\(\int \frac {(c+d x)^7}{(a+b x)^{15}} \, dx\) [91]

Optimal result

Integrand size = 15, antiderivative size = 200 \[ \int \frac {(c+d x)^7}{(a+b x)^{15}} \, dx=-\frac {(b c-a d)^7}{14 b^8 (a+b x)^{14}}-\frac {7 d (b c-a d)^6}{13 b^8 (a+b x)^{13}}-\frac {7 d^2 (b c-a d)^5}{4 b^8 (a+b x)^{12}}-\frac {35 d^3 (b c-a d)^4}{11 b^8 (a+b x)^{11}}-\frac {7 d^4 (b c-a d)^3}{2 b^8 (a+b x)^{10}}-\frac {7 d^5 (b c-a d)^2}{3 b^8 (a+b x)^9}-\frac {7 d^6 (b c-a d)}{8 b^8 (a+b x)^8}-\frac {d^7}{7 b^8 (a+b x)^7} \] Output:

-1/14*(-a*d+b*c)^7/b^8/(b*x+a)^14-7/13*d*(-a*d+b*c)^6/b^8/(b*x+a)^13-7/4*d 
^2*(-a*d+b*c)^5/b^8/(b*x+a)^12-35/11*d^3*(-a*d+b*c)^4/b^8/(b*x+a)^11-7/2*d 
^4*(-a*d+b*c)^3/b^8/(b*x+a)^10-7/3*d^5*(-a*d+b*c)^2/b^8/(b*x+a)^9-7/8*d^6* 
(-a*d+b*c)/b^8/(b*x+a)^8-1/7*d^7/b^8/(b*x+a)^7
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.86 \[ \int \frac {(c+d x)^7}{(a+b x)^{15}} \, dx=-\frac {a^7 d^7+7 a^6 b d^6 (c+2 d x)+7 a^5 b^2 d^5 \left (4 c^2+14 c d x+13 d^2 x^2\right )+7 a^4 b^3 d^4 \left (12 c^3+56 c^2 d x+91 c d^2 x^2+52 d^3 x^3\right )+7 a^3 b^4 d^3 \left (30 c^4+168 c^3 d x+364 c^2 d^2 x^2+364 c d^3 x^3+143 d^4 x^4\right )+7 a^2 b^5 d^2 \left (66 c^5+420 c^4 d x+1092 c^3 d^2 x^2+1456 c^2 d^3 x^3+1001 c d^4 x^4+286 d^5 x^5\right )+7 a b^6 d \left (132 c^6+924 c^5 d x+2730 c^4 d^2 x^2+4368 c^3 d^3 x^3+4004 c^2 d^4 x^4+2002 c d^5 x^5+429 d^6 x^6\right )+b^7 \left (1716 c^7+12936 c^6 d x+42042 c^5 d^2 x^2+76440 c^4 d^3 x^3+84084 c^3 d^4 x^4+56056 c^2 d^5 x^5+21021 c d^6 x^6+3432 d^7 x^7\right )}{24024 b^8 (a+b x)^{14}} \] Input:

Integrate[(c + d*x)^7/(a + b*x)^15,x]
 

Output:

-1/24024*(a^7*d^7 + 7*a^6*b*d^6*(c + 2*d*x) + 7*a^5*b^2*d^5*(4*c^2 + 14*c* 
d*x + 13*d^2*x^2) + 7*a^4*b^3*d^4*(12*c^3 + 56*c^2*d*x + 91*c*d^2*x^2 + 52 
*d^3*x^3) + 7*a^3*b^4*d^3*(30*c^4 + 168*c^3*d*x + 364*c^2*d^2*x^2 + 364*c* 
d^3*x^3 + 143*d^4*x^4) + 7*a^2*b^5*d^2*(66*c^5 + 420*c^4*d*x + 1092*c^3*d^ 
2*x^2 + 1456*c^2*d^3*x^3 + 1001*c*d^4*x^4 + 286*d^5*x^5) + 7*a*b^6*d*(132* 
c^6 + 924*c^5*d*x + 2730*c^4*d^2*x^2 + 4368*c^3*d^3*x^3 + 4004*c^2*d^4*x^4 
 + 2002*c*d^5*x^5 + 429*d^6*x^6) + b^7*(1716*c^7 + 12936*c^6*d*x + 42042*c 
^5*d^2*x^2 + 76440*c^4*d^3*x^3 + 84084*c^3*d^4*x^4 + 56056*c^2*d^5*x^5 + 2 
1021*c*d^6*x^6 + 3432*d^7*x^7))/(b^8*(a + b*x)^14)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 200, 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 = {53, 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.

\(\displaystyle \int \frac {(c+d x)^7}{(a+b x)^{15}} \, dx\)

\(\Big \downarrow \) 53

\(\displaystyle \int \left (\frac {7 d^6 (b c-a d)}{b^7 (a+b x)^9}+\frac {21 d^5 (b c-a d)^2}{b^7 (a+b x)^{10}}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)^{11}}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^{12}}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^{13}}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^{14}}+\frac {(b c-a d)^7}{b^7 (a+b x)^{15}}+\frac {d^7}{b^7 (a+b x)^8}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {7 d^6 (b c-a d)}{8 b^8 (a+b x)^8}-\frac {7 d^5 (b c-a d)^2}{3 b^8 (a+b x)^9}-\frac {7 d^4 (b c-a d)^3}{2 b^8 (a+b x)^{10}}-\frac {35 d^3 (b c-a d)^4}{11 b^8 (a+b x)^{11}}-\frac {7 d^2 (b c-a d)^5}{4 b^8 (a+b x)^{12}}-\frac {7 d (b c-a d)^6}{13 b^8 (a+b x)^{13}}-\frac {(b c-a d)^7}{14 b^8 (a+b x)^{14}}-\frac {d^7}{7 b^8 (a+b x)^7}\)

Input:

Int[(c + d*x)^7/(a + b*x)^15,x]
 

Output:

-1/14*(b*c - a*d)^7/(b^8*(a + b*x)^14) - (7*d*(b*c - a*d)^6)/(13*b^8*(a + 
b*x)^13) - (7*d^2*(b*c - a*d)^5)/(4*b^8*(a + b*x)^12) - (35*d^3*(b*c - a*d 
)^4)/(11*b^8*(a + b*x)^11) - (7*d^4*(b*c - a*d)^3)/(2*b^8*(a + b*x)^10) - 
(7*d^5*(b*c - a*d)^2)/(3*b^8*(a + b*x)^9) - (7*d^6*(b*c - a*d))/(8*b^8*(a 
+ b*x)^8) - d^7/(7*b^8*(a + b*x)^7)
 

Defintions of rubi rules used

rule 53

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, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[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. \(437\) vs. \(2(184)=368\).

Time = 0.19 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.19

method	result	size
risch	\(\frac {-\frac {d^{7} x^{7}}{7 b}-\frac {d^{6} \left (a d +7 b c \right ) x^{6}}{8 b^{2}}-\frac {d^{5} \left (a^{2} d^{2}+7 a b c d +28 b^{2} c^{2}\right ) x^{5}}{12 b^{3}}-\frac {d^{4} \left (a^{3} d^{3}+7 a^{2} b c \,d^{2}+28 a \,b^{2} c^{2} d +84 b^{3} c^{3}\right ) x^{4}}{24 b^{4}}-\frac {d^{3} \left (a^{4} d^{4}+7 a^{3} b c \,d^{3}+28 a^{2} b^{2} c^{2} d^{2}+84 a \,b^{3} c^{3} d +210 c^{4} b^{4}\right ) x^{3}}{66 b^{5}}-\frac {d^{2} \left (a^{5} d^{5}+7 a^{4} b c \,d^{4}+28 a^{3} b^{2} c^{2} d^{3}+84 a^{2} b^{3} c^{3} d^{2}+210 a \,b^{4} c^{4} d +462 c^{5} b^{5}\right ) x^{2}}{264 b^{6}}-\frac {d \left (a^{6} d^{6}+7 a^{5} b c \,d^{5}+28 a^{4} b^{2} c^{2} d^{4}+84 a^{3} b^{3} c^{3} d^{3}+210 a^{2} b^{4} c^{4} d^{2}+462 a \,b^{5} c^{5} d +924 c^{6} b^{6}\right ) x}{1716 b^{7}}-\frac {a^{7} d^{7}+7 a^{6} b c \,d^{6}+28 a^{5} b^{2} c^{2} d^{5}+84 a^{4} b^{3} c^{3} d^{4}+210 a^{3} b^{4} c^{4} d^{3}+462 a^{2} b^{5} c^{5} d^{2}+924 a \,b^{6} c^{6} d +1716 b^{7} c^{7}}{24024 b^{8}}}{\left (b x +a \right )^{14}}\)	\(438\)
default	\(-\frac {7 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{3 b^{8} \left (b x +a \right )^{9}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +c^{6} b^{6}\right )}{13 b^{8} \left (b x +a \right )^{13}}-\frac {d^{7}}{7 b^{8} \left (b x +a \right )^{7}}+\frac {7 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}{4 b^{8} \left (b x +a \right )^{12}}+\frac {7 d^{6} \left (a d -b c \right )}{8 b^{8} \left (b x +a \right )^{8}}+\frac {7 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{2 b^{8} \left (b x +a \right )^{10}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{14 b^{8} \left (b x +a \right )^{14}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}{11 b^{8} \left (b x +a \right )^{11}}\)	\(464\)
norman	\(\frac {\frac {-a^{7} b^{6} d^{7}-7 a^{6} b^{7} c \,d^{6}-28 a^{5} b^{8} c^{2} d^{5}-84 a^{4} b^{9} c^{3} d^{4}-210 a^{3} b^{10} c^{4} d^{3}-462 a^{2} c^{5} d^{2} b^{11}-924 a \,b^{12} c^{6} d -1716 b^{13} c^{7}}{24024 b^{14}}+\frac {\left (-a^{6} b^{6} d^{7}-7 a^{5} b^{7} c \,d^{6}-28 a^{4} b^{8} c^{2} d^{5}-84 a^{3} b^{9} c^{3} d^{4}-210 a^{2} b^{10} c^{4} d^{3}-462 a \,c^{5} d^{2} b^{11}-924 b^{12} c^{6} d \right ) x}{1716 b^{13}}+\frac {\left (-a^{5} b^{6} d^{7}-7 a^{4} b^{7} c \,d^{6}-28 a^{3} b^{8} c^{2} d^{5}-84 a^{2} b^{9} c^{3} d^{4}-210 a \,b^{10} c^{4} d^{3}-462 c^{5} d^{2} b^{11}\right ) x^{2}}{264 b^{12}}+\frac {\left (-a^{4} b^{6} d^{7}-7 a^{3} b^{7} c \,d^{6}-28 a^{2} b^{8} c^{2} d^{5}-84 a \,b^{9} c^{3} d^{4}-210 b^{10} c^{4} d^{3}\right ) x^{3}}{66 b^{11}}+\frac {\left (-a^{3} b^{6} d^{7}-7 a^{2} b^{7} c \,d^{6}-28 a \,b^{8} c^{2} d^{5}-84 b^{9} c^{3} d^{4}\right ) x^{4}}{24 b^{10}}+\frac {\left (-a^{2} b^{6} d^{7}-7 a \,b^{7} c \,d^{6}-28 b^{8} c^{2} d^{5}\right ) x^{5}}{12 b^{9}}+\frac {\left (-a \,b^{6} d^{7}-7 b^{7} c \,d^{6}\right ) x^{6}}{8 b^{8}}-\frac {d^{7} x^{7}}{7 b}}{\left (b x +a \right )^{14}}\)	\(492\)
gosper	\(-\frac {3432 x^{7} d^{7} b^{7}+3003 x^{6} a \,b^{6} d^{7}+21021 x^{6} b^{7} c \,d^{6}+2002 x^{5} a^{2} b^{5} d^{7}+14014 x^{5} a \,b^{6} c \,d^{6}+56056 x^{5} b^{7} c^{2} d^{5}+1001 x^{4} a^{3} b^{4} d^{7}+7007 x^{4} a^{2} b^{5} c \,d^{6}+28028 x^{4} a \,b^{6} c^{2} d^{5}+84084 x^{4} b^{7} c^{3} d^{4}+364 x^{3} a^{4} b^{3} d^{7}+2548 x^{3} a^{3} b^{4} c \,d^{6}+10192 x^{3} a^{2} b^{5} c^{2} d^{5}+30576 x^{3} a \,b^{6} c^{3} d^{4}+76440 x^{3} b^{7} c^{4} d^{3}+91 x^{2} a^{5} b^{2} d^{7}+637 x^{2} a^{4} b^{3} c \,d^{6}+2548 x^{2} a^{3} b^{4} c^{2} d^{5}+7644 x^{2} a^{2} b^{5} c^{3} d^{4}+19110 x^{2} a \,b^{6} c^{4} d^{3}+42042 x^{2} b^{7} c^{5} d^{2}+14 x \,a^{6} b \,d^{7}+98 x \,a^{5} b^{2} c \,d^{6}+392 x \,a^{4} b^{3} c^{2} d^{5}+1176 x \,a^{3} b^{4} c^{3} d^{4}+2940 x \,a^{2} b^{5} c^{4} d^{3}+6468 x a \,b^{6} c^{5} d^{2}+12936 x \,b^{7} c^{6} d +a^{7} d^{7}+7 a^{6} b c \,d^{6}+28 a^{5} b^{2} c^{2} d^{5}+84 a^{4} b^{3} c^{3} d^{4}+210 a^{3} b^{4} c^{4} d^{3}+462 a^{2} b^{5} c^{5} d^{2}+924 a \,b^{6} c^{6} d +1716 b^{7} c^{7}}{24024 b^{8} \left (b x +a \right )^{14}}\)	\(497\)
orering	\(-\frac {3432 x^{7} d^{7} b^{7}+3003 x^{6} a \,b^{6} d^{7}+21021 x^{6} b^{7} c \,d^{6}+2002 x^{5} a^{2} b^{5} d^{7}+14014 x^{5} a \,b^{6} c \,d^{6}+56056 x^{5} b^{7} c^{2} d^{5}+1001 x^{4} a^{3} b^{4} d^{7}+7007 x^{4} a^{2} b^{5} c \,d^{6}+28028 x^{4} a \,b^{6} c^{2} d^{5}+84084 x^{4} b^{7} c^{3} d^{4}+364 x^{3} a^{4} b^{3} d^{7}+2548 x^{3} a^{3} b^{4} c \,d^{6}+10192 x^{3} a^{2} b^{5} c^{2} d^{5}+30576 x^{3} a \,b^{6} c^{3} d^{4}+76440 x^{3} b^{7} c^{4} d^{3}+91 x^{2} a^{5} b^{2} d^{7}+637 x^{2} a^{4} b^{3} c \,d^{6}+2548 x^{2} a^{3} b^{4} c^{2} d^{5}+7644 x^{2} a^{2} b^{5} c^{3} d^{4}+19110 x^{2} a \,b^{6} c^{4} d^{3}+42042 x^{2} b^{7} c^{5} d^{2}+14 x \,a^{6} b \,d^{7}+98 x \,a^{5} b^{2} c \,d^{6}+392 x \,a^{4} b^{3} c^{2} d^{5}+1176 x \,a^{3} b^{4} c^{3} d^{4}+2940 x \,a^{2} b^{5} c^{4} d^{3}+6468 x a \,b^{6} c^{5} d^{2}+12936 x \,b^{7} c^{6} d +a^{7} d^{7}+7 a^{6} b c \,d^{6}+28 a^{5} b^{2} c^{2} d^{5}+84 a^{4} b^{3} c^{3} d^{4}+210 a^{3} b^{4} c^{4} d^{3}+462 a^{2} b^{5} c^{5} d^{2}+924 a \,b^{6} c^{6} d +1716 b^{7} c^{7}}{24024 b^{8} \left (b x +a \right )^{14}}\)	\(497\)
parallelrisch	\(\frac {-3432 d^{7} x^{7} b^{13}-3003 a \,b^{12} d^{7} x^{6}-21021 b^{13} c \,d^{6} x^{6}-2002 a^{2} b^{11} d^{7} x^{5}-14014 a \,b^{12} c \,d^{6} x^{5}-56056 b^{13} c^{2} d^{5} x^{5}-1001 a^{3} b^{10} d^{7} x^{4}-7007 a^{2} b^{11} c \,d^{6} x^{4}-28028 a \,b^{12} c^{2} d^{5} x^{4}-84084 b^{13} c^{3} d^{4} x^{4}-364 a^{4} b^{9} d^{7} x^{3}-2548 a^{3} b^{10} c \,d^{6} x^{3}-10192 a^{2} b^{11} c^{2} d^{5} x^{3}-30576 a \,b^{12} c^{3} d^{4} x^{3}-76440 b^{13} c^{4} d^{3} x^{3}-91 a^{5} b^{8} d^{7} x^{2}-637 a^{4} b^{9} c \,d^{6} x^{2}-2548 a^{3} b^{10} c^{2} d^{5} x^{2}-7644 a^{2} b^{11} c^{3} d^{4} x^{2}-19110 a \,b^{12} c^{4} d^{3} x^{2}-42042 b^{13} c^{5} d^{2} x^{2}-14 a^{6} b^{7} d^{7} x -98 a^{5} b^{8} c \,d^{6} x -392 a^{4} b^{9} c^{2} d^{5} x -1176 a^{3} b^{10} c^{3} d^{4} x -2940 a^{2} b^{11} c^{4} d^{3} x -6468 a \,b^{12} c^{5} d^{2} x -12936 b^{13} c^{6} d x -a^{7} b^{6} d^{7}-7 a^{6} b^{7} c \,d^{6}-28 a^{5} b^{8} c^{2} d^{5}-84 a^{4} b^{9} c^{3} d^{4}-210 a^{3} b^{10} c^{4} d^{3}-462 a^{2} c^{5} d^{2} b^{11}-924 a \,b^{12} c^{6} d -1716 b^{13} c^{7}}{24024 b^{14} \left (b x +a \right )^{14}}\)	\(505\)

Input:

int((d*x+c)^7/(b*x+a)^15,x,method=_RETURNVERBOSE)
 

Output:

(-1/7/b*d^7*x^7-1/8/b^2*d^6*(a*d+7*b*c)*x^6-1/12/b^3*d^5*(a^2*d^2+7*a*b*c* 
d+28*b^2*c^2)*x^5-1/24/b^4*d^4*(a^3*d^3+7*a^2*b*c*d^2+28*a*b^2*c^2*d+84*b^ 
3*c^3)*x^4-1/66/b^5*d^3*(a^4*d^4+7*a^3*b*c*d^3+28*a^2*b^2*c^2*d^2+84*a*b^3 
*c^3*d+210*b^4*c^4)*x^3-1/264/b^6*d^2*(a^5*d^5+7*a^4*b*c*d^4+28*a^3*b^2*c^ 
2*d^3+84*a^2*b^3*c^3*d^2+210*a*b^4*c^4*d+462*b^5*c^5)*x^2-1/1716/b^7*d*(a^ 
6*d^6+7*a^5*b*c*d^5+28*a^4*b^2*c^2*d^4+84*a^3*b^3*c^3*d^3+210*a^2*b^4*c^4* 
d^2+462*a*b^5*c^5*d+924*b^6*c^6)*x-1/24024/b^8*(a^7*d^7+7*a^6*b*c*d^6+28*a 
^5*b^2*c^2*d^5+84*a^4*b^3*c^3*d^4+210*a^3*b^4*c^4*d^3+462*a^2*b^5*c^5*d^2+ 
924*a*b^6*c^6*d+1716*b^7*c^7))/(b*x+a)^14
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (184) = 368\).

Time = 0.08 (sec) , antiderivative size = 603, normalized size of antiderivative = 3.02 \[ \int \frac {(c+d x)^7}{(a+b x)^{15}} \, dx=-\frac {3432 \, b^{7} d^{7} x^{7} + 1716 \, b^{7} c^{7} + 924 \, a b^{6} c^{6} d + 462 \, a^{2} b^{5} c^{5} d^{2} + 210 \, a^{3} b^{4} c^{4} d^{3} + 84 \, a^{4} b^{3} c^{3} d^{4} + 28 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} + a^{7} d^{7} + 3003 \, {\left (7 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 2002 \, {\left (28 \, b^{7} c^{2} d^{5} + 7 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1001 \, {\left (84 \, b^{7} c^{3} d^{4} + 28 \, a b^{6} c^{2} d^{5} + 7 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 364 \, {\left (210 \, b^{7} c^{4} d^{3} + 84 \, a b^{6} c^{3} d^{4} + 28 \, a^{2} b^{5} c^{2} d^{5} + 7 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 91 \, {\left (462 \, b^{7} c^{5} d^{2} + 210 \, a b^{6} c^{4} d^{3} + 84 \, a^{2} b^{5} c^{3} d^{4} + 28 \, a^{3} b^{4} c^{2} d^{5} + 7 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 14 \, {\left (924 \, b^{7} c^{6} d + 462 \, a b^{6} c^{5} d^{2} + 210 \, a^{2} b^{5} c^{4} d^{3} + 84 \, a^{3} b^{4} c^{3} d^{4} + 28 \, a^{4} b^{3} c^{2} d^{5} + 7 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{24024 \, {\left (b^{22} x^{14} + 14 \, a b^{21} x^{13} + 91 \, a^{2} b^{20} x^{12} + 364 \, a^{3} b^{19} x^{11} + 1001 \, a^{4} b^{18} x^{10} + 2002 \, a^{5} b^{17} x^{9} + 3003 \, a^{6} b^{16} x^{8} + 3432 \, a^{7} b^{15} x^{7} + 3003 \, a^{8} b^{14} x^{6} + 2002 \, a^{9} b^{13} x^{5} + 1001 \, a^{10} b^{12} x^{4} + 364 \, a^{11} b^{11} x^{3} + 91 \, a^{12} b^{10} x^{2} + 14 \, a^{13} b^{9} x + a^{14} b^{8}\right )}} \] Input:

integrate((d*x+c)^7/(b*x+a)^15,x, algorithm="fricas")
 

Output:

-1/24024*(3432*b^7*d^7*x^7 + 1716*b^7*c^7 + 924*a*b^6*c^6*d + 462*a^2*b^5* 
c^5*d^2 + 210*a^3*b^4*c^4*d^3 + 84*a^4*b^3*c^3*d^4 + 28*a^5*b^2*c^2*d^5 + 
7*a^6*b*c*d^6 + a^7*d^7 + 3003*(7*b^7*c*d^6 + a*b^6*d^7)*x^6 + 2002*(28*b^ 
7*c^2*d^5 + 7*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 1001*(84*b^7*c^3*d^4 + 28*a 
*b^6*c^2*d^5 + 7*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 364*(210*b^7*c^4*d^3 + 
 84*a*b^6*c^3*d^4 + 28*a^2*b^5*c^2*d^5 + 7*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^ 
3 + 91*(462*b^7*c^5*d^2 + 210*a*b^6*c^4*d^3 + 84*a^2*b^5*c^3*d^4 + 28*a^3* 
b^4*c^2*d^5 + 7*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 14*(924*b^7*c^6*d + 462 
*a*b^6*c^5*d^2 + 210*a^2*b^5*c^4*d^3 + 84*a^3*b^4*c^3*d^4 + 28*a^4*b^3*c^2 
*d^5 + 7*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^22*x^14 + 14*a*b^21*x^13 + 91*a^ 
2*b^20*x^12 + 364*a^3*b^19*x^11 + 1001*a^4*b^18*x^10 + 2002*a^5*b^17*x^9 + 
 3003*a^6*b^16*x^8 + 3432*a^7*b^15*x^7 + 3003*a^8*b^14*x^6 + 2002*a^9*b^13 
*x^5 + 1001*a^10*b^12*x^4 + 364*a^11*b^11*x^3 + 91*a^12*b^10*x^2 + 14*a^13 
*b^9*x + a^14*b^8)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^{15}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**7/(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. 603 vs. \(2 (184) = 368\).

Time = 0.06 (sec) , antiderivative size = 603, normalized size of antiderivative = 3.02 \[ \int \frac {(c+d x)^7}{(a+b x)^{15}} \, dx=-\frac {3432 \, b^{7} d^{7} x^{7} + 1716 \, b^{7} c^{7} + 924 \, a b^{6} c^{6} d + 462 \, a^{2} b^{5} c^{5} d^{2} + 210 \, a^{3} b^{4} c^{4} d^{3} + 84 \, a^{4} b^{3} c^{3} d^{4} + 28 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} + a^{7} d^{7} + 3003 \, {\left (7 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 2002 \, {\left (28 \, b^{7} c^{2} d^{5} + 7 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1001 \, {\left (84 \, b^{7} c^{3} d^{4} + 28 \, a b^{6} c^{2} d^{5} + 7 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 364 \, {\left (210 \, b^{7} c^{4} d^{3} + 84 \, a b^{6} c^{3} d^{4} + 28 \, a^{2} b^{5} c^{2} d^{5} + 7 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 91 \, {\left (462 \, b^{7} c^{5} d^{2} + 210 \, a b^{6} c^{4} d^{3} + 84 \, a^{2} b^{5} c^{3} d^{4} + 28 \, a^{3} b^{4} c^{2} d^{5} + 7 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 14 \, {\left (924 \, b^{7} c^{6} d + 462 \, a b^{6} c^{5} d^{2} + 210 \, a^{2} b^{5} c^{4} d^{3} + 84 \, a^{3} b^{4} c^{3} d^{4} + 28 \, a^{4} b^{3} c^{2} d^{5} + 7 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{24024 \, {\left (b^{22} x^{14} + 14 \, a b^{21} x^{13} + 91 \, a^{2} b^{20} x^{12} + 364 \, a^{3} b^{19} x^{11} + 1001 \, a^{4} b^{18} x^{10} + 2002 \, a^{5} b^{17} x^{9} + 3003 \, a^{6} b^{16} x^{8} + 3432 \, a^{7} b^{15} x^{7} + 3003 \, a^{8} b^{14} x^{6} + 2002 \, a^{9} b^{13} x^{5} + 1001 \, a^{10} b^{12} x^{4} + 364 \, a^{11} b^{11} x^{3} + 91 \, a^{12} b^{10} x^{2} + 14 \, a^{13} b^{9} x + a^{14} b^{8}\right )}} \] Input:

integrate((d*x+c)^7/(b*x+a)^15,x, algorithm="maxima")
 

Output:

-1/24024*(3432*b^7*d^7*x^7 + 1716*b^7*c^7 + 924*a*b^6*c^6*d + 462*a^2*b^5* 
c^5*d^2 + 210*a^3*b^4*c^4*d^3 + 84*a^4*b^3*c^3*d^4 + 28*a^5*b^2*c^2*d^5 + 
7*a^6*b*c*d^6 + a^7*d^7 + 3003*(7*b^7*c*d^6 + a*b^6*d^7)*x^6 + 2002*(28*b^ 
7*c^2*d^5 + 7*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 1001*(84*b^7*c^3*d^4 + 28*a 
*b^6*c^2*d^5 + 7*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 364*(210*b^7*c^4*d^3 + 
 84*a*b^6*c^3*d^4 + 28*a^2*b^5*c^2*d^5 + 7*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^ 
3 + 91*(462*b^7*c^5*d^2 + 210*a*b^6*c^4*d^3 + 84*a^2*b^5*c^3*d^4 + 28*a^3* 
b^4*c^2*d^5 + 7*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 14*(924*b^7*c^6*d + 462 
*a*b^6*c^5*d^2 + 210*a^2*b^5*c^4*d^3 + 84*a^3*b^4*c^3*d^4 + 28*a^4*b^3*c^2 
*d^5 + 7*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^22*x^14 + 14*a*b^21*x^13 + 91*a^ 
2*b^20*x^12 + 364*a^3*b^19*x^11 + 1001*a^4*b^18*x^10 + 2002*a^5*b^17*x^9 + 
 3003*a^6*b^16*x^8 + 3432*a^7*b^15*x^7 + 3003*a^8*b^14*x^6 + 2002*a^9*b^13 
*x^5 + 1001*a^10*b^12*x^4 + 364*a^11*b^11*x^3 + 91*a^12*b^10*x^2 + 14*a^13 
*b^9*x + a^14*b^8)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (184) = 368\).

Time = 0.12 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.48 \[ \int \frac {(c+d x)^7}{(a+b x)^{15}} \, dx=-\frac {3432 \, b^{7} d^{7} x^{7} + 21021 \, b^{7} c d^{6} x^{6} + 3003 \, a b^{6} d^{7} x^{6} + 56056 \, b^{7} c^{2} d^{5} x^{5} + 14014 \, a b^{6} c d^{6} x^{5} + 2002 \, a^{2} b^{5} d^{7} x^{5} + 84084 \, b^{7} c^{3} d^{4} x^{4} + 28028 \, a b^{6} c^{2} d^{5} x^{4} + 7007 \, a^{2} b^{5} c d^{6} x^{4} + 1001 \, a^{3} b^{4} d^{7} x^{4} + 76440 \, b^{7} c^{4} d^{3} x^{3} + 30576 \, a b^{6} c^{3} d^{4} x^{3} + 10192 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 2548 \, a^{3} b^{4} c d^{6} x^{3} + 364 \, a^{4} b^{3} d^{7} x^{3} + 42042 \, b^{7} c^{5} d^{2} x^{2} + 19110 \, a b^{6} c^{4} d^{3} x^{2} + 7644 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 2548 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 637 \, a^{4} b^{3} c d^{6} x^{2} + 91 \, a^{5} b^{2} d^{7} x^{2} + 12936 \, b^{7} c^{6} d x + 6468 \, a b^{6} c^{5} d^{2} x + 2940 \, a^{2} b^{5} c^{4} d^{3} x + 1176 \, a^{3} b^{4} c^{3} d^{4} x + 392 \, a^{4} b^{3} c^{2} d^{5} x + 98 \, a^{5} b^{2} c d^{6} x + 14 \, a^{6} b d^{7} x + 1716 \, b^{7} c^{7} + 924 \, a b^{6} c^{6} d + 462 \, a^{2} b^{5} c^{5} d^{2} + 210 \, a^{3} b^{4} c^{4} d^{3} + 84 \, a^{4} b^{3} c^{3} d^{4} + 28 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} + a^{7} d^{7}}{24024 \, {\left (b x + a\right )}^{14} b^{8}} \] Input:

integrate((d*x+c)^7/(b*x+a)^15,x, algorithm="giac")
 

Output:

-1/24024*(3432*b^7*d^7*x^7 + 21021*b^7*c*d^6*x^6 + 3003*a*b^6*d^7*x^6 + 56 
056*b^7*c^2*d^5*x^5 + 14014*a*b^6*c*d^6*x^5 + 2002*a^2*b^5*d^7*x^5 + 84084 
*b^7*c^3*d^4*x^4 + 28028*a*b^6*c^2*d^5*x^4 + 7007*a^2*b^5*c*d^6*x^4 + 1001 
*a^3*b^4*d^7*x^4 + 76440*b^7*c^4*d^3*x^3 + 30576*a*b^6*c^3*d^4*x^3 + 10192 
*a^2*b^5*c^2*d^5*x^3 + 2548*a^3*b^4*c*d^6*x^3 + 364*a^4*b^3*d^7*x^3 + 4204 
2*b^7*c^5*d^2*x^2 + 19110*a*b^6*c^4*d^3*x^2 + 7644*a^2*b^5*c^3*d^4*x^2 + 2 
548*a^3*b^4*c^2*d^5*x^2 + 637*a^4*b^3*c*d^6*x^2 + 91*a^5*b^2*d^7*x^2 + 129 
36*b^7*c^6*d*x + 6468*a*b^6*c^5*d^2*x + 2940*a^2*b^5*c^4*d^3*x + 1176*a^3* 
b^4*c^3*d^4*x + 392*a^4*b^3*c^2*d^5*x + 98*a^5*b^2*c*d^6*x + 14*a^6*b*d^7* 
x + 1716*b^7*c^7 + 924*a*b^6*c^6*d + 462*a^2*b^5*c^5*d^2 + 210*a^3*b^4*c^4 
*d^3 + 84*a^4*b^3*c^3*d^4 + 28*a^5*b^2*c^2*d^5 + 7*a^6*b*c*d^6 + a^7*d^7)/ 
((b*x + a)^14*b^8)
 

Mupad [B] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 581, normalized size of antiderivative = 2.90 \[ \int \frac {(c+d x)^7}{(a+b x)^{15}} \, dx=-\frac {\frac {a^7\,d^7+7\,a^6\,b\,c\,d^6+28\,a^5\,b^2\,c^2\,d^5+84\,a^4\,b^3\,c^3\,d^4+210\,a^3\,b^4\,c^4\,d^3+462\,a^2\,b^5\,c^5\,d^2+924\,a\,b^6\,c^6\,d+1716\,b^7\,c^7}{24024\,b^8}+\frac {d^7\,x^7}{7\,b}+\frac {d^2\,x^2\,\left (a^5\,d^5+7\,a^4\,b\,c\,d^4+28\,a^3\,b^2\,c^2\,d^3+84\,a^2\,b^3\,c^3\,d^2+210\,a\,b^4\,c^4\,d+462\,b^5\,c^5\right )}{264\,b^6}+\frac {d^4\,x^4\,\left (a^3\,d^3+7\,a^2\,b\,c\,d^2+28\,a\,b^2\,c^2\,d+84\,b^3\,c^3\right )}{24\,b^4}+\frac {d^6\,x^6\,\left (a\,d+7\,b\,c\right )}{8\,b^2}+\frac {d^3\,x^3\,\left (a^4\,d^4+7\,a^3\,b\,c\,d^3+28\,a^2\,b^2\,c^2\,d^2+84\,a\,b^3\,c^3\,d+210\,b^4\,c^4\right )}{66\,b^5}+\frac {d\,x\,\left (a^6\,d^6+7\,a^5\,b\,c\,d^5+28\,a^4\,b^2\,c^2\,d^4+84\,a^3\,b^3\,c^3\,d^3+210\,a^2\,b^4\,c^4\,d^2+462\,a\,b^5\,c^5\,d+924\,b^6\,c^6\right )}{1716\,b^7}+\frac {d^5\,x^5\,\left (a^2\,d^2+7\,a\,b\,c\,d+28\,b^2\,c^2\right )}{12\,b^3}}{a^{14}+14\,a^{13}\,b\,x+91\,a^{12}\,b^2\,x^2+364\,a^{11}\,b^3\,x^3+1001\,a^{10}\,b^4\,x^4+2002\,a^9\,b^5\,x^5+3003\,a^8\,b^6\,x^6+3432\,a^7\,b^7\,x^7+3003\,a^6\,b^8\,x^8+2002\,a^5\,b^9\,x^9+1001\,a^4\,b^{10}\,x^{10}+364\,a^3\,b^{11}\,x^{11}+91\,a^2\,b^{12}\,x^{12}+14\,a\,b^{13}\,x^{13}+b^{14}\,x^{14}} \] Input:

int((c + d*x)^7/(a + b*x)^15,x)
 

Output:

-((a^7*d^7 + 1716*b^7*c^7 + 462*a^2*b^5*c^5*d^2 + 210*a^3*b^4*c^4*d^3 + 84 
*a^4*b^3*c^3*d^4 + 28*a^5*b^2*c^2*d^5 + 924*a*b^6*c^6*d + 7*a^6*b*c*d^6)/( 
24024*b^8) + (d^7*x^7)/(7*b) + (d^2*x^2*(a^5*d^5 + 462*b^5*c^5 + 84*a^2*b^ 
3*c^3*d^2 + 28*a^3*b^2*c^2*d^3 + 210*a*b^4*c^4*d + 7*a^4*b*c*d^4))/(264*b^ 
6) + (d^4*x^4*(a^3*d^3 + 84*b^3*c^3 + 28*a*b^2*c^2*d + 7*a^2*b*c*d^2))/(24 
*b^4) + (d^6*x^6*(a*d + 7*b*c))/(8*b^2) + (d^3*x^3*(a^4*d^4 + 210*b^4*c^4 
+ 28*a^2*b^2*c^2*d^2 + 84*a*b^3*c^3*d + 7*a^3*b*c*d^3))/(66*b^5) + (d*x*(a 
^6*d^6 + 924*b^6*c^6 + 210*a^2*b^4*c^4*d^2 + 84*a^3*b^3*c^3*d^3 + 28*a^4*b 
^2*c^2*d^4 + 462*a*b^5*c^5*d + 7*a^5*b*c*d^5))/(1716*b^7) + (d^5*x^5*(a^2* 
d^2 + 28*b^2*c^2 + 7*a*b*c*d))/(12*b^3))/(a^14 + b^14*x^14 + 14*a*b^13*x^1 
3 + 91*a^12*b^2*x^2 + 364*a^11*b^3*x^3 + 1001*a^10*b^4*x^4 + 2002*a^9*b^5* 
x^5 + 3003*a^8*b^6*x^6 + 3432*a^7*b^7*x^7 + 3003*a^6*b^8*x^8 + 2002*a^5*b^ 
9*x^9 + 1001*a^4*b^10*x^10 + 364*a^3*b^11*x^11 + 91*a^2*b^12*x^12 + 14*a^1 
3*b*x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 640, normalized size of antiderivative = 3.20 \[ \int \frac {(c+d x)^7}{(a+b x)^{15}} \, dx=\frac {-3432 b^{7} d^{7} x^{7}-3003 a \,b^{6} d^{7} x^{6}-21021 b^{7} c \,d^{6} x^{6}-2002 a^{2} b^{5} d^{7} x^{5}-14014 a \,b^{6} c \,d^{6} x^{5}-56056 b^{7} c^{2} d^{5} x^{5}-1001 a^{3} b^{4} d^{7} x^{4}-7007 a^{2} b^{5} c \,d^{6} x^{4}-28028 a \,b^{6} c^{2} d^{5} x^{4}-84084 b^{7} c^{3} d^{4} x^{4}-364 a^{4} b^{3} d^{7} x^{3}-2548 a^{3} b^{4} c \,d^{6} x^{3}-10192 a^{2} b^{5} c^{2} d^{5} x^{3}-30576 a \,b^{6} c^{3} d^{4} x^{3}-76440 b^{7} c^{4} d^{3} x^{3}-91 a^{5} b^{2} d^{7} x^{2}-637 a^{4} b^{3} c \,d^{6} x^{2}-2548 a^{3} b^{4} c^{2} d^{5} x^{2}-7644 a^{2} b^{5} c^{3} d^{4} x^{2}-19110 a \,b^{6} c^{4} d^{3} x^{2}-42042 b^{7} c^{5} d^{2} x^{2}-14 a^{6} b \,d^{7} x -98 a^{5} b^{2} c \,d^{6} x -392 a^{4} b^{3} c^{2} d^{5} x -1176 a^{3} b^{4} c^{3} d^{4} x -2940 a^{2} b^{5} c^{4} d^{3} x -6468 a \,b^{6} c^{5} d^{2} x -12936 b^{7} c^{6} d x -a^{7} d^{7}-7 a^{6} b c \,d^{6}-28 a^{5} b^{2} c^{2} d^{5}-84 a^{4} b^{3} c^{3} d^{4}-210 a^{3} b^{4} c^{4} d^{3}-462 a^{2} b^{5} c^{5} d^{2}-924 a \,b^{6} c^{6} d -1716 b^{7} c^{7}}{24024 b^{8} \left (b^{14} x^{14}+14 a \,b^{13} x^{13}+91 a^{2} b^{12} x^{12}+364 a^{3} b^{11} x^{11}+1001 a^{4} b^{10} x^{10}+2002 a^{5} b^{9} x^{9}+3003 a^{6} b^{8} x^{8}+3432 a^{7} b^{7} x^{7}+3003 a^{8} b^{6} x^{6}+2002 a^{9} b^{5} x^{5}+1001 a^{10} b^{4} x^{4}+364 a^{11} b^{3} x^{3}+91 a^{12} b^{2} x^{2}+14 a^{13} b x +a^{14}\right )} \] Input:

int((d*x+c)^7/(b*x+a)^15,x)
 

Output:

( - a**7*d**7 - 7*a**6*b*c*d**6 - 14*a**6*b*d**7*x - 28*a**5*b**2*c**2*d** 
5 - 98*a**5*b**2*c*d**6*x - 91*a**5*b**2*d**7*x**2 - 84*a**4*b**3*c**3*d** 
4 - 392*a**4*b**3*c**2*d**5*x - 637*a**4*b**3*c*d**6*x**2 - 364*a**4*b**3* 
d**7*x**3 - 210*a**3*b**4*c**4*d**3 - 1176*a**3*b**4*c**3*d**4*x - 2548*a* 
*3*b**4*c**2*d**5*x**2 - 2548*a**3*b**4*c*d**6*x**3 - 1001*a**3*b**4*d**7* 
x**4 - 462*a**2*b**5*c**5*d**2 - 2940*a**2*b**5*c**4*d**3*x - 7644*a**2*b* 
*5*c**3*d**4*x**2 - 10192*a**2*b**5*c**2*d**5*x**3 - 7007*a**2*b**5*c*d**6 
*x**4 - 2002*a**2*b**5*d**7*x**5 - 924*a*b**6*c**6*d - 6468*a*b**6*c**5*d* 
*2*x - 19110*a*b**6*c**4*d**3*x**2 - 30576*a*b**6*c**3*d**4*x**3 - 28028*a 
*b**6*c**2*d**5*x**4 - 14014*a*b**6*c*d**6*x**5 - 3003*a*b**6*d**7*x**6 - 
1716*b**7*c**7 - 12936*b**7*c**6*d*x - 42042*b**7*c**5*d**2*x**2 - 76440*b 
**7*c**4*d**3*x**3 - 84084*b**7*c**3*d**4*x**4 - 56056*b**7*c**2*d**5*x**5 
 - 21021*b**7*c*d**6*x**6 - 3432*b**7*d**7*x**7)/(24024*b**8*(a**14 + 14*a 
**13*b*x + 91*a**12*b**2*x**2 + 364*a**11*b**3*x**3 + 1001*a**10*b**4*x**4 
 + 2002*a**9*b**5*x**5 + 3003*a**8*b**6*x**6 + 3432*a**7*b**7*x**7 + 3003* 
a**6*b**8*x**8 + 2002*a**5*b**9*x**9 + 1001*a**4*b**10*x**10 + 364*a**3*b* 
*11*x**11 + 91*a**2*b**12*x**12 + 14*a*b**13*x**13 + b**14*x**14))