\(\int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx\) [112]

Optimal result

Integrand size = 15, antiderivative size = 262 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=\frac {120 d^7 (b c-a d)^3 x}{b^{10}}-\frac {(b c-a d)^{10}}{6 b^{11} (a+b x)^6}-\frac {2 d (b c-a d)^9}{b^{11} (a+b x)^5}-\frac {45 d^2 (b c-a d)^8}{4 b^{11} (a+b x)^4}-\frac {40 d^3 (b c-a d)^7}{b^{11} (a+b x)^3}-\frac {105 d^4 (b c-a d)^6}{b^{11} (a+b x)^2}-\frac {252 d^5 (b c-a d)^5}{b^{11} (a+b x)}+\frac {45 d^8 (b c-a d)^2 (a+b x)^2}{2 b^{11}}+\frac {10 d^9 (b c-a d) (a+b x)^3}{3 b^{11}}+\frac {d^{10} (a+b x)^4}{4 b^{11}}+\frac {210 d^6 (b c-a d)^4 \log (a+b x)}{b^{11}} \] Output:

120*d^7*(-a*d+b*c)^3*x/b^10-1/6*(-a*d+b*c)^10/b^11/(b*x+a)^6-2*d*(-a*d+b*c 
)^9/b^11/(b*x+a)^5-45/4*d^2*(-a*d+b*c)^8/b^11/(b*x+a)^4-40*d^3*(-a*d+b*c)^ 
7/b^11/(b*x+a)^3-105*d^4*(-a*d+b*c)^6/b^11/(b*x+a)^2-252*d^5*(-a*d+b*c)^5/ 
b^11/(b*x+a)+45/2*d^8*(-a*d+b*c)^2*(b*x+a)^2/b^11+10/3*d^9*(-a*d+b*c)*(b*x 
+a)^3/b^11+1/4*d^10*(b*x+a)^4/b^11+210*d^6*(-a*d+b*c)^4*ln(b*x+a)/b^11
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.01 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=\frac {12 b d^7 \left (120 b^3 c^3-315 a b^2 c^2 d+280 a^2 b c d^2-84 a^3 d^3\right ) x+6 b^2 d^8 \left (45 b^2 c^2-70 a b c d+28 a^2 d^2\right ) x^2+4 b^3 d^9 (10 b c-7 a d) x^3+3 b^4 d^{10} x^4-\frac {2 (b c-a d)^{10}}{(a+b x)^6}+\frac {24 d (-b c+a d)^9}{(a+b x)^5}-\frac {135 d^2 (b c-a d)^8}{(a+b x)^4}+\frac {480 d^3 (-b c+a d)^7}{(a+b x)^3}-\frac {1260 d^4 (b c-a d)^6}{(a+b x)^2}+\frac {3024 d^5 (-b c+a d)^5}{a+b x}+2520 d^6 (b c-a d)^4 \log (a+b x)}{12 b^{11}} \] Input:

Integrate[(c + d*x)^10/(a + b*x)^7,x]
 

Output:

(12*b*d^7*(120*b^3*c^3 - 315*a*b^2*c^2*d + 280*a^2*b*c*d^2 - 84*a^3*d^3)*x 
 + 6*b^2*d^8*(45*b^2*c^2 - 70*a*b*c*d + 28*a^2*d^2)*x^2 + 4*b^3*d^9*(10*b* 
c - 7*a*d)*x^3 + 3*b^4*d^10*x^4 - (2*(b*c - a*d)^10)/(a + b*x)^6 + (24*d*( 
-(b*c) + a*d)^9)/(a + b*x)^5 - (135*d^2*(b*c - a*d)^8)/(a + b*x)^4 + (480* 
d^3*(-(b*c) + a*d)^7)/(a + b*x)^3 - (1260*d^4*(b*c - a*d)^6)/(a + b*x)^2 + 
 (3024*d^5*(-(b*c) + a*d)^5)/(a + b*x) + 2520*d^6*(b*c - a*d)^4*Log[a + b* 
x])/(12*b^11)
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 262, 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.

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {10 d^9 (a+b x)^3 (b c-a d)}{3 b^{11}}+\frac {45 d^8 (a+b x)^2 (b c-a d)^2}{2 b^{11}}+\frac {210 d^6 (b c-a d)^4 \log (a+b x)}{b^{11}}-\frac {252 d^5 (b c-a d)^5}{b^{11} (a+b x)}-\frac {105 d^4 (b c-a d)^6}{b^{11} (a+b x)^2}-\frac {40 d^3 (b c-a d)^7}{b^{11} (a+b x)^3}-\frac {45 d^2 (b c-a d)^8}{4 b^{11} (a+b x)^4}-\frac {2 d (b c-a d)^9}{b^{11} (a+b x)^5}-\frac {(b c-a d)^{10}}{6 b^{11} (a+b x)^6}+\frac {d^{10} (a+b x)^4}{4 b^{11}}+\frac {120 d^7 x (b c-a d)^3}{b^{10}}\)

Input:

Int[(c + d*x)^10/(a + b*x)^7,x]
 

Output:

(120*d^7*(b*c - a*d)^3*x)/b^10 - (b*c - a*d)^10/(6*b^11*(a + b*x)^6) - (2* 
d*(b*c - a*d)^9)/(b^11*(a + b*x)^5) - (45*d^2*(b*c - a*d)^8)/(4*b^11*(a + 
b*x)^4) - (40*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^3) - (105*d^4*(b*c - a*d) 
^6)/(b^11*(a + b*x)^2) - (252*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)) + (45*d^ 
8*(b*c - a*d)^2*(a + b*x)^2)/(2*b^11) + (10*d^9*(b*c - a*d)*(a + b*x)^3)/( 
3*b^11) + (d^10*(a + b*x)^4)/(4*b^11) + (210*d^6*(b*c - a*d)^4*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. \(849\) vs. \(2(252)=504\).

Time = 0.14 (sec) , antiderivative size = 850, normalized size of antiderivative = 3.24

method	result	size
norman	\(\frac {\frac {6174 a^{10} d^{10}-24696 a^{9} b c \,d^{9}+37044 a^{8} b^{2} c^{2} d^{8}-24696 a^{7} b^{3} c^{3} d^{7}+6174 a^{6} b^{4} c^{4} d^{6}-504 a^{5} b^{5} c^{5} d^{5}-84 a^{4} b^{6} c^{6} d^{4}-24 a^{3} b^{7} c^{7} d^{3}-9 a^{2} b^{8} c^{8} d^{2}-4 a \,b^{9} c^{9} d -2 b^{10} c^{10}}{12 b^{11}}+\frac {d^{10} x^{10}}{4 b}+\frac {6 \left (210 a^{5} d^{10}-840 a^{4} b c \,d^{9}+1260 a^{3} b^{2} c^{2} d^{8}-840 a^{2} b^{3} c^{3} d^{7}+210 a \,b^{4} c^{4} d^{6}-42 b^{5} c^{5} d^{5}\right ) x^{5}}{b^{6}}+\frac {15 \left (315 a^{6} d^{10}-1260 a^{5} b c \,d^{9}+1890 a^{4} b^{2} c^{2} d^{8}-1260 a^{3} b^{3} c^{3} d^{7}+315 a^{2} b^{4} c^{4} d^{6}-42 a \,b^{5} c^{5} d^{5}-7 b^{6} c^{6} d^{4}\right ) x^{4}}{b^{7}}+\frac {20 \left (385 a^{7} d^{10}-1540 a^{6} b c \,d^{9}+2310 a^{5} b^{2} c^{2} d^{8}-1540 a^{4} b^{3} c^{3} d^{7}+385 a^{3} b^{4} c^{4} d^{6}-42 a^{2} b^{5} c^{5} d^{5}-7 a \,b^{6} c^{6} d^{4}-2 b^{7} c^{7} d^{3}\right ) x^{3}}{b^{8}}+\frac {15 \left (1750 a^{8} d^{10}-7000 a^{7} b c \,d^{9}+10500 a^{6} b^{2} c^{2} d^{8}-7000 a^{5} b^{3} c^{3} d^{7}+1750 a^{4} b^{4} c^{4} d^{6}-168 a^{3} b^{5} c^{5} d^{5}-28 a^{2} b^{6} c^{6} d^{4}-8 a \,b^{7} c^{7} d^{3}-3 b^{8} c^{8} d^{2}\right ) x^{2}}{4 b^{9}}+\frac {\left (5754 a^{9} d^{10}-23016 a^{8} b c \,d^{9}+34524 a^{7} b^{2} c^{2} d^{8}-23016 a^{6} b^{3} c^{3} d^{7}+5754 a^{5} b^{4} c^{4} d^{6}-504 a^{4} b^{5} c^{5} d^{5}-84 a^{3} b^{6} c^{6} d^{4}-24 a^{2} b^{7} c^{7} d^{3}-9 a \,b^{8} c^{8} d^{2}-4 b^{9} c^{9} d \right ) x}{2 b^{10}}-\frac {30 d^{7} \left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) x^{7}}{b^{4}}+\frac {15 d^{8} \left (a^{2} d^{2}-4 a b c d +6 b^{2} c^{2}\right ) x^{8}}{4 b^{3}}-\frac {5 d^{9} \left (a d -4 b c \right ) x^{9}}{6 b^{2}}}{\left (b x +a \right )^{6}}+\frac {210 d^{6} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) \ln \left (b x +a \right )}{b^{11}}\)	\(850\)
default	\(-\frac {d^{7} \left (-\frac {1}{4} d^{3} x^{4} b^{3}+\frac {7}{3} x^{3} a \,b^{2} d^{3}-\frac {10}{3} x^{3} b^{3} c \,d^{2}-14 x^{2} a^{2} b \,d^{3}+35 x^{2} a \,b^{2} c \,d^{2}-\frac {45}{2} x^{2} b^{3} c^{2} d +84 a^{3} d^{3} x -280 a^{2} b c \,d^{2} x +315 a \,b^{2} c^{2} d x -120 b^{3} c^{3} x \right )}{b^{10}}+\frac {2 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -c^{9} b^{9}\right )}{b^{11} \left (b x +a \right )^{5}}-\frac {45 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +c^{8} b^{8}\right )}{4 b^{11} \left (b x +a \right )^{4}}-\frac {105 d^{4} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +c^{6} b^{6}\right )}{b^{11} \left (b x +a \right )^{2}}+\frac {252 d^{5} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}{b^{11} \left (b x +a \right )}+\frac {210 d^{6} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) \ln \left (b x +a \right )}{b^{11}}+\frac {40 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{b^{11} \left (b x +a \right )^{3}}-\frac {a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}}{6 b^{11} \left (b x +a \right )^{6}}\)	\(862\)
risch	\(\frac {d^{10} x^{4}}{4 b^{7}}-\frac {7 d^{10} x^{3} a}{3 b^{8}}+\frac {10 d^{9} x^{3} c}{3 b^{7}}+\frac {14 d^{10} x^{2} a^{2}}{b^{9}}-\frac {35 d^{9} x^{2} a c}{b^{8}}+\frac {45 d^{8} x^{2} c^{2}}{2 b^{7}}-\frac {84 d^{10} a^{3} x}{b^{10}}+\frac {280 d^{9} a^{2} c x}{b^{9}}-\frac {315 d^{8} a \,c^{2} x}{b^{8}}+\frac {120 d^{7} c^{3} x}{b^{7}}+\frac {\left (252 a^{5} b^{4} d^{10}-1260 a^{4} b^{5} c \,d^{9}+2520 a^{3} b^{6} c^{2} d^{8}-2520 a^{2} b^{7} c^{3} d^{7}+1260 a \,b^{8} c^{4} d^{6}-252 b^{9} c^{5} d^{5}\right ) x^{5}+105 b^{3} d^{4} \left (11 a^{6} d^{6}-54 a^{5} b c \,d^{5}+105 a^{4} b^{2} c^{2} d^{4}-100 a^{3} b^{3} c^{3} d^{3}+45 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d -c^{6} b^{6}\right ) x^{4}+20 b^{2} d^{3} \left (107 a^{7} d^{7}-518 a^{6} b c \,d^{6}+987 a^{5} b^{2} c^{2} d^{5}-910 a^{4} b^{3} c^{3} d^{4}+385 a^{3} b^{4} c^{4} d^{3}-42 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d -2 b^{7} c^{7}\right ) x^{3}+\frac {15 b \,d^{2} \left (533 a^{8} d^{8}-2552 a^{7} b c \,d^{7}+4788 a^{6} b^{2} c^{2} d^{6}-4312 a^{5} b^{3} c^{3} d^{5}+1750 a^{4} b^{4} c^{4} d^{4}-168 a^{3} b^{5} c^{5} d^{3}-28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d -3 c^{8} b^{8}\right ) x^{2}}{4}+\frac {d \left (1879 a^{9} d^{9}-8916 a^{8} b c \,d^{8}+16524 a^{7} b^{2} c^{2} d^{7}-14616 a^{6} b^{3} c^{3} d^{6}+5754 a^{5} b^{4} c^{4} d^{5}-504 a^{4} b^{5} c^{5} d^{4}-84 a^{3} b^{6} c^{6} d^{3}-24 a^{2} b^{7} c^{7} d^{2}-9 a \,b^{8} c^{8} d -4 c^{9} b^{9}\right ) x}{2}+\frac {2131 a^{10} d^{10}-10036 a^{9} b c \,d^{9}+18414 a^{8} b^{2} c^{2} d^{8}-16056 a^{7} b^{3} c^{3} d^{7}+6174 a^{6} b^{4} c^{4} d^{6}-504 a^{5} b^{5} c^{5} d^{5}-84 a^{4} b^{6} c^{6} d^{4}-24 a^{3} b^{7} c^{7} d^{3}-9 a^{2} b^{8} c^{8} d^{2}-4 a \,b^{9} c^{9} d -2 b^{10} c^{10}}{12 b}}{b^{10} \left (b x +a \right )^{6}}+\frac {210 d^{10} \ln \left (b x +a \right ) a^{4}}{b^{11}}-\frac {840 d^{9} \ln \left (b x +a \right ) a^{3} c}{b^{10}}+\frac {1260 d^{8} \ln \left (b x +a \right ) a^{2} c^{2}}{b^{9}}-\frac {840 d^{7} \ln \left (b x +a \right ) a \,c^{3}}{b^{8}}+\frac {210 d^{6} \ln \left (b x +a \right ) c^{4}}{b^{7}}\)	\(884\)
parallelrisch	\(\text {Expression too large to display}\)	\(1619\)

Input:

int((d*x+c)^10/(b*x+a)^7,x,method=_RETURNVERBOSE)
 

Output:

(1/12*(6174*a^10*d^10-24696*a^9*b*c*d^9+37044*a^8*b^2*c^2*d^8-24696*a^7*b^ 
3*c^3*d^7+6174*a^6*b^4*c^4*d^6-504*a^5*b^5*c^5*d^5-84*a^4*b^6*c^6*d^4-24*a 
^3*b^7*c^7*d^3-9*a^2*b^8*c^8*d^2-4*a*b^9*c^9*d-2*b^10*c^10)/b^11+1/4/b*d^1 
0*x^10+6*(210*a^5*d^10-840*a^4*b*c*d^9+1260*a^3*b^2*c^2*d^8-840*a^2*b^3*c^ 
3*d^7+210*a*b^4*c^4*d^6-42*b^5*c^5*d^5)/b^6*x^5+15*(315*a^6*d^10-1260*a^5* 
b*c*d^9+1890*a^4*b^2*c^2*d^8-1260*a^3*b^3*c^3*d^7+315*a^2*b^4*c^4*d^6-42*a 
*b^5*c^5*d^5-7*b^6*c^6*d^4)/b^7*x^4+20*(385*a^7*d^10-1540*a^6*b*c*d^9+2310 
*a^5*b^2*c^2*d^8-1540*a^4*b^3*c^3*d^7+385*a^3*b^4*c^4*d^6-42*a^2*b^5*c^5*d 
^5-7*a*b^6*c^6*d^4-2*b^7*c^7*d^3)/b^8*x^3+15/4*(1750*a^8*d^10-7000*a^7*b*c 
*d^9+10500*a^6*b^2*c^2*d^8-7000*a^5*b^3*c^3*d^7+1750*a^4*b^4*c^4*d^6-168*a 
^3*b^5*c^5*d^5-28*a^2*b^6*c^6*d^4-8*a*b^7*c^7*d^3-3*b^8*c^8*d^2)/b^9*x^2+1 
/2*(5754*a^9*d^10-23016*a^8*b*c*d^9+34524*a^7*b^2*c^2*d^8-23016*a^6*b^3*c^ 
3*d^7+5754*a^5*b^4*c^4*d^6-504*a^4*b^5*c^5*d^5-84*a^3*b^6*c^6*d^4-24*a^2*b 
^7*c^7*d^3-9*a*b^8*c^8*d^2-4*b^9*c^9*d)/b^10*x-30*d^7*(a^3*d^3-4*a^2*b*c*d 
^2+6*a*b^2*c^2*d-4*b^3*c^3)/b^4*x^7+15/4*d^8*(a^2*d^2-4*a*b*c*d+6*b^2*c^2) 
/b^3*x^8-5/6*d^9*(a*d-4*b*c)/b^2*x^9)/(b*x+a)^6+210/b^11*d^6*(a^4*d^4-4*a^ 
3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*ln(b*x+a)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1386 vs. \(2 (252) = 504\).

Time = 0.09 (sec) , antiderivative size = 1386, normalized size of antiderivative = 5.29 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^7,x, algorithm="fricas")
 

Output:

1/12*(3*b^10*d^10*x^10 - 2*b^10*c^10 - 4*a*b^9*c^9*d - 9*a^2*b^8*c^8*d^2 - 
 24*a^3*b^7*c^7*d^3 - 84*a^4*b^6*c^6*d^4 - 504*a^5*b^5*c^5*d^5 + 6174*a^6* 
b^4*c^4*d^6 - 16056*a^7*b^3*c^3*d^7 + 18414*a^8*b^2*c^2*d^8 - 10036*a^9*b* 
c*d^9 + 2131*a^10*d^10 + 10*(4*b^10*c*d^9 - a*b^9*d^10)*x^9 + 45*(6*b^10*c 
^2*d^8 - 4*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 360*(4*b^10*c^3*d^7 - 6*a*b^9 
*c^2*d^8 + 4*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + (8640*a*b^9*c^3*d^7 - 186 
30*a^2*b^8*c^2*d^8 + 14660*a^3*b^7*c*d^9 - 4043*a^4*b^6*d^10)*x^6 - 6*(504 
*b^10*c^5*d^5 - 2520*a*b^9*c^4*d^6 + 1440*a^2*b^8*c^3*d^7 + 3510*a^3*b^7*c 
^2*d^8 - 4580*a^4*b^6*c*d^9 + 1523*a^5*b^5*d^10)*x^5 - 15*(84*b^10*c^6*d^4 
 + 504*a*b^9*c^5*d^5 - 3780*a^2*b^8*c^4*d^6 + 6480*a^3*b^7*c^3*d^7 - 4050* 
a^4*b^6*c^2*d^8 + 460*a^5*b^5*c*d^9 + 263*a^6*b^4*d^10)*x^4 - 20*(24*b^10* 
c^7*d^3 + 84*a*b^9*c^6*d^4 + 504*a^2*b^8*c^5*d^5 - 4620*a^3*b^7*c^4*d^6 + 
9840*a^4*b^6*c^3*d^7 - 9090*a^5*b^5*c^2*d^8 + 3820*a^6*b^4*c*d^9 - 577*a^7 
*b^3*d^10)*x^3 - 15*(9*b^10*c^8*d^2 + 24*a*b^9*c^7*d^3 + 84*a^2*b^8*c^6*d^ 
4 + 504*a^3*b^7*c^5*d^5 - 5250*a^4*b^6*c^4*d^6 + 12360*a^5*b^5*c^3*d^7 - 1 
2870*a^6*b^4*c^2*d^8 + 6340*a^7*b^3*c*d^9 - 1207*a^8*b^2*d^10)*x^2 - 6*(4* 
b^10*c^9*d + 9*a*b^9*c^8*d^2 + 24*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 5 
04*a^4*b^6*c^5*d^5 - 5754*a^5*b^5*c^4*d^6 + 14376*a^6*b^4*c^3*d^7 - 15894* 
a^7*b^3*c^2*d^8 + 8356*a^8*b^2*c*d^9 - 1711*a^9*b*d^10)*x + 2520*(a^6*b^4* 
c^4*d^6 - 4*a^7*b^3*c^3*d^7 + 6*a^8*b^2*c^2*d^8 - 4*a^9*b*c*d^9 + a^10*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**10/(b*x+a)**7,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 925 vs. \(2 (252) = 504\).

Time = 0.10 (sec) , antiderivative size = 925, normalized size of antiderivative = 3.53 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^7,x, algorithm="maxima")
 

Output:

-1/12*(2*b^10*c^10 + 4*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^ 
3 + 84*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^5*d^5 - 6174*a^6*b^4*c^4*d^6 + 1605 
6*a^7*b^3*c^3*d^7 - 18414*a^8*b^2*c^2*d^8 + 10036*a^9*b*c*d^9 - 2131*a^10* 
d^10 + 3024*(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)*x^5 + 1260*(b^10*c^6*d^4 + 6 
*a*b^9*c^5*d^5 - 45*a^2*b^8*c^4*d^6 + 100*a^3*b^7*c^3*d^7 - 105*a^4*b^6*c^ 
2*d^8 + 54*a^5*b^5*c*d^9 - 11*a^6*b^4*d^10)*x^4 + 240*(2*b^10*c^7*d^3 + 7* 
a*b^9*c^6*d^4 + 42*a^2*b^8*c^5*d^5 - 385*a^3*b^7*c^4*d^6 + 910*a^4*b^6*c^3 
*d^7 - 987*a^5*b^5*c^2*d^8 + 518*a^6*b^4*c*d^9 - 107*a^7*b^3*d^10)*x^3 + 4 
5*(3*b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 168*a^3*b^7*c^5 
*d^5 - 1750*a^4*b^6*c^4*d^6 + 4312*a^5*b^5*c^3*d^7 - 4788*a^6*b^4*c^2*d^8 
+ 2552*a^7*b^3*c*d^9 - 533*a^8*b^2*d^10)*x^2 + 6*(4*b^10*c^9*d + 9*a*b^9*c 
^8*d^2 + 24*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 504*a^4*b^6*c^5*d^5 - 5 
754*a^5*b^5*c^4*d^6 + 14616*a^6*b^4*c^3*d^7 - 16524*a^7*b^3*c^2*d^8 + 8916 
*a^8*b^2*c*d^9 - 1879*a^9*b*d^10)*x)/(b^17*x^6 + 6*a*b^16*x^5 + 15*a^2*b^1 
5*x^4 + 20*a^3*b^14*x^3 + 15*a^4*b^13*x^2 + 6*a^5*b^12*x + a^6*b^11) + 1/1 
2*(3*b^3*d^10*x^4 + 4*(10*b^3*c*d^9 - 7*a*b^2*d^10)*x^3 + 6*(45*b^3*c^2*d^ 
8 - 70*a*b^2*c*d^9 + 28*a^2*b*d^10)*x^2 + 12*(120*b^3*c^3*d^7 - 315*a*b^2* 
c^2*d^8 + 280*a^2*b*c*d^9 - 84*a^3*d^10)*x)/b^10 + 210*(b^4*c^4*d^6 - 4*a* 
b^3*c^3*d^7 + 6*a^2*b^2*c^2*d^8 - 4*a^3*b*c*d^9 + a^4*d^10)*log(b*x + a...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 878 vs. \(2 (252) = 504\).

Time = 0.12 (sec) , antiderivative size = 878, normalized size of antiderivative = 3.35 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^7,x, algorithm="giac")
 

Output:

210*(b^4*c^4*d^6 - 4*a*b^3*c^3*d^7 + 6*a^2*b^2*c^2*d^8 - 4*a^3*b*c*d^9 + a 
^4*d^10)*log(abs(b*x + a))/b^11 - 1/12*(2*b^10*c^10 + 4*a*b^9*c^9*d + 9*a^ 
2*b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^5* 
d^5 - 6174*a^6*b^4*c^4*d^6 + 16056*a^7*b^3*c^3*d^7 - 18414*a^8*b^2*c^2*d^8 
 + 10036*a^9*b*c*d^9 - 2131*a^10*d^10 + 3024*(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)*x^5 + 1260*(b^10*c^6*d^4 + 6*a*b^9*c^5*d^5 - 45*a^2*b^8*c^4*d^6 + 100 
*a^3*b^7*c^3*d^7 - 105*a^4*b^6*c^2*d^8 + 54*a^5*b^5*c*d^9 - 11*a^6*b^4*d^1 
0)*x^4 + 240*(2*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 42*a^2*b^8*c^5*d^5 - 385* 
a^3*b^7*c^4*d^6 + 910*a^4*b^6*c^3*d^7 - 987*a^5*b^5*c^2*d^8 + 518*a^6*b^4* 
c*d^9 - 107*a^7*b^3*d^10)*x^3 + 45*(3*b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 + 28* 
a^2*b^8*c^6*d^4 + 168*a^3*b^7*c^5*d^5 - 1750*a^4*b^6*c^4*d^6 + 4312*a^5*b^ 
5*c^3*d^7 - 4788*a^6*b^4*c^2*d^8 + 2552*a^7*b^3*c*d^9 - 533*a^8*b^2*d^10)* 
x^2 + 6*(4*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 24*a^2*b^8*c^7*d^3 + 84*a^3*b^7* 
c^6*d^4 + 504*a^4*b^6*c^5*d^5 - 5754*a^5*b^5*c^4*d^6 + 14616*a^6*b^4*c^3*d 
^7 - 16524*a^7*b^3*c^2*d^8 + 8916*a^8*b^2*c*d^9 - 1879*a^9*b*d^10)*x)/((b* 
x + a)^6*b^11) + 1/12*(3*b^21*d^10*x^4 + 40*b^21*c*d^9*x^3 - 28*a*b^20*d^1 
0*x^3 + 270*b^21*c^2*d^8*x^2 - 420*a*b^20*c*d^9*x^2 + 168*a^2*b^19*d^10*x^ 
2 + 1440*b^21*c^3*d^7*x - 3780*a*b^20*c^2*d^8*x + 3360*a^2*b^19*c*d^9*x - 
1008*a^3*b^18*d^10*x)/b^28
 

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 997, normalized size of antiderivative = 3.81 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx =\text {Too large to display} \] Input:

int((c + d*x)^10/(a + b*x)^7,x)
 

Output:

x^2*((7*a*((7*a*d^10)/b^8 - (10*c*d^9)/b^7))/(2*b) - (21*a^2*d^10)/(2*b^9) 
 + (45*c^2*d^8)/(2*b^7)) - (x^4*(105*b^9*c^6*d^4 - 1155*a^6*b^3*d^10 + 630 
*a*b^8*c^5*d^5 + 5670*a^5*b^4*c*d^9 - 4725*a^2*b^7*c^4*d^6 + 10500*a^3*b^6 
*c^3*d^7 - 11025*a^4*b^5*c^2*d^8) + (2*b^10*c^10 - 2131*a^10*d^10 + 9*a^2* 
b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^5*d^ 
5 - 6174*a^6*b^4*c^4*d^6 + 16056*a^7*b^3*c^3*d^7 - 18414*a^8*b^2*c^2*d^8 + 
 4*a*b^9*c^9*d + 10036*a^9*b*c*d^9)/(12*b) + x*(2*b^9*c^9*d - (1879*a^9*d^ 
10)/2 + (9*a*b^8*c^8*d^2)/2 + 12*a^2*b^7*c^7*d^3 + 42*a^3*b^6*c^6*d^4 + 25 
2*a^4*b^5*c^5*d^5 - 2877*a^5*b^4*c^4*d^6 + 7308*a^6*b^3*c^3*d^7 - 8262*a^7 
*b^2*c^2*d^8 + 4458*a^8*b*c*d^9) + x^3*(40*b^9*c^7*d^3 - 2140*a^7*b^2*d^10 
 + 140*a*b^8*c^6*d^4 + 10360*a^6*b^3*c*d^9 + 840*a^2*b^7*c^5*d^5 - 7700*a^ 
3*b^6*c^4*d^6 + 18200*a^4*b^5*c^3*d^7 - 19740*a^5*b^4*c^2*d^8) + x^2*((45* 
b^9*c^8*d^2)/4 - (7995*a^8*b*d^10)/4 + 30*a*b^8*c^7*d^3 + 9570*a^7*b^2*c*d 
^9 + 105*a^2*b^7*c^6*d^4 + 630*a^3*b^6*c^5*d^5 - (13125*a^4*b^5*c^4*d^6)/2 
 + 16170*a^5*b^4*c^3*d^7 - 17955*a^6*b^3*c^2*d^8) - x^5*(252*a^5*b^4*d^10 
- 252*b^9*c^5*d^5 + 1260*a*b^8*c^4*d^6 - 1260*a^4*b^5*c*d^9 - 2520*a^2*b^7 
*c^3*d^7 + 2520*a^3*b^6*c^2*d^8))/(a^6*b^10 + b^16*x^6 + 6*a^5*b^11*x + 6* 
a*b^15*x^5 + 15*a^4*b^12*x^2 + 20*a^3*b^13*x^3 + 15*a^2*b^14*x^4) - x^3*(( 
7*a*d^10)/(3*b^8) - (10*c*d^9)/(3*b^7)) - x*((7*a*((7*a*((7*a*d^10)/b^8 - 
(10*c*d^9)/b^7))/b - (21*a^2*d^10)/b^9 + (45*c^2*d^8)/b^7))/b + (35*a^3...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1626, normalized size of antiderivative = 6.21 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^10/(b*x+a)^7,x)
                                                                                    
                                                                                    
 

Output:

(2520*log(a + b*x)*a**11*d**10 - 10080*log(a + b*x)*a**10*b*c*d**9 + 15120 
*log(a + b*x)*a**10*b*d**10*x + 15120*log(a + b*x)*a**9*b**2*c**2*d**8 - 6 
0480*log(a + b*x)*a**9*b**2*c*d**9*x + 37800*log(a + b*x)*a**9*b**2*d**10* 
x**2 - 10080*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 - 151200*log(a + b*x)*a**8*b**3*c*d**9*x**2 + 50400*log(a + 
 b*x)*a**8*b**3*d**10*x**3 + 2520*log(a + b*x)*a**7*b**4*c**4*d**6 - 60480 
*log(a + b*x)*a**7*b**4*c**3*d**7*x + 226800*log(a + b*x)*a**7*b**4*c**2*d 
**8*x**2 - 201600*log(a + b*x)*a**7*b**4*c*d**9*x**3 + 37800*log(a + b*x)* 
a**7*b**4*d**10*x**4 + 15120*log(a + b*x)*a**6*b**5*c**4*d**6*x - 151200*l 
og(a + b*x)*a**6*b**5*c**3*d**7*x**2 + 302400*log(a + b*x)*a**6*b**5*c**2* 
d**8*x**3 - 151200*log(a + b*x)*a**6*b**5*c*d**9*x**4 + 15120*log(a + b*x) 
*a**6*b**5*d**10*x**5 + 37800*log(a + b*x)*a**5*b**6*c**4*d**6*x**2 - 2016 
00*log(a + b*x)*a**5*b**6*c**3*d**7*x**3 + 226800*log(a + b*x)*a**5*b**6*c 
**2*d**8*x**4 - 60480*log(a + b*x)*a**5*b**6*c*d**9*x**5 + 2520*log(a + b* 
x)*a**5*b**6*d**10*x**6 + 50400*log(a + b*x)*a**4*b**7*c**4*d**6*x**3 - 15 
1200*log(a + b*x)*a**4*b**7*c**3*d**7*x**4 + 90720*log(a + b*x)*a**4*b**7* 
c**2*d**8*x**5 - 10080*log(a + b*x)*a**4*b**7*c*d**9*x**6 + 37800*log(a + 
b*x)*a**3*b**8*c**4*d**6*x**4 - 60480*log(a + b*x)*a**3*b**8*c**3*d**7*x** 
5 + 15120*log(a + b*x)*a**3*b**8*c**2*d**8*x**6 + 15120*log(a + b*x)*a**2* 
b**9*c**4*d**6*x**5 - 10080*log(a + b*x)*a**2*b**9*c**3*d**7*x**6 + 252...