\(\int \frac {(c+d x)^{10}}{(a+b x)^{18}} \, dx\) [1329]

Optimal result

Integrand size = 15, antiderivative size = 213 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{18}} \, dx=-\frac {(c+d x)^{11}}{17 (b c-a d) (a+b x)^{17}}+\frac {3 d (c+d x)^{11}}{136 (b c-a d)^2 (a+b x)^{16}}-\frac {d^2 (c+d x)^{11}}{136 (b c-a d)^3 (a+b x)^{15}}+\frac {d^3 (c+d x)^{11}}{476 (b c-a d)^4 (a+b x)^{14}}-\frac {3 d^4 (c+d x)^{11}}{6188 (b c-a d)^5 (a+b x)^{13}}+\frac {d^5 (c+d x)^{11}}{12376 (b c-a d)^6 (a+b x)^{12}}-\frac {d^6 (c+d x)^{11}}{136136 (b c-a d)^7 (a+b x)^{11}} \]

[Out]

-1/17*(d*x+c)^11/(-a*d+b*c)/(b*x+a)^17+3/136*d*(d*x+c)^11/(-a*d+b*c)^2/(b*x+a)^16-1/136*d^2*(d*x+c)^11/(-a*d+b
*c)^3/(b*x+a)^15+1/476*d^3*(d*x+c)^11/(-a*d+b*c)^4/(b*x+a)^14-3/6188*d^4*(d*x+c)^11/(-a*d+b*c)^5/(b*x+a)^13+1/
12376*d^5*(d*x+c)^11/(-a*d+b*c)^6/(b*x+a)^12-1/136136*d^6*(d*x+c)^11/(-a*d+b*c)^7/(b*x+a)^11

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {(c+d x)^{10}}{(a+b x)^{18}} \, dx=-\frac {d^6 (c+d x)^{11}}{136136 (a+b x)^{11} (b c-a d)^7}+\frac {d^5 (c+d x)^{11}}{12376 (a+b x)^{12} (b c-a d)^6}-\frac {3 d^4 (c+d x)^{11}}{6188 (a+b x)^{13} (b c-a d)^5}+\frac {d^3 (c+d x)^{11}}{476 (a+b x)^{14} (b c-a d)^4}-\frac {d^2 (c+d x)^{11}}{136 (a+b x)^{15} (b c-a d)^3}+\frac {3 d (c+d x)^{11}}{136 (a+b x)^{16} (b c-a d)^2}-\frac {(c+d x)^{11}}{17 (a+b x)^{17} (b c-a d)} \]

[In]

Int[(c + d*x)^10/(a + b*x)^18,x]

[Out]

-1/17*(c + d*x)^11/((b*c - a*d)*(a + b*x)^17) + (3*d*(c + d*x)^11)/(136*(b*c - a*d)^2*(a + b*x)^16) - (d^2*(c
+ d*x)^11)/(136*(b*c - a*d)^3*(a + b*x)^15) + (d^3*(c + d*x)^11)/(476*(b*c - a*d)^4*(a + b*x)^14) - (3*d^4*(c
+ d*x)^11)/(6188*(b*c - a*d)^5*(a + b*x)^13) + (d^5*(c + d*x)^11)/(12376*(b*c - a*d)^6*(a + b*x)^12) - (d^6*(c
+ d*x)^11)/(136136*(b*c - a*d)^7*(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}}{17 (b c-a d) (a+b x)^{17}}-\frac {(6 d) \int \frac {(c+d x)^{10}}{(a+b x)^{17}} \, dx}{17 (b c-a d)} \\ & = -\frac {(c+d x)^{11}}{17 (b c-a d) (a+b x)^{17}}+\frac {3 d (c+d x)^{11}}{136 (b c-a d)^2 (a+b x)^{16}}+\frac {\left (15 d^2\right ) \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx}{136 (b c-a d)^2} \\ & = -\frac {(c+d x)^{11}}{17 (b c-a d) (a+b x)^{17}}+\frac {3 d (c+d x)^{11}}{136 (b c-a d)^2 (a+b x)^{16}}-\frac {d^2 (c+d x)^{11}}{136 (b c-a d)^3 (a+b x)^{15}}-\frac {d^3 \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx}{34 (b c-a d)^3} \\ & = -\frac {(c+d x)^{11}}{17 (b c-a d) (a+b x)^{17}}+\frac {3 d (c+d x)^{11}}{136 (b c-a d)^2 (a+b x)^{16}}-\frac {d^2 (c+d x)^{11}}{136 (b c-a d)^3 (a+b x)^{15}}+\frac {d^3 (c+d x)^{11}}{476 (b c-a d)^4 (a+b x)^{14}}+\frac {\left (3 d^4\right ) \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx}{476 (b c-a d)^4} \\ & = -\frac {(c+d x)^{11}}{17 (b c-a d) (a+b x)^{17}}+\frac {3 d (c+d x)^{11}}{136 (b c-a d)^2 (a+b x)^{16}}-\frac {d^2 (c+d x)^{11}}{136 (b c-a d)^3 (a+b x)^{15}}+\frac {d^3 (c+d x)^{11}}{476 (b c-a d)^4 (a+b x)^{14}}-\frac {3 d^4 (c+d x)^{11}}{6188 (b c-a d)^5 (a+b x)^{13}}-\frac {\left (3 d^5\right ) \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx}{3094 (b c-a d)^5} \\ & = -\frac {(c+d x)^{11}}{17 (b c-a d) (a+b x)^{17}}+\frac {3 d (c+d x)^{11}}{136 (b c-a d)^2 (a+b x)^{16}}-\frac {d^2 (c+d x)^{11}}{136 (b c-a d)^3 (a+b x)^{15}}+\frac {d^3 (c+d x)^{11}}{476 (b c-a d)^4 (a+b x)^{14}}-\frac {3 d^4 (c+d x)^{11}}{6188 (b c-a d)^5 (a+b x)^{13}}+\frac {d^5 (c+d x)^{11}}{12376 (b c-a d)^6 (a+b x)^{12}}+\frac {d^6 \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx}{12376 (b c-a d)^6} \\ & = -\frac {(c+d x)^{11}}{17 (b c-a d) (a+b x)^{17}}+\frac {3 d (c+d x)^{11}}{136 (b c-a d)^2 (a+b x)^{16}}-\frac {d^2 (c+d x)^{11}}{136 (b c-a d)^3 (a+b x)^{15}}+\frac {d^3 (c+d x)^{11}}{476 (b c-a d)^4 (a+b x)^{14}}-\frac {3 d^4 (c+d x)^{11}}{6188 (b c-a d)^5 (a+b x)^{13}}+\frac {d^5 (c+d x)^{11}}{12376 (b c-a d)^6 (a+b x)^{12}}-\frac {d^6 (c+d x)^{11}}{136136 (b c-a d)^7 (a+b x)^{11}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(690\) vs. \(2(213)=426\).

Time = 0.18 (sec) , antiderivative size = 690, normalized size of antiderivative = 3.24 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{18}} \, dx=-\frac {a^{10} d^{10}+a^9 b d^9 (7 c+17 d x)+a^8 b^2 d^8 \left (28 c^2+119 c d x+136 d^2 x^2\right )+4 a^7 b^3 d^7 \left (21 c^3+119 c^2 d x+238 c d^2 x^2+170 d^3 x^3\right )+14 a^6 b^4 d^6 \left (15 c^4+102 c^3 d x+272 c^2 d^2 x^2+340 c d^3 x^3+170 d^4 x^4\right )+14 a^5 b^5 d^5 \left (33 c^5+255 c^4 d x+816 c^3 d^2 x^2+1360 c^2 d^3 x^3+1190 c d^4 x^4+442 d^5 x^5\right )+14 a^4 b^6 d^4 \left (66 c^6+561 c^5 d x+2040 c^4 d^2 x^2+4080 c^3 d^3 x^3+4760 c^2 d^4 x^4+3094 c d^5 x^5+884 d^6 x^6\right )+4 a^3 b^7 d^3 \left (429 c^7+3927 c^6 d x+15708 c^5 d^2 x^2+35700 c^4 d^3 x^3+49980 c^3 d^4 x^4+43316 c^2 d^5 x^5+21658 c d^6 x^6+4862 d^7 x^7\right )+a^2 b^8 d^2 \left (3003 c^8+29172 c^7 d x+125664 c^6 d^2 x^2+314160 c^5 d^3 x^3+499800 c^4 d^4 x^4+519792 c^3 d^5 x^5+346528 c^2 d^6 x^6+136136 c d^7 x^7+24310 d^8 x^8\right )+a b^9 d \left (5005 c^9+51051 c^8 d x+233376 c^7 d^2 x^2+628320 c^6 d^3 x^3+1099560 c^5 d^4 x^4+1299480 c^4 d^5 x^5+1039584 c^3 d^6 x^6+544544 c^2 d^7 x^7+170170 c d^8 x^8+24310 d^9 x^9\right )+b^{10} \left (8008 c^{10}+85085 c^9 d x+408408 c^8 d^2 x^2+1166880 c^7 d^3 x^3+2199120 c^6 d^4 x^4+2858856 c^5 d^5 x^5+2598960 c^4 d^6 x^6+1633632 c^3 d^7 x^7+680680 c^2 d^8 x^8+170170 c d^9 x^9+19448 d^{10} x^{10}\right )}{136136 b^{11} (a+b x)^{17}} \]

[In]

Integrate[(c + d*x)^10/(a + b*x)^18,x]

[Out]

-1/136136*(a^10*d^10 + a^9*b*d^9*(7*c + 17*d*x) + a^8*b^2*d^8*(28*c^2 + 119*c*d*x + 136*d^2*x^2) + 4*a^7*b^3*d
^7*(21*c^3 + 119*c^2*d*x + 238*c*d^2*x^2 + 170*d^3*x^3) + 14*a^6*b^4*d^6*(15*c^4 + 102*c^3*d*x + 272*c^2*d^2*x
^2 + 340*c*d^3*x^3 + 170*d^4*x^4) + 14*a^5*b^5*d^5*(33*c^5 + 255*c^4*d*x + 816*c^3*d^2*x^2 + 1360*c^2*d^3*x^3
+ 1190*c*d^4*x^4 + 442*d^5*x^5) + 14*a^4*b^6*d^4*(66*c^6 + 561*c^5*d*x + 2040*c^4*d^2*x^2 + 4080*c^3*d^3*x^3 +
4760*c^2*d^4*x^4 + 3094*c*d^5*x^5 + 884*d^6*x^6) + 4*a^3*b^7*d^3*(429*c^7 + 3927*c^6*d*x + 15708*c^5*d^2*x^2
+ 35700*c^4*d^3*x^3 + 49980*c^3*d^4*x^4 + 43316*c^2*d^5*x^5 + 21658*c*d^6*x^6 + 4862*d^7*x^7) + a^2*b^8*d^2*(3
003*c^8 + 29172*c^7*d*x + 125664*c^6*d^2*x^2 + 314160*c^5*d^3*x^3 + 499800*c^4*d^4*x^4 + 519792*c^3*d^5*x^5 +
346528*c^2*d^6*x^6 + 136136*c*d^7*x^7 + 24310*d^8*x^8) + a*b^9*d*(5005*c^9 + 51051*c^8*d*x + 233376*c^7*d^2*x^
2 + 628320*c^6*d^3*x^3 + 1099560*c^5*d^4*x^4 + 1299480*c^4*d^5*x^5 + 1039584*c^3*d^6*x^6 + 544544*c^2*d^7*x^7
+ 170170*c*d^8*x^8 + 24310*d^9*x^9) + b^10*(8008*c^10 + 85085*c^9*d*x + 408408*c^8*d^2*x^2 + 1166880*c^7*d^3*x
^3 + 2199120*c^6*d^4*x^4 + 2858856*c^5*d^5*x^5 + 2598960*c^4*d^6*x^6 + 1633632*c^3*d^7*x^7 + 680680*c^2*d^8*x^
8 + 170170*c*d^9*x^9 + 19448*d^10*x^10))/(b^11*(a + b*x)^17)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(830\) vs. \(2(199)=398\).

Time = 0.25 (sec) , antiderivative size = 831, normalized size of antiderivative = 3.90

method	result	size
risch	\(\frac {-\frac {a^{10} d^{10}+7 a^{9} b c \,d^{9}+28 a^{8} b^{2} c^{2} d^{8}+84 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}+462 a^{5} b^{5} c^{5} d^{5}+924 a^{4} b^{6} c^{6} d^{4}+1716 a^{3} b^{7} c^{7} d^{3}+3003 a^{2} b^{8} c^{8} d^{2}+5005 a \,b^{9} c^{9} d +8008 b^{10} c^{10}}{136136 b^{11}}-\frac {d \left (a^{9} d^{9}+7 a^{8} b c \,d^{8}+28 a^{7} b^{2} c^{2} d^{7}+84 a^{6} b^{3} c^{3} d^{6}+210 a^{5} b^{4} c^{4} d^{5}+462 a^{4} b^{5} c^{5} d^{4}+924 a^{3} b^{6} c^{6} d^{3}+1716 a^{2} b^{7} c^{7} d^{2}+3003 a \,b^{8} c^{8} d +5005 b^{9} c^{9}\right ) x}{8008 b^{10}}-\frac {d^{2} \left (a^{8} d^{8}+7 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}+84 a^{5} b^{3} c^{3} d^{5}+210 a^{4} b^{4} c^{4} d^{4}+462 a^{3} b^{5} c^{5} d^{3}+924 a^{2} b^{6} c^{6} d^{2}+1716 a \,b^{7} c^{7} d +3003 b^{8} c^{8}\right ) x^{2}}{1001 b^{9}}-\frac {5 d^{3} \left (a^{7} d^{7}+7 a^{6} b c \,d^{6}+28 a^{5} b^{2} c^{2} d^{5}+84 a^{4} b^{3} c^{3} d^{4}+210 a^{3} b^{4} c^{4} d^{3}+462 a^{2} b^{5} c^{5} d^{2}+924 a \,b^{6} c^{6} d +1716 b^{7} c^{7}\right ) x^{3}}{1001 b^{8}}-\frac {5 d^{4} \left (a^{6} d^{6}+7 a^{5} b c \,d^{5}+28 a^{4} b^{2} c^{2} d^{4}+84 a^{3} b^{3} c^{3} d^{3}+210 a^{2} b^{4} c^{4} d^{2}+462 a \,b^{5} c^{5} d +924 b^{6} c^{6}\right ) x^{4}}{286 b^{7}}-\frac {d^{5} \left (a^{5} d^{5}+7 a^{4} b c \,d^{4}+28 a^{3} b^{2} c^{2} d^{3}+84 a^{2} b^{3} c^{3} d^{2}+210 a \,b^{4} c^{4} d +462 b^{5} c^{5}\right ) x^{5}}{22 b^{6}}-\frac {d^{6} \left (a^{4} d^{4}+7 a^{3} b c \,d^{3}+28 a^{2} b^{2} c^{2} d^{2}+84 a \,b^{3} c^{3} d +210 b^{4} c^{4}\right ) x^{6}}{11 b^{5}}-\frac {d^{7} \left (a^{3} d^{3}+7 a^{2} b c \,d^{2}+28 a \,b^{2} c^{2} d +84 b^{3} c^{3}\right ) x^{7}}{7 b^{4}}-\frac {5 d^{8} \left (a^{2} d^{2}+7 a b c d +28 b^{2} c^{2}\right ) x^{8}}{28 b^{3}}-\frac {5 d^{9} \left (a d +7 b c \right ) x^{9}}{28 b^{2}}-\frac {d^{10} x^{10}}{7 b}}{\left (b x +a \right )^{17}}\)	\(831\)
default	\(-\frac {210 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 )}{11 b^{11} \left (b x +a \right )^{11}}-\frac {210 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 )}{13 b^{11} \left (b x +a \right )^{13}}-\frac {5 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{11} \left (b x +a \right )^{9}}+\frac {5 d^{9} \left (a d -b c \right )}{4 b^{11} \left (b x +a \right )^{8}}+\frac {60 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 )}{7 b^{11} \left (b x +a \right )^{14}}+\frac {21 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 )}{b^{11} \left (b x +a \right )^{12}}-\frac {d^{10}}{7 b^{11} \left (b x +a \right )^{7}}+\frac {12 d^{7} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{b^{11} \left (b x +a \right )^{10}}+\frac {5 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}\right )}{8 b^{11} \left (b x +a \right )^{16}}-\frac {3 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 )}{b^{11} \left (b x +a \right )^{15}}-\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}}{17 b^{11} \left (b x +a \right )^{17}}\)	\(867\)
norman	\(\frac {\frac {-a^{10} b^{6} d^{10}-7 a^{9} b^{7} c \,d^{9}-28 a^{8} b^{8} c^{2} d^{8}-84 a^{7} b^{9} c^{3} d^{7}-210 a^{6} b^{10} c^{4} d^{6}-462 a^{5} b^{11} c^{5} d^{5}-924 a^{4} b^{12} c^{6} d^{4}-1716 a^{3} c^{7} d^{3} b^{13}-3003 a^{2} b^{14} c^{8} d^{2}-5005 a \,b^{15} c^{9} d -8008 b^{16} c^{10}}{136136 b^{17}}+\frac {\left (-a^{9} b^{6} d^{10}-7 a^{8} b^{7} c \,d^{9}-28 a^{7} b^{8} c^{2} d^{8}-84 a^{6} b^{9} c^{3} d^{7}-210 a^{5} b^{10} c^{4} d^{6}-462 a^{4} b^{11} c^{5} d^{5}-924 a^{3} b^{12} c^{6} d^{4}-1716 a^{2} b^{13} c^{7} d^{3}-3003 a \,b^{14} c^{8} d^{2}-5005 b^{15} c^{9} d \right ) x}{8008 b^{16}}+\frac {\left (-a^{8} b^{6} d^{10}-7 a^{7} b^{7} c \,d^{9}-28 a^{6} b^{8} c^{2} d^{8}-84 a^{5} b^{9} c^{3} d^{7}-210 a^{4} b^{10} c^{4} d^{6}-462 a^{3} b^{11} c^{5} d^{5}-924 a^{2} b^{12} c^{6} d^{4}-1716 a \,b^{13} c^{7} d^{3}-3003 b^{14} c^{8} d^{2}\right ) x^{2}}{1001 b^{15}}+\frac {5 \left (-a^{7} b^{6} d^{10}-7 a^{6} b^{7} c \,d^{9}-28 a^{5} b^{8} c^{2} d^{8}-84 a^{4} b^{9} c^{3} d^{7}-210 a^{3} b^{10} c^{4} d^{6}-462 a^{2} b^{11} c^{5} d^{5}-924 a \,b^{12} c^{6} d^{4}-1716 b^{13} c^{7} d^{3}\right ) x^{3}}{1001 b^{14}}+\frac {5 \left (-a^{6} b^{6} d^{10}-7 a^{5} b^{7} c \,d^{9}-28 a^{4} b^{8} c^{2} d^{8}-84 a^{3} b^{9} c^{3} d^{7}-210 a^{2} b^{10} c^{4} d^{6}-462 a \,b^{11} c^{5} d^{5}-924 b^{12} c^{6} d^{4}\right ) x^{4}}{286 b^{13}}+\frac {\left (-a^{5} b^{6} d^{10}-7 a^{4} b^{7} c \,d^{9}-28 a^{3} b^{8} c^{2} d^{8}-84 a^{2} b^{9} c^{3} d^{7}-210 a \,b^{10} c^{4} d^{6}-462 b^{11} c^{5} d^{5}\right ) x^{5}}{22 b^{12}}+\frac {\left (-a^{4} b^{6} d^{10}-7 a^{3} b^{7} c \,d^{9}-28 a^{2} b^{8} c^{2} d^{8}-84 a \,b^{9} c^{3} d^{7}-210 b^{10} c^{4} d^{6}\right ) x^{6}}{11 b^{11}}+\frac {\left (-a^{3} b^{6} d^{10}-7 a^{2} b^{7} c \,d^{9}-28 a \,b^{8} c^{2} d^{8}-84 b^{9} c^{3} d^{7}\right ) x^{7}}{7 b^{10}}+\frac {5 \left (-a^{2} b^{6} d^{10}-7 a \,b^{7} c \,d^{9}-28 b^{8} c^{2} d^{8}\right ) x^{8}}{28 b^{9}}+\frac {5 \left (-a \,b^{6} d^{10}-7 b^{7} c \,d^{9}\right ) x^{9}}{28 b^{8}}-\frac {d^{10} x^{10}}{7 b}}{\left (b x +a \right )^{17}}\)	\(909\)
gosper	\(-\frac {19448 x^{10} d^{10} b^{10}+24310 x^{9} a \,b^{9} d^{10}+170170 x^{9} b^{10} c \,d^{9}+24310 x^{8} a^{2} b^{8} d^{10}+170170 x^{8} a \,b^{9} c \,d^{9}+680680 x^{8} b^{10} c^{2} d^{8}+19448 x^{7} a^{3} b^{7} d^{10}+136136 x^{7} a^{2} b^{8} c \,d^{9}+544544 x^{7} a \,b^{9} c^{2} d^{8}+1633632 x^{7} b^{10} c^{3} d^{7}+12376 x^{6} a^{4} b^{6} d^{10}+86632 x^{6} a^{3} b^{7} c \,d^{9}+346528 x^{6} a^{2} b^{8} c^{2} d^{8}+1039584 x^{6} a \,b^{9} c^{3} d^{7}+2598960 x^{6} b^{10} c^{4} d^{6}+6188 x^{5} a^{5} b^{5} d^{10}+43316 x^{5} a^{4} b^{6} c \,d^{9}+173264 x^{5} a^{3} b^{7} c^{2} d^{8}+519792 x^{5} a^{2} b^{8} c^{3} d^{7}+1299480 x^{5} a \,b^{9} c^{4} d^{6}+2858856 x^{5} b^{10} c^{5} d^{5}+2380 x^{4} a^{6} b^{4} d^{10}+16660 x^{4} a^{5} b^{5} c \,d^{9}+66640 x^{4} a^{4} b^{6} c^{2} d^{8}+199920 x^{4} a^{3} b^{7} c^{3} d^{7}+499800 x^{4} a^{2} b^{8} c^{4} d^{6}+1099560 x^{4} a \,b^{9} c^{5} d^{5}+2199120 x^{4} b^{10} c^{6} d^{4}+680 x^{3} a^{7} b^{3} d^{10}+4760 x^{3} a^{6} b^{4} c \,d^{9}+19040 x^{3} a^{5} b^{5} c^{2} d^{8}+57120 x^{3} a^{4} b^{6} c^{3} d^{7}+142800 x^{3} a^{3} b^{7} c^{4} d^{6}+314160 x^{3} a^{2} b^{8} c^{5} d^{5}+628320 x^{3} a \,b^{9} c^{6} d^{4}+1166880 x^{3} b^{10} c^{7} d^{3}+136 x^{2} a^{8} b^{2} d^{10}+952 x^{2} a^{7} b^{3} c \,d^{9}+3808 x^{2} a^{6} b^{4} c^{2} d^{8}+11424 x^{2} a^{5} b^{5} c^{3} d^{7}+28560 x^{2} a^{4} b^{6} c^{4} d^{6}+62832 x^{2} a^{3} b^{7} c^{5} d^{5}+125664 x^{2} a^{2} b^{8} c^{6} d^{4}+233376 x^{2} a \,b^{9} c^{7} d^{3}+408408 x^{2} b^{10} c^{8} d^{2}+17 x \,a^{9} b \,d^{10}+119 x \,a^{8} b^{2} c \,d^{9}+476 x \,a^{7} b^{3} c^{2} d^{8}+1428 x \,a^{6} b^{4} c^{3} d^{7}+3570 x \,a^{5} b^{5} c^{4} d^{6}+7854 x \,a^{4} b^{6} c^{5} d^{5}+15708 x \,a^{3} b^{7} c^{6} d^{4}+29172 x \,a^{2} b^{8} c^{7} d^{3}+51051 x a \,b^{9} c^{8} d^{2}+85085 x \,b^{10} c^{9} d +a^{10} d^{10}+7 a^{9} b c \,d^{9}+28 a^{8} b^{2} c^{2} d^{8}+84 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}+462 a^{5} b^{5} c^{5} d^{5}+924 a^{4} b^{6} c^{6} d^{4}+1716 a^{3} b^{7} c^{7} d^{3}+3003 a^{2} b^{8} c^{8} d^{2}+5005 a \,b^{9} c^{9} d +8008 b^{10} c^{10}}{136136 b^{11} \left (b x +a \right )^{17}}\)	\(962\)
parallelrisch	\(\frac {-19448 d^{10} x^{10} b^{16}-24310 a \,b^{15} d^{10} x^{9}-170170 b^{16} c \,d^{9} x^{9}-24310 a^{2} b^{14} d^{10} x^{8}-170170 a \,b^{15} c \,d^{9} x^{8}-680680 b^{16} c^{2} d^{8} x^{8}-19448 a^{3} b^{13} d^{10} x^{7}-136136 a^{2} b^{14} c \,d^{9} x^{7}-544544 a \,b^{15} c^{2} d^{8} x^{7}-1633632 b^{16} c^{3} d^{7} x^{7}-12376 a^{4} b^{12} d^{10} x^{6}-86632 a^{3} b^{13} c \,d^{9} x^{6}-346528 a^{2} b^{14} c^{2} d^{8} x^{6}-1039584 a \,b^{15} c^{3} d^{7} x^{6}-2598960 b^{16} c^{4} d^{6} x^{6}-6188 a^{5} b^{11} d^{10} x^{5}-43316 a^{4} b^{12} c \,d^{9} x^{5}-173264 a^{3} b^{13} c^{2} d^{8} x^{5}-519792 a^{2} b^{14} c^{3} d^{7} x^{5}-1299480 a \,b^{15} c^{4} d^{6} x^{5}-2858856 b^{16} c^{5} d^{5} x^{5}-2380 a^{6} b^{10} d^{10} x^{4}-16660 a^{5} b^{11} c \,d^{9} x^{4}-66640 a^{4} b^{12} c^{2} d^{8} x^{4}-199920 a^{3} b^{13} c^{3} d^{7} x^{4}-499800 a^{2} b^{14} c^{4} d^{6} x^{4}-1099560 a \,b^{15} c^{5} d^{5} x^{4}-2199120 b^{16} c^{6} d^{4} x^{4}-680 a^{7} b^{9} d^{10} x^{3}-4760 a^{6} b^{10} c \,d^{9} x^{3}-19040 a^{5} b^{11} c^{2} d^{8} x^{3}-57120 a^{4} b^{12} c^{3} d^{7} x^{3}-142800 a^{3} b^{13} c^{4} d^{6} x^{3}-314160 a^{2} b^{14} c^{5} d^{5} x^{3}-628320 a \,b^{15} c^{6} d^{4} x^{3}-1166880 b^{16} c^{7} d^{3} x^{3}-136 a^{8} b^{8} d^{10} x^{2}-952 a^{7} b^{9} c \,d^{9} x^{2}-3808 a^{6} b^{10} c^{2} d^{8} x^{2}-11424 a^{5} b^{11} c^{3} d^{7} x^{2}-28560 a^{4} b^{12} c^{4} d^{6} x^{2}-62832 a^{3} b^{13} c^{5} d^{5} x^{2}-125664 a^{2} b^{14} c^{6} d^{4} x^{2}-233376 a \,b^{15} c^{7} d^{3} x^{2}-408408 b^{16} c^{8} d^{2} x^{2}-17 a^{9} b^{7} d^{10} x -119 a^{8} b^{8} c \,d^{9} x -476 a^{7} b^{9} c^{2} d^{8} x -1428 a^{6} b^{10} c^{3} d^{7} x -3570 a^{5} b^{11} c^{4} d^{6} x -7854 a^{4} b^{12} c^{5} d^{5} x -15708 a^{3} b^{13} c^{6} d^{4} x -29172 a^{2} b^{14} c^{7} d^{3} x -51051 a \,b^{15} c^{8} d^{2} x -85085 b^{16} c^{9} d x -a^{10} b^{6} d^{10}-7 a^{9} b^{7} c \,d^{9}-28 a^{8} b^{8} c^{2} d^{8}-84 a^{7} b^{9} c^{3} d^{7}-210 a^{6} b^{10} c^{4} d^{6}-462 a^{5} b^{11} c^{5} d^{5}-924 a^{4} b^{12} c^{6} d^{4}-1716 a^{3} c^{7} d^{3} b^{13}-3003 a^{2} b^{14} c^{8} d^{2}-5005 a \,b^{15} c^{9} d -8008 b^{16} c^{10}}{136136 b^{17} \left (b x +a \right )^{17}}\)	\(970\)

[In]

int((d*x+c)^10/(b*x+a)^18,x,method=_RETURNVERBOSE)

[Out]

(-1/136136/b^11*(a^10*d^10+7*a^9*b*c*d^9+28*a^8*b^2*c^2*d^8+84*a^7*b^3*c^3*d^7+210*a^6*b^4*c^4*d^6+462*a^5*b^5
*c^5*d^5+924*a^4*b^6*c^6*d^4+1716*a^3*b^7*c^7*d^3+3003*a^2*b^8*c^8*d^2+5005*a*b^9*c^9*d+8008*b^10*c^10)-1/8008
/b^10*d*(a^9*d^9+7*a^8*b*c*d^8+28*a^7*b^2*c^2*d^7+84*a^6*b^3*c^3*d^6+210*a^5*b^4*c^4*d^5+462*a^4*b^5*c^5*d^4+9
24*a^3*b^6*c^6*d^3+1716*a^2*b^7*c^7*d^2+3003*a*b^8*c^8*d+5005*b^9*c^9)*x-1/1001/b^9*d^2*(a^8*d^8+7*a^7*b*c*d^7
+28*a^6*b^2*c^2*d^6+84*a^5*b^3*c^3*d^5+210*a^4*b^4*c^4*d^4+462*a^3*b^5*c^5*d^3+924*a^2*b^6*c^6*d^2+1716*a*b^7*
c^7*d+3003*b^8*c^8)*x^2-5/1001/b^8*d^3*(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)*x^3-5/286/b^7*d^4*(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^4-1/22/b^6*d^5*(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^5-1/11/b^5*d^6*(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^6-1/7/b^4*d^7*(a^3*d^3+7*a^2*b*c*d^2+28*a*b^2*c^2*d+84*b
^3*c^3)*x^7-5/28/b^3*d^8*(a^2*d^2+7*a*b*c*d+28*b^2*c^2)*x^8-5/28/b^2*d^9*(a*d+7*b*c)*x^9-1/7/b*d^10*x^10)/(b*x
+a)^17

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1041 vs. \(2 (199) = 398\).

Time = 0.23 (sec) , antiderivative size = 1041, normalized size of antiderivative = 4.89 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{18}} \, dx=-\frac {19448 \, b^{10} d^{10} x^{10} + 8008 \, b^{10} c^{10} + 5005 \, a b^{9} c^{9} d + 3003 \, a^{2} b^{8} c^{8} d^{2} + 1716 \, a^{3} b^{7} c^{7} d^{3} + 924 \, a^{4} b^{6} c^{6} d^{4} + 462 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} + 84 \, a^{7} b^{3} c^{3} d^{7} + 28 \, a^{8} b^{2} c^{2} d^{8} + 7 \, a^{9} b c d^{9} + a^{10} d^{10} + 24310 \, {\left (7 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 24310 \, {\left (28 \, b^{10} c^{2} d^{8} + 7 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 19448 \, {\left (84 \, b^{10} c^{3} d^{7} + 28 \, a b^{9} c^{2} d^{8} + 7 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 12376 \, {\left (210 \, b^{10} c^{4} d^{6} + 84 \, a b^{9} c^{3} d^{7} + 28 \, a^{2} b^{8} c^{2} d^{8} + 7 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 6188 \, {\left (462 \, b^{10} c^{5} d^{5} + 210 \, a b^{9} c^{4} d^{6} + 84 \, a^{2} b^{8} c^{3} d^{7} + 28 \, a^{3} b^{7} c^{2} d^{8} + 7 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 2380 \, {\left (924 \, b^{10} c^{6} d^{4} + 462 \, a b^{9} c^{5} d^{5} + 210 \, a^{2} b^{8} c^{4} d^{6} + 84 \, a^{3} b^{7} c^{3} d^{7} + 28 \, a^{4} b^{6} c^{2} d^{8} + 7 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 680 \, {\left (1716 \, b^{10} c^{7} d^{3} + 924 \, a b^{9} c^{6} d^{4} + 462 \, a^{2} b^{8} c^{5} d^{5} + 210 \, a^{3} b^{7} c^{4} d^{6} + 84 \, a^{4} b^{6} c^{3} d^{7} + 28 \, a^{5} b^{5} c^{2} d^{8} + 7 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 136 \, {\left (3003 \, b^{10} c^{8} d^{2} + 1716 \, a b^{9} c^{7} d^{3} + 924 \, a^{2} b^{8} c^{6} d^{4} + 462 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} + 84 \, a^{5} b^{5} c^{3} d^{7} + 28 \, a^{6} b^{4} c^{2} d^{8} + 7 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 17 \, {\left (5005 \, b^{10} c^{9} d + 3003 \, a b^{9} c^{8} d^{2} + 1716 \, a^{2} b^{8} c^{7} d^{3} + 924 \, a^{3} b^{7} c^{6} d^{4} + 462 \, a^{4} b^{6} c^{5} d^{5} + 210 \, a^{5} b^{5} c^{4} d^{6} + 84 \, a^{6} b^{4} c^{3} d^{7} + 28 \, a^{7} b^{3} c^{2} d^{8} + 7 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{136136 \, {\left (b^{28} x^{17} + 17 \, a b^{27} x^{16} + 136 \, a^{2} b^{26} x^{15} + 680 \, a^{3} b^{25} x^{14} + 2380 \, a^{4} b^{24} x^{13} + 6188 \, a^{5} b^{23} x^{12} + 12376 \, a^{6} b^{22} x^{11} + 19448 \, a^{7} b^{21} x^{10} + 24310 \, a^{8} b^{20} x^{9} + 24310 \, a^{9} b^{19} x^{8} + 19448 \, a^{10} b^{18} x^{7} + 12376 \, a^{11} b^{17} x^{6} + 6188 \, a^{12} b^{16} x^{5} + 2380 \, a^{13} b^{15} x^{4} + 680 \, a^{14} b^{14} x^{3} + 136 \, a^{15} b^{13} x^{2} + 17 \, a^{16} b^{12} x + a^{17} b^{11}\right )}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^18,x, algorithm="fricas")

[Out]

-1/136136*(19448*b^10*d^10*x^10 + 8008*b^10*c^10 + 5005*a*b^9*c^9*d + 3003*a^2*b^8*c^8*d^2 + 1716*a^3*b^7*c^7*
d^3 + 924*a^4*b^6*c^6*d^4 + 462*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 84*a^7*b^3*c^3*d^7 + 28*a^8*b^2*c^2*d^
8 + 7*a^9*b*c*d^9 + a^10*d^10 + 24310*(7*b^10*c*d^9 + a*b^9*d^10)*x^9 + 24310*(28*b^10*c^2*d^8 + 7*a*b^9*c*d^9
+ a^2*b^8*d^10)*x^8 + 19448*(84*b^10*c^3*d^7 + 28*a*b^9*c^2*d^8 + 7*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 12376
*(210*b^10*c^4*d^6 + 84*a*b^9*c^3*d^7 + 28*a^2*b^8*c^2*d^8 + 7*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 6188*(462*b
^10*c^5*d^5 + 210*a*b^9*c^4*d^6 + 84*a^2*b^8*c^3*d^7 + 28*a^3*b^7*c^2*d^8 + 7*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^
5 + 2380*(924*b^10*c^6*d^4 + 462*a*b^9*c^5*d^5 + 210*a^2*b^8*c^4*d^6 + 84*a^3*b^7*c^3*d^7 + 28*a^4*b^6*c^2*d^8
+ 7*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 680*(1716*b^10*c^7*d^3 + 924*a*b^9*c^6*d^4 + 462*a^2*b^8*c^5*d^5 + 21
0*a^3*b^7*c^4*d^6 + 84*a^4*b^6*c^3*d^7 + 28*a^5*b^5*c^2*d^8 + 7*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 136*(3003*
b^10*c^8*d^2 + 1716*a*b^9*c^7*d^3 + 924*a^2*b^8*c^6*d^4 + 462*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 + 84*a^5*b
^5*c^3*d^7 + 28*a^6*b^4*c^2*d^8 + 7*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 17*(5005*b^10*c^9*d + 3003*a*b^9*c^8*d
^2 + 1716*a^2*b^8*c^7*d^3 + 924*a^3*b^7*c^6*d^4 + 462*a^4*b^6*c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 84*a^6*b^4*c^3*d
^7 + 28*a^7*b^3*c^2*d^8 + 7*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^28*x^17 + 17*a*b^27*x^16 + 136*a^2*b^26*x^15 + 6
80*a^3*b^25*x^14 + 2380*a^4*b^24*x^13 + 6188*a^5*b^23*x^12 + 12376*a^6*b^22*x^11 + 19448*a^7*b^21*x^10 + 24310
*a^8*b^20*x^9 + 24310*a^9*b^19*x^8 + 19448*a^10*b^18*x^7 + 12376*a^11*b^17*x^6 + 6188*a^12*b^16*x^5 + 2380*a^1
3*b^15*x^4 + 680*a^14*b^14*x^3 + 136*a^15*b^13*x^2 + 17*a^16*b^12*x + a^17*b^11)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{18}} \, dx=\text {Timed out} \]

[In]

integrate((d*x+c)**10/(b*x+a)**18,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1041 vs. \(2 (199) = 398\).

Time = 0.26 (sec) , antiderivative size = 1041, normalized size of antiderivative = 4.89 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{18}} \, dx=-\frac {19448 \, b^{10} d^{10} x^{10} + 8008 \, b^{10} c^{10} + 5005 \, a b^{9} c^{9} d + 3003 \, a^{2} b^{8} c^{8} d^{2} + 1716 \, a^{3} b^{7} c^{7} d^{3} + 924 \, a^{4} b^{6} c^{6} d^{4} + 462 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} + 84 \, a^{7} b^{3} c^{3} d^{7} + 28 \, a^{8} b^{2} c^{2} d^{8} + 7 \, a^{9} b c d^{9} + a^{10} d^{10} + 24310 \, {\left (7 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 24310 \, {\left (28 \, b^{10} c^{2} d^{8} + 7 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 19448 \, {\left (84 \, b^{10} c^{3} d^{7} + 28 \, a b^{9} c^{2} d^{8} + 7 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 12376 \, {\left (210 \, b^{10} c^{4} d^{6} + 84 \, a b^{9} c^{3} d^{7} + 28 \, a^{2} b^{8} c^{2} d^{8} + 7 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 6188 \, {\left (462 \, b^{10} c^{5} d^{5} + 210 \, a b^{9} c^{4} d^{6} + 84 \, a^{2} b^{8} c^{3} d^{7} + 28 \, a^{3} b^{7} c^{2} d^{8} + 7 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 2380 \, {\left (924 \, b^{10} c^{6} d^{4} + 462 \, a b^{9} c^{5} d^{5} + 210 \, a^{2} b^{8} c^{4} d^{6} + 84 \, a^{3} b^{7} c^{3} d^{7} + 28 \, a^{4} b^{6} c^{2} d^{8} + 7 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 680 \, {\left (1716 \, b^{10} c^{7} d^{3} + 924 \, a b^{9} c^{6} d^{4} + 462 \, a^{2} b^{8} c^{5} d^{5} + 210 \, a^{3} b^{7} c^{4} d^{6} + 84 \, a^{4} b^{6} c^{3} d^{7} + 28 \, a^{5} b^{5} c^{2} d^{8} + 7 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 136 \, {\left (3003 \, b^{10} c^{8} d^{2} + 1716 \, a b^{9} c^{7} d^{3} + 924 \, a^{2} b^{8} c^{6} d^{4} + 462 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} + 84 \, a^{5} b^{5} c^{3} d^{7} + 28 \, a^{6} b^{4} c^{2} d^{8} + 7 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 17 \, {\left (5005 \, b^{10} c^{9} d + 3003 \, a b^{9} c^{8} d^{2} + 1716 \, a^{2} b^{8} c^{7} d^{3} + 924 \, a^{3} b^{7} c^{6} d^{4} + 462 \, a^{4} b^{6} c^{5} d^{5} + 210 \, a^{5} b^{5} c^{4} d^{6} + 84 \, a^{6} b^{4} c^{3} d^{7} + 28 \, a^{7} b^{3} c^{2} d^{8} + 7 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{136136 \, {\left (b^{28} x^{17} + 17 \, a b^{27} x^{16} + 136 \, a^{2} b^{26} x^{15} + 680 \, a^{3} b^{25} x^{14} + 2380 \, a^{4} b^{24} x^{13} + 6188 \, a^{5} b^{23} x^{12} + 12376 \, a^{6} b^{22} x^{11} + 19448 \, a^{7} b^{21} x^{10} + 24310 \, a^{8} b^{20} x^{9} + 24310 \, a^{9} b^{19} x^{8} + 19448 \, a^{10} b^{18} x^{7} + 12376 \, a^{11} b^{17} x^{6} + 6188 \, a^{12} b^{16} x^{5} + 2380 \, a^{13} b^{15} x^{4} + 680 \, a^{14} b^{14} x^{3} + 136 \, a^{15} b^{13} x^{2} + 17 \, a^{16} b^{12} x + a^{17} b^{11}\right )}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^18,x, algorithm="maxima")

[Out]

-1/136136*(19448*b^10*d^10*x^10 + 8008*b^10*c^10 + 5005*a*b^9*c^9*d + 3003*a^2*b^8*c^8*d^2 + 1716*a^3*b^7*c^7*
d^3 + 924*a^4*b^6*c^6*d^4 + 462*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 84*a^7*b^3*c^3*d^7 + 28*a^8*b^2*c^2*d^
8 + 7*a^9*b*c*d^9 + a^10*d^10 + 24310*(7*b^10*c*d^9 + a*b^9*d^10)*x^9 + 24310*(28*b^10*c^2*d^8 + 7*a*b^9*c*d^9
+ a^2*b^8*d^10)*x^8 + 19448*(84*b^10*c^3*d^7 + 28*a*b^9*c^2*d^8 + 7*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 12376
*(210*b^10*c^4*d^6 + 84*a*b^9*c^3*d^7 + 28*a^2*b^8*c^2*d^8 + 7*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 6188*(462*b
^10*c^5*d^5 + 210*a*b^9*c^4*d^6 + 84*a^2*b^8*c^3*d^7 + 28*a^3*b^7*c^2*d^8 + 7*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^
5 + 2380*(924*b^10*c^6*d^4 + 462*a*b^9*c^5*d^5 + 210*a^2*b^8*c^4*d^6 + 84*a^3*b^7*c^3*d^7 + 28*a^4*b^6*c^2*d^8
+ 7*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 680*(1716*b^10*c^7*d^3 + 924*a*b^9*c^6*d^4 + 462*a^2*b^8*c^5*d^5 + 21
0*a^3*b^7*c^4*d^6 + 84*a^4*b^6*c^3*d^7 + 28*a^5*b^5*c^2*d^8 + 7*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 136*(3003*
b^10*c^8*d^2 + 1716*a*b^9*c^7*d^3 + 924*a^2*b^8*c^6*d^4 + 462*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 + 84*a^5*b
^5*c^3*d^7 + 28*a^6*b^4*c^2*d^8 + 7*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 17*(5005*b^10*c^9*d + 3003*a*b^9*c^8*d
^2 + 1716*a^2*b^8*c^7*d^3 + 924*a^3*b^7*c^6*d^4 + 462*a^4*b^6*c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 84*a^6*b^4*c^3*d
^7 + 28*a^7*b^3*c^2*d^8 + 7*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^28*x^17 + 17*a*b^27*x^16 + 136*a^2*b^26*x^15 + 6
80*a^3*b^25*x^14 + 2380*a^4*b^24*x^13 + 6188*a^5*b^23*x^12 + 12376*a^6*b^22*x^11 + 19448*a^7*b^21*x^10 + 24310
*a^8*b^20*x^9 + 24310*a^9*b^19*x^8 + 19448*a^10*b^18*x^7 + 12376*a^11*b^17*x^6 + 6188*a^12*b^16*x^5 + 2380*a^1
3*b^15*x^4 + 680*a^14*b^14*x^3 + 136*a^15*b^13*x^2 + 17*a^16*b^12*x + a^17*b^11)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (199) = 398\).

Time = 0.30 (sec) , antiderivative size = 961, normalized size of antiderivative = 4.51 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{18}} \, dx=-\frac {19448 \, b^{10} d^{10} x^{10} + 170170 \, b^{10} c d^{9} x^{9} + 24310 \, a b^{9} d^{10} x^{9} + 680680 \, b^{10} c^{2} d^{8} x^{8} + 170170 \, a b^{9} c d^{9} x^{8} + 24310 \, a^{2} b^{8} d^{10} x^{8} + 1633632 \, b^{10} c^{3} d^{7} x^{7} + 544544 \, a b^{9} c^{2} d^{8} x^{7} + 136136 \, a^{2} b^{8} c d^{9} x^{7} + 19448 \, a^{3} b^{7} d^{10} x^{7} + 2598960 \, b^{10} c^{4} d^{6} x^{6} + 1039584 \, a b^{9} c^{3} d^{7} x^{6} + 346528 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 86632 \, a^{3} b^{7} c d^{9} x^{6} + 12376 \, a^{4} b^{6} d^{10} x^{6} + 2858856 \, b^{10} c^{5} d^{5} x^{5} + 1299480 \, a b^{9} c^{4} d^{6} x^{5} + 519792 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 173264 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 43316 \, a^{4} b^{6} c d^{9} x^{5} + 6188 \, a^{5} b^{5} d^{10} x^{5} + 2199120 \, b^{10} c^{6} d^{4} x^{4} + 1099560 \, a b^{9} c^{5} d^{5} x^{4} + 499800 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 199920 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 66640 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 16660 \, a^{5} b^{5} c d^{9} x^{4} + 2380 \, a^{6} b^{4} d^{10} x^{4} + 1166880 \, b^{10} c^{7} d^{3} x^{3} + 628320 \, a b^{9} c^{6} d^{4} x^{3} + 314160 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 142800 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 57120 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 19040 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 4760 \, a^{6} b^{4} c d^{9} x^{3} + 680 \, a^{7} b^{3} d^{10} x^{3} + 408408 \, b^{10} c^{8} d^{2} x^{2} + 233376 \, a b^{9} c^{7} d^{3} x^{2} + 125664 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 62832 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 28560 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 11424 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 3808 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 952 \, a^{7} b^{3} c d^{9} x^{2} + 136 \, a^{8} b^{2} d^{10} x^{2} + 85085 \, b^{10} c^{9} d x + 51051 \, a b^{9} c^{8} d^{2} x + 29172 \, a^{2} b^{8} c^{7} d^{3} x + 15708 \, a^{3} b^{7} c^{6} d^{4} x + 7854 \, a^{4} b^{6} c^{5} d^{5} x + 3570 \, a^{5} b^{5} c^{4} d^{6} x + 1428 \, a^{6} b^{4} c^{3} d^{7} x + 476 \, a^{7} b^{3} c^{2} d^{8} x + 119 \, a^{8} b^{2} c d^{9} x + 17 \, a^{9} b d^{10} x + 8008 \, b^{10} c^{10} + 5005 \, a b^{9} c^{9} d + 3003 \, a^{2} b^{8} c^{8} d^{2} + 1716 \, a^{3} b^{7} c^{7} d^{3} + 924 \, a^{4} b^{6} c^{6} d^{4} + 462 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} + 84 \, a^{7} b^{3} c^{3} d^{7} + 28 \, a^{8} b^{2} c^{2} d^{8} + 7 \, a^{9} b c d^{9} + a^{10} d^{10}}{136136 \, {\left (b x + a\right )}^{17} b^{11}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^18,x, algorithm="giac")

[Out]

-1/136136*(19448*b^10*d^10*x^10 + 170170*b^10*c*d^9*x^9 + 24310*a*b^9*d^10*x^9 + 680680*b^10*c^2*d^8*x^8 + 170
170*a*b^9*c*d^9*x^8 + 24310*a^2*b^8*d^10*x^8 + 1633632*b^10*c^3*d^7*x^7 + 544544*a*b^9*c^2*d^8*x^7 + 136136*a^
2*b^8*c*d^9*x^7 + 19448*a^3*b^7*d^10*x^7 + 2598960*b^10*c^4*d^6*x^6 + 1039584*a*b^9*c^3*d^7*x^6 + 346528*a^2*b
^8*c^2*d^8*x^6 + 86632*a^3*b^7*c*d^9*x^6 + 12376*a^4*b^6*d^10*x^6 + 2858856*b^10*c^5*d^5*x^5 + 1299480*a*b^9*c
^4*d^6*x^5 + 519792*a^2*b^8*c^3*d^7*x^5 + 173264*a^3*b^7*c^2*d^8*x^5 + 43316*a^4*b^6*c*d^9*x^5 + 6188*a^5*b^5*
d^10*x^5 + 2199120*b^10*c^6*d^4*x^4 + 1099560*a*b^9*c^5*d^5*x^4 + 499800*a^2*b^8*c^4*d^6*x^4 + 199920*a^3*b^7*
c^3*d^7*x^4 + 66640*a^4*b^6*c^2*d^8*x^4 + 16660*a^5*b^5*c*d^9*x^4 + 2380*a^6*b^4*d^10*x^4 + 1166880*b^10*c^7*d
^3*x^3 + 628320*a*b^9*c^6*d^4*x^3 + 314160*a^2*b^8*c^5*d^5*x^3 + 142800*a^3*b^7*c^4*d^6*x^3 + 57120*a^4*b^6*c^
3*d^7*x^3 + 19040*a^5*b^5*c^2*d^8*x^3 + 4760*a^6*b^4*c*d^9*x^3 + 680*a^7*b^3*d^10*x^3 + 408408*b^10*c^8*d^2*x^
2 + 233376*a*b^9*c^7*d^3*x^2 + 125664*a^2*b^8*c^6*d^4*x^2 + 62832*a^3*b^7*c^5*d^5*x^2 + 28560*a^4*b^6*c^4*d^6*
x^2 + 11424*a^5*b^5*c^3*d^7*x^2 + 3808*a^6*b^4*c^2*d^8*x^2 + 952*a^7*b^3*c*d^9*x^2 + 136*a^8*b^2*d^10*x^2 + 85
085*b^10*c^9*d*x + 51051*a*b^9*c^8*d^2*x + 29172*a^2*b^8*c^7*d^3*x + 15708*a^3*b^7*c^6*d^4*x + 7854*a^4*b^6*c^
5*d^5*x + 3570*a^5*b^5*c^4*d^6*x + 1428*a^6*b^4*c^3*d^7*x + 476*a^7*b^3*c^2*d^8*x + 119*a^8*b^2*c*d^9*x + 17*a
^9*b*d^10*x + 8008*b^10*c^10 + 5005*a*b^9*c^9*d + 3003*a^2*b^8*c^8*d^2 + 1716*a^3*b^7*c^7*d^3 + 924*a^4*b^6*c^
6*d^4 + 462*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 84*a^7*b^3*c^3*d^7 + 28*a^8*b^2*c^2*d^8 + 7*a^9*b*c*d^9 +
a^10*d^10)/((b*x + a)^17*b^11)

Mupad [B] (verification not implemented)

Time = 0.96 (sec) , antiderivative size = 1142, normalized size of antiderivative = 5.36 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{18}} \, dx=-\frac {a^{10}\,d^{10}+7\,a^9\,b\,c\,d^9+17\,a^9\,b\,d^{10}\,x+28\,a^8\,b^2\,c^2\,d^8+119\,a^8\,b^2\,c\,d^9\,x+136\,a^8\,b^2\,d^{10}\,x^2+84\,a^7\,b^3\,c^3\,d^7+476\,a^7\,b^3\,c^2\,d^8\,x+952\,a^7\,b^3\,c\,d^9\,x^2+680\,a^7\,b^3\,d^{10}\,x^3+210\,a^6\,b^4\,c^4\,d^6+1428\,a^6\,b^4\,c^3\,d^7\,x+3808\,a^6\,b^4\,c^2\,d^8\,x^2+4760\,a^6\,b^4\,c\,d^9\,x^3+2380\,a^6\,b^4\,d^{10}\,x^4+462\,a^5\,b^5\,c^5\,d^5+3570\,a^5\,b^5\,c^4\,d^6\,x+11424\,a^5\,b^5\,c^3\,d^7\,x^2+19040\,a^5\,b^5\,c^2\,d^8\,x^3+16660\,a^5\,b^5\,c\,d^9\,x^4+6188\,a^5\,b^5\,d^{10}\,x^5+924\,a^4\,b^6\,c^6\,d^4+7854\,a^4\,b^6\,c^5\,d^5\,x+28560\,a^4\,b^6\,c^4\,d^6\,x^2+57120\,a^4\,b^6\,c^3\,d^7\,x^3+66640\,a^4\,b^6\,c^2\,d^8\,x^4+43316\,a^4\,b^6\,c\,d^9\,x^5+12376\,a^4\,b^6\,d^{10}\,x^6+1716\,a^3\,b^7\,c^7\,d^3+15708\,a^3\,b^7\,c^6\,d^4\,x+62832\,a^3\,b^7\,c^5\,d^5\,x^2+142800\,a^3\,b^7\,c^4\,d^6\,x^3+199920\,a^3\,b^7\,c^3\,d^7\,x^4+173264\,a^3\,b^7\,c^2\,d^8\,x^5+86632\,a^3\,b^7\,c\,d^9\,x^6+19448\,a^3\,b^7\,d^{10}\,x^7+3003\,a^2\,b^8\,c^8\,d^2+29172\,a^2\,b^8\,c^7\,d^3\,x+125664\,a^2\,b^8\,c^6\,d^4\,x^2+314160\,a^2\,b^8\,c^5\,d^5\,x^3+499800\,a^2\,b^8\,c^4\,d^6\,x^4+519792\,a^2\,b^8\,c^3\,d^7\,x^5+346528\,a^2\,b^8\,c^2\,d^8\,x^6+136136\,a^2\,b^8\,c\,d^9\,x^7+24310\,a^2\,b^8\,d^{10}\,x^8+5005\,a\,b^9\,c^9\,d+51051\,a\,b^9\,c^8\,d^2\,x+233376\,a\,b^9\,c^7\,d^3\,x^2+628320\,a\,b^9\,c^6\,d^4\,x^3+1099560\,a\,b^9\,c^5\,d^5\,x^4+1299480\,a\,b^9\,c^4\,d^6\,x^5+1039584\,a\,b^9\,c^3\,d^7\,x^6+544544\,a\,b^9\,c^2\,d^8\,x^7+170170\,a\,b^9\,c\,d^9\,x^8+24310\,a\,b^9\,d^{10}\,x^9+8008\,b^{10}\,c^{10}+85085\,b^{10}\,c^9\,d\,x+408408\,b^{10}\,c^8\,d^2\,x^2+1166880\,b^{10}\,c^7\,d^3\,x^3+2199120\,b^{10}\,c^6\,d^4\,x^4+2858856\,b^{10}\,c^5\,d^5\,x^5+2598960\,b^{10}\,c^4\,d^6\,x^6+1633632\,b^{10}\,c^3\,d^7\,x^7+680680\,b^{10}\,c^2\,d^8\,x^8+170170\,b^{10}\,c\,d^9\,x^9+19448\,b^{10}\,d^{10}\,x^{10}}{136136\,a^{17}\,b^{11}+2314312\,a^{16}\,b^{12}\,x+18514496\,a^{15}\,b^{13}\,x^2+92572480\,a^{14}\,b^{14}\,x^3+324003680\,a^{13}\,b^{15}\,x^4+842409568\,a^{12}\,b^{16}\,x^5+1684819136\,a^{11}\,b^{17}\,x^6+2647572928\,a^{10}\,b^{18}\,x^7+3309466160\,a^9\,b^{19}\,x^8+3309466160\,a^8\,b^{20}\,x^9+2647572928\,a^7\,b^{21}\,x^{10}[phone]\,a^6\,b^{22}\,x^{11}[phone]\,a^5\,b^{23}\,x^{12}[phone]\,a^4\,b^{24}\,x^{13}[phone]\,a^3\,b^{25}\,x^{14}[phone]\,a^2\,b^{26}\,x^{15}+2314312\,a\,b^{27}\,x^{16}+136136\,b^{28}\,x^{17}} \]

[In]

int((c + d*x)^10/(a + b*x)^18,x)

[Out]

-(a^10*d^10 + 8008*b^10*c^10 + 19448*b^10*d^10*x^10 + 24310*a*b^9*d^10*x^9 + 170170*b^10*c*d^9*x^9 + 3003*a^2*
b^8*c^8*d^2 + 1716*a^3*b^7*c^7*d^3 + 924*a^4*b^6*c^6*d^4 + 462*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 84*a^7*
b^3*c^3*d^7 + 28*a^8*b^2*c^2*d^8 + 136*a^8*b^2*d^10*x^2 + 680*a^7*b^3*d^10*x^3 + 2380*a^6*b^4*d^10*x^4 + 6188*
a^5*b^5*d^10*x^5 + 12376*a^4*b^6*d^10*x^6 + 19448*a^3*b^7*d^10*x^7 + 24310*a^2*b^8*d^10*x^8 + 408408*b^10*c^8*
d^2*x^2 + 1166880*b^10*c^7*d^3*x^3 + 2199120*b^10*c^6*d^4*x^4 + 2858856*b^10*c^5*d^5*x^5 + 2598960*b^10*c^4*d^
6*x^6 + 1633632*b^10*c^3*d^7*x^7 + 680680*b^10*c^2*d^8*x^8 + 5005*a*b^9*c^9*d + 7*a^9*b*c*d^9 + 17*a^9*b*d^10*
x + 85085*b^10*c^9*d*x + 125664*a^2*b^8*c^6*d^4*x^2 + 62832*a^3*b^7*c^5*d^5*x^2 + 28560*a^4*b^6*c^4*d^6*x^2 +
11424*a^5*b^5*c^3*d^7*x^2 + 3808*a^6*b^4*c^2*d^8*x^2 + 314160*a^2*b^8*c^5*d^5*x^3 + 142800*a^3*b^7*c^4*d^6*x^3
+ 57120*a^4*b^6*c^3*d^7*x^3 + 19040*a^5*b^5*c^2*d^8*x^3 + 499800*a^2*b^8*c^4*d^6*x^4 + 199920*a^3*b^7*c^3*d^7
*x^4 + 66640*a^4*b^6*c^2*d^8*x^4 + 519792*a^2*b^8*c^3*d^7*x^5 + 173264*a^3*b^7*c^2*d^8*x^5 + 346528*a^2*b^8*c^
2*d^8*x^6 + 51051*a*b^9*c^8*d^2*x + 119*a^8*b^2*c*d^9*x + 170170*a*b^9*c*d^9*x^8 + 29172*a^2*b^8*c^7*d^3*x + 1
5708*a^3*b^7*c^6*d^4*x + 7854*a^4*b^6*c^5*d^5*x + 3570*a^5*b^5*c^4*d^6*x + 1428*a^6*b^4*c^3*d^7*x + 476*a^7*b^
3*c^2*d^8*x + 233376*a*b^9*c^7*d^3*x^2 + 952*a^7*b^3*c*d^9*x^2 + 628320*a*b^9*c^6*d^4*x^3 + 4760*a^6*b^4*c*d^9
*x^3 + 1099560*a*b^9*c^5*d^5*x^4 + 16660*a^5*b^5*c*d^9*x^4 + 1299480*a*b^9*c^4*d^6*x^5 + 43316*a^4*b^6*c*d^9*x
^5 + 1039584*a*b^9*c^3*d^7*x^6 + 86632*a^3*b^7*c*d^9*x^6 + 544544*a*b^9*c^2*d^8*x^7 + 136136*a^2*b^8*c*d^9*x^7
)/(136136*a^17*b^11 + 136136*b^28*x^17 + 2314312*a^16*b^12*x + 2314312*a*b^27*x^16 + 18514496*a^15*b^13*x^2 +
92572480*a^14*b^14*x^3 + 324003680*a^13*b^15*x^4 + 842409568*a^12*b^16*x^5 + 1684819136*a^11*b^17*x^6 + 264757
2928*a^10*b^18*x^7 + 3309466160*a^9*b^19*x^8 + 3309466160*a^8*b^20*x^9 + 2647572928*a^7*b^21*x^10 + 1684819136
*a^6*b^22*x^11 + 842409568*a^5*b^23*x^12 + 324003680*a^4*b^24*x^13 + 92572480*a^3*b^25*x^14 + 18514496*a^2*b^2
6*x^15)