\(\int \frac {(c+d x)^{10}}{(a+b x)^2} \, dx\) [107]

Optimal result

Integrand size = 15, antiderivative size = 258 \[ \int \frac {(c+d x)^{10}}{(a+b x)^2} \, dx=\frac {45 d^2 (b c-a d)^8 x}{b^{10}}-\frac {(b c-a d)^{10}}{b^{11} (a+b x)}+\frac {60 d^3 (b c-a d)^7 (a+b x)^2}{b^{11}}+\frac {70 d^4 (b c-a d)^6 (a+b x)^3}{b^{11}}+\frac {63 d^5 (b c-a d)^5 (a+b x)^4}{b^{11}}+\frac {42 d^6 (b c-a d)^4 (a+b x)^5}{b^{11}}+\frac {20 d^7 (b c-a d)^3 (a+b x)^6}{b^{11}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^7}{7 b^{11}}+\frac {5 d^9 (b c-a d) (a+b x)^8}{4 b^{11}}+\frac {d^{10} (a+b x)^9}{9 b^{11}}+\frac {10 d (b c-a d)^9 \log (a+b x)}{b^{11}} \] Output:

45*d^2*(-a*d+b*c)^8*x/b^10-(-a*d+b*c)^10/b^11/(b*x+a)+60*d^3*(-a*d+b*c)^7* 
(b*x+a)^2/b^11+70*d^4*(-a*d+b*c)^6*(b*x+a)^3/b^11+63*d^5*(-a*d+b*c)^5*(b*x 
+a)^4/b^11+42*d^6*(-a*d+b*c)^4*(b*x+a)^5/b^11+20*d^7*(-a*d+b*c)^3*(b*x+a)^ 
6/b^11+45/7*d^8*(-a*d+b*c)^2*(b*x+a)^7/b^11+5/4*d^9*(-a*d+b*c)*(b*x+a)^8/b 
^11+1/9*d^10*(b*x+a)^9/b^11+10*d*(-a*d+b*c)^9*ln(b*x+a)/b^11
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(708\) vs. \(2(258)=516\).

Time = 0.14 (sec) , antiderivative size = 708, normalized size of antiderivative = 2.74 \[ \int \frac {(c+d x)^{10}}{(a+b x)^2} \, dx=\frac {-252 a^{10} d^{10}+252 a^9 b d^9 (10 c+9 d x)+1260 a^8 b^2 d^8 \left (-9 c^2-16 c d x+d^2 x^2\right )-420 a^7 b^3 d^7 \left (-72 c^3-189 c^2 d x+27 c d^2 x^2+d^3 x^3\right )+210 a^6 b^4 d^6 \left (-252 c^4-864 c^3 d x+216 c^2 d^2 x^2+18 c d^3 x^3+d^4 x^4\right )-126 a^5 b^5 d^5 \left (-504 c^5-2100 c^4 d x+840 c^3 d^2 x^2+120 c^2 d^3 x^3+15 c d^4 x^4+d^5 x^5\right )+42 a^4 b^6 d^4 \left (-1260 c^6-6048 c^5 d x+3780 c^4 d^2 x^2+840 c^3 d^3 x^3+180 c^2 d^4 x^4+27 c d^5 x^5+2 d^6 x^6\right )-12 a^3 b^7 d^3 \left (-2520 c^7-13230 c^6 d x+13230 c^5 d^2 x^2+4410 c^4 d^3 x^3+1470 c^3 d^4 x^4+378 c^2 d^5 x^5+63 c d^6 x^6+5 d^7 x^7\right )+9 a^2 b^8 d^2 \left (-1260 c^8-6720 c^7 d x+11760 c^6 d^2 x^2+5880 c^5 d^3 x^3+2940 c^4 d^4 x^4+1176 c^3 d^5 x^5+336 c^2 d^6 x^6+60 c d^7 x^7+5 d^8 x^8\right )-a b^9 d \left (-2520 c^9-11340 c^8 d x+45360 c^7 d^2 x^2+35280 c^6 d^3 x^3+26460 c^5 d^4 x^4+15876 c^4 d^5 x^5+7056 c^3 d^6 x^6+2160 c^2 d^7 x^7+405 c d^8 x^8+35 d^9 x^9\right )+b^{10} \left (-252 c^{10}+11340 c^8 d^2 x^2+15120 c^7 d^3 x^3+17640 c^6 d^4 x^4+15876 c^5 d^5 x^5+10584 c^4 d^6 x^6+5040 c^3 d^7 x^7+1620 c^2 d^8 x^8+315 c d^9 x^9+28 d^{10} x^{10}\right )-2520 d (-b c+a d)^9 (a+b x) \log (a+b x)}{252 b^{11} (a+b x)} \] Input:

Integrate[(c + d*x)^10/(a + b*x)^2,x]
 

Output:

(-252*a^10*d^10 + 252*a^9*b*d^9*(10*c + 9*d*x) + 1260*a^8*b^2*d^8*(-9*c^2 
- 16*c*d*x + d^2*x^2) - 420*a^7*b^3*d^7*(-72*c^3 - 189*c^2*d*x + 27*c*d^2* 
x^2 + d^3*x^3) + 210*a^6*b^4*d^6*(-252*c^4 - 864*c^3*d*x + 216*c^2*d^2*x^2 
 + 18*c*d^3*x^3 + d^4*x^4) - 126*a^5*b^5*d^5*(-504*c^5 - 2100*c^4*d*x + 84 
0*c^3*d^2*x^2 + 120*c^2*d^3*x^3 + 15*c*d^4*x^4 + d^5*x^5) + 42*a^4*b^6*d^4 
*(-1260*c^6 - 6048*c^5*d*x + 3780*c^4*d^2*x^2 + 840*c^3*d^3*x^3 + 180*c^2* 
d^4*x^4 + 27*c*d^5*x^5 + 2*d^6*x^6) - 12*a^3*b^7*d^3*(-2520*c^7 - 13230*c^ 
6*d*x + 13230*c^5*d^2*x^2 + 4410*c^4*d^3*x^3 + 1470*c^3*d^4*x^4 + 378*c^2* 
d^5*x^5 + 63*c*d^6*x^6 + 5*d^7*x^7) + 9*a^2*b^8*d^2*(-1260*c^8 - 6720*c^7* 
d*x + 11760*c^6*d^2*x^2 + 5880*c^5*d^3*x^3 + 2940*c^4*d^4*x^4 + 1176*c^3*d 
^5*x^5 + 336*c^2*d^6*x^6 + 60*c*d^7*x^7 + 5*d^8*x^8) - a*b^9*d*(-2520*c^9 
- 11340*c^8*d*x + 45360*c^7*d^2*x^2 + 35280*c^6*d^3*x^3 + 26460*c^5*d^4*x^ 
4 + 15876*c^4*d^5*x^5 + 7056*c^3*d^6*x^6 + 2160*c^2*d^7*x^7 + 405*c*d^8*x^ 
8 + 35*d^9*x^9) + b^10*(-252*c^10 + 11340*c^8*d^2*x^2 + 15120*c^7*d^3*x^3 
+ 17640*c^6*d^4*x^4 + 15876*c^5*d^5*x^5 + 10584*c^4*d^6*x^6 + 5040*c^3*d^7 
*x^7 + 1620*c^2*d^8*x^8 + 315*c*d^9*x^9 + 28*d^10*x^10) - 2520*d*(-(b*c) + 
 a*d)^9*(a + b*x)*Log[a + b*x])/(252*b^11*(a + b*x))
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {49, 2009}

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.

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

\(\Big \downarrow \) 49

\(\displaystyle \int \left (\frac {10 d^9 (a+b x)^7 (b c-a d)}{b^{10}}+\frac {45 d^8 (a+b x)^6 (b c-a d)^2}{b^{10}}+\frac {120 d^7 (a+b x)^5 (b c-a d)^3}{b^{10}}+\frac {210 d^6 (a+b x)^4 (b c-a d)^4}{b^{10}}+\frac {252 d^5 (a+b x)^3 (b c-a d)^5}{b^{10}}+\frac {210 d^4 (a+b x)^2 (b c-a d)^6}{b^{10}}+\frac {120 d^3 (a+b x) (b c-a d)^7}{b^{10}}+\frac {45 d^2 (b c-a d)^8}{b^{10}}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^2}+\frac {d^{10} (a+b x)^8}{b^{10}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5 d^9 (a+b x)^8 (b c-a d)}{4 b^{11}}+\frac {45 d^8 (a+b x)^7 (b c-a d)^2}{7 b^{11}}+\frac {20 d^7 (a+b x)^6 (b c-a d)^3}{b^{11}}+\frac {42 d^6 (a+b x)^5 (b c-a d)^4}{b^{11}}+\frac {63 d^5 (a+b x)^4 (b c-a d)^5}{b^{11}}+\frac {70 d^4 (a+b x)^3 (b c-a d)^6}{b^{11}}+\frac {60 d^3 (a+b x)^2 (b c-a d)^7}{b^{11}}-\frac {(b c-a d)^{10}}{b^{11} (a+b x)}+\frac {10 d (b c-a d)^9 \log (a+b x)}{b^{11}}+\frac {d^{10} (a+b x)^9}{9 b^{11}}+\frac {45 d^2 x (b c-a d)^8}{b^{10}}\)

Input:

Int[(c + d*x)^10/(a + b*x)^2,x]
 

Output:

(45*d^2*(b*c - a*d)^8*x)/b^10 - (b*c - a*d)^10/(b^11*(a + b*x)) + (60*d^3* 
(b*c - a*d)^7*(a + b*x)^2)/b^11 + (70*d^4*(b*c - a*d)^6*(a + b*x)^3)/b^11 
+ (63*d^5*(b*c - a*d)^5*(a + b*x)^4)/b^11 + (42*d^6*(b*c - a*d)^4*(a + b*x 
)^5)/b^11 + (20*d^7*(b*c - a*d)^3*(a + b*x)^6)/b^11 + (45*d^8*(b*c - a*d)^ 
2*(a + b*x)^7)/(7*b^11) + (5*d^9*(b*c - a*d)*(a + b*x)^8)/(4*b^11) + (d^10 
*(a + b*x)^9)/(9*b^11) + (10*d*(b*c - a*d)^9*Log[a + b*x])/b^11
 

Defintions of rubi rules used

rule 49

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(839\) vs. \(2(252)=504\).

Time = 0.13 (sec) , antiderivative size = 840, normalized size of antiderivative = 3.26

method	result	size
norman	\(\frac {\frac {\left (10 a^{10} d^{10}-90 a^{9} b c \,d^{9}+360 a^{8} b^{2} c^{2} d^{8}-840 a^{7} b^{3} c^{3} d^{7}+1260 a^{6} b^{4} c^{4} d^{6}-1260 a^{5} b^{5} c^{5} d^{5}+840 a^{4} b^{6} c^{6} d^{4}-360 a^{3} b^{7} c^{7} d^{3}+90 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}\right ) x}{a \,b^{10}}+\frac {d^{10} x^{10}}{9 b}+\frac {5 d^{2} \left (a^{8} d^{8}-9 a^{7} b c \,d^{7}+36 a^{6} b^{2} c^{2} d^{6}-84 a^{5} b^{3} c^{3} d^{5}+126 a^{4} b^{4} c^{4} d^{4}-126 a^{3} b^{5} c^{5} d^{3}+84 a^{2} b^{6} c^{6} d^{2}-36 a \,b^{7} c^{7} d +9 c^{8} b^{8}\right ) x^{2}}{b^{9}}-\frac {5 d^{3} \left (a^{7} d^{7}-9 a^{6} b c \,d^{6}+36 a^{5} b^{2} c^{2} d^{5}-84 a^{4} b^{3} c^{3} d^{4}+126 a^{3} b^{4} c^{4} d^{3}-126 a^{2} b^{5} c^{5} d^{2}+84 a \,b^{6} c^{6} d -36 b^{7} c^{7}\right ) x^{3}}{3 b^{8}}+\frac {5 d^{4} \left (a^{6} d^{6}-9 a^{5} b c \,d^{5}+36 a^{4} b^{2} c^{2} d^{4}-84 a^{3} b^{3} c^{3} d^{3}+126 a^{2} b^{4} c^{4} d^{2}-126 a \,b^{5} c^{5} d +84 c^{6} b^{6}\right ) x^{4}}{6 b^{7}}-\frac {d^{5} \left (a^{5} d^{5}-9 a^{4} b c \,d^{4}+36 a^{3} b^{2} c^{2} d^{3}-84 a^{2} b^{3} c^{3} d^{2}+126 a \,b^{4} c^{4} d -126 c^{5} b^{5}\right ) x^{5}}{2 b^{6}}+\frac {d^{6} \left (d^{4} a^{4}-9 a^{3} b c \,d^{3}+36 a^{2} b^{2} c^{2} d^{2}-84 a \,b^{3} c^{3} d +126 c^{4} b^{4}\right ) x^{6}}{3 b^{5}}-\frac {5 d^{7} \left (a^{3} d^{3}-9 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -84 b^{3} c^{3}\right ) x^{7}}{21 b^{4}}+\frac {5 d^{8} \left (a^{2} d^{2}-9 a b c d +36 b^{2} c^{2}\right ) x^{8}}{28 b^{3}}-\frac {5 d^{9} \left (a d -9 b c \right ) x^{9}}{36 b^{2}}}{b x +a}-\frac {10 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -c^{9} b^{9}\right ) \ln \left (b x +a \right )}{b^{11}}\)	\(840\)
default	\(\frac {d^{2} \left (a^{4} b^{4} d^{8} x^{5}+70 b^{8} c^{6} d^{2} x^{3}-4 a^{7} b \,d^{8} x^{2}+60 b^{8} c^{7} d \,x^{2}+210 a^{2} b^{6} c^{4} d^{4} x^{3}-168 a \,b^{7} c^{5} d^{3} x^{3}+35 a^{6} b^{2} c \,d^{7} x^{2}-135 a^{5} b^{3} c^{2} d^{6} x^{2}+300 a^{4} b^{4} c^{3} d^{5} x^{2}-420 a^{3} b^{5} c^{4} d^{4} x^{2}+378 a^{2} b^{6} c^{5} d^{3} x^{2}-210 a \,b^{7} c^{6} d^{2} x^{2}-80 a^{7} b c \,d^{7} x +315 a^{6} b^{2} c^{2} d^{6} x -720 a^{5} b^{3} c^{3} d^{5} x +1050 a^{4} b^{4} c^{4} d^{4} x -1008 a^{3} b^{5} c^{5} d^{3} x +630 a^{2} b^{6} c^{6} d^{2} x -240 a \,b^{7} c^{7} d x +\frac {1}{9} d^{8} x^{9} b^{8}-\frac {2}{3} a^{3} b^{5} d^{8} x^{6}+20 b^{8} c^{3} d^{5} x^{6}+42 b^{8} c^{4} d^{4} x^{5}-\frac {3}{2} a^{5} b^{3} d^{8} x^{4}+63 b^{8} c^{5} d^{3} x^{4}+\frac {7}{3} a^{6} b^{2} d^{8} x^{3}-\frac {1}{4} a \,b^{7} d^{8} x^{8}+\frac {5}{4} b^{8} c \,d^{7} x^{8}+\frac {3}{7} a^{2} b^{6} d^{8} x^{7}+\frac {45}{7} b^{8} c^{2} d^{6} x^{7}+27 a^{2} b^{6} c^{2} d^{6} x^{5}-48 a \,b^{7} c^{3} d^{5} x^{5}+\frac {25}{2} a^{4} b^{4} c \,d^{7} x^{4}-45 a^{3} b^{5} c^{2} d^{6} x^{4}+90 a^{2} b^{6} c^{3} d^{5} x^{4}-105 a \,b^{7} c^{4} d^{4} x^{4}-20 a^{5} b^{3} c \,d^{7} x^{3}+75 a^{4} b^{4} c^{2} d^{6} x^{3}-160 a^{3} b^{5} c^{3} d^{5} x^{3}-8 a^{3} b^{5} c \,d^{7} x^{5}+45 c^{8} b^{8} x -\frac {20}{7} a \,b^{7} c \,d^{7} x^{7}+5 a^{2} b^{6} c \,d^{7} x^{6}-15 a \,b^{7} c^{2} d^{6} x^{6}+9 a^{8} d^{8} x \right )}{b^{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}}{b^{11} \left (b x +a \right )}-\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 ) \ln \left (b x +a \right )}{b^{11}}\)	\(933\)
risch	\(\text {Expression too large to display}\)	\(1066\)
parallelrisch	\(\text {Expression too large to display}\)	\(1209\)

Input:

int((d*x+c)^10/(b*x+a)^2,x,method=_RETURNVERBOSE)
 

Output:

((10*a^10*d^10-90*a^9*b*c*d^9+360*a^8*b^2*c^2*d^8-840*a^7*b^3*c^3*d^7+1260 
*a^6*b^4*c^4*d^6-1260*a^5*b^5*c^5*d^5+840*a^4*b^6*c^6*d^4-360*a^3*b^7*c^7* 
d^3+90*a^2*b^8*c^8*d^2-10*a*b^9*c^9*d+b^10*c^10)/a/b^10*x+1/9/b*d^10*x^10+ 
5*d^2*(a^8*d^8-9*a^7*b*c*d^7+36*a^6*b^2*c^2*d^6-84*a^5*b^3*c^3*d^5+126*a^4 
*b^4*c^4*d^4-126*a^3*b^5*c^5*d^3+84*a^2*b^6*c^6*d^2-36*a*b^7*c^7*d+9*b^8*c 
^8)/b^9*x^2-5/3*d^3*(a^7*d^7-9*a^6*b*c*d^6+36*a^5*b^2*c^2*d^5-84*a^4*b^3*c 
^3*d^4+126*a^3*b^4*c^4*d^3-126*a^2*b^5*c^5*d^2+84*a*b^6*c^6*d-36*b^7*c^7)/ 
b^8*x^3+5/6*d^4*(a^6*d^6-9*a^5*b*c*d^5+36*a^4*b^2*c^2*d^4-84*a^3*b^3*c^3*d 
^3+126*a^2*b^4*c^4*d^2-126*a*b^5*c^5*d+84*b^6*c^6)/b^7*x^4-1/2*d^5*(a^5*d^ 
5-9*a^4*b*c*d^4+36*a^3*b^2*c^2*d^3-84*a^2*b^3*c^3*d^2+126*a*b^4*c^4*d-126* 
b^5*c^5)/b^6*x^5+1/3*d^6*(a^4*d^4-9*a^3*b*c*d^3+36*a^2*b^2*c^2*d^2-84*a*b^ 
3*c^3*d+126*b^4*c^4)/b^5*x^6-5/21*d^7*(a^3*d^3-9*a^2*b*c*d^2+36*a*b^2*c^2* 
d-84*b^3*c^3)/b^4*x^7+5/28*d^8*(a^2*d^2-9*a*b*c*d+36*b^2*c^2)/b^3*x^8-5/36 
*d^9*(a*d-9*b*c)/b^2*x^9)/(b*x+a)-10*d/b^11*(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-b^9*c^9)*ln(b*x+a)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1124 vs. \(2 (252) = 504\).

Time = 0.11 (sec) , antiderivative size = 1124, normalized size of antiderivative = 4.36 \[ \int \frac {(c+d x)^{10}}{(a+b x)^2} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^2,x, algorithm="fricas")
 

Output:

1/252*(28*b^10*d^10*x^10 - 252*b^10*c^10 + 2520*a*b^9*c^9*d - 11340*a^2*b^ 
8*c^8*d^2 + 30240*a^3*b^7*c^7*d^3 - 52920*a^4*b^6*c^6*d^4 + 63504*a^5*b^5* 
c^5*d^5 - 52920*a^6*b^4*c^4*d^6 + 30240*a^7*b^3*c^3*d^7 - 11340*a^8*b^2*c^ 
2*d^8 + 2520*a^9*b*c*d^9 - 252*a^10*d^10 + 35*(9*b^10*c*d^9 - a*b^9*d^10)* 
x^9 + 45*(36*b^10*c^2*d^8 - 9*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 60*(84*b^1 
0*c^3*d^7 - 36*a*b^9*c^2*d^8 + 9*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 84*(1 
26*b^10*c^4*d^6 - 84*a*b^9*c^3*d^7 + 36*a^2*b^8*c^2*d^8 - 9*a^3*b^7*c*d^9 
+ a^4*b^6*d^10)*x^6 + 126*(126*b^10*c^5*d^5 - 126*a*b^9*c^4*d^6 + 84*a^2*b 
^8*c^3*d^7 - 36*a^3*b^7*c^2*d^8 + 9*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 + 21 
0*(84*b^10*c^6*d^4 - 126*a*b^9*c^5*d^5 + 126*a^2*b^8*c^4*d^6 - 84*a^3*b^7* 
c^3*d^7 + 36*a^4*b^6*c^2*d^8 - 9*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 420*( 
36*b^10*c^7*d^3 - 84*a*b^9*c^6*d^4 + 126*a^2*b^8*c^5*d^5 - 126*a^3*b^7*c^4 
*d^6 + 84*a^4*b^6*c^3*d^7 - 36*a^5*b^5*c^2*d^8 + 9*a^6*b^4*c*d^9 - a^7*b^3 
*d^10)*x^3 + 1260*(9*b^10*c^8*d^2 - 36*a*b^9*c^7*d^3 + 84*a^2*b^8*c^6*d^4 
- 126*a^3*b^7*c^5*d^5 + 126*a^4*b^6*c^4*d^6 - 84*a^5*b^5*c^3*d^7 + 36*a^6* 
b^4*c^2*d^8 - 9*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 252*(45*a*b^9*c^8*d^2 
- 240*a^2*b^8*c^7*d^3 + 630*a^3*b^7*c^6*d^4 - 1008*a^4*b^6*c^5*d^5 + 1050* 
a^5*b^5*c^4*d^6 - 720*a^6*b^4*c^3*d^7 + 315*a^7*b^3*c^2*d^8 - 80*a^8*b^2*c 
*d^9 + 9*a^9*b*d^10)*x + 2520*(a*b^9*c^9*d - 9*a^2*b^8*c^8*d^2 + 36*a^3*b^ 
7*c^7*d^3 - 84*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 - 126*a^6*b^4*c^4*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 816 vs. \(2 (240) = 480\).

Time = 1.21 (sec) , antiderivative size = 816, normalized size of antiderivative = 3.16 \[ \int \frac {(c+d x)^{10}}{(a+b x)^2} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)**10/(b*x+a)**2,x)
 

Output:

x**8*(-a*d**10/(4*b**3) + 5*c*d**9/(4*b**2)) + x**7*(3*a**2*d**10/(7*b**4) 
 - 20*a*c*d**9/(7*b**3) + 45*c**2*d**8/(7*b**2)) + x**6*(-2*a**3*d**10/(3* 
b**5) + 5*a**2*c*d**9/b**4 - 15*a*c**2*d**8/b**3 + 20*c**3*d**7/b**2) + x* 
*5*(a**4*d**10/b**6 - 8*a**3*c*d**9/b**5 + 27*a**2*c**2*d**8/b**4 - 48*a*c 
**3*d**7/b**3 + 42*c**4*d**6/b**2) + x**4*(-3*a**5*d**10/(2*b**7) + 25*a** 
4*c*d**9/(2*b**6) - 45*a**3*c**2*d**8/b**5 + 90*a**2*c**3*d**7/b**4 - 105* 
a*c**4*d**6/b**3 + 63*c**5*d**5/b**2) + x**3*(7*a**6*d**10/(3*b**8) - 20*a 
**5*c*d**9/b**7 + 75*a**4*c**2*d**8/b**6 - 160*a**3*c**3*d**7/b**5 + 210*a 
**2*c**4*d**6/b**4 - 168*a*c**5*d**5/b**3 + 70*c**6*d**4/b**2) + x**2*(-4* 
a**7*d**10/b**9 + 35*a**6*c*d**9/b**8 - 135*a**5*c**2*d**8/b**7 + 300*a**4 
*c**3*d**7/b**6 - 420*a**3*c**4*d**6/b**5 + 378*a**2*c**5*d**5/b**4 - 210* 
a*c**6*d**4/b**3 + 60*c**7*d**3/b**2) + x*(9*a**8*d**10/b**10 - 80*a**7*c* 
d**9/b**9 + 315*a**6*c**2*d**8/b**8 - 720*a**5*c**3*d**7/b**7 + 1050*a**4* 
c**4*d**6/b**6 - 1008*a**3*c**5*d**5/b**5 + 630*a**2*c**6*d**4/b**4 - 240* 
a*c**7*d**3/b**3 + 45*c**8*d**2/b**2) + (-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)/(a*b**11 
+ b**12*x) + d**10*x**9/(9*b**2) - 10*d*(a*d - b*c)**9*log(a + b*x)/b**11
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 874 vs. \(2 (252) = 504\).

Time = 0.04 (sec) , antiderivative size = 874, normalized size of antiderivative = 3.39 \[ \int \frac {(c+d x)^{10}}{(a+b x)^2} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^2,x, algorithm="maxima")
 

Output:

-(b^10*c^10 - 10*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 - 120*a^3*b^7*c^7*d^3 + 
210*a^4*b^6*c^6*d^4 - 252*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 - 120*a^7* 
b^3*c^3*d^7 + 45*a^8*b^2*c^2*d^8 - 10*a^9*b*c*d^9 + a^10*d^10)/(b^12*x + a 
*b^11) + 1/252*(28*b^8*d^10*x^9 + 63*(5*b^8*c*d^9 - a*b^7*d^10)*x^8 + 36*( 
45*b^8*c^2*d^8 - 20*a*b^7*c*d^9 + 3*a^2*b^6*d^10)*x^7 + 84*(60*b^8*c^3*d^7 
 - 45*a*b^7*c^2*d^8 + 15*a^2*b^6*c*d^9 - 2*a^3*b^5*d^10)*x^6 + 252*(42*b^8 
*c^4*d^6 - 48*a*b^7*c^3*d^7 + 27*a^2*b^6*c^2*d^8 - 8*a^3*b^5*c*d^9 + a^4*b 
^4*d^10)*x^5 + 126*(126*b^8*c^5*d^5 - 210*a*b^7*c^4*d^6 + 180*a^2*b^6*c^3* 
d^7 - 90*a^3*b^5*c^2*d^8 + 25*a^4*b^4*c*d^9 - 3*a^5*b^3*d^10)*x^4 + 84*(21 
0*b^8*c^6*d^4 - 504*a*b^7*c^5*d^5 + 630*a^2*b^6*c^4*d^6 - 480*a^3*b^5*c^3* 
d^7 + 225*a^4*b^4*c^2*d^8 - 60*a^5*b^3*c*d^9 + 7*a^6*b^2*d^10)*x^3 + 252*( 
60*b^8*c^7*d^3 - 210*a*b^7*c^6*d^4 + 378*a^2*b^6*c^5*d^5 - 420*a^3*b^5*c^4 
*d^6 + 300*a^4*b^4*c^3*d^7 - 135*a^5*b^3*c^2*d^8 + 35*a^6*b^2*c*d^9 - 4*a^ 
7*b*d^10)*x^2 + 252*(45*b^8*c^8*d^2 - 240*a*b^7*c^7*d^3 + 630*a^2*b^6*c^6* 
d^4 - 1008*a^3*b^5*c^5*d^5 + 1050*a^4*b^4*c^4*d^6 - 720*a^5*b^3*c^3*d^7 + 
315*a^6*b^2*c^2*d^8 - 80*a^7*b*c*d^9 + 9*a^8*d^10)*x)/b^10 + 10*(b^9*c^9*d 
 - 9*a*b^8*c^8*d^2 + 36*a^2*b^7*c^7*d^3 - 84*a^3*b^6*c^6*d^4 + 126*a^4*b^5 
*c^5*d^5 - 126*a^5*b^4*c^4*d^6 + 84*a^6*b^3*c^3*d^7 - 36*a^7*b^2*c^2*d^8 + 
 9*a^8*b*c*d^9 - a^9*d^10)*log(b*x + a)/b^11
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1012 vs. \(2 (252) = 504\).

Time = 0.13 (sec) , antiderivative size = 1012, normalized size of antiderivative = 3.92 \[ \int \frac {(c+d x)^{10}}{(a+b x)^2} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^2,x, algorithm="giac")
 

Output:

1/252*(28*d^10 + 315*(b^2*c*d^9 - a*b*d^10)/((b*x + a)*b) + 1620*(b^4*c^2* 
d^8 - 2*a*b^3*c*d^9 + a^2*b^2*d^10)/((b*x + a)^2*b^2) + 5040*(b^6*c^3*d^7 
- 3*a*b^5*c^2*d^8 + 3*a^2*b^4*c*d^9 - a^3*b^3*d^10)/((b*x + a)^3*b^3) + 10 
584*(b^8*c^4*d^6 - 4*a*b^7*c^3*d^7 + 6*a^2*b^6*c^2*d^8 - 4*a^3*b^5*c*d^9 + 
 a^4*b^4*d^10)/((b*x + a)^4*b^4) + 15876*(b^10*c^5*d^5 - 5*a*b^9*c^4*d^6 + 
 10*a^2*b^8*c^3*d^7 - 10*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 - a^5*b^5*d^10) 
/((b*x + a)^5*b^5) + 17640*(b^12*c^6*d^4 - 6*a*b^11*c^5*d^5 + 15*a^2*b^10* 
c^4*d^6 - 20*a^3*b^9*c^3*d^7 + 15*a^4*b^8*c^2*d^8 - 6*a^5*b^7*c*d^9 + a^6* 
b^6*d^10)/((b*x + a)^6*b^6) + 15120*(b^14*c^7*d^3 - 7*a*b^13*c^6*d^4 + 21* 
a^2*b^12*c^5*d^5 - 35*a^3*b^11*c^4*d^6 + 35*a^4*b^10*c^3*d^7 - 21*a^5*b^9* 
c^2*d^8 + 7*a^6*b^8*c*d^9 - a^7*b^7*d^10)/((b*x + a)^7*b^7) + 11340*(b^16* 
c^8*d^2 - 8*a*b^15*c^7*d^3 + 28*a^2*b^14*c^6*d^4 - 56*a^3*b^13*c^5*d^5 + 7 
0*a^4*b^12*c^4*d^6 - 56*a^5*b^11*c^3*d^7 + 28*a^6*b^10*c^2*d^8 - 8*a^7*b^9 
*c*d^9 + a^8*b^8*d^10)/((b*x + a)^8*b^8))*(b*x + a)^9/b^11 - 10*(b^9*c^9*d 
 - 9*a*b^8*c^8*d^2 + 36*a^2*b^7*c^7*d^3 - 84*a^3*b^6*c^6*d^4 + 126*a^4*b^5 
*c^5*d^5 - 126*a^5*b^4*c^4*d^6 + 84*a^6*b^3*c^3*d^7 - 36*a^7*b^2*c^2*d^8 + 
 9*a^8*b*c*d^9 - a^9*d^10)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^11 - ( 
b^19*c^10/(b*x + a) - 10*a*b^18*c^9*d/(b*x + a) + 45*a^2*b^17*c^8*d^2/(b*x 
 + a) - 120*a^3*b^16*c^7*d^3/(b*x + a) + 210*a^4*b^15*c^6*d^4/(b*x + a) - 
252*a^5*b^14*c^5*d^5/(b*x + a) + 210*a^6*b^13*c^4*d^6/(b*x + a) - 120*a...
 

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 3475, normalized size of antiderivative = 13.47 \[ \int \frac {(c+d x)^{10}}{(a+b x)^2} \, dx=\text {Too large to display} \] Input:

int((c + d*x)^10/(a + b*x)^2,x)
 

Output:

x^7*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/(7*b) - (a^2*d^10)/(7*b^4) + 
(45*c^2*d^8)/(7*b^2)) - x^5*((2*a*((120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^ 
10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^ 
2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/(5*b) - (42*c^4*d^6)/b^2 + (a^2 
*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8 
)/b^2))/(5*b^2)) - x^8*((a*d^10)/(4*b^3) - (5*c*d^9)/(4*b^2)) + x^3*((70*c 
^6*d^4)/b^2 - (2*a*((2*a*((2*a*((120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10) 
/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*( 
(2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/b - (210*c^4*d^6)/b^2 + (a^2*((2*a 
*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2) 
)/b^2))/b - (a^2*((120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 - (10*c*d 
^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*((2*a*d^10)/b^3 
 - (10*c*d^9)/b^2))/b^2))/b^2 + (252*c^5*d^5)/b^2))/(3*b) + (a^2*((2*a*((1 
20*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d 
^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/ 
b^2))/b - (210*c^4*d^6)/b^2 + (a^2*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2) 
)/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b^2))/(3*b^2)) - x^2*((a*((210*c 
^6*d^4)/b^2 - (2*a*((2*a*((2*a*((120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10) 
/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*( 
(2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/b - (210*c^4*d^6)/b^2 + (a^2*((...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1255, normalized size of antiderivative = 4.86 \[ \int \frac {(c+d x)^{10}}{(a+b x)^2} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^10/(b*x+a)^2,x)
 

Output:

( - 2520*log(a + b*x)*a**11*d**10 + 22680*log(a + b*x)*a**10*b*c*d**9 - 25 
20*log(a + b*x)*a**10*b*d**10*x - 90720*log(a + b*x)*a**9*b**2*c**2*d**8 + 
 22680*log(a + b*x)*a**9*b**2*c*d**9*x + 211680*log(a + b*x)*a**8*b**3*c** 
3*d**7 - 90720*log(a + b*x)*a**8*b**3*c**2*d**8*x - 317520*log(a + b*x)*a* 
*7*b**4*c**4*d**6 + 211680*log(a + b*x)*a**7*b**4*c**3*d**7*x + 317520*log 
(a + b*x)*a**6*b**5*c**5*d**5 - 317520*log(a + b*x)*a**6*b**5*c**4*d**6*x 
- 211680*log(a + b*x)*a**5*b**6*c**6*d**4 + 317520*log(a + b*x)*a**5*b**6* 
c**5*d**5*x + 90720*log(a + b*x)*a**4*b**7*c**7*d**3 - 211680*log(a + b*x) 
*a**4*b**7*c**6*d**4*x - 22680*log(a + b*x)*a**3*b**8*c**8*d**2 + 90720*lo 
g(a + b*x)*a**3*b**8*c**7*d**3*x + 2520*log(a + b*x)*a**2*b**9*c**9*d - 22 
680*log(a + b*x)*a**2*b**9*c**8*d**2*x + 2520*log(a + b*x)*a*b**10*c**9*d* 
x + 2520*a**10*b*d**10*x - 22680*a**9*b**2*c*d**9*x + 1260*a**9*b**2*d**10 
*x**2 + 90720*a**8*b**3*c**2*d**8*x - 11340*a**8*b**3*c*d**9*x**2 - 420*a* 
*8*b**3*d**10*x**3 - 211680*a**7*b**4*c**3*d**7*x + 45360*a**7*b**4*c**2*d 
**8*x**2 + 3780*a**7*b**4*c*d**9*x**3 + 210*a**7*b**4*d**10*x**4 + 317520* 
a**6*b**5*c**4*d**6*x - 105840*a**6*b**5*c**3*d**7*x**2 - 15120*a**6*b**5* 
c**2*d**8*x**3 - 1890*a**6*b**5*c*d**9*x**4 - 126*a**6*b**5*d**10*x**5 - 3 
17520*a**5*b**6*c**5*d**5*x + 158760*a**5*b**6*c**4*d**6*x**2 + 35280*a**5 
*b**6*c**3*d**7*x**3 + 7560*a**5*b**6*c**2*d**8*x**4 + 1134*a**5*b**6*c*d* 
*9*x**5 + 84*a**5*b**6*d**10*x**6 + 211680*a**4*b**7*c**6*d**4*x - 1587...