\(\int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx\) [119]

Optimal result

Integrand size = 15, antiderivative size = 89 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx=-\frac {(c+d x)^{11}}{13 (b c-a d) (a+b x)^{13}}+\frac {d (c+d x)^{11}}{78 (b c-a d)^2 (a+b x)^{12}}-\frac {d^2 (c+d x)^{11}}{858 (b c-a d)^3 (a+b x)^{11}} \] Output:

-1/13*(d*x+c)^11/(-a*d+b*c)/(b*x+a)^13+1/78*d*(d*x+c)^11/(-a*d+b*c)^2/(b*x 
+a)^12-1/858*d^2*(d*x+c)^11/(-a*d+b*c)^3/(b*x+a)^11
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(690\) vs. \(2(89)=178\).

Time = 0.16 (sec) , antiderivative size = 690, normalized size of antiderivative = 7.75 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx=-\frac {a^{10} d^{10}+a^9 b d^9 (3 c+13 d x)+3 a^8 b^2 d^8 \left (2 c^2+13 c d x+26 d^2 x^2\right )+2 a^7 b^3 d^7 \left (5 c^3+39 c^2 d x+117 c d^2 x^2+143 d^3 x^3\right )+a^6 b^4 d^6 \left (15 c^4+130 c^3 d x+468 c^2 d^2 x^2+858 c d^3 x^3+715 d^4 x^4\right )+3 a^5 b^5 d^5 \left (7 c^5+65 c^4 d x+260 c^3 d^2 x^2+572 c^2 d^3 x^3+715 c d^4 x^4+429 d^5 x^5\right )+a^4 b^6 d^4 \left (28 c^6+273 c^5 d x+1170 c^4 d^2 x^2+2860 c^3 d^3 x^3+4290 c^2 d^4 x^4+3861 c d^5 x^5+1716 d^6 x^6\right )+2 a^3 b^7 d^3 \left (18 c^7+182 c^6 d x+819 c^5 d^2 x^2+2145 c^4 d^3 x^3+3575 c^3 d^4 x^4+3861 c^2 d^5 x^5+2574 c d^6 x^6+858 d^7 x^7\right )+3 a^2 b^8 d^2 \left (15 c^8+156 c^7 d x+728 c^6 d^2 x^2+2002 c^5 d^3 x^3+3575 c^4 d^4 x^4+4290 c^3 d^5 x^5+3432 c^2 d^6 x^6+1716 c d^7 x^7+429 d^8 x^8\right )+a b^9 d \left (55 c^9+585 c^8 d x+2808 c^7 d^2 x^2+8008 c^6 d^3 x^3+15015 c^5 d^4 x^4+19305 c^4 d^5 x^5+17160 c^3 d^6 x^6+10296 c^2 d^7 x^7+3861 c d^8 x^8+715 d^9 x^9\right )+b^{10} \left (66 c^{10}+715 c^9 d x+3510 c^8 d^2 x^2+10296 c^7 d^3 x^3+20020 c^6 d^4 x^4+27027 c^5 d^5 x^5+25740 c^4 d^6 x^6+17160 c^3 d^7 x^7+7722 c^2 d^8 x^8+2145 c d^9 x^9+286 d^{10} x^{10}\right )}{858 b^{11} (a+b x)^{13}} \] Input:

Integrate[(c + d*x)^10/(a + b*x)^14,x]
 

Output:

-1/858*(a^10*d^10 + a^9*b*d^9*(3*c + 13*d*x) + 3*a^8*b^2*d^8*(2*c^2 + 13*c 
*d*x + 26*d^2*x^2) + 2*a^7*b^3*d^7*(5*c^3 + 39*c^2*d*x + 117*c*d^2*x^2 + 1 
43*d^3*x^3) + a^6*b^4*d^6*(15*c^4 + 130*c^3*d*x + 468*c^2*d^2*x^2 + 858*c* 
d^3*x^3 + 715*d^4*x^4) + 3*a^5*b^5*d^5*(7*c^5 + 65*c^4*d*x + 260*c^3*d^2*x 
^2 + 572*c^2*d^3*x^3 + 715*c*d^4*x^4 + 429*d^5*x^5) + a^4*b^6*d^4*(28*c^6 
+ 273*c^5*d*x + 1170*c^4*d^2*x^2 + 2860*c^3*d^3*x^3 + 4290*c^2*d^4*x^4 + 3 
861*c*d^5*x^5 + 1716*d^6*x^6) + 2*a^3*b^7*d^3*(18*c^7 + 182*c^6*d*x + 819* 
c^5*d^2*x^2 + 2145*c^4*d^3*x^3 + 3575*c^3*d^4*x^4 + 3861*c^2*d^5*x^5 + 257 
4*c*d^6*x^6 + 858*d^7*x^7) + 3*a^2*b^8*d^2*(15*c^8 + 156*c^7*d*x + 728*c^6 
*d^2*x^2 + 2002*c^5*d^3*x^3 + 3575*c^4*d^4*x^4 + 4290*c^3*d^5*x^5 + 3432*c 
^2*d^6*x^6 + 1716*c*d^7*x^7 + 429*d^8*x^8) + a*b^9*d*(55*c^9 + 585*c^8*d*x 
 + 2808*c^7*d^2*x^2 + 8008*c^6*d^3*x^3 + 15015*c^5*d^4*x^4 + 19305*c^4*d^5 
*x^5 + 17160*c^3*d^6*x^6 + 10296*c^2*d^7*x^7 + 3861*c*d^8*x^8 + 715*d^9*x^ 
9) + b^10*(66*c^10 + 715*c^9*d*x + 3510*c^8*d^2*x^2 + 10296*c^7*d^3*x^3 + 
20020*c^6*d^4*x^4 + 27027*c^5*d^5*x^5 + 25740*c^4*d^6*x^6 + 17160*c^3*d^7* 
x^7 + 7722*c^2*d^8*x^8 + 2145*c*d^9*x^9 + 286*d^10*x^10))/(b^11*(a + b*x)^ 
13)
 

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {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.

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 48

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

Input:

Int[(c + d*x)^10/(a + b*x)^14,x]
 

Output:

-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))
 

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. \(2(83)=166\).

Time = 0.18 (sec) , antiderivative size = 831, normalized size of antiderivative = 9.34

method	result	size
risch	\(\frac {-\frac {d^{10} x^{10}}{3 b}-\frac {5 d^{9} \left (a d +3 b c \right ) x^{9}}{6 b^{2}}-\frac {3 d^{8} \left (a^{2} d^{2}+3 a b c d +6 b^{2} c^{2}\right ) x^{8}}{2 b^{3}}-\frac {2 d^{7} \left (a^{3} d^{3}+3 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d +10 b^{3} c^{3}\right ) x^{7}}{b^{4}}-\frac {2 d^{6} \left (d^{4} a^{4}+3 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+10 a \,b^{3} c^{3} d +15 c^{4} b^{4}\right ) x^{6}}{b^{5}}-\frac {3 d^{5} \left (a^{5} d^{5}+3 a^{4} b c \,d^{4}+6 a^{3} b^{2} c^{2} d^{3}+10 a^{2} b^{3} c^{3} d^{2}+15 a \,b^{4} c^{4} d +21 c^{5} b^{5}\right ) x^{5}}{2 b^{6}}-\frac {5 d^{4} \left (a^{6} d^{6}+3 a^{5} b c \,d^{5}+6 a^{4} b^{2} c^{2} d^{4}+10 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}+21 a \,b^{5} c^{5} d +28 c^{6} b^{6}\right ) x^{4}}{6 b^{7}}-\frac {d^{3} \left (a^{7} d^{7}+3 a^{6} b c \,d^{6}+6 a^{5} b^{2} c^{2} d^{5}+10 a^{4} b^{3} c^{3} d^{4}+15 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+28 a \,b^{6} c^{6} d +36 b^{7} c^{7}\right ) x^{3}}{3 b^{8}}-\frac {d^{2} \left (a^{8} d^{8}+3 a^{7} b c \,d^{7}+6 a^{6} b^{2} c^{2} d^{6}+10 a^{5} b^{3} c^{3} d^{5}+15 a^{4} b^{4} c^{4} d^{4}+21 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}+36 a \,b^{7} c^{7} d +45 c^{8} b^{8}\right ) x^{2}}{11 b^{9}}-\frac {d \left (a^{9} d^{9}+3 a^{8} b c \,d^{8}+6 a^{7} b^{2} c^{2} d^{7}+10 a^{6} b^{3} c^{3} d^{6}+15 a^{5} b^{4} c^{4} d^{5}+21 a^{4} b^{5} c^{5} d^{4}+28 a^{3} b^{6} c^{6} d^{3}+36 a^{2} b^{7} c^{7} d^{2}+45 a \,b^{8} c^{8} d +55 c^{9} b^{9}\right ) x}{66 b^{10}}-\frac {a^{10} d^{10}+3 a^{9} b c \,d^{9}+6 a^{8} b^{2} c^{2} d^{8}+10 a^{7} b^{3} c^{3} d^{7}+15 a^{6} b^{4} c^{4} d^{6}+21 a^{5} b^{5} c^{5} d^{5}+28 a^{4} b^{6} c^{6} d^{4}+36 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}+55 a \,b^{9} c^{9} d +66 b^{10} c^{10}}{858 b^{11}}}{\left (b x +a \right )^{13}}\)	\(831\)
default	\(-\frac {70 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 )}{3 b^{11} \left (b x +a \right )^{9}}-\frac {9 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{11} \left (b x +a \right )^{5}}+\frac {5 d^{9} \left (a d -b c \right )}{2 b^{11} \left (b x +a \right )^{4}}-\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}}{13 b^{11} \left (b x +a \right )^{13}}-\frac {30 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 )}{b^{11} \left (b x +a \right )^{7}}+\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 )}{6 b^{11} \left (b x +a \right )^{12}}+\frac {63 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 )}{2 b^{11} \left (b x +a \right )^{8}}+\frac {12 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 )^{10}}-\frac {d^{10}}{3 b^{11} \left (b x +a \right )^{3}}+\frac {20 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 )^{6}}-\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 )}{11 b^{11} \left (b x +a \right )^{11}}\)	\(867\)
norman	\(\frac {-\frac {d^{10} x^{10}}{3 b}+\frac {5 \left (-a \,b^{2} d^{10}-3 b^{3} c \,d^{9}\right ) x^{9}}{6 b^{4}}+\frac {3 \left (-a^{2} b^{2} d^{10}-3 a \,b^{3} c \,d^{9}-6 b^{4} c^{2} d^{8}\right ) x^{8}}{2 b^{5}}+\frac {2 \left (-a^{3} b^{2} d^{10}-3 a^{2} b^{3} c \,d^{9}-6 a \,b^{4} c^{2} d^{8}-10 b^{5} c^{3} d^{7}\right ) x^{7}}{b^{6}}+\frac {2 \left (-a^{4} b^{2} d^{10}-3 a^{3} b^{3} c \,d^{9}-6 a^{2} b^{4} c^{2} d^{8}-10 a \,b^{5} c^{3} d^{7}-15 b^{6} c^{4} d^{6}\right ) x^{6}}{b^{7}}+\frac {3 \left (-a^{5} b^{2} d^{10}-3 a^{4} b^{3} c \,d^{9}-6 a^{3} b^{4} c^{2} d^{8}-10 a^{2} b^{5} c^{3} d^{7}-15 a \,b^{6} c^{4} d^{6}-21 b^{7} c^{5} d^{5}\right ) x^{5}}{2 b^{8}}+\frac {5 \left (-a^{6} b^{2} d^{10}-3 a^{5} b^{3} c \,d^{9}-6 a^{4} b^{4} c^{2} d^{8}-10 a^{3} b^{5} c^{3} d^{7}-15 a^{2} b^{6} c^{4} d^{6}-21 a \,b^{7} c^{5} d^{5}-28 b^{8} c^{6} d^{4}\right ) x^{4}}{6 b^{9}}+\frac {\left (-a^{7} b^{2} d^{10}-3 a^{6} b^{3} c \,d^{9}-6 a^{5} b^{4} c^{2} d^{8}-10 a^{4} b^{5} c^{3} d^{7}-15 a^{3} b^{6} c^{4} d^{6}-21 a^{2} b^{7} c^{5} d^{5}-28 a \,b^{8} c^{6} d^{4}-36 b^{9} c^{7} d^{3}\right ) x^{3}}{3 b^{10}}+\frac {\left (-a^{8} b^{2} d^{10}-3 a^{7} b^{3} c \,d^{9}-6 a^{6} b^{4} c^{2} d^{8}-10 a^{5} b^{5} c^{3} d^{7}-15 a^{4} b^{6} c^{4} d^{6}-21 a^{3} b^{7} c^{5} d^{5}-28 a^{2} b^{8} c^{6} d^{4}-36 a \,b^{9} c^{7} d^{3}-45 b^{10} c^{8} d^{2}\right ) x^{2}}{11 b^{11}}+\frac {\left (-b^{2} a^{9} d^{10}-3 b^{3} a^{8} c \,d^{9}-6 a^{7} b^{4} c^{2} d^{8}-10 a^{6} b^{5} c^{3} d^{7}-15 a^{5} b^{6} c^{4} d^{6}-21 a^{4} b^{7} c^{5} d^{5}-28 a^{3} b^{8} c^{6} d^{4}-36 a^{2} b^{9} c^{7} d^{3}-45 a \,b^{10} c^{8} d^{2}-55 b^{11} c^{9} d \right ) x}{66 b^{12}}+\frac {-a^{10} b^{2} d^{10}-3 a^{9} b^{3} c \,d^{9}-6 a^{8} b^{4} c^{2} d^{8}-10 a^{7} b^{5} c^{3} d^{7}-15 a^{6} b^{6} c^{4} d^{6}-21 a^{5} b^{7} c^{5} d^{5}-28 a^{4} b^{8} c^{6} d^{4}-36 a^{3} b^{9} c^{7} d^{3}-45 a^{2} b^{10} c^{8} d^{2}-55 a \,b^{11} c^{9} d -66 b^{12} c^{10}}{858 b^{13}}}{\left (b x +a \right )^{13}}\)	\(909\)
gosper	\(-\frac {286 x^{10} d^{10} b^{10}+715 x^{9} a \,b^{9} d^{10}+2145 x^{9} b^{10} c \,d^{9}+1287 x^{8} a^{2} b^{8} d^{10}+3861 x^{8} a \,b^{9} c \,d^{9}+7722 x^{8} b^{10} c^{2} d^{8}+1716 x^{7} a^{3} b^{7} d^{10}+5148 x^{7} a^{2} b^{8} c \,d^{9}+10296 x^{7} a \,b^{9} c^{2} d^{8}+17160 x^{7} b^{10} c^{3} d^{7}+1716 x^{6} a^{4} b^{6} d^{10}+5148 x^{6} a^{3} b^{7} c \,d^{9}+10296 x^{6} a^{2} b^{8} c^{2} d^{8}+17160 x^{6} a \,b^{9} c^{3} d^{7}+25740 x^{6} b^{10} c^{4} d^{6}+1287 x^{5} a^{5} b^{5} d^{10}+3861 x^{5} a^{4} b^{6} c \,d^{9}+7722 x^{5} a^{3} b^{7} c^{2} d^{8}+12870 x^{5} a^{2} b^{8} c^{3} d^{7}+19305 x^{5} a \,b^{9} c^{4} d^{6}+27027 x^{5} b^{10} c^{5} d^{5}+715 x^{4} a^{6} b^{4} d^{10}+2145 x^{4} a^{5} b^{5} c \,d^{9}+4290 x^{4} a^{4} b^{6} c^{2} d^{8}+7150 x^{4} a^{3} b^{7} c^{3} d^{7}+10725 x^{4} a^{2} b^{8} c^{4} d^{6}+15015 x^{4} a \,b^{9} c^{5} d^{5}+20020 x^{4} b^{10} c^{6} d^{4}+286 x^{3} a^{7} b^{3} d^{10}+858 x^{3} a^{6} b^{4} c \,d^{9}+1716 x^{3} a^{5} b^{5} c^{2} d^{8}+2860 x^{3} a^{4} b^{6} c^{3} d^{7}+4290 x^{3} a^{3} b^{7} c^{4} d^{6}+6006 x^{3} a^{2} b^{8} c^{5} d^{5}+8008 x^{3} a \,b^{9} c^{6} d^{4}+10296 x^{3} b^{10} c^{7} d^{3}+78 x^{2} a^{8} b^{2} d^{10}+234 x^{2} a^{7} b^{3} c \,d^{9}+468 x^{2} a^{6} b^{4} c^{2} d^{8}+780 x^{2} a^{5} b^{5} c^{3} d^{7}+1170 x^{2} a^{4} b^{6} c^{4} d^{6}+1638 x^{2} a^{3} b^{7} c^{5} d^{5}+2184 x^{2} a^{2} b^{8} c^{6} d^{4}+2808 x^{2} a \,b^{9} c^{7} d^{3}+3510 x^{2} b^{10} c^{8} d^{2}+13 x \,a^{9} b \,d^{10}+39 x \,a^{8} b^{2} c \,d^{9}+78 x \,a^{7} b^{3} c^{2} d^{8}+130 x \,a^{6} b^{4} c^{3} d^{7}+195 x \,a^{5} b^{5} c^{4} d^{6}+273 x \,a^{4} b^{6} c^{5} d^{5}+364 x \,a^{3} b^{7} c^{6} d^{4}+468 x \,a^{2} b^{8} c^{7} d^{3}+585 x a \,b^{9} c^{8} d^{2}+715 x \,b^{10} c^{9} d +a^{10} d^{10}+3 a^{9} b c \,d^{9}+6 a^{8} b^{2} c^{2} d^{8}+10 a^{7} b^{3} c^{3} d^{7}+15 a^{6} b^{4} c^{4} d^{6}+21 a^{5} b^{5} c^{5} d^{5}+28 a^{4} b^{6} c^{6} d^{4}+36 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}+55 a \,b^{9} c^{9} d +66 b^{10} c^{10}}{858 b^{11} \left (b x +a \right )^{13}}\)	\(962\)
orering	\(-\frac {286 x^{10} d^{10} b^{10}+715 x^{9} a \,b^{9} d^{10}+2145 x^{9} b^{10} c \,d^{9}+1287 x^{8} a^{2} b^{8} d^{10}+3861 x^{8} a \,b^{9} c \,d^{9}+7722 x^{8} b^{10} c^{2} d^{8}+1716 x^{7} a^{3} b^{7} d^{10}+5148 x^{7} a^{2} b^{8} c \,d^{9}+10296 x^{7} a \,b^{9} c^{2} d^{8}+17160 x^{7} b^{10} c^{3} d^{7}+1716 x^{6} a^{4} b^{6} d^{10}+5148 x^{6} a^{3} b^{7} c \,d^{9}+10296 x^{6} a^{2} b^{8} c^{2} d^{8}+17160 x^{6} a \,b^{9} c^{3} d^{7}+25740 x^{6} b^{10} c^{4} d^{6}+1287 x^{5} a^{5} b^{5} d^{10}+3861 x^{5} a^{4} b^{6} c \,d^{9}+7722 x^{5} a^{3} b^{7} c^{2} d^{8}+12870 x^{5} a^{2} b^{8} c^{3} d^{7}+19305 x^{5} a \,b^{9} c^{4} d^{6}+27027 x^{5} b^{10} c^{5} d^{5}+715 x^{4} a^{6} b^{4} d^{10}+2145 x^{4} a^{5} b^{5} c \,d^{9}+4290 x^{4} a^{4} b^{6} c^{2} d^{8}+7150 x^{4} a^{3} b^{7} c^{3} d^{7}+10725 x^{4} a^{2} b^{8} c^{4} d^{6}+15015 x^{4} a \,b^{9} c^{5} d^{5}+20020 x^{4} b^{10} c^{6} d^{4}+286 x^{3} a^{7} b^{3} d^{10}+858 x^{3} a^{6} b^{4} c \,d^{9}+1716 x^{3} a^{5} b^{5} c^{2} d^{8}+2860 x^{3} a^{4} b^{6} c^{3} d^{7}+4290 x^{3} a^{3} b^{7} c^{4} d^{6}+6006 x^{3} a^{2} b^{8} c^{5} d^{5}+8008 x^{3} a \,b^{9} c^{6} d^{4}+10296 x^{3} b^{10} c^{7} d^{3}+78 x^{2} a^{8} b^{2} d^{10}+234 x^{2} a^{7} b^{3} c \,d^{9}+468 x^{2} a^{6} b^{4} c^{2} d^{8}+780 x^{2} a^{5} b^{5} c^{3} d^{7}+1170 x^{2} a^{4} b^{6} c^{4} d^{6}+1638 x^{2} a^{3} b^{7} c^{5} d^{5}+2184 x^{2} a^{2} b^{8} c^{6} d^{4}+2808 x^{2} a \,b^{9} c^{7} d^{3}+3510 x^{2} b^{10} c^{8} d^{2}+13 x \,a^{9} b \,d^{10}+39 x \,a^{8} b^{2} c \,d^{9}+78 x \,a^{7} b^{3} c^{2} d^{8}+130 x \,a^{6} b^{4} c^{3} d^{7}+195 x \,a^{5} b^{5} c^{4} d^{6}+273 x \,a^{4} b^{6} c^{5} d^{5}+364 x \,a^{3} b^{7} c^{6} d^{4}+468 x \,a^{2} b^{8} c^{7} d^{3}+585 x a \,b^{9} c^{8} d^{2}+715 x \,b^{10} c^{9} d +a^{10} d^{10}+3 a^{9} b c \,d^{9}+6 a^{8} b^{2} c^{2} d^{8}+10 a^{7} b^{3} c^{3} d^{7}+15 a^{6} b^{4} c^{4} d^{6}+21 a^{5} b^{5} c^{5} d^{5}+28 a^{4} b^{6} c^{6} d^{4}+36 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}+55 a \,b^{9} c^{9} d +66 b^{10} c^{10}}{858 b^{11} \left (b x +a \right )^{13}}\)	\(962\)
parallelrisch	\(\frac {-286 d^{10} x^{10} b^{12}-715 a \,b^{11} d^{10} x^{9}-2145 b^{12} c \,d^{9} x^{9}-1287 a^{2} b^{10} d^{10} x^{8}-3861 a \,b^{11} c \,d^{9} x^{8}-7722 b^{12} c^{2} d^{8} x^{8}-1716 a^{3} b^{9} d^{10} x^{7}-5148 a^{2} b^{10} c \,d^{9} x^{7}-10296 a \,b^{11} c^{2} d^{8} x^{7}-17160 b^{12} c^{3} d^{7} x^{7}-1716 a^{4} b^{8} d^{10} x^{6}-5148 a^{3} b^{9} c \,d^{9} x^{6}-10296 a^{2} b^{10} c^{2} d^{8} x^{6}-17160 a \,b^{11} c^{3} d^{7} x^{6}-25740 b^{12} c^{4} d^{6} x^{6}-1287 a^{5} b^{7} d^{10} x^{5}-3861 a^{4} b^{8} c \,d^{9} x^{5}-7722 a^{3} b^{9} c^{2} d^{8} x^{5}-12870 a^{2} b^{10} c^{3} d^{7} x^{5}-19305 a \,b^{11} c^{4} d^{6} x^{5}-27027 b^{12} c^{5} d^{5} x^{5}-715 a^{6} b^{6} d^{10} x^{4}-2145 a^{5} b^{7} c \,d^{9} x^{4}-4290 a^{4} b^{8} c^{2} d^{8} x^{4}-7150 a^{3} b^{9} c^{3} d^{7} x^{4}-10725 a^{2} b^{10} c^{4} d^{6} x^{4}-15015 a \,b^{11} c^{5} d^{5} x^{4}-20020 b^{12} c^{6} d^{4} x^{4}-286 a^{7} b^{5} d^{10} x^{3}-858 a^{6} b^{6} c \,d^{9} x^{3}-1716 a^{5} b^{7} c^{2} d^{8} x^{3}-2860 a^{4} b^{8} c^{3} d^{7} x^{3}-4290 a^{3} b^{9} c^{4} d^{6} x^{3}-6006 a^{2} b^{10} c^{5} d^{5} x^{3}-8008 a \,b^{11} c^{6} d^{4} x^{3}-10296 b^{12} c^{7} d^{3} x^{3}-78 a^{8} b^{4} d^{10} x^{2}-234 a^{7} b^{5} c \,d^{9} x^{2}-468 a^{6} b^{6} c^{2} d^{8} x^{2}-780 a^{5} b^{7} c^{3} d^{7} x^{2}-1170 a^{4} b^{8} c^{4} d^{6} x^{2}-1638 a^{3} b^{9} c^{5} d^{5} x^{2}-2184 a^{2} b^{10} c^{6} d^{4} x^{2}-2808 a \,b^{11} c^{7} d^{3} x^{2}-3510 b^{12} c^{8} d^{2} x^{2}-13 a^{9} b^{3} d^{10} x -39 a^{8} b^{4} c \,d^{9} x -78 a^{7} b^{5} c^{2} d^{8} x -130 a^{6} b^{6} c^{3} d^{7} x -195 a^{5} b^{7} c^{4} d^{6} x -273 a^{4} b^{8} c^{5} d^{5} x -364 a^{3} b^{9} c^{6} d^{4} x -468 a^{2} b^{10} c^{7} d^{3} x -585 a \,b^{11} c^{8} d^{2} x -715 b^{12} c^{9} d x -a^{10} b^{2} d^{10}-3 a^{9} b^{3} c \,d^{9}-6 a^{8} b^{4} c^{2} d^{8}-10 a^{7} b^{5} c^{3} d^{7}-15 a^{6} b^{6} c^{4} d^{6}-21 a^{5} b^{7} c^{5} d^{5}-28 a^{4} b^{8} c^{6} d^{4}-36 a^{3} b^{9} c^{7} d^{3}-45 a^{2} b^{10} c^{8} d^{2}-55 a \,b^{11} c^{9} d -66 b^{12} c^{10}}{858 b^{13} \left (b x +a \right )^{13}}\)	\(970\)

Input:

int((d*x+c)^10/(b*x+a)^14,x,method=_RETURNVERBOSE)
 

Output:

(-1/3/b*d^10*x^10-5/6/b^2*d^9*(a*d+3*b*c)*x^9-3/2/b^3*d^8*(a^2*d^2+3*a*b*c 
*d+6*b^2*c^2)*x^8-2/b^4*d^7*(a^3*d^3+3*a^2*b*c*d^2+6*a*b^2*c^2*d+10*b^3*c^ 
3)*x^7-2/b^5*d^6*(a^4*d^4+3*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2+10*a*b^3*c^3*d+1 
5*b^4*c^4)*x^6-3/2/b^6*d^5*(a^5*d^5+3*a^4*b*c*d^4+6*a^3*b^2*c^2*d^3+10*a^2 
*b^3*c^3*d^2+15*a*b^4*c^4*d+21*b^5*c^5)*x^5-5/6/b^7*d^4*(a^6*d^6+3*a^5*b*c 
*d^5+6*a^4*b^2*c^2*d^4+10*a^3*b^3*c^3*d^3+15*a^2*b^4*c^4*d^2+21*a*b^5*c^5* 
d+28*b^6*c^6)*x^4-1/3/b^8*d^3*(a^7*d^7+3*a^6*b*c*d^6+6*a^5*b^2*c^2*d^5+10* 
a^4*b^3*c^3*d^4+15*a^3*b^4*c^4*d^3+21*a^2*b^5*c^5*d^2+28*a*b^6*c^6*d+36*b^ 
7*c^7)*x^3-1/11/b^9*d^2*(a^8*d^8+3*a^7*b*c*d^7+6*a^6*b^2*c^2*d^6+10*a^5*b^ 
3*c^3*d^5+15*a^4*b^4*c^4*d^4+21*a^3*b^5*c^5*d^3+28*a^2*b^6*c^6*d^2+36*a*b^ 
7*c^7*d+45*b^8*c^8)*x^2-1/66*d/b^10*(a^9*d^9+3*a^8*b*c*d^8+6*a^7*b^2*c^2*d 
^7+10*a^6*b^3*c^3*d^6+15*a^5*b^4*c^4*d^5+21*a^4*b^5*c^5*d^4+28*a^3*b^6*c^6 
*d^3+36*a^2*b^7*c^7*d^2+45*a*b^8*c^8*d+55*b^9*c^9)*x-1/858/b^11*(a^10*d^10 
+3*a^9*b*c*d^9+6*a^8*b^2*c^2*d^8+10*a^7*b^3*c^3*d^7+15*a^6*b^4*c^4*d^6+21* 
a^5*b^5*c^5*d^5+28*a^4*b^6*c^6*d^4+36*a^3*b^7*c^7*d^3+45*a^2*b^8*c^8*d^2+5 
5*a*b^9*c^9*d+66*b^10*c^10))/(b*x+a)^13
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (83) = 166\).

Time = 0.09 (sec) , antiderivative size = 997, normalized size of antiderivative = 11.20 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^14,x, algorithm="fricas")
 

Output:

-1/858*(286*b^10*d^10*x^10 + 66*b^10*c^10 + 55*a*b^9*c^9*d + 45*a^2*b^8*c^ 
8*d^2 + 36*a^3*b^7*c^7*d^3 + 28*a^4*b^6*c^6*d^4 + 21*a^5*b^5*c^5*d^5 + 15* 
a^6*b^4*c^4*d^6 + 10*a^7*b^3*c^3*d^7 + 6*a^8*b^2*c^2*d^8 + 3*a^9*b*c*d^9 + 
 a^10*d^10 + 715*(3*b^10*c*d^9 + a*b^9*d^10)*x^9 + 1287*(6*b^10*c^2*d^8 + 
3*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 1716*(10*b^10*c^3*d^7 + 6*a*b^9*c^2*d^ 
8 + 3*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 1716*(15*b^10*c^4*d^6 + 10*a*b^9 
*c^3*d^7 + 6*a^2*b^8*c^2*d^8 + 3*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 1287* 
(21*b^10*c^5*d^5 + 15*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 + 6*a^3*b^7*c^2*d 
^8 + 3*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 715*(28*b^10*c^6*d^4 + 21*a*b^9 
*c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 10*a^3*b^7*c^3*d^7 + 6*a^4*b^6*c^2*d^8 + 3 
*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 286*(36*b^10*c^7*d^3 + 28*a*b^9*c^6*d 
^4 + 21*a^2*b^8*c^5*d^5 + 15*a^3*b^7*c^4*d^6 + 10*a^4*b^6*c^3*d^7 + 6*a^5* 
b^5*c^2*d^8 + 3*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 78*(45*b^10*c^8*d^2 + 
36*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 21*a^3*b^7*c^5*d^5 + 15*a^4*b^6*c^ 
4*d^6 + 10*a^5*b^5*c^3*d^7 + 6*a^6*b^4*c^2*d^8 + 3*a^7*b^3*c*d^9 + a^8*b^2 
*d^10)*x^2 + 13*(55*b^10*c^9*d + 45*a*b^9*c^8*d^2 + 36*a^2*b^8*c^7*d^3 + 2 
8*a^3*b^7*c^6*d^4 + 21*a^4*b^6*c^5*d^5 + 15*a^5*b^5*c^4*d^6 + 10*a^6*b^4*c 
^3*d^7 + 6*a^7*b^3*c^2*d^8 + 3*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^24*x^13 + 
 13*a*b^23*x^12 + 78*a^2*b^22*x^11 + 286*a^3*b^21*x^10 + 715*a^4*b^20*x^9 
+ 1287*a^5*b^19*x^8 + 1716*a^6*b^18*x^7 + 1716*a^7*b^17*x^6 + 1287*a^8*...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**10/(b*x+a)**14,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (83) = 166\).

Time = 0.09 (sec) , antiderivative size = 997, normalized size of antiderivative = 11.20 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^14,x, algorithm="maxima")
 

Output:

-1/858*(286*b^10*d^10*x^10 + 66*b^10*c^10 + 55*a*b^9*c^9*d + 45*a^2*b^8*c^ 
8*d^2 + 36*a^3*b^7*c^7*d^3 + 28*a^4*b^6*c^6*d^4 + 21*a^5*b^5*c^5*d^5 + 15* 
a^6*b^4*c^4*d^6 + 10*a^7*b^3*c^3*d^7 + 6*a^8*b^2*c^2*d^8 + 3*a^9*b*c*d^9 + 
 a^10*d^10 + 715*(3*b^10*c*d^9 + a*b^9*d^10)*x^9 + 1287*(6*b^10*c^2*d^8 + 
3*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 1716*(10*b^10*c^3*d^7 + 6*a*b^9*c^2*d^ 
8 + 3*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 1716*(15*b^10*c^4*d^6 + 10*a*b^9 
*c^3*d^7 + 6*a^2*b^8*c^2*d^8 + 3*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 1287* 
(21*b^10*c^5*d^5 + 15*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 + 6*a^3*b^7*c^2*d 
^8 + 3*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 715*(28*b^10*c^6*d^4 + 21*a*b^9 
*c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 10*a^3*b^7*c^3*d^7 + 6*a^4*b^6*c^2*d^8 + 3 
*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 286*(36*b^10*c^7*d^3 + 28*a*b^9*c^6*d 
^4 + 21*a^2*b^8*c^5*d^5 + 15*a^3*b^7*c^4*d^6 + 10*a^4*b^6*c^3*d^7 + 6*a^5* 
b^5*c^2*d^8 + 3*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 78*(45*b^10*c^8*d^2 + 
36*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 21*a^3*b^7*c^5*d^5 + 15*a^4*b^6*c^ 
4*d^6 + 10*a^5*b^5*c^3*d^7 + 6*a^6*b^4*c^2*d^8 + 3*a^7*b^3*c*d^9 + a^8*b^2 
*d^10)*x^2 + 13*(55*b^10*c^9*d + 45*a*b^9*c^8*d^2 + 36*a^2*b^8*c^7*d^3 + 2 
8*a^3*b^7*c^6*d^4 + 21*a^4*b^6*c^5*d^5 + 15*a^5*b^5*c^4*d^6 + 10*a^6*b^4*c 
^3*d^7 + 6*a^7*b^3*c^2*d^8 + 3*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^24*x^13 + 
 13*a*b^23*x^12 + 78*a^2*b^22*x^11 + 286*a^3*b^21*x^10 + 715*a^4*b^20*x^9 
+ 1287*a^5*b^19*x^8 + 1716*a^6*b^18*x^7 + 1716*a^7*b^17*x^6 + 1287*a^8*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (83) = 166\).

Time = 0.27 (sec) , antiderivative size = 961, normalized size of antiderivative = 10.80 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^14,x, algorithm="giac")
 

Output:

-1/858*(286*b^10*d^10*x^10 + 2145*b^10*c*d^9*x^9 + 715*a*b^9*d^10*x^9 + 77 
22*b^10*c^2*d^8*x^8 + 3861*a*b^9*c*d^9*x^8 + 1287*a^2*b^8*d^10*x^8 + 17160 
*b^10*c^3*d^7*x^7 + 10296*a*b^9*c^2*d^8*x^7 + 5148*a^2*b^8*c*d^9*x^7 + 171 
6*a^3*b^7*d^10*x^7 + 25740*b^10*c^4*d^6*x^6 + 17160*a*b^9*c^3*d^7*x^6 + 10 
296*a^2*b^8*c^2*d^8*x^6 + 5148*a^3*b^7*c*d^9*x^6 + 1716*a^4*b^6*d^10*x^6 + 
 27027*b^10*c^5*d^5*x^5 + 19305*a*b^9*c^4*d^6*x^5 + 12870*a^2*b^8*c^3*d^7* 
x^5 + 7722*a^3*b^7*c^2*d^8*x^5 + 3861*a^4*b^6*c*d^9*x^5 + 1287*a^5*b^5*d^1 
0*x^5 + 20020*b^10*c^6*d^4*x^4 + 15015*a*b^9*c^5*d^5*x^4 + 10725*a^2*b^8*c 
^4*d^6*x^4 + 7150*a^3*b^7*c^3*d^7*x^4 + 4290*a^4*b^6*c^2*d^8*x^4 + 2145*a^ 
5*b^5*c*d^9*x^4 + 715*a^6*b^4*d^10*x^4 + 10296*b^10*c^7*d^3*x^3 + 8008*a*b 
^9*c^6*d^4*x^3 + 6006*a^2*b^8*c^5*d^5*x^3 + 4290*a^3*b^7*c^4*d^6*x^3 + 286 
0*a^4*b^6*c^3*d^7*x^3 + 1716*a^5*b^5*c^2*d^8*x^3 + 858*a^6*b^4*c*d^9*x^3 + 
 286*a^7*b^3*d^10*x^3 + 3510*b^10*c^8*d^2*x^2 + 2808*a*b^9*c^7*d^3*x^2 + 2 
184*a^2*b^8*c^6*d^4*x^2 + 1638*a^3*b^7*c^5*d^5*x^2 + 1170*a^4*b^6*c^4*d^6* 
x^2 + 780*a^5*b^5*c^3*d^7*x^2 + 468*a^6*b^4*c^2*d^8*x^2 + 234*a^7*b^3*c*d^ 
9*x^2 + 78*a^8*b^2*d^10*x^2 + 715*b^10*c^9*d*x + 585*a*b^9*c^8*d^2*x + 468 
*a^2*b^8*c^7*d^3*x + 364*a^3*b^7*c^6*d^4*x + 273*a^4*b^6*c^5*d^5*x + 195*a 
^5*b^5*c^4*d^6*x + 130*a^6*b^4*c^3*d^7*x + 78*a^7*b^3*c^2*d^8*x + 39*a^8*b 
^2*c*d^9*x + 13*a^9*b*d^10*x + 66*b^10*c^10 + 55*a*b^9*c^9*d + 45*a^2*b^8* 
c^8*d^2 + 36*a^3*b^7*c^7*d^3 + 28*a^4*b^6*c^6*d^4 + 21*a^5*b^5*c^5*d^5 ...
 

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 1098, normalized size of antiderivative = 12.34 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx =\text {Too large to display} \] Input:

int((c + d*x)^10/(a + b*x)^14,x)
 

Output:

-(a^10*d^10 + 66*b^10*c^10 + 286*b^10*d^10*x^10 + 715*a*b^9*d^10*x^9 + 214 
5*b^10*c*d^9*x^9 + 45*a^2*b^8*c^8*d^2 + 36*a^3*b^7*c^7*d^3 + 28*a^4*b^6*c^ 
6*d^4 + 21*a^5*b^5*c^5*d^5 + 15*a^6*b^4*c^4*d^6 + 10*a^7*b^3*c^3*d^7 + 6*a 
^8*b^2*c^2*d^8 + 78*a^8*b^2*d^10*x^2 + 286*a^7*b^3*d^10*x^3 + 715*a^6*b^4* 
d^10*x^4 + 1287*a^5*b^5*d^10*x^5 + 1716*a^4*b^6*d^10*x^6 + 1716*a^3*b^7*d^ 
10*x^7 + 1287*a^2*b^8*d^10*x^8 + 3510*b^10*c^8*d^2*x^2 + 10296*b^10*c^7*d^ 
3*x^3 + 20020*b^10*c^6*d^4*x^4 + 27027*b^10*c^5*d^5*x^5 + 25740*b^10*c^4*d 
^6*x^6 + 17160*b^10*c^3*d^7*x^7 + 7722*b^10*c^2*d^8*x^8 + 55*a*b^9*c^9*d + 
 3*a^9*b*c*d^9 + 13*a^9*b*d^10*x + 715*b^10*c^9*d*x + 2184*a^2*b^8*c^6*d^4 
*x^2 + 1638*a^3*b^7*c^5*d^5*x^2 + 1170*a^4*b^6*c^4*d^6*x^2 + 780*a^5*b^5*c 
^3*d^7*x^2 + 468*a^6*b^4*c^2*d^8*x^2 + 6006*a^2*b^8*c^5*d^5*x^3 + 4290*a^3 
*b^7*c^4*d^6*x^3 + 2860*a^4*b^6*c^3*d^7*x^3 + 1716*a^5*b^5*c^2*d^8*x^3 + 1 
0725*a^2*b^8*c^4*d^6*x^4 + 7150*a^3*b^7*c^3*d^7*x^4 + 4290*a^4*b^6*c^2*d^8 
*x^4 + 12870*a^2*b^8*c^3*d^7*x^5 + 7722*a^3*b^7*c^2*d^8*x^5 + 10296*a^2*b^ 
8*c^2*d^8*x^6 + 585*a*b^9*c^8*d^2*x + 39*a^8*b^2*c*d^9*x + 3861*a*b^9*c*d^ 
9*x^8 + 468*a^2*b^8*c^7*d^3*x + 364*a^3*b^7*c^6*d^4*x + 273*a^4*b^6*c^5*d^ 
5*x + 195*a^5*b^5*c^4*d^6*x + 130*a^6*b^4*c^3*d^7*x + 78*a^7*b^3*c^2*d^8*x 
 + 2808*a*b^9*c^7*d^3*x^2 + 234*a^7*b^3*c*d^9*x^2 + 8008*a*b^9*c^6*d^4*x^3 
 + 858*a^6*b^4*c*d^9*x^3 + 15015*a*b^9*c^5*d^5*x^4 + 2145*a^5*b^5*c*d^9*x^ 
4 + 19305*a*b^9*c^4*d^6*x^5 + 3861*a^4*b^6*c*d^9*x^5 + 17160*a*b^9*c^3*...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1094, normalized size of antiderivative = 12.29 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^10/(b*x+a)^14,x)
 

Output:

( - a**10*d**10 - 3*a**9*b*c*d**9 - 13*a**9*b*d**10*x - 6*a**8*b**2*c**2*d 
**8 - 39*a**8*b**2*c*d**9*x - 78*a**8*b**2*d**10*x**2 - 10*a**7*b**3*c**3* 
d**7 - 78*a**7*b**3*c**2*d**8*x - 234*a**7*b**3*c*d**9*x**2 - 286*a**7*b** 
3*d**10*x**3 - 15*a**6*b**4*c**4*d**6 - 130*a**6*b**4*c**3*d**7*x - 468*a* 
*6*b**4*c**2*d**8*x**2 - 858*a**6*b**4*c*d**9*x**3 - 715*a**6*b**4*d**10*x 
**4 - 21*a**5*b**5*c**5*d**5 - 195*a**5*b**5*c**4*d**6*x - 780*a**5*b**5*c 
**3*d**7*x**2 - 1716*a**5*b**5*c**2*d**8*x**3 - 2145*a**5*b**5*c*d**9*x**4 
 - 1287*a**5*b**5*d**10*x**5 - 28*a**4*b**6*c**6*d**4 - 273*a**4*b**6*c**5 
*d**5*x - 1170*a**4*b**6*c**4*d**6*x**2 - 2860*a**4*b**6*c**3*d**7*x**3 - 
4290*a**4*b**6*c**2*d**8*x**4 - 3861*a**4*b**6*c*d**9*x**5 - 1716*a**4*b** 
6*d**10*x**6 - 36*a**3*b**7*c**7*d**3 - 364*a**3*b**7*c**6*d**4*x - 1638*a 
**3*b**7*c**5*d**5*x**2 - 4290*a**3*b**7*c**4*d**6*x**3 - 7150*a**3*b**7*c 
**3*d**7*x**4 - 7722*a**3*b**7*c**2*d**8*x**5 - 5148*a**3*b**7*c*d**9*x**6 
 - 1716*a**3*b**7*d**10*x**7 - 45*a**2*b**8*c**8*d**2 - 468*a**2*b**8*c**7 
*d**3*x - 2184*a**2*b**8*c**6*d**4*x**2 - 6006*a**2*b**8*c**5*d**5*x**3 - 
10725*a**2*b**8*c**4*d**6*x**4 - 12870*a**2*b**8*c**3*d**7*x**5 - 10296*a* 
*2*b**8*c**2*d**8*x**6 - 5148*a**2*b**8*c*d**9*x**7 - 1287*a**2*b**8*d**10 
*x**8 - 55*a*b**9*c**9*d - 585*a*b**9*c**8*d**2*x - 2808*a*b**9*c**7*d**3* 
x**2 - 8008*a*b**9*c**6*d**4*x**3 - 15015*a*b**9*c**5*d**5*x**4 - 19305*a* 
b**9*c**4*d**6*x**5 - 17160*a*b**9*c**3*d**7*x**6 - 10296*a*b**9*c**2*d...