c^{6} d^{4}-120 b^{10} c^{7} d^{3}\right ) x^{3}}{11 b^{11}}+\frac {\left (-a^{6} b^{3} d^{10}-4 a^{5} b^{4} c \,d^{9}-10 a^{4} b^{5} c^{2} d^{8}-20 a^{3} b^{6} c^{3} d^{7}-35 a^{2} b^{7} c^{4} d^{6}-56 a \,b^{8} c^{5} d^{5}-84 b^{9} c^{6} d^{4}\right ) x^{4}}{4 b^{10}}+\frac {\left (-a^{5} b^{3} d^{10}-4 a^{4} b^{4} c \,d^{9}-10 a^{3} b^{5} c^{2} d^{8}-20 a^{2} b^{6} c^{3} d^{7}-35 a \,b^{7} c^{4} d^{6}-56 b^{8} c^{5} d^{5}\right ) x^{5}}{2 b^{9}}+\frac {3 \left (-a^{4} b^{3} d^{10}-4 a^{3} b^{4} c \,d^{9}-10 a^{2} b^{5} c^{2} d^{8}-20 a \,b^{6} c^{3} d^{7}-35 b^{7} c^{4} d^{6}\right ) x^{6}}{4 b^{8}}+\frac {6 \left (-a^{3} b^{3} d^{10}-4 a^{2} b^{4} c \,d^{9}-10 a \,b^{5} c^{2} d^{8}-20 b^{6} c^{3} d^{7}\right ) x^{7}}{7 b^{7}}+\frac {3 \left (-a^{2} b^{3} d^{10}-4 a \,b^{4} c \,d^{9}-10 b^{5} c^{2} d^{8}\right ) x^{8}}{4 b^{6}}+\frac {\left (-a \,b^{3} d^{10}-4 b^{4} c \,d^{9}\right ) x^{9}}{2 b^{5}}-\frac {d^{10} x^{10}}{4 b}}{\left (b x +a \right )^{14}}\)	\(909\)
gosper	\(-\frac {1001 x^{10} d^{10} b^{10}+2002 x^{9} a \,b^{9} d^{10}+8008 x^{9} b^{10} c \,d^{9}+3003 x^{8} a^{2} b^{8} d^{10}+12012 x^{8} a \,b^{9} c \,d^{9}+30030 x^{8} b^{10} c^{2} d^{8}+3432 x^{7} a^{3} b^{7} d^{10}+13728 x^{7} a^{2} b^{8} c \,d^{9}+34320 x^{7} a \,b^{9} c^{2} d^{8}+68640 x^{7} b^{10} c^{3} d^{7}+3003 x^{6} a^{4} b^{6} d^{10}+12012 x^{6} a^{3} b^{7} c \,d^{9}+30030 x^{6} a^{2} b^{8} c^{2} d^{8}+60060 x^{6} a \,b^{9} c^{3} d^{7}+105105 x^{6} b^{10} c^{4} d^{6}+2002 x^{5} a^{5} b^{5} d^{10}+8008 x^{5} a^{4} b^{6} c \,d^{9}+20020 x^{5} a^{3} b^{7} c^{2} d^{8}+40040 x^{5} a^{2} b^{8} c^{3} d^{7}+70070 x^{5} a \,b^{9} c^{4} d^{6}+112112 x^{5} b^{10} c^{5} d^{5}+1001 x^{4} a^{6} b^{4} d^{10}+4004 x^{4} a^{5} b^{5} c \,d^{9}+10010 x^{4} a^{4} b^{6} c^{2} d^{8}+20020 x^{4} a^{3} b^{7} c^{3} d^{7}+35035 x^{4} a^{2} b^{8} c^{4} d^{6}+56056 x^{4} a \,b^{9} c^{5} d^{5}+84084 x^{4} b^{10} c^{6} d^{4}+364 x^{3} a^{7} b^{3} d^{10}+1456 x^{3} a^{6} b^{4} c \,d^{9}+3640 x^{3} a^{5} b^{5} c^{2} d^{8}+7280 x^{3} a^{4} b^{6} c^{3} d^{7}+12740 x^{3} a^{3} b^{7} c^{4} d^{6}+20384 x^{3} a^{2} b^{8} c^{5} d^{5}+30576 x^{3} a \,b^{9} c^{6} d^{4}+43680 x^{3} b^{10} c^{7} d^{3}+91 x^{2} a^{8} b^{2} d^{10}+364 x^{2} a^{7} b^{3} c \,d^{9}+910 x^{2} a^{6} b^{4} c^{2} d^{8}+1820 x^{2} a^{5} b^{5} c^{3} d^{7}+3185 x^{2} a^{4} b^{6} c^{4} d^{6}+5096 x^{2} a^{3} b^{7} c^{5} d^{5}+7644 x^{2} a^{2} b^{8} c^{6} d^{4}+10920 x^{2} a \,b^{9} c^{7} d^{3}+15015 x^{2} b^{10} c^{8} d^{2}+14 x \,a^{9} b \,d^{10}+56 x \,a^{8} b^{2} c \,d^{9}+140 x \,a^{7} b^{3} c^{2} d^{8}+280 x \,a^{6} b^{4} c^{3} d^{7}+490 x \,a^{5} b^{5} c^{4} d^{6}+784 x \,a^{4} b^{6} c^{5} d^{5}+1176 x \,a^{3} b^{7} c^{6} d^{4}+1680 x \,a^{2} b^{8} c^{7} d^{3}+2310 x a \,b^{9} c^{8} d^{2}+3080 x \,b^{10} c^{9} d +a^{10} d^{10}+4 a^{9} b c \,d^{9}+10 a^{8} b^{2} c^{2} d^{8}+20 a^{7} b^{3} c^{3} d^{7}+35 a^{6} b^{4} c^{4} d^{6}+56 a^{5} b^{5} c^{5} d^{5}+84 a^{4} b^{6} c^{6} d^{4}+120 a^{3} b^{7} c^{7} d^{3}+165 a^{2} b^{8} c^{8} d^{2}+220 a \,b^{9} c^{9} d +286 b^{10} c^{10}}{4004 b^{11} \left (b x +a \right )^{14}}\)	\(962\)
orering	\(-\frac {1001 x^{10} d^{10} b^{10}+2002 x^{9} a \,b^{9} d^{10}+8008 x^{9} b^{10} c \,d^{9}+3003 x^{8} a^{2} b^{8} d^{10}+12012 x^{8} a \,b^{9} c \,d^{9}+30030 x^{8} b^{10} c^{2} d^{8}+3432 x^{7} a^{3} b^{7} d^{10}+13728 x^{7} a^{2} b^{8} c \,d^{9}+34320 x^{7} a \,b^{9} c^{2} d^{8}+68640 x^{7} b^{10} c^{3} d^{7}+3003 x^{6} a^{4} b^{6} d^{10}+12012 x^{6} a^{3} b^{7} c \,d^{9}+30030 x^{6} a^{2} b^{8} c^{2} d^{8}+60060 x^{6} a \,b^{9} c^{3} d^{7}+105105 x^{6} b^{10} c^{4} d^{6}+2002 x^{5} a^{5} b^{5} d^{10}+8008 x^{5} a^{4} b^{6} c \,d^{9}+20020 x^{5} a^{3} b^{7} c^{2} d^{8}+40040 x^{5} a^{2} b^{8} c^{3} d^{7}+70070 x^{5} a \,b^{9} c^{4} d^{6}+112112 x^{5} b^{10} c^{5} d^{5}+1001 x^{4} a^{6} b^{4} d^{10}+4004 x^{4} a^{5} b^{5} c \,d^{9}+10010 x^{4} a^{4} b^{6} c^{2} d^{8}+20020 x^{4} a^{3} b^{7} c^{3} d^{7}+35035 x^{4} a^{2} b^{8} c^{4} d^{6}+56056 x^{4} a \,b^{9} c^{5} d^{5}+84084 x^{4} b^{10} c^{6} d^{4}+364 x^{3} a^{7} b^{3} d^{10}+1456 x^{3} a^{6} b^{4} c \,d^{9}+3640 x^{3} a^{5} b^{5} c^{2} d^{8}+7280 x^{3} a^{4} b^{6} c^{3} d^{7}+12740 x^{3} a^{3} b^{7} c^{4} d^{6}+20384 x^{3} a^{2} b^{8} c^{5} d^{5}+30576 x^{3} a \,b^{9} c^{6} d^{4}+43680 x^{3} b^{10} c^{7} d^{3}+91 x^{2} a^{8} b^{2} d^{10}+364 x^{2} a^{7} b^{3} c \,d^{9}+910 x^{2} a^{6} b^{4} c^{2} d^{8}+1820 x^{2} a^{5} b^{5} c^{3} d^{7}+3185 x^{2} a^{4} b^{6} c^{4} d^{6}+5096 x^{2} a^{3} b^{7} c^{5} d^{5}+7644 x^{2} a^{2} b^{8} c^{6} d^{4}+10920 x^{2} a \,b^{9} c^{7} d^{3}+15015 x^{2} b^{10} c^{8} d^{2}+14 x \,a^{9} b \,d^{10}+56 x \,a^{8} b^{2} c \,d^{9}+140 x \,a^{7} b^{3} c^{2} d^{8}+280 x \,a^{6} b^{4} c^{3} d^{7}+490 x \,a^{5} b^{5} c^{4} d^{6}+784 x \,a^{4} b^{6} c^{5} d^{5}+1176 x \,a^{3} b^{7} c^{6} d^{4}+1680 x \,a^{2} b^{8} c^{7} d^{3}+2310 x a \,b^{9} c^{8} d^{2}+3080 x \,b^{10} c^{9} d +a^{10} d^{10}+4 a^{9} b c \,d^{9}+10 a^{8} b^{2} c^{2} d^{8}+20 a^{7} b^{3} c^{3} d^{7}+35 a^{6} b^{4} c^{4} d^{6}+56 a^{5} b^{5} c^{5} d^{5}+84 a^{4} b^{6} c^{6} d^{4}+120 a^{3} b^{7} c^{7} d^{3}+165 a^{2} b^{8} c^{8} d^{2}+220 a \,b^{9} c^{9} d +286 b^{10} c^{10}}{4004 b^{11} \left (b x +a \right )^{14}}\)	\(962\)
parallelrisch	\(\frac {-1001 d^{10} x^{10} b^{13}-2002 a \,b^{12} d^{10} x^{9}-8008 b^{13} c \,d^{9} x^{9}-3003 a^{2} b^{11} d^{10} x^{8}-12012 a \,b^{12} c \,d^{9} x^{8}-30030 b^{13} c^{2} d^{8} x^{8}-3432 a^{3} b^{10} d^{10} x^{7}-13728 a^{2} b^{11} c \,d^{9} x^{7}-34320 a \,b^{12} c^{2} d^{8} x^{7}-68640 b^{13} c^{3} d^{7} x^{7}-3003 a^{4} b^{9} d^{10} x^{6}-12012 a^{3} b^{10} c \,d^{9} x^{6}-30030 a^{2} b^{11} c^{2} d^{8} x^{6}-60060 a \,b^{12} c^{3} d^{7} x^{6}-105105 b^{13} c^{4} d^{6} x^{6}-2002 a^{5} b^{8} d^{10} x^{5}-8008 a^{4} b^{9} c \,d^{9} x^{5}-20020 a^{3} b^{10} c^{2} d^{8} x^{5}-40040 a^{2} b^{11} c^{3} d^{7} x^{5}-70070 a \,b^{12} c^{4} d^{6} x^{5}-112112 b^{13} c^{5} d^{5} x^{5}-1001 a^{6} b^{7} d^{10} x^{4}-4004 a^{5} b^{8} c \,d^{9} x^{4}-10010 a^{4} b^{9} c^{2} d^{8} x^{4}-20020 a^{3} b^{10} c^{3} d^{7} x^{4}-35035 a^{2} b^{11} c^{4} d^{6} x^{4}-56056 a \,b^{12} c^{5} d^{5} x^{4}-84084 b^{13} c^{6} d^{4} x^{4}-364 a^{7} b^{6} d^{10} x^{3}-1456 a^{6} b^{7} c \,d^{9} x^{3}-3640 a^{5} b^{8} c^{2} d^{8} x^{3}-7280 a^{4} b^{9} c^{3} d^{7} x^{3}-12740 a^{3} b^{10} c^{4} d^{6} x^{3}-20384 a^{2} b^{11} c^{5} d^{5} x^{3}-30576 a \,b^{12} c^{6} d^{4} x^{3}-43680 b^{13} c^{7} d^{3} x^{3}-91 a^{8} b^{5} d^{10} x^{2}-364 a^{7} b^{6} c \,d^{9} x^{2}-910 a^{6} b^{7} c^{2} d^{8} x^{2}-1820 a^{5} b^{8} c^{3} d^{7} x^{2}-3185 a^{4} b^{9} c^{4} d^{6} x^{2}-5096 a^{3} b^{10} c^{5} d^{5} x^{2}-7644 a^{2} b^{11} c^{6} d^{4} x^{2}-10920 a \,b^{12} c^{7} d^{3} x^{2}-15015 b^{13} c^{8} d^{2} x^{2}-14 a^{9} b^{4} d^{10} x -56 a^{8} b^{5} c \,d^{9} x -140 a^{7} b^{6} c^{2} d^{8} x -280 a^{6} b^{7} c^{3} d^{7} x -490 a^{5} b^{8} c^{4} d^{6} x -784 a^{4} b^{9} c^{5} d^{5} x -1176 a^{3} b^{10} c^{6} d^{4} x -1680 a^{2} b^{11} c^{7} d^{3} x -2310 a \,b^{12} c^{8} d^{2} x -3080 b^{13} c^{9} d x -a^{10} b^{3} d^{10}-4 a^{9} b^{4} c \,d^{9}-10 a^{8} b^{5} c^{2} d^{8}-20 a^{7} b^{6} c^{3} d^{7}-35 a^{6} b^{7} c^{4} d^{6}-56 a^{5} b^{8} c^{5} d^{5}-84 a^{4} b^{9} c^{6} d^{4}-120 a^{3} b^{10} c^{7} d^{3}-165 a^{2} c^{8} d^{2} b^{11}-220 a \,b^{12} c^{9} d -286 b^{13} c^{10}}{4004 b^{14} \left (b x +a \right )^{14}}\)	\(970\)

Input:

int((d*x+c)^10/(b*x+a)^15,x,method=_RETURNVERBOSE)
 

Output:

(-1/4/b*d^10*x^10-1/2/b^2*d^9*(a*d+4*b*c)*x^9-3/4/b^3*d^8*(a^2*d^2+4*a*b*c 
*d+10*b^2*c^2)*x^8-6/7/b^4*d^7*(a^3*d^3+4*a^2*b*c*d^2+10*a*b^2*c^2*d+20*b^ 
3*c^3)*x^7-3/4/b^5*d^6*(a^4*d^4+4*a^3*b*c*d^3+10*a^2*b^2*c^2*d^2+20*a*b^3* 
c^3*d+35*b^4*c^4)*x^6-1/2/b^6*d^5*(a^5*d^5+4*a^4*b*c*d^4+10*a^3*b^2*c^2*d^ 
3+20*a^2*b^3*c^3*d^2+35*a*b^4*c^4*d+56*b^5*c^5)*x^5-1/4/b^7*d^4*(a^6*d^6+4 
*a^5*b*c*d^5+10*a^4*b^2*c^2*d^4+20*a^3*b^3*c^3*d^3+35*a^2*b^4*c^4*d^2+56*a 
*b^5*c^5*d+84*b^6*c^6)*x^4-1/11/b^8*d^3*(a^7*d^7+4*a^6*b*c*d^6+10*a^5*b^2* 
c^2*d^5+20*a^4*b^3*c^3*d^4+35*a^3*b^4*c^4*d^3+56*a^2*b^5*c^5*d^2+84*a*b^6* 
c^6*d+120*b^7*c^7)*x^3-1/44/b^9*d^2*(a^8*d^8+4*a^7*b*c*d^7+10*a^6*b^2*c^2* 
d^6+20*a^5*b^3*c^3*d^5+35*a^4*b^4*c^4*d^4+56*a^3*b^5*c^5*d^3+84*a^2*b^6*c^ 
6*d^2+120*a*b^7*c^7*d+165*b^8*c^8)*x^2-1/286/b^10*d*(a^9*d^9+4*a^8*b*c*d^8 
+10*a^7*b^2*c^2*d^7+20*a^6*b^3*c^3*d^6+35*a^5*b^4*c^4*d^5+56*a^4*b^5*c^5*d 
^4+84*a^3*b^6*c^6*d^3+120*a^2*b^7*c^7*d^2+165*a*b^8*c^8*d+220*b^9*c^9)*x-1 
/4004/b^11*(a^10*d^10+4*a^9*b*c*d^9+10*a^8*b^2*c^2*d^8+20*a^7*b^3*c^3*d^7+ 
35*a^6*b^4*c^4*d^6+56*a^5*b^5*c^5*d^5+84*a^4*b^6*c^6*d^4+120*a^3*b^7*c^7*d 
^3+165*a^2*b^8*c^8*d^2+220*a*b^9*c^9*d+286*b^10*c^10))/(b*x+a)^14
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1008 vs. \(2 (112) = 224\).

Time = 0.08 (sec) , antiderivative size = 1008, normalized size of antiderivative = 8.40 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^15,x, algorithm="fricas")
 

Output:

-1/4004*(1001*b^10*d^10*x^10 + 286*b^10*c^10 + 220*a*b^9*c^9*d + 165*a^2*b 
^8*c^8*d^2 + 120*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 56*a^5*b^5*c^5*d^5 
 + 35*a^6*b^4*c^4*d^6 + 20*a^7*b^3*c^3*d^7 + 10*a^8*b^2*c^2*d^8 + 4*a^9*b* 
c*d^9 + a^10*d^10 + 2002*(4*b^10*c*d^9 + a*b^9*d^10)*x^9 + 3003*(10*b^10*c 
^2*d^8 + 4*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 3432*(20*b^10*c^3*d^7 + 10*a* 
b^9*c^2*d^8 + 4*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 3003*(35*b^10*c^4*d^6 
+ 20*a*b^9*c^3*d^7 + 10*a^2*b^8*c^2*d^8 + 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)* 
x^6 + 2002*(56*b^10*c^5*d^5 + 35*a*b^9*c^4*d^6 + 20*a^2*b^8*c^3*d^7 + 10*a 
^3*b^7*c^2*d^8 + 4*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 1001*(84*b^10*c^6*d 
^4 + 56*a*b^9*c^5*d^5 + 35*a^2*b^8*c^4*d^6 + 20*a^3*b^7*c^3*d^7 + 10*a^4*b 
^6*c^2*d^8 + 4*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 364*(120*b^10*c^7*d^3 + 
 84*a*b^9*c^6*d^4 + 56*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6 + 20*a^4*b^6*c 
^3*d^7 + 10*a^5*b^5*c^2*d^8 + 4*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 91*(16 
5*b^10*c^8*d^2 + 120*a*b^9*c^7*d^3 + 84*a^2*b^8*c^6*d^4 + 56*a^3*b^7*c^5*d 
^5 + 35*a^4*b^6*c^4*d^6 + 20*a^5*b^5*c^3*d^7 + 10*a^6*b^4*c^2*d^8 + 4*a^7* 
b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 14*(220*b^10*c^9*d + 165*a*b^9*c^8*d^2 + 1 
20*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 56*a^4*b^6*c^5*d^5 + 35*a^5*b^5* 
c^4*d^6 + 20*a^6*b^4*c^3*d^7 + 10*a^7*b^3*c^2*d^8 + 4*a^8*b^2*c*d^9 + a^9* 
b*d^10)*x)/(b^25*x^14 + 14*a*b^24*x^13 + 91*a^2*b^23*x^12 + 364*a^3*b^22*x 
^11 + 1001*a^4*b^21*x^10 + 2002*a^5*b^20*x^9 + 3003*a^6*b^19*x^8 + 3432...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**10/(b*x+a)**15,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1008 vs. \(2 (112) = 224\).

Time = 0.09 (sec) , antiderivative size = 1008, normalized size of antiderivative = 8.40 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^15,x, algorithm="maxima")
 

Output:

-1/4004*(1001*b^10*d^10*x^10 + 286*b^10*c^10 + 220*a*b^9*c^9*d + 165*a^2*b 
^8*c^8*d^2 + 120*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 56*a^5*b^5*c^5*d^5 
 + 35*a^6*b^4*c^4*d^6 + 20*a^7*b^3*c^3*d^7 + 10*a^8*b^2*c^2*d^8 + 4*a^9*b* 
c*d^9 + a^10*d^10 + 2002*(4*b^10*c*d^9 + a*b^9*d^10)*x^9 + 3003*(10*b^10*c 
^2*d^8 + 4*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 3432*(20*b^10*c^3*d^7 + 10*a* 
b^9*c^2*d^8 + 4*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 3003*(35*b^10*c^4*d^6 
+ 20*a*b^9*c^3*d^7 + 10*a^2*b^8*c^2*d^8 + 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)* 
x^6 + 2002*(56*b^10*c^5*d^5 + 35*a*b^9*c^4*d^6 + 20*a^2*b^8*c^3*d^7 + 10*a 
^3*b^7*c^2*d^8 + 4*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 1001*(84*b^10*c^6*d 
^4 + 56*a*b^9*c^5*d^5 + 35*a^2*b^8*c^4*d^6 + 20*a^3*b^7*c^3*d^7 + 10*a^4*b 
^6*c^2*d^8 + 4*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 364*(120*b^10*c^7*d^3 + 
 84*a*b^9*c^6*d^4 + 56*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6 + 20*a^4*b^6*c 
^3*d^7 + 10*a^5*b^5*c^2*d^8 + 4*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 91*(16 
5*b^10*c^8*d^2 + 120*a*b^9*c^7*d^3 + 84*a^2*b^8*c^6*d^4 + 56*a^3*b^7*c^5*d 
^5 + 35*a^4*b^6*c^4*d^6 + 20*a^5*b^5*c^3*d^7 + 10*a^6*b^4*c^2*d^8 + 4*a^7* 
b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 14*(220*b^10*c^9*d + 165*a*b^9*c^8*d^2 + 1 
20*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 56*a^4*b^6*c^5*d^5 + 35*a^5*b^5* 
c^4*d^6 + 20*a^6*b^4*c^3*d^7 + 10*a^7*b^3*c^2*d^8 + 4*a^8*b^2*c*d^9 + a^9* 
b*d^10)*x)/(b^25*x^14 + 14*a*b^24*x^13 + 91*a^2*b^23*x^12 + 364*a^3*b^22*x 
^11 + 1001*a^4*b^21*x^10 + 2002*a^5*b^20*x^9 + 3003*a^6*b^19*x^8 + 3432...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (112) = 224\).

Time = 0.12 (sec) , antiderivative size = 961, normalized size of antiderivative = 8.01 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^15,x, algorithm="giac")
 

Output:

-1/4004*(1001*b^10*d^10*x^10 + 8008*b^10*c*d^9*x^9 + 2002*a*b^9*d^10*x^9 + 
 30030*b^10*c^2*d^8*x^8 + 12012*a*b^9*c*d^9*x^8 + 3003*a^2*b^8*d^10*x^8 + 
68640*b^10*c^3*d^7*x^7 + 34320*a*b^9*c^2*d^8*x^7 + 13728*a^2*b^8*c*d^9*x^7 
 + 3432*a^3*b^7*d^10*x^7 + 105105*b^10*c^4*d^6*x^6 + 60060*a*b^9*c^3*d^7*x 
^6 + 30030*a^2*b^8*c^2*d^8*x^6 + 12012*a^3*b^7*c*d^9*x^6 + 3003*a^4*b^6*d^ 
10*x^6 + 112112*b^10*c^5*d^5*x^5 + 70070*a*b^9*c^4*d^6*x^5 + 40040*a^2*b^8 
*c^3*d^7*x^5 + 20020*a^3*b^7*c^2*d^8*x^5 + 8008*a^4*b^6*c*d^9*x^5 + 2002*a 
^5*b^5*d^10*x^5 + 84084*b^10*c^6*d^4*x^4 + 56056*a*b^9*c^5*d^5*x^4 + 35035 
*a^2*b^8*c^4*d^6*x^4 + 20020*a^3*b^7*c^3*d^7*x^4 + 10010*a^4*b^6*c^2*d^8*x 
^4 + 4004*a^5*b^5*c*d^9*x^4 + 1001*a^6*b^4*d^10*x^4 + 43680*b^10*c^7*d^3*x 
^3 + 30576*a*b^9*c^6*d^4*x^3 + 20384*a^2*b^8*c^5*d^5*x^3 + 12740*a^3*b^7*c 
^4*d^6*x^3 + 7280*a^4*b^6*c^3*d^7*x^3 + 3640*a^5*b^5*c^2*d^8*x^3 + 1456*a^ 
6*b^4*c*d^9*x^3 + 364*a^7*b^3*d^10*x^3 + 15015*b^10*c^8*d^2*x^2 + 10920*a* 
b^9*c^7*d^3*x^2 + 7644*a^2*b^8*c^6*d^4*x^2 + 5096*a^3*b^7*c^5*d^5*x^2 + 31 
85*a^4*b^6*c^4*d^6*x^2 + 1820*a^5*b^5*c^3*d^7*x^2 + 910*a^6*b^4*c^2*d^8*x^ 
2 + 364*a^7*b^3*c*d^9*x^2 + 91*a^8*b^2*d^10*x^2 + 3080*b^10*c^9*d*x + 2310 
*a*b^9*c^8*d^2*x + 1680*a^2*b^8*c^7*d^3*x + 1176*a^3*b^7*c^6*d^4*x + 784*a 
^4*b^6*c^5*d^5*x + 490*a^5*b^5*c^4*d^6*x + 280*a^6*b^4*c^3*d^7*x + 140*a^7 
*b^3*c^2*d^8*x + 56*a^8*b^2*c*d^9*x + 14*a^9*b*d^10*x + 286*b^10*c^10 + 22 
0*a*b^9*c^9*d + 165*a^2*b^8*c^8*d^2 + 120*a^3*b^7*c^7*d^3 + 84*a^4*b^6*...
 

Mupad [B] (verification not implemented)

Time = 0.81 (sec) , antiderivative size = 1109, normalized size of antiderivative = 9.24 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx =\text {Too large to display} \] Input:

int((c + d*x)^10/(a + b*x)^15,x)
 

Output:

-(a^10*d^10 + 286*b^10*c^10 + 1001*b^10*d^10*x^10 + 2002*a*b^9*d^10*x^9 + 
8008*b^10*c*d^9*x^9 + 165*a^2*b^8*c^8*d^2 + 120*a^3*b^7*c^7*d^3 + 84*a^4*b 
^6*c^6*d^4 + 56*a^5*b^5*c^5*d^5 + 35*a^6*b^4*c^4*d^6 + 20*a^7*b^3*c^3*d^7 
+ 10*a^8*b^2*c^2*d^8 + 91*a^8*b^2*d^10*x^2 + 364*a^7*b^3*d^10*x^3 + 1001*a 
^6*b^4*d^10*x^4 + 2002*a^5*b^5*d^10*x^5 + 3003*a^4*b^6*d^10*x^6 + 3432*a^3 
*b^7*d^10*x^7 + 3003*a^2*b^8*d^10*x^8 + 15015*b^10*c^8*d^2*x^2 + 43680*b^1 
0*c^7*d^3*x^3 + 84084*b^10*c^6*d^4*x^4 + 112112*b^10*c^5*d^5*x^5 + 105105* 
b^10*c^4*d^6*x^6 + 68640*b^10*c^3*d^7*x^7 + 30030*b^10*c^2*d^8*x^8 + 220*a 
*b^9*c^9*d + 4*a^9*b*c*d^9 + 14*a^9*b*d^10*x + 3080*b^10*c^9*d*x + 7644*a^ 
2*b^8*c^6*d^4*x^2 + 5096*a^3*b^7*c^5*d^5*x^2 + 3185*a^4*b^6*c^4*d^6*x^2 + 
1820*a^5*b^5*c^3*d^7*x^2 + 910*a^6*b^4*c^2*d^8*x^2 + 20384*a^2*b^8*c^5*d^5 
*x^3 + 12740*a^3*b^7*c^4*d^6*x^3 + 7280*a^4*b^6*c^3*d^7*x^3 + 3640*a^5*b^5 
*c^2*d^8*x^3 + 35035*a^2*b^8*c^4*d^6*x^4 + 20020*a^3*b^7*c^3*d^7*x^4 + 100 
10*a^4*b^6*c^2*d^8*x^4 + 40040*a^2*b^8*c^3*d^7*x^5 + 20020*a^3*b^7*c^2*d^8 
*x^5 + 30030*a^2*b^8*c^2*d^8*x^6 + 2310*a*b^9*c^8*d^2*x + 56*a^8*b^2*c*d^9 
*x + 12012*a*b^9*c*d^9*x^8 + 1680*a^2*b^8*c^7*d^3*x + 1176*a^3*b^7*c^6*d^4 
*x + 784*a^4*b^6*c^5*d^5*x + 490*a^5*b^5*c^4*d^6*x + 280*a^6*b^4*c^3*d^7*x 
 + 140*a^7*b^3*c^2*d^8*x + 10920*a*b^9*c^7*d^3*x^2 + 364*a^7*b^3*c*d^9*x^2 
 + 30576*a*b^9*c^6*d^4*x^3 + 1456*a^6*b^4*c*d^9*x^3 + 56056*a*b^9*c^5*d^5* 
x^4 + 4004*a^5*b^5*c*d^9*x^4 + 70070*a*b^9*c^4*d^6*x^5 + 8008*a^4*b^6*c...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1105, normalized size of antiderivative = 9.21 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^10/(b*x+a)^15,x)
 

Output:

( - a**10*d**10 - 4*a**9*b*c*d**9 - 14*a**9*b*d**10*x - 10*a**8*b**2*c**2* 
d**8 - 56*a**8*b**2*c*d**9*x - 91*a**8*b**2*d**10*x**2 - 20*a**7*b**3*c**3 
*d**7 - 140*a**7*b**3*c**2*d**8*x - 364*a**7*b**3*c*d**9*x**2 - 364*a**7*b 
**3*d**10*x**3 - 35*a**6*b**4*c**4*d**6 - 280*a**6*b**4*c**3*d**7*x - 910* 
a**6*b**4*c**2*d**8*x**2 - 1456*a**6*b**4*c*d**9*x**3 - 1001*a**6*b**4*d** 
10*x**4 - 56*a**5*b**5*c**5*d**5 - 490*a**5*b**5*c**4*d**6*x - 1820*a**5*b 
**5*c**3*d**7*x**2 - 3640*a**5*b**5*c**2*d**8*x**3 - 4004*a**5*b**5*c*d**9 
*x**4 - 2002*a**5*b**5*d**10*x**5 - 84*a**4*b**6*c**6*d**4 - 784*a**4*b**6 
*c**5*d**5*x - 3185*a**4*b**6*c**4*d**6*x**2 - 7280*a**4*b**6*c**3*d**7*x* 
*3 - 10010*a**4*b**6*c**2*d**8*x**4 - 8008*a**4*b**6*c*d**9*x**5 - 3003*a* 
*4*b**6*d**10*x**6 - 120*a**3*b**7*c**7*d**3 - 1176*a**3*b**7*c**6*d**4*x 
- 5096*a**3*b**7*c**5*d**5*x**2 - 12740*a**3*b**7*c**4*d**6*x**3 - 20020*a 
**3*b**7*c**3*d**7*x**4 - 20020*a**3*b**7*c**2*d**8*x**5 - 12012*a**3*b**7 
*c*d**9*x**6 - 3432*a**3*b**7*d**10*x**7 - 165*a**2*b**8*c**8*d**2 - 1680* 
a**2*b**8*c**7*d**3*x - 7644*a**2*b**8*c**6*d**4*x**2 - 20384*a**2*b**8*c* 
*5*d**5*x**3 - 35035*a**2*b**8*c**4*d**6*x**4 - 40040*a**2*b**8*c**3*d**7* 
x**5 - 30030*a**2*b**8*c**2*d**8*x**6 - 13728*a**2*b**8*c*d**9*x**7 - 3003 
*a**2*b**8*d**10*x**8 - 220*a*b**9*c**9*d - 2310*a*b**9*c**8*d**2*x - 1092 
0*a*b**9*c**7*d**3*x**2 - 30576*a*b**9*c**6*d**4*x**3 - 56056*a*b**9*c**5* 
d**5*x**4 - 70070*a*b**9*c**4*d**6*x**5 - 60060*a*b**9*c**3*d**7*x**6 -...