\(\int \frac {(c+d x)^{10}}{(a+b x)^{17}} \, dx\) [1328]

Optimal result

Integrand size = 15, antiderivative size = 182 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{17}} \, dx=-\frac {(c+d x)^{11}}{16 (b c-a d) (a+b x)^{16}}+\frac {d (c+d x)^{11}}{48 (b c-a d)^2 (a+b x)^{15}}-\frac {d^2 (c+d x)^{11}}{168 (b c-a d)^3 (a+b x)^{14}}+\frac {d^3 (c+d x)^{11}}{728 (b c-a d)^4 (a+b x)^{13}}-\frac {d^4 (c+d x)^{11}}{4368 (b c-a d)^5 (a+b x)^{12}}+\frac {d^5 (c+d x)^{11}}{48048 (b c-a d)^6 (a+b x)^{11}} \]

[Out]

-1/16*(d*x+c)^11/(-a*d+b*c)/(b*x+a)^16+1/48*d*(d*x+c)^11/(-a*d+b*c)^2/(b*x+a)^15-1/168*d^2*(d*x+c)^11/(-a*d+b*
c)^3/(b*x+a)^14+1/728*d^3*(d*x+c)^11/(-a*d+b*c)^4/(b*x+a)^13-1/4368*d^4*(d*x+c)^11/(-a*d+b*c)^5/(b*x+a)^12+1/4
8048*d^5*(d*x+c)^11/(-a*d+b*c)^6/(b*x+a)^11

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {(c+d x)^{10}}{(a+b x)^{17}} \, dx=\frac {d^5 (c+d x)^{11}}{48048 (a+b x)^{11} (b c-a d)^6}-\frac {d^4 (c+d x)^{11}}{4368 (a+b x)^{12} (b c-a d)^5}+\frac {d^3 (c+d x)^{11}}{728 (a+b x)^{13} (b c-a d)^4}-\frac {d^2 (c+d x)^{11}}{168 (a+b x)^{14} (b c-a d)^3}+\frac {d (c+d x)^{11}}{48 (a+b x)^{15} (b c-a d)^2}-\frac {(c+d x)^{11}}{16 (a+b x)^{16} (b c-a d)} \]

[In]

Int[(c + d*x)^10/(a + b*x)^17,x]

[Out]

-1/16*(c + d*x)^11/((b*c - a*d)*(a + b*x)^16) + (d*(c + d*x)^11)/(48*(b*c - a*d)^2*(a + b*x)^15) - (d^2*(c + d
*x)^11)/(168*(b*c - a*d)^3*(a + b*x)^14) + (d^3*(c + d*x)^11)/(728*(b*c - a*d)^4*(a + b*x)^13) - (d^4*(c + d*x
)^11)/(4368*(b*c - a*d)^5*(a + b*x)^12) + (d^5*(c + d*x)^11)/(48048*(b*c - a*d)^6*(a + b*x)^11)

Rule 37

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] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

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] - Dist[d*(Simplify[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] && NeQ[b*c - a*d, 0] && 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] ||  !SumSimplerQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^{11}}{16 (b c-a d) (a+b x)^{16}}-\frac {(5 d) \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx}{16 (b c-a d)} \\ & = -\frac {(c+d x)^{11}}{16 (b c-a d) (a+b x)^{16}}+\frac {d (c+d x)^{11}}{48 (b c-a d)^2 (a+b x)^{15}}+\frac {d^2 \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx}{12 (b c-a d)^2} \\ & = -\frac {(c+d x)^{11}}{16 (b c-a d) (a+b x)^{16}}+\frac {d (c+d x)^{11}}{48 (b c-a d)^2 (a+b x)^{15}}-\frac {d^2 (c+d x)^{11}}{168 (b c-a d)^3 (a+b x)^{14}}-\frac {d^3 \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx}{56 (b c-a d)^3} \\ & = -\frac {(c+d x)^{11}}{16 (b c-a d) (a+b x)^{16}}+\frac {d (c+d x)^{11}}{48 (b c-a d)^2 (a+b x)^{15}}-\frac {d^2 (c+d x)^{11}}{168 (b c-a d)^3 (a+b x)^{14}}+\frac {d^3 (c+d x)^{11}}{728 (b c-a d)^4 (a+b x)^{13}}+\frac {d^4 \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx}{364 (b c-a d)^4} \\ & = -\frac {(c+d x)^{11}}{16 (b c-a d) (a+b x)^{16}}+\frac {d (c+d x)^{11}}{48 (b c-a d)^2 (a+b x)^{15}}-\frac {d^2 (c+d x)^{11}}{168 (b c-a d)^3 (a+b x)^{14}}+\frac {d^3 (c+d x)^{11}}{728 (b c-a d)^4 (a+b x)^{13}}-\frac {d^4 (c+d x)^{11}}{4368 (b c-a d)^5 (a+b x)^{12}}-\frac {d^5 \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx}{4368 (b c-a d)^5} \\ & = -\frac {(c+d x)^{11}}{16 (b c-a d) (a+b x)^{16}}+\frac {d (c+d x)^{11}}{48 (b c-a d)^2 (a+b x)^{15}}-\frac {d^2 (c+d x)^{11}}{168 (b c-a d)^3 (a+b x)^{14}}+\frac {d^3 (c+d x)^{11}}{728 (b c-a d)^4 (a+b x)^{13}}-\frac {d^4 (c+d x)^{11}}{4368 (b c-a d)^5 (a+b x)^{12}}+\frac {d^5 (c+d x)^{11}}{48048 (b c-a d)^6 (a+b x)^{11}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(694\) vs. \(2(182)=364\).

Time = 0.18 (sec) , antiderivative size = 694, normalized size of antiderivative = 3.81 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{17}} \, dx=-\frac {a^{10} d^{10}+2 a^9 b d^9 (3 c+8 d x)+3 a^8 b^2 d^8 \left (7 c^2+32 c d x+40 d^2 x^2\right )+8 a^7 b^3 d^7 \left (7 c^3+42 c^2 d x+90 c d^2 x^2+70 d^3 x^3\right )+14 a^6 b^4 d^6 \left (9 c^4+64 c^3 d x+180 c^2 d^2 x^2+240 c d^3 x^3+130 d^4 x^4\right )+84 a^5 b^5 d^5 \left (3 c^5+24 c^4 d x+80 c^3 d^2 x^2+140 c^2 d^3 x^3+130 c d^4 x^4+52 d^5 x^5\right )+14 a^4 b^6 d^4 \left (33 c^6+288 c^5 d x+1080 c^4 d^2 x^2+2240 c^3 d^3 x^3+2730 c^2 d^4 x^4+1872 c d^5 x^5+572 d^6 x^6\right )+8 a^3 b^7 d^3 \left (99 c^7+924 c^6 d x+3780 c^5 d^2 x^2+8820 c^4 d^3 x^3+12740 c^3 d^4 x^4+11466 c^2 d^5 x^5+6006 c d^6 x^6+1430 d^7 x^7\right )+3 a^2 b^8 d^2 \left (429 c^8+4224 c^7 d x+18480 c^6 d^2 x^2+47040 c^5 d^3 x^3+76440 c^4 d^4 x^4+81536 c^3 d^5 x^5+56056 c^2 d^6 x^6+22880 c d^7 x^7+4290 d^8 x^8\right )+2 a b^9 d \left (1001 c^9+10296 c^8 d x+47520 c^7 d^2 x^2+129360 c^6 d^3 x^3+229320 c^5 d^4 x^4+275184 c^4 d^5 x^5+224224 c^3 d^6 x^6+120120 c^2 d^7 x^7+38610 c d^8 x^8+5720 d^9 x^9\right )+b^{10} \left (3003 c^{10}+32032 c^9 d x+154440 c^8 d^2 x^2+443520 c^7 d^3 x^3+840840 c^6 d^4 x^4+1100736 c^5 d^5 x^5+1009008 c^4 d^6 x^6+640640 c^3 d^7 x^7+270270 c^2 d^8 x^8+68640 c d^9 x^9+8008 d^{10} x^{10}\right )}{48048 b^{11} (a+b x)^{16}} \]

[In]

Integrate[(c + d*x)^10/(a + b*x)^17,x]

[Out]

-1/48048*(a^10*d^10 + 2*a^9*b*d^9*(3*c + 8*d*x) + 3*a^8*b^2*d^8*(7*c^2 + 32*c*d*x + 40*d^2*x^2) + 8*a^7*b^3*d^
7*(7*c^3 + 42*c^2*d*x + 90*c*d^2*x^2 + 70*d^3*x^3) + 14*a^6*b^4*d^6*(9*c^4 + 64*c^3*d*x + 180*c^2*d^2*x^2 + 24
0*c*d^3*x^3 + 130*d^4*x^4) + 84*a^5*b^5*d^5*(3*c^5 + 24*c^4*d*x + 80*c^3*d^2*x^2 + 140*c^2*d^3*x^3 + 130*c*d^4
*x^4 + 52*d^5*x^5) + 14*a^4*b^6*d^4*(33*c^6 + 288*c^5*d*x + 1080*c^4*d^2*x^2 + 2240*c^3*d^3*x^3 + 2730*c^2*d^4
*x^4 + 1872*c*d^5*x^5 + 572*d^6*x^6) + 8*a^3*b^7*d^3*(99*c^7 + 924*c^6*d*x + 3780*c^5*d^2*x^2 + 8820*c^4*d^3*x
^3 + 12740*c^3*d^4*x^4 + 11466*c^2*d^5*x^5 + 6006*c*d^6*x^6 + 1430*d^7*x^7) + 3*a^2*b^8*d^2*(429*c^8 + 4224*c^
7*d*x + 18480*c^6*d^2*x^2 + 47040*c^5*d^3*x^3 + 76440*c^4*d^4*x^4 + 81536*c^3*d^5*x^5 + 56056*c^2*d^6*x^6 + 22
880*c*d^7*x^7 + 4290*d^8*x^8) + 2*a*b^9*d*(1001*c^9 + 10296*c^8*d*x + 47520*c^7*d^2*x^2 + 129360*c^6*d^3*x^3 +
229320*c^5*d^4*x^4 + 275184*c^4*d^5*x^5 + 224224*c^3*d^6*x^6 + 120120*c^2*d^7*x^7 + 38610*c*d^8*x^8 + 5720*d^
9*x^9) + b^10*(3003*c^10 + 32032*c^9*d*x + 154440*c^8*d^2*x^2 + 443520*c^7*d^3*x^3 + 840840*c^6*d^4*x^4 + 1100
736*c^5*d^5*x^5 + 1009008*c^4*d^6*x^6 + 640640*c^3*d^7*x^7 + 270270*c^2*d^8*x^8 + 68640*c*d^9*x^9 + 8008*d^10*
x^10))/(b^11*(a + b*x)^16)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(830\) vs. \(2(170)=340\).

Time = 0.25 (sec) , antiderivative size = 831, normalized size of antiderivative = 4.57

method	result	size
risch	\(\frac {-\frac {a^{10} d^{10}+6 a^{9} b c \,d^{9}+21 a^{8} b^{2} c^{2} d^{8}+56 a^{7} b^{3} c^{3} d^{7}+126 a^{6} b^{4} c^{4} d^{6}+252 a^{5} b^{5} c^{5} d^{5}+462 a^{4} b^{6} c^{6} d^{4}+792 a^{3} b^{7} c^{7} d^{3}+1287 a^{2} b^{8} c^{8} d^{2}+2002 a \,b^{9} c^{9} d +3003 b^{10} c^{10}}{48048 b^{11}}-\frac {d \left (a^{9} d^{9}+6 a^{8} b c \,d^{8}+21 a^{7} b^{2} c^{2} d^{7}+56 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}+252 a^{4} b^{5} c^{5} d^{4}+462 a^{3} b^{6} c^{6} d^{3}+792 a^{2} b^{7} c^{7} d^{2}+1287 a \,b^{8} c^{8} d +2002 b^{9} c^{9}\right ) x}{3003 b^{10}}-\frac {5 d^{2} \left (a^{8} d^{8}+6 a^{7} b c \,d^{7}+21 a^{6} b^{2} c^{2} d^{6}+56 a^{5} b^{3} c^{3} d^{5}+126 a^{4} b^{4} c^{4} d^{4}+252 a^{3} b^{5} c^{5} d^{3}+462 a^{2} b^{6} c^{6} d^{2}+792 a \,b^{7} c^{7} d +1287 b^{8} c^{8}\right ) x^{2}}{2002 b^{9}}-\frac {5 d^{3} \left (a^{7} d^{7}+6 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}+56 a^{4} b^{3} c^{3} d^{4}+126 a^{3} b^{4} c^{4} d^{3}+252 a^{2} b^{5} c^{5} d^{2}+462 a \,b^{6} c^{6} d +792 b^{7} c^{7}\right ) x^{3}}{429 b^{8}}-\frac {5 d^{4} \left (a^{6} d^{6}+6 a^{5} b c \,d^{5}+21 a^{4} b^{2} c^{2} d^{4}+56 a^{3} b^{3} c^{3} d^{3}+126 a^{2} b^{4} c^{4} d^{2}+252 a \,b^{5} c^{5} d +462 b^{6} c^{6}\right ) x^{4}}{132 b^{7}}-\frac {d^{5} \left (a^{5} d^{5}+6 a^{4} b c \,d^{4}+21 a^{3} b^{2} c^{2} d^{3}+56 a^{2} b^{3} c^{3} d^{2}+126 a \,b^{4} c^{4} d +252 b^{5} c^{5}\right ) x^{5}}{11 b^{6}}-\frac {d^{6} \left (a^{4} d^{4}+6 a^{3} b c \,d^{3}+21 a^{2} b^{2} c^{2} d^{2}+56 a \,b^{3} c^{3} d +126 b^{4} c^{4}\right ) x^{6}}{6 b^{5}}-\frac {5 d^{7} \left (a^{3} d^{3}+6 a^{2} b c \,d^{2}+21 a \,b^{2} c^{2} d +56 b^{3} c^{3}\right ) x^{7}}{21 b^{4}}-\frac {15 d^{8} \left (a^{2} d^{2}+6 a b c d +21 b^{2} c^{2}\right ) x^{8}}{56 b^{3}}-\frac {5 d^{9} \left (a d +6 b c \right ) x^{9}}{21 b^{2}}-\frac {d^{10} x^{10}}{6 b}}{\left (b x +a \right )^{16}}\)	\(831\)
default	\(\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 -b^{5} c^{5}\right )}{11 b^{11} \left (b x +a \right )^{11}}+\frac {120 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{13 b^{11} \left (b x +a \right )^{13}}+\frac {40 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 )}{3 b^{11} \left (b x +a \right )^{9}}-\frac {d^{10}}{6 b^{11} \left (b x +a \right )^{6}}-\frac {45 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{8 b^{11} \left (b x +a \right )^{8}}-\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 +b^{8} c^{8}\right )}{14 b^{11} \left (b x +a \right )^{14}}-\frac {35 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 +b^{6} c^{6}\right )}{2 b^{11} \left (b x +a \right )^{12}}+\frac {10 d^{9} \left (a d -b c \right )}{7 b^{11} \left (b x +a \right )^{7}}-\frac {21 d^{6} \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 +b^{4} c^{4}\right )}{b^{11} \left (b x +a \right )^{10}}-\frac {a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}}{16 b^{11} \left (b x +a \right )^{16}}+\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 -b^{9} c^{9}\right )}{3 b^{11} \left (b x +a \right )^{15}}\)	\(867\)
norman	\(\frac {\frac {-a^{10} b^{5} d^{10}-6 a^{9} b^{6} c \,d^{9}-21 a^{8} b^{7} c^{2} d^{8}-56 a^{7} b^{8} c^{3} d^{7}-126 a^{6} b^{9} c^{4} d^{6}-252 a^{5} b^{10} c^{5} d^{5}-462 a^{4} b^{11} c^{6} d^{4}-792 a^{3} c^{7} d^{3} b^{12}-1287 a^{2} b^{13} c^{8} d^{2}-2002 a \,b^{14} c^{9} d -3003 b^{15} c^{10}}{48048 b^{16}}+\frac {\left (-a^{9} b^{5} d^{10}-6 a^{8} b^{6} c \,d^{9}-21 a^{7} b^{7} c^{2} d^{8}-56 a^{6} b^{8} c^{3} d^{7}-126 a^{5} b^{9} c^{4} d^{6}-252 a^{4} b^{10} c^{5} d^{5}-462 a^{3} b^{11} c^{6} d^{4}-792 a^{2} c^{7} d^{3} b^{12}-1287 a \,b^{13} c^{8} d^{2}-2002 b^{14} c^{9} d \right ) x}{3003 b^{15}}+\frac {5 \left (-a^{8} b^{5} d^{10}-6 a^{7} b^{6} c \,d^{9}-21 a^{6} b^{7} c^{2} d^{8}-56 a^{5} b^{8} c^{3} d^{7}-126 a^{4} b^{9} c^{4} d^{6}-252 a^{3} b^{10} c^{5} d^{5}-462 a^{2} b^{11} c^{6} d^{4}-792 a \,c^{7} d^{3} b^{12}-1287 b^{13} c^{8} d^{2}\right ) x^{2}}{2002 b^{14}}+\frac {5 \left (-a^{7} b^{5} d^{10}-6 a^{6} b^{6} c \,d^{9}-21 a^{5} b^{7} c^{2} d^{8}-56 a^{4} b^{8} c^{3} d^{7}-126 a^{3} b^{9} c^{4} d^{6}-252 a^{2} b^{10} c^{5} d^{5}-462 a \,b^{11} c^{6} d^{4}-792 b^{12} c^{7} d^{3}\right ) x^{3}}{429 b^{13}}+\frac {5 \left (-a^{6} b^{5} d^{10}-6 a^{5} b^{6} c \,d^{9}-21 a^{4} b^{7} c^{2} d^{8}-56 a^{3} b^{8} c^{3} d^{7}-126 a^{2} b^{9} c^{4} d^{6}-252 a \,b^{10} c^{5} d^{5}-462 b^{11} c^{6} d^{4}\right ) x^{4}}{132 b^{12}}+\frac {\left (-a^{5} b^{5} d^{10}-6 a^{4} b^{6} c \,d^{9}-21 a^{3} b^{7} c^{2} d^{8}-56 a^{2} b^{8} c^{3} d^{7}-126 a \,b^{9} c^{4} d^{6}-252 b^{10} c^{5} d^{5}\right ) x^{5}}{11 b^{11}}+\frac {\left (-a^{4} b^{5} d^{10}-6 a^{3} b^{6} c \,d^{9}-21 a^{2} b^{7} c^{2} d^{8}-56 a \,b^{8} c^{3} d^{7}-126 b^{9} c^{4} d^{6}\right ) x^{6}}{6 b^{10}}+\frac {5 \left (-a^{3} b^{5} d^{10}-6 a^{2} b^{6} c \,d^{9}-21 a \,b^{7} c^{2} d^{8}-56 b^{8} c^{3} d^{7}\right ) x^{7}}{21 b^{9}}+\frac {15 \left (-a^{2} b^{5} d^{10}-6 a \,b^{6} c \,d^{9}-21 b^{7} c^{2} d^{8}\right ) x^{8}}{56 b^{8}}+\frac {5 \left (-a \,b^{5} d^{10}-6 b^{6} c \,d^{9}\right ) x^{9}}{21 b^{7}}-\frac {d^{10} x^{10}}{6 b}}{\left (b x +a \right )^{16}}\)	\(909\)
gosper	\(-\frac {8008 x^{10} d^{10} b^{10}+11440 x^{9} a \,b^{9} d^{10}+68640 x^{9} b^{10} c \,d^{9}+12870 x^{8} a^{2} b^{8} d^{10}+77220 x^{8} a \,b^{9} c \,d^{9}+270270 x^{8} b^{10} c^{2} d^{8}+11440 x^{7} a^{3} b^{7} d^{10}+68640 x^{7} a^{2} b^{8} c \,d^{9}+240240 x^{7} a \,b^{9} c^{2} d^{8}+640640 x^{7} b^{10} c^{3} d^{7}+8008 x^{6} a^{4} b^{6} d^{10}+48048 x^{6} a^{3} b^{7} c \,d^{9}+168168 x^{6} a^{2} b^{8} c^{2} d^{8}+448448 x^{6} a \,b^{9} c^{3} d^{7}+1009008 x^{6} b^{10} c^{4} d^{6}+4368 x^{5} a^{5} b^{5} d^{10}+26208 x^{5} a^{4} b^{6} c \,d^{9}+91728 x^{5} a^{3} b^{7} c^{2} d^{8}+244608 x^{5} a^{2} b^{8} c^{3} d^{7}+550368 x^{5} a \,b^{9} c^{4} d^{6}+1100736 x^{5} b^{10} c^{5} d^{5}+1820 x^{4} a^{6} b^{4} d^{10}+10920 x^{4} a^{5} b^{5} c \,d^{9}+38220 x^{4} a^{4} b^{6} c^{2} d^{8}+101920 x^{4} a^{3} b^{7} c^{3} d^{7}+229320 x^{4} a^{2} b^{8} c^{4} d^{6}+458640 x^{4} a \,b^{9} c^{5} d^{5}+840840 x^{4} b^{10} c^{6} d^{4}+560 x^{3} a^{7} b^{3} d^{10}+3360 x^{3} a^{6} b^{4} c \,d^{9}+11760 x^{3} a^{5} b^{5} c^{2} d^{8}+31360 x^{3} a^{4} b^{6} c^{3} d^{7}+70560 x^{3} a^{3} b^{7} c^{4} d^{6}+141120 x^{3} a^{2} b^{8} c^{5} d^{5}+258720 x^{3} a \,b^{9} c^{6} d^{4}+443520 x^{3} b^{10} c^{7} d^{3}+120 x^{2} a^{8} b^{2} d^{10}+720 x^{2} a^{7} b^{3} c \,d^{9}+2520 x^{2} a^{6} b^{4} c^{2} d^{8}+6720 x^{2} a^{5} b^{5} c^{3} d^{7}+15120 x^{2} a^{4} b^{6} c^{4} d^{6}+30240 x^{2} a^{3} b^{7} c^{5} d^{5}+55440 x^{2} a^{2} b^{8} c^{6} d^{4}+95040 x^{2} a \,b^{9} c^{7} d^{3}+154440 x^{2} b^{10} c^{8} d^{2}+16 x \,a^{9} b \,d^{10}+96 x \,a^{8} b^{2} c \,d^{9}+336 x \,a^{7} b^{3} c^{2} d^{8}+896 x \,a^{6} b^{4} c^{3} d^{7}+2016 x \,a^{5} b^{5} c^{4} d^{6}+4032 x \,a^{4} b^{6} c^{5} d^{5}+7392 x \,a^{3} b^{7} c^{6} d^{4}+12672 x \,a^{2} b^{8} c^{7} d^{3}+20592 x a \,b^{9} c^{8} d^{2}+32032 x \,b^{10} c^{9} d +a^{10} d^{10}+6 a^{9} b c \,d^{9}+21 a^{8} b^{2} c^{2} d^{8}+56 a^{7} b^{3} c^{3} d^{7}+126 a^{6} b^{4} c^{4} d^{6}+252 a^{5} b^{5} c^{5} d^{5}+462 a^{4} b^{6} c^{6} d^{4}+792 a^{3} b^{7} c^{7} d^{3}+1287 a^{2} b^{8} c^{8} d^{2}+2002 a \,b^{9} c^{9} d +3003 b^{10} c^{10}}{48048 b^{11} \left (b x +a \right )^{16}}\)	\(962\)
parallelrisch	\(\frac {-8008 d^{10} x^{10} b^{15}-11440 a \,b^{14} d^{10} x^{9}-68640 b^{15} c \,d^{9} x^{9}-12870 a^{2} b^{13} d^{10} x^{8}-77220 a \,b^{14} c \,d^{9} x^{8}-270270 b^{15} c^{2} d^{8} x^{8}-11440 a^{3} b^{12} d^{10} x^{7}-68640 a^{2} b^{13} c \,d^{9} x^{7}-240240 a \,b^{14} c^{2} d^{8} x^{7}-640640 b^{15} c^{3} d^{7} x^{7}-8008 a^{4} b^{11} d^{10} x^{6}-48048 a^{3} b^{12} c \,d^{9} x^{6}-168168 a^{2} b^{13} c^{2} d^{8} x^{6}-448448 a \,b^{14} c^{3} d^{7} x^{6}-1009008 b^{15} c^{4} d^{6} x^{6}-4368 a^{5} b^{10} d^{10} x^{5}-26208 a^{4} b^{11} c \,d^{9} x^{5}-91728 a^{3} b^{12} c^{2} d^{8} x^{5}-244608 a^{2} b^{13} c^{3} d^{7} x^{5}-550368 a \,b^{14} c^{4} d^{6} x^{5}-1100736 b^{15} c^{5} d^{5} x^{5}-1820 a^{6} b^{9} d^{10} x^{4}-10920 a^{5} b^{10} c \,d^{9} x^{4}-38220 a^{4} b^{11} c^{2} d^{8} x^{4}-101920 a^{3} b^{12} c^{3} d^{7} x^{4}-229320 a^{2} b^{13} c^{4} d^{6} x^{4}-458640 a \,b^{14} c^{5} d^{5} x^{4}-840840 b^{15} c^{6} d^{4} x^{4}-560 a^{7} b^{8} d^{10} x^{3}-3360 a^{6} b^{9} c \,d^{9} x^{3}-11760 a^{5} b^{10} c^{2} d^{8} x^{3}-31360 a^{4} b^{11} c^{3} d^{7} x^{3}-70560 a^{3} b^{12} c^{4} d^{6} x^{3}-141120 a^{2} b^{13} c^{5} d^{5} x^{3}-258720 a \,b^{14} c^{6} d^{4} x^{3}-443520 b^{15} c^{7} d^{3} x^{3}-120 a^{8} b^{7} d^{10} x^{2}-720 a^{7} b^{8} c \,d^{9} x^{2}-2520 a^{6} b^{9} c^{2} d^{8} x^{2}-6720 a^{5} b^{10} c^{3} d^{7} x^{2}-15120 a^{4} b^{11} c^{4} d^{6} x^{2}-30240 a^{3} b^{12} c^{5} d^{5} x^{2}-55440 a^{2} b^{13} c^{6} d^{4} x^{2}-95040 a \,b^{14} c^{7} d^{3} x^{2}-154440 b^{15} c^{8} d^{2} x^{2}-16 a^{9} b^{6} d^{10} x -96 a^{8} b^{7} c \,d^{9} x -336 a^{7} b^{8} c^{2} d^{8} x -896 a^{6} b^{9} c^{3} d^{7} x -2016 a^{5} b^{10} c^{4} d^{6} x -4032 a^{4} b^{11} c^{5} d^{5} x -7392 a^{3} b^{12} c^{6} d^{4} x -12672 a^{2} b^{13} c^{7} d^{3} x -20592 a \,b^{14} c^{8} d^{2} x -32032 b^{15} c^{9} d x -a^{10} b^{5} d^{10}-6 a^{9} b^{6} c \,d^{9}-21 a^{8} b^{7} c^{2} d^{8}-56 a^{7} b^{8} c^{3} d^{7}-126 a^{6} b^{9} c^{4} d^{6}-252 a^{5} b^{10} c^{5} d^{5}-462 a^{4} b^{11} c^{6} d^{4}-792 a^{3} c^{7} d^{3} b^{12}-1287 a^{2} b^{13} c^{8} d^{2}-2002 a \,b^{14} c^{9} d -3003 b^{15} c^{10}}{48048 b^{16} \left (b x +a \right )^{16}}\)	\(970\)

[In]

int((d*x+c)^10/(b*x+a)^17,x,method=_RETURNVERBOSE)

[Out]

(-1/48048/b^11*(a^10*d^10+6*a^9*b*c*d^9+21*a^8*b^2*c^2*d^8+56*a^7*b^3*c^3*d^7+126*a^6*b^4*c^4*d^6+252*a^5*b^5*
c^5*d^5+462*a^4*b^6*c^6*d^4+792*a^3*b^7*c^7*d^3+1287*a^2*b^8*c^8*d^2+2002*a*b^9*c^9*d+3003*b^10*c^10)-1/3003/b
^10*d*(a^9*d^9+6*a^8*b*c*d^8+21*a^7*b^2*c^2*d^7+56*a^6*b^3*c^3*d^6+126*a^5*b^4*c^4*d^5+252*a^4*b^5*c^5*d^4+462
*a^3*b^6*c^6*d^3+792*a^2*b^7*c^7*d^2+1287*a*b^8*c^8*d+2002*b^9*c^9)*x-5/2002/b^9*d^2*(a^8*d^8+6*a^7*b*c*d^7+21
*a^6*b^2*c^2*d^6+56*a^5*b^3*c^3*d^5+126*a^4*b^4*c^4*d^4+252*a^3*b^5*c^5*d^3+462*a^2*b^6*c^6*d^2+792*a*b^7*c^7*
d+1287*b^8*c^8)*x^2-5/429/b^8*d^3*(a^7*d^7+6*a^6*b*c*d^6+21*a^5*b^2*c^2*d^5+56*a^4*b^3*c^3*d^4+126*a^3*b^4*c^4
*d^3+252*a^2*b^5*c^5*d^2+462*a*b^6*c^6*d+792*b^7*c^7)*x^3-5/132/b^7*d^4*(a^6*d^6+6*a^5*b*c*d^5+21*a^4*b^2*c^2*
d^4+56*a^3*b^3*c^3*d^3+126*a^2*b^4*c^4*d^2+252*a*b^5*c^5*d+462*b^6*c^6)*x^4-1/11/b^6*d^5*(a^5*d^5+6*a^4*b*c*d^
4+21*a^3*b^2*c^2*d^3+56*a^2*b^3*c^3*d^2+126*a*b^4*c^4*d+252*b^5*c^5)*x^5-1/6/b^5*d^6*(a^4*d^4+6*a^3*b*c*d^3+21
*a^2*b^2*c^2*d^2+56*a*b^3*c^3*d+126*b^4*c^4)*x^6-5/21/b^4*d^7*(a^3*d^3+6*a^2*b*c*d^2+21*a*b^2*c^2*d+56*b^3*c^3
)*x^7-15/56/b^3*d^8*(a^2*d^2+6*a*b*c*d+21*b^2*c^2)*x^8-5/21/b^2*d^9*(a*d+6*b*c)*x^9-1/6/b*d^10*x^10)/(b*x+a)^1
6

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1030 vs. \(2 (170) = 340\).

Time = 0.24 (sec) , antiderivative size = 1030, normalized size of antiderivative = 5.66 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{17}} \, dx=-\frac {8008 \, b^{10} d^{10} x^{10} + 3003 \, b^{10} c^{10} + 2002 \, a b^{9} c^{9} d + 1287 \, a^{2} b^{8} c^{8} d^{2} + 792 \, a^{3} b^{7} c^{7} d^{3} + 462 \, a^{4} b^{6} c^{6} d^{4} + 252 \, a^{5} b^{5} c^{5} d^{5} + 126 \, a^{6} b^{4} c^{4} d^{6} + 56 \, a^{7} b^{3} c^{3} d^{7} + 21 \, a^{8} b^{2} c^{2} d^{8} + 6 \, a^{9} b c d^{9} + a^{10} d^{10} + 11440 \, {\left (6 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 12870 \, {\left (21 \, b^{10} c^{2} d^{8} + 6 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 11440 \, {\left (56 \, b^{10} c^{3} d^{7} + 21 \, a b^{9} c^{2} d^{8} + 6 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 8008 \, {\left (126 \, b^{10} c^{4} d^{6} + 56 \, a b^{9} c^{3} d^{7} + 21 \, a^{2} b^{8} c^{2} d^{8} + 6 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 4368 \, {\left (252 \, b^{10} c^{5} d^{5} + 126 \, a b^{9} c^{4} d^{6} + 56 \, a^{2} b^{8} c^{3} d^{7} + 21 \, a^{3} b^{7} c^{2} d^{8} + 6 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 1820 \, {\left (462 \, b^{10} c^{6} d^{4} + 252 \, a b^{9} c^{5} d^{5} + 126 \, a^{2} b^{8} c^{4} d^{6} + 56 \, a^{3} b^{7} c^{3} d^{7} + 21 \, a^{4} b^{6} c^{2} d^{8} + 6 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 560 \, {\left (792 \, b^{10} c^{7} d^{3} + 462 \, a b^{9} c^{6} d^{4} + 252 \, a^{2} b^{8} c^{5} d^{5} + 126 \, a^{3} b^{7} c^{4} d^{6} + 56 \, a^{4} b^{6} c^{3} d^{7} + 21 \, a^{5} b^{5} c^{2} d^{8} + 6 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 120 \, {\left (1287 \, b^{10} c^{8} d^{2} + 792 \, a b^{9} c^{7} d^{3} + 462 \, a^{2} b^{8} c^{6} d^{4} + 252 \, a^{3} b^{7} c^{5} d^{5} + 126 \, a^{4} b^{6} c^{4} d^{6} + 56 \, a^{5} b^{5} c^{3} d^{7} + 21 \, a^{6} b^{4} c^{2} d^{8} + 6 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 16 \, {\left (2002 \, b^{10} c^{9} d + 1287 \, a b^{9} c^{8} d^{2} + 792 \, a^{2} b^{8} c^{7} d^{3} + 462 \, a^{3} b^{7} c^{6} d^{4} + 252 \, a^{4} b^{6} c^{5} d^{5} + 126 \, a^{5} b^{5} c^{4} d^{6} + 56 \, a^{6} b^{4} c^{3} d^{7} + 21 \, a^{7} b^{3} c^{2} d^{8} + 6 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{48048 \, {\left (b^{27} x^{16} + 16 \, a b^{26} x^{15} + 120 \, a^{2} b^{25} x^{14} + 560 \, a^{3} b^{24} x^{13} + 1820 \, a^{4} b^{23} x^{12} + 4368 \, a^{5} b^{22} x^{11} + 8008 \, a^{6} b^{21} x^{10} + 11440 \, a^{7} b^{20} x^{9} + 12870 \, a^{8} b^{19} x^{8} + 11440 \, a^{9} b^{18} x^{7} + 8008 \, a^{10} b^{17} x^{6} + 4368 \, a^{11} b^{16} x^{5} + 1820 \, a^{12} b^{15} x^{4} + 560 \, a^{13} b^{14} x^{3} + 120 \, a^{14} b^{13} x^{2} + 16 \, a^{15} b^{12} x + a^{16} b^{11}\right )}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^17,x, algorithm="fricas")

[Out]

-1/48048*(8008*b^10*d^10*x^10 + 3003*b^10*c^10 + 2002*a*b^9*c^9*d + 1287*a^2*b^8*c^8*d^2 + 792*a^3*b^7*c^7*d^3
+ 462*a^4*b^6*c^6*d^4 + 252*a^5*b^5*c^5*d^5 + 126*a^6*b^4*c^4*d^6 + 56*a^7*b^3*c^3*d^7 + 21*a^8*b^2*c^2*d^8 +
6*a^9*b*c*d^9 + a^10*d^10 + 11440*(6*b^10*c*d^9 + a*b^9*d^10)*x^9 + 12870*(21*b^10*c^2*d^8 + 6*a*b^9*c*d^9 +
a^2*b^8*d^10)*x^8 + 11440*(56*b^10*c^3*d^7 + 21*a*b^9*c^2*d^8 + 6*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 8008*(12
6*b^10*c^4*d^6 + 56*a*b^9*c^3*d^7 + 21*a^2*b^8*c^2*d^8 + 6*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 4368*(252*b^10*
c^5*d^5 + 126*a*b^9*c^4*d^6 + 56*a^2*b^8*c^3*d^7 + 21*a^3*b^7*c^2*d^8 + 6*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 +
1820*(462*b^10*c^6*d^4 + 252*a*b^9*c^5*d^5 + 126*a^2*b^8*c^4*d^6 + 56*a^3*b^7*c^3*d^7 + 21*a^4*b^6*c^2*d^8 + 6
*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 560*(792*b^10*c^7*d^3 + 462*a*b^9*c^6*d^4 + 252*a^2*b^8*c^5*d^5 + 126*a^3
*b^7*c^4*d^6 + 56*a^4*b^6*c^3*d^7 + 21*a^5*b^5*c^2*d^8 + 6*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 120*(1287*b^10*
c^8*d^2 + 792*a*b^9*c^7*d^3 + 462*a^2*b^8*c^6*d^4 + 252*a^3*b^7*c^5*d^5 + 126*a^4*b^6*c^4*d^6 + 56*a^5*b^5*c^3
*d^7 + 21*a^6*b^4*c^2*d^8 + 6*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 16*(2002*b^10*c^9*d + 1287*a*b^9*c^8*d^2 + 7
92*a^2*b^8*c^7*d^3 + 462*a^3*b^7*c^6*d^4 + 252*a^4*b^6*c^5*d^5 + 126*a^5*b^5*c^4*d^6 + 56*a^6*b^4*c^3*d^7 + 21
*a^7*b^3*c^2*d^8 + 6*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^27*x^16 + 16*a*b^26*x^15 + 120*a^2*b^25*x^14 + 560*a^3*
b^24*x^13 + 1820*a^4*b^23*x^12 + 4368*a^5*b^22*x^11 + 8008*a^6*b^21*x^10 + 11440*a^7*b^20*x^9 + 12870*a^8*b^19
*x^8 + 11440*a^9*b^18*x^7 + 8008*a^10*b^17*x^6 + 4368*a^11*b^16*x^5 + 1820*a^12*b^15*x^4 + 560*a^13*b^14*x^3 +
120*a^14*b^13*x^2 + 16*a^15*b^12*x + a^16*b^11)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{17}} \, dx=\text {Timed out} \]

[In]

integrate((d*x+c)**10/(b*x+a)**17,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1030 vs. \(2 (170) = 340\).

Time = 0.25 (sec) , antiderivative size = 1030, normalized size of antiderivative = 5.66 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{17}} \, dx=-\frac {8008 \, b^{10} d^{10} x^{10} + 3003 \, b^{10} c^{10} + 2002 \, a b^{9} c^{9} d + 1287 \, a^{2} b^{8} c^{8} d^{2} + 792 \, a^{3} b^{7} c^{7} d^{3} + 462 \, a^{4} b^{6} c^{6} d^{4} + 252 \, a^{5} b^{5} c^{5} d^{5} + 126 \, a^{6} b^{4} c^{4} d^{6} + 56 \, a^{7} b^{3} c^{3} d^{7} + 21 \, a^{8} b^{2} c^{2} d^{8} + 6 \, a^{9} b c d^{9} + a^{10} d^{10} + 11440 \, {\left (6 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 12870 \, {\left (21 \, b^{10} c^{2} d^{8} + 6 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 11440 \, {\left (56 \, b^{10} c^{3} d^{7} + 21 \, a b^{9} c^{2} d^{8} + 6 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 8008 \, {\left (126 \, b^{10} c^{4} d^{6} + 56 \, a b^{9} c^{3} d^{7} + 21 \, a^{2} b^{8} c^{2} d^{8} + 6 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 4368 \, {\left (252 \, b^{10} c^{5} d^{5} + 126 \, a b^{9} c^{4} d^{6} + 56 \, a^{2} b^{8} c^{3} d^{7} + 21 \, a^{3} b^{7} c^{2} d^{8} + 6 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 1820 \, {\left (462 \, b^{10} c^{6} d^{4} + 252 \, a b^{9} c^{5} d^{5} + 126 \, a^{2} b^{8} c^{4} d^{6} + 56 \, a^{3} b^{7} c^{3} d^{7} + 21 \, a^{4} b^{6} c^{2} d^{8} + 6 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 560 \, {\left (792 \, b^{10} c^{7} d^{3} + 462 \, a b^{9} c^{6} d^{4} + 252 \, a^{2} b^{8} c^{5} d^{5} + 126 \, a^{3} b^{7} c^{4} d^{6} + 56 \, a^{4} b^{6} c^{3} d^{7} + 21 \, a^{5} b^{5} c^{2} d^{8} + 6 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 120 \, {\left (1287 \, b^{10} c^{8} d^{2} + 792 \, a b^{9} c^{7} d^{3} + 462 \, a^{2} b^{8} c^{6} d^{4} + 252 \, a^{3} b^{7} c^{5} d^{5} + 126 \, a^{4} b^{6} c^{4} d^{6} + 56 \, a^{5} b^{5} c^{3} d^{7} + 21 \, a^{6} b^{4} c^{2} d^{8} + 6 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 16 \, {\left (2002 \, b^{10} c^{9} d + 1287 \, a b^{9} c^{8} d^{2} + 792 \, a^{2} b^{8} c^{7} d^{3} + 462 \, a^{3} b^{7} c^{6} d^{4} + 252 \, a^{4} b^{6} c^{5} d^{5} + 126 \, a^{5} b^{5} c^{4} d^{6} + 56 \, a^{6} b^{4} c^{3} d^{7} + 21 \, a^{7} b^{3} c^{2} d^{8} + 6 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{48048 \, {\left (b^{27} x^{16} + 16 \, a b^{26} x^{15} + 120 \, a^{2} b^{25} x^{14} + 560 \, a^{3} b^{24} x^{13} + 1820 \, a^{4} b^{23} x^{12} + 4368 \, a^{5} b^{22} x^{11} + 8008 \, a^{6} b^{21} x^{10} + 11440 \, a^{7} b^{20} x^{9} + 12870 \, a^{8} b^{19} x^{8} + 11440 \, a^{9} b^{18} x^{7} + 8008 \, a^{10} b^{17} x^{6} + 4368 \, a^{11} b^{16} x^{5} + 1820 \, a^{12} b^{15} x^{4} + 560 \, a^{13} b^{14} x^{3} + 120 \, a^{14} b^{13} x^{2} + 16 \, a^{15} b^{12} x + a^{16} b^{11}\right )}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^17,x, algorithm="maxima")

[Out]

-1/48048*(8008*b^10*d^10*x^10 + 3003*b^10*c^10 + 2002*a*b^9*c^9*d + 1287*a^2*b^8*c^8*d^2 + 792*a^3*b^7*c^7*d^3
+ 462*a^4*b^6*c^6*d^4 + 252*a^5*b^5*c^5*d^5 + 126*a^6*b^4*c^4*d^6 + 56*a^7*b^3*c^3*d^7 + 21*a^8*b^2*c^2*d^8 +
6*a^9*b*c*d^9 + a^10*d^10 + 11440*(6*b^10*c*d^9 + a*b^9*d^10)*x^9 + 12870*(21*b^10*c^2*d^8 + 6*a*b^9*c*d^9 +
a^2*b^8*d^10)*x^8 + 11440*(56*b^10*c^3*d^7 + 21*a*b^9*c^2*d^8 + 6*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 8008*(12
6*b^10*c^4*d^6 + 56*a*b^9*c^3*d^7 + 21*a^2*b^8*c^2*d^8 + 6*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 4368*(252*b^10*
c^5*d^5 + 126*a*b^9*c^4*d^6 + 56*a^2*b^8*c^3*d^7 + 21*a^3*b^7*c^2*d^8 + 6*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 +
1820*(462*b^10*c^6*d^4 + 252*a*b^9*c^5*d^5 + 126*a^2*b^8*c^4*d^6 + 56*a^3*b^7*c^3*d^7 + 21*a^4*b^6*c^2*d^8 + 6
*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 560*(792*b^10*c^7*d^3 + 462*a*b^9*c^6*d^4 + 252*a^2*b^8*c^5*d^5 + 126*a^3
*b^7*c^4*d^6 + 56*a^4*b^6*c^3*d^7 + 21*a^5*b^5*c^2*d^8 + 6*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 120*(1287*b^10*
c^8*d^2 + 792*a*b^9*c^7*d^3 + 462*a^2*b^8*c^6*d^4 + 252*a^3*b^7*c^5*d^5 + 126*a^4*b^6*c^4*d^6 + 56*a^5*b^5*c^3
*d^7 + 21*a^6*b^4*c^2*d^8 + 6*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 16*(2002*b^10*c^9*d + 1287*a*b^9*c^8*d^2 + 7
92*a^2*b^8*c^7*d^3 + 462*a^3*b^7*c^6*d^4 + 252*a^4*b^6*c^5*d^5 + 126*a^5*b^5*c^4*d^6 + 56*a^6*b^4*c^3*d^7 + 21
*a^7*b^3*c^2*d^8 + 6*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^27*x^16 + 16*a*b^26*x^15 + 120*a^2*b^25*x^14 + 560*a^3*
b^24*x^13 + 1820*a^4*b^23*x^12 + 4368*a^5*b^22*x^11 + 8008*a^6*b^21*x^10 + 11440*a^7*b^20*x^9 + 12870*a^8*b^19
*x^8 + 11440*a^9*b^18*x^7 + 8008*a^10*b^17*x^6 + 4368*a^11*b^16*x^5 + 1820*a^12*b^15*x^4 + 560*a^13*b^14*x^3 +
120*a^14*b^13*x^2 + 16*a^15*b^12*x + a^16*b^11)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (170) = 340\).

Time = 0.32 (sec) , antiderivative size = 961, normalized size of antiderivative = 5.28 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{17}} \, dx=-\frac {8008 \, b^{10} d^{10} x^{10} + 68640 \, b^{10} c d^{9} x^{9} + 11440 \, a b^{9} d^{10} x^{9} + 270270 \, b^{10} c^{2} d^{8} x^{8} + 77220 \, a b^{9} c d^{9} x^{8} + 12870 \, a^{2} b^{8} d^{10} x^{8} + 640640 \, b^{10} c^{3} d^{7} x^{7} + 240240 \, a b^{9} c^{2} d^{8} x^{7} + 68640 \, a^{2} b^{8} c d^{9} x^{7} + 11440 \, a^{3} b^{7} d^{10} x^{7} + 1009008 \, b^{10} c^{4} d^{6} x^{6} + 448448 \, a b^{9} c^{3} d^{7} x^{6} + 168168 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 48048 \, a^{3} b^{7} c d^{9} x^{6} + 8008 \, a^{4} b^{6} d^{10} x^{6} + 1100736 \, b^{10} c^{5} d^{5} x^{5} + 550368 \, a b^{9} c^{4} d^{6} x^{5} + 244608 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 91728 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 26208 \, a^{4} b^{6} c d^{9} x^{5} + 4368 \, a^{5} b^{5} d^{10} x^{5} + 840840 \, b^{10} c^{6} d^{4} x^{4} + 458640 \, a b^{9} c^{5} d^{5} x^{4} + 229320 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 101920 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 38220 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 10920 \, a^{5} b^{5} c d^{9} x^{4} + 1820 \, a^{6} b^{4} d^{10} x^{4} + 443520 \, b^{10} c^{7} d^{3} x^{3} + 258720 \, a b^{9} c^{6} d^{4} x^{3} + 141120 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 70560 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 31360 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 11760 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 3360 \, a^{6} b^{4} c d^{9} x^{3} + 560 \, a^{7} b^{3} d^{10} x^{3} + 154440 \, b^{10} c^{8} d^{2} x^{2} + 95040 \, a b^{9} c^{7} d^{3} x^{2} + 55440 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 30240 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 15120 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 6720 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 2520 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 720 \, a^{7} b^{3} c d^{9} x^{2} + 120 \, a^{8} b^{2} d^{10} x^{2} + 32032 \, b^{10} c^{9} d x + 20592 \, a b^{9} c^{8} d^{2} x + 12672 \, a^{2} b^{8} c^{7} d^{3} x + 7392 \, a^{3} b^{7} c^{6} d^{4} x + 4032 \, a^{4} b^{6} c^{5} d^{5} x + 2016 \, a^{5} b^{5} c^{4} d^{6} x + 896 \, a^{6} b^{4} c^{3} d^{7} x + 336 \, a^{7} b^{3} c^{2} d^{8} x + 96 \, a^{8} b^{2} c d^{9} x + 16 \, a^{9} b d^{10} x + 3003 \, b^{10} c^{10} + 2002 \, a b^{9} c^{9} d + 1287 \, a^{2} b^{8} c^{8} d^{2} + 792 \, a^{3} b^{7} c^{7} d^{3} + 462 \, a^{4} b^{6} c^{6} d^{4} + 252 \, a^{5} b^{5} c^{5} d^{5} + 126 \, a^{6} b^{4} c^{4} d^{6} + 56 \, a^{7} b^{3} c^{3} d^{7} + 21 \, a^{8} b^{2} c^{2} d^{8} + 6 \, a^{9} b c d^{9} + a^{10} d^{10}}{48048 \, {\left (b x + a\right )}^{16} b^{11}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^17,x, algorithm="giac")

[Out]

-1/48048*(8008*b^10*d^10*x^10 + 68640*b^10*c*d^9*x^9 + 11440*a*b^9*d^10*x^9 + 270270*b^10*c^2*d^8*x^8 + 77220*
a*b^9*c*d^9*x^8 + 12870*a^2*b^8*d^10*x^8 + 640640*b^10*c^3*d^7*x^7 + 240240*a*b^9*c^2*d^8*x^7 + 68640*a^2*b^8*
c*d^9*x^7 + 11440*a^3*b^7*d^10*x^7 + 1009008*b^10*c^4*d^6*x^6 + 448448*a*b^9*c^3*d^7*x^6 + 168168*a^2*b^8*c^2*
d^8*x^6 + 48048*a^3*b^7*c*d^9*x^6 + 8008*a^4*b^6*d^10*x^6 + 1100736*b^10*c^5*d^5*x^5 + 550368*a*b^9*c^4*d^6*x^
5 + 244608*a^2*b^8*c^3*d^7*x^5 + 91728*a^3*b^7*c^2*d^8*x^5 + 26208*a^4*b^6*c*d^9*x^5 + 4368*a^5*b^5*d^10*x^5 +
840840*b^10*c^6*d^4*x^4 + 458640*a*b^9*c^5*d^5*x^4 + 229320*a^2*b^8*c^4*d^6*x^4 + 101920*a^3*b^7*c^3*d^7*x^4
+ 38220*a^4*b^6*c^2*d^8*x^4 + 10920*a^5*b^5*c*d^9*x^4 + 1820*a^6*b^4*d^10*x^4 + 443520*b^10*c^7*d^3*x^3 + 2587
20*a*b^9*c^6*d^4*x^3 + 141120*a^2*b^8*c^5*d^5*x^3 + 70560*a^3*b^7*c^4*d^6*x^3 + 31360*a^4*b^6*c^3*d^7*x^3 + 11
760*a^5*b^5*c^2*d^8*x^3 + 3360*a^6*b^4*c*d^9*x^3 + 560*a^7*b^3*d^10*x^3 + 154440*b^10*c^8*d^2*x^2 + 95040*a*b^
9*c^7*d^3*x^2 + 55440*a^2*b^8*c^6*d^4*x^2 + 30240*a^3*b^7*c^5*d^5*x^2 + 15120*a^4*b^6*c^4*d^6*x^2 + 6720*a^5*b
^5*c^3*d^7*x^2 + 2520*a^6*b^4*c^2*d^8*x^2 + 720*a^7*b^3*c*d^9*x^2 + 120*a^8*b^2*d^10*x^2 + 32032*b^10*c^9*d*x
+ 20592*a*b^9*c^8*d^2*x + 12672*a^2*b^8*c^7*d^3*x + 7392*a^3*b^7*c^6*d^4*x + 4032*a^4*b^6*c^5*d^5*x + 2016*a^5
*b^5*c^4*d^6*x + 896*a^6*b^4*c^3*d^7*x + 336*a^7*b^3*c^2*d^8*x + 96*a^8*b^2*c*d^9*x + 16*a^9*b*d^10*x + 3003*b
^10*c^10 + 2002*a*b^9*c^9*d + 1287*a^2*b^8*c^8*d^2 + 792*a^3*b^7*c^7*d^3 + 462*a^4*b^6*c^6*d^4 + 252*a^5*b^5*c
^5*d^5 + 126*a^6*b^4*c^4*d^6 + 56*a^7*b^3*c^3*d^7 + 21*a^8*b^2*c^2*d^8 + 6*a^9*b*c*d^9 + a^10*d^10)/((b*x + a)
^16*b^11)

Mupad [B] (verification not implemented)

Time = 0.81 (sec) , antiderivative size = 1131, normalized size of antiderivative = 6.21 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{17}} \, dx=-\frac {a^{10}\,d^{10}+6\,a^9\,b\,c\,d^9+16\,a^9\,b\,d^{10}\,x+21\,a^8\,b^2\,c^2\,d^8+96\,a^8\,b^2\,c\,d^9\,x+120\,a^8\,b^2\,d^{10}\,x^2+56\,a^7\,b^3\,c^3\,d^7+336\,a^7\,b^3\,c^2\,d^8\,x+720\,a^7\,b^3\,c\,d^9\,x^2+560\,a^7\,b^3\,d^{10}\,x^3+126\,a^6\,b^4\,c^4\,d^6+896\,a^6\,b^4\,c^3\,d^7\,x+2520\,a^6\,b^4\,c^2\,d^8\,x^2+3360\,a^6\,b^4\,c\,d^9\,x^3+1820\,a^6\,b^4\,d^{10}\,x^4+252\,a^5\,b^5\,c^5\,d^5+2016\,a^5\,b^5\,c^4\,d^6\,x+6720\,a^5\,b^5\,c^3\,d^7\,x^2+11760\,a^5\,b^5\,c^2\,d^8\,x^3+10920\,a^5\,b^5\,c\,d^9\,x^4+4368\,a^5\,b^5\,d^{10}\,x^5+462\,a^4\,b^6\,c^6\,d^4+4032\,a^4\,b^6\,c^5\,d^5\,x+15120\,a^4\,b^6\,c^4\,d^6\,x^2+31360\,a^4\,b^6\,c^3\,d^7\,x^3+38220\,a^4\,b^6\,c^2\,d^8\,x^4+26208\,a^4\,b^6\,c\,d^9\,x^5+8008\,a^4\,b^6\,d^{10}\,x^6+792\,a^3\,b^7\,c^7\,d^3+7392\,a^3\,b^7\,c^6\,d^4\,x+30240\,a^3\,b^7\,c^5\,d^5\,x^2+70560\,a^3\,b^7\,c^4\,d^6\,x^3+101920\,a^3\,b^7\,c^3\,d^7\,x^4+91728\,a^3\,b^7\,c^2\,d^8\,x^5+48048\,a^3\,b^7\,c\,d^9\,x^6+11440\,a^3\,b^7\,d^{10}\,x^7+1287\,a^2\,b^8\,c^8\,d^2+12672\,a^2\,b^8\,c^7\,d^3\,x+55440\,a^2\,b^8\,c^6\,d^4\,x^2+141120\,a^2\,b^8\,c^5\,d^5\,x^3+229320\,a^2\,b^8\,c^4\,d^6\,x^4+244608\,a^2\,b^8\,c^3\,d^7\,x^5+168168\,a^2\,b^8\,c^2\,d^8\,x^6+68640\,a^2\,b^8\,c\,d^9\,x^7+12870\,a^2\,b^8\,d^{10}\,x^8+2002\,a\,b^9\,c^9\,d+20592\,a\,b^9\,c^8\,d^2\,x+95040\,a\,b^9\,c^7\,d^3\,x^2+258720\,a\,b^9\,c^6\,d^4\,x^3+458640\,a\,b^9\,c^5\,d^5\,x^4+550368\,a\,b^9\,c^4\,d^6\,x^5+448448\,a\,b^9\,c^3\,d^7\,x^6+240240\,a\,b^9\,c^2\,d^8\,x^7+77220\,a\,b^9\,c\,d^9\,x^8+11440\,a\,b^9\,d^{10}\,x^9+3003\,b^{10}\,c^{10}+32032\,b^{10}\,c^9\,d\,x+154440\,b^{10}\,c^8\,d^2\,x^2+443520\,b^{10}\,c^7\,d^3\,x^3+840840\,b^{10}\,c^6\,d^4\,x^4+1100736\,b^{10}\,c^5\,d^5\,x^5+1009008\,b^{10}\,c^4\,d^6\,x^6+640640\,b^{10}\,c^3\,d^7\,x^7+270270\,b^{10}\,c^2\,d^8\,x^8+68640\,b^{10}\,c\,d^9\,x^9+8008\,b^{10}\,d^{10}\,x^{10}}{48048\,a^{16}\,b^{11}+768768\,a^{15}\,b^{12}\,x+5765760\,a^{14}\,b^{13}\,x^2+26906880\,a^{13}\,b^{14}\,x^3+87447360\,a^{12}\,b^{15}\,x^4+209873664\,a^{11}\,b^{16}\,x^5+384768384\,a^{10}\,b^{17}\,x^6+549669120\,a^9\,b^{18}\,x^7+618377760\,a^8\,b^{19}\,x^8+549669120\,a^7\,b^{20}\,x^9+384768384\,a^6\,b^{21}\,x^{10}[phone]\,a^5\,b^{22}\,x^{11}[phone]\,a^4\,b^{23}\,x^{12}[phone]\,a^3\,b^{24}\,x^{13}+5765760\,a^2\,b^{25}\,x^{14}+768768\,a\,b^{26}\,x^{15}+48048\,b^{27}\,x^{16}} \]

[In]

int((c + d*x)^10/(a + b*x)^17,x)

[Out]

-(a^10*d^10 + 3003*b^10*c^10 + 8008*b^10*d^10*x^10 + 11440*a*b^9*d^10*x^9 + 68640*b^10*c*d^9*x^9 + 1287*a^2*b^
8*c^8*d^2 + 792*a^3*b^7*c^7*d^3 + 462*a^4*b^6*c^6*d^4 + 252*a^5*b^5*c^5*d^5 + 126*a^6*b^4*c^4*d^6 + 56*a^7*b^3
*c^3*d^7 + 21*a^8*b^2*c^2*d^8 + 120*a^8*b^2*d^10*x^2 + 560*a^7*b^3*d^10*x^3 + 1820*a^6*b^4*d^10*x^4 + 4368*a^5
*b^5*d^10*x^5 + 8008*a^4*b^6*d^10*x^6 + 11440*a^3*b^7*d^10*x^7 + 12870*a^2*b^8*d^10*x^8 + 154440*b^10*c^8*d^2*
x^2 + 443520*b^10*c^7*d^3*x^3 + 840840*b^10*c^6*d^4*x^4 + 1100736*b^10*c^5*d^5*x^5 + 1009008*b^10*c^4*d^6*x^6
+ 640640*b^10*c^3*d^7*x^7 + 270270*b^10*c^2*d^8*x^8 + 2002*a*b^9*c^9*d + 6*a^9*b*c*d^9 + 16*a^9*b*d^10*x + 320
32*b^10*c^9*d*x + 55440*a^2*b^8*c^6*d^4*x^2 + 30240*a^3*b^7*c^5*d^5*x^2 + 15120*a^4*b^6*c^4*d^6*x^2 + 6720*a^5
*b^5*c^3*d^7*x^2 + 2520*a^6*b^4*c^2*d^8*x^2 + 141120*a^2*b^8*c^5*d^5*x^3 + 70560*a^3*b^7*c^4*d^6*x^3 + 31360*a
^4*b^6*c^3*d^7*x^3 + 11760*a^5*b^5*c^2*d^8*x^3 + 229320*a^2*b^8*c^4*d^6*x^4 + 101920*a^3*b^7*c^3*d^7*x^4 + 382
20*a^4*b^6*c^2*d^8*x^4 + 244608*a^2*b^8*c^3*d^7*x^5 + 91728*a^3*b^7*c^2*d^8*x^5 + 168168*a^2*b^8*c^2*d^8*x^6 +
20592*a*b^9*c^8*d^2*x + 96*a^8*b^2*c*d^9*x + 77220*a*b^9*c*d^9*x^8 + 12672*a^2*b^8*c^7*d^3*x + 7392*a^3*b^7*c
^6*d^4*x + 4032*a^4*b^6*c^5*d^5*x + 2016*a^5*b^5*c^4*d^6*x + 896*a^6*b^4*c^3*d^7*x + 336*a^7*b^3*c^2*d^8*x + 9
5040*a*b^9*c^7*d^3*x^2 + 720*a^7*b^3*c*d^9*x^2 + 258720*a*b^9*c^6*d^4*x^3 + 3360*a^6*b^4*c*d^9*x^3 + 458640*a*
b^9*c^5*d^5*x^4 + 10920*a^5*b^5*c*d^9*x^4 + 550368*a*b^9*c^4*d^6*x^5 + 26208*a^4*b^6*c*d^9*x^5 + 448448*a*b^9*
c^3*d^7*x^6 + 48048*a^3*b^7*c*d^9*x^6 + 240240*a*b^9*c^2*d^8*x^7 + 68640*a^2*b^8*c*d^9*x^7)/(48048*a^16*b^11 +
48048*b^27*x^16 + 768768*a^15*b^12*x + 768768*a*b^26*x^15 + 5765760*a^14*b^13*x^2 + 26906880*a^13*b^14*x^3 +
87447360*a^12*b^15*x^4 + 209873664*a^11*b^16*x^5 + 384768384*a^10*b^17*x^6 + 549669120*a^9*b^18*x^7 + 61837776
0*a^8*b^19*x^8 + 549669120*a^7*b^20*x^9 + 384768384*a^6*b^21*x^10 + 209873664*a^5*b^22*x^11 + 87447360*a^4*b^2
3*x^12 + 26906880*a^3*b^24*x^13 + 5765760*a^2*b^25*x^14)