\(\int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx\) [1326]

Optimal result

Integrand size = 15, antiderivative size = 120 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx=-\frac {(c+d x)^{11}}{14 (b c-a d) (a+b x)^{14}}+\frac {3 d (c+d x)^{11}}{182 (b c-a d)^2 (a+b x)^{13}}-\frac {d^2 (c+d x)^{11}}{364 (b c-a d)^3 (a+b x)^{12}}+\frac {d^3 (c+d x)^{11}}{4004 (b c-a d)^4 (a+b x)^{11}} \]

[Out]

-1/14*(d*x+c)^11/(-a*d+b*c)/(b*x+a)^14+3/182*d*(d*x+c)^11/(-a*d+b*c)^2/(b*x+a)^13-1/364*d^2*(d*x+c)^11/(-a*d+b
*c)^3/(b*x+a)^12+1/4004*d^3*(d*x+c)^11/(-a*d+b*c)^4/(b*x+a)^11

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx=\frac {d^3 (c+d x)^{11}}{4004 (a+b x)^{11} (b c-a d)^4}-\frac {d^2 (c+d x)^{11}}{364 (a+b x)^{12} (b c-a d)^3}+\frac {3 d (c+d x)^{11}}{182 (a+b x)^{13} (b c-a d)^2}-\frac {(c+d x)^{11}}{14 (a+b x)^{14} (b c-a d)} \]

[In]

Int[(c + d*x)^10/(a + b*x)^15,x]

[Out]

-1/14*(c + d*x)^11/((b*c - a*d)*(a + b*x)^14) + (3*d*(c + d*x)^11)/(182*(b*c - a*d)^2*(a + b*x)^13) - (d^2*(c
+ d*x)^11)/(364*(b*c - a*d)^3*(a + b*x)^12) + (d^3*(c + d*x)^11)/(4004*(b*c - a*d)^4*(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}}{14 (b c-a d) (a+b x)^{14}}-\frac {(3 d) \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx}{14 (b c-a d)} \\ & = -\frac {(c+d x)^{11}}{14 (b c-a d) (a+b x)^{14}}+\frac {3 d (c+d x)^{11}}{182 (b c-a d)^2 (a+b x)^{13}}+\frac {\left (3 d^2\right ) \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx}{91 (b c-a d)^2} \\ & = -\frac {(c+d x)^{11}}{14 (b c-a d) (a+b x)^{14}}+\frac {3 d (c+d x)^{11}}{182 (b c-a d)^2 (a+b x)^{13}}-\frac {d^2 (c+d x)^{11}}{364 (b c-a d)^3 (a+b x)^{12}}-\frac {d^3 \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx}{364 (b c-a d)^3} \\ & = -\frac {(c+d x)^{11}}{14 (b c-a d) (a+b x)^{14}}+\frac {3 d (c+d x)^{11}}{182 (b c-a d)^2 (a+b x)^{13}}-\frac {d^2 (c+d x)^{11}}{364 (b c-a d)^3 (a+b x)^{12}}+\frac {d^3 (c+d x)^{11}}{4004 (b c-a d)^4 (a+b x)^{11}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(692\) vs. \(2(120)=240\).

Time = 0.18 (sec) , antiderivative size = 692, normalized size of antiderivative = 5.77 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx=-\frac {a^{10} d^{10}+2 a^9 b d^9 (2 c+7 d x)+a^8 b^2 d^8 \left (10 c^2+56 c d x+91 d^2 x^2\right )+4 a^7 b^3 d^7 \left (5 c^3+35 c^2 d x+91 c d^2 x^2+91 d^3 x^3\right )+7 a^6 b^4 d^6 \left (5 c^4+40 c^3 d x+130 c^2 d^2 x^2+208 c d^3 x^3+143 d^4 x^4\right )+14 a^5 b^5 d^5 \left (4 c^5+35 c^4 d x+130 c^3 d^2 x^2+260 c^2 d^3 x^3+286 c d^4 x^4+143 d^5 x^5\right )+7 a^4 b^6 d^4 \left (12 c^6+112 c^5 d x+455 c^4 d^2 x^2+1040 c^3 d^3 x^3+1430 c^2 d^4 x^4+1144 c d^5 x^5+429 d^6 x^6\right )+4 a^3 b^7 d^3 \left (30 c^7+294 c^6 d x+1274 c^5 d^2 x^2+3185 c^4 d^3 x^3+5005 c^3 d^4 x^4+5005 c^2 d^5 x^5+3003 c d^6 x^6+858 d^7 x^7\right )+a^2 b^8 d^2 \left (165 c^8+1680 c^7 d x+7644 c^6 d^2 x^2+20384 c^5 d^3 x^3+35035 c^4 d^4 x^4+40040 c^3 d^5 x^5+30030 c^2 d^6 x^6+13728 c d^7 x^7+3003 d^8 x^8\right )+2 a b^9 d \left (110 c^9+1155 c^8 d x+5460 c^7 d^2 x^2+15288 c^6 d^3 x^3+28028 c^5 d^4 x^4+35035 c^4 d^5 x^5+30030 c^3 d^6 x^6+17160 c^2 d^7 x^7+6006 c d^8 x^8+1001 d^9 x^9\right )+b^{10} \left (286 c^{10}+3080 c^9 d x+15015 c^8 d^2 x^2+43680 c^7 d^3 x^3+84084 c^6 d^4 x^4+112112 c^5 d^5 x^5+105105 c^4 d^6 x^6+68640 c^3 d^7 x^7+30030 c^2 d^8 x^8+8008 c d^9 x^9+1001 d^{10} x^{10}\right )}{4004 b^{11} (a+b x)^{14}} \]

[In]

Integrate[(c + d*x)^10/(a + b*x)^15,x]

[Out]

-1/4004*(a^10*d^10 + 2*a^9*b*d^9*(2*c + 7*d*x) + a^8*b^2*d^8*(10*c^2 + 56*c*d*x + 91*d^2*x^2) + 4*a^7*b^3*d^7*
(5*c^3 + 35*c^2*d*x + 91*c*d^2*x^2 + 91*d^3*x^3) + 7*a^6*b^4*d^6*(5*c^4 + 40*c^3*d*x + 130*c^2*d^2*x^2 + 208*c
*d^3*x^3 + 143*d^4*x^4) + 14*a^5*b^5*d^5*(4*c^5 + 35*c^4*d*x + 130*c^3*d^2*x^2 + 260*c^2*d^3*x^3 + 286*c*d^4*x
^4 + 143*d^5*x^5) + 7*a^4*b^6*d^4*(12*c^6 + 112*c^5*d*x + 455*c^4*d^2*x^2 + 1040*c^3*d^3*x^3 + 1430*c^2*d^4*x^
4 + 1144*c*d^5*x^5 + 429*d^6*x^6) + 4*a^3*b^7*d^3*(30*c^7 + 294*c^6*d*x + 1274*c^5*d^2*x^2 + 3185*c^4*d^3*x^3
+ 5005*c^3*d^4*x^4 + 5005*c^2*d^5*x^5 + 3003*c*d^6*x^6 + 858*d^7*x^7) + a^2*b^8*d^2*(165*c^8 + 1680*c^7*d*x +
7644*c^6*d^2*x^2 + 20384*c^5*d^3*x^3 + 35035*c^4*d^4*x^4 + 40040*c^3*d^5*x^5 + 30030*c^2*d^6*x^6 + 13728*c*d^7
*x^7 + 3003*d^8*x^8) + 2*a*b^9*d*(110*c^9 + 1155*c^8*d*x + 5460*c^7*d^2*x^2 + 15288*c^6*d^3*x^3 + 28028*c^5*d^
4*x^4 + 35035*c^4*d^5*x^5 + 30030*c^3*d^6*x^6 + 17160*c^2*d^7*x^7 + 6006*c*d^8*x^8 + 1001*d^9*x^9) + b^10*(286
*c^10 + 3080*c^9*d*x + 15015*c^8*d^2*x^2 + 43680*c^7*d^3*x^3 + 84084*c^6*d^4*x^4 + 112112*c^5*d^5*x^5 + 105105
*c^4*d^6*x^6 + 68640*c^3*d^7*x^7 + 30030*c^2*d^8*x^8 + 8008*c*d^9*x^9 + 1001*d^10*x^10))/(b^11*(a + b*x)^14)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(830\) vs. \(2(112)=224\).

Time = 0.25 (sec) , antiderivative size = 831, normalized size of antiderivative = 6.92

method	result	size
risch	\(\frac {-\frac {d^{10} x^{10}}{4 b}-\frac {d^{9} \left (a d +4 b c \right ) x^{9}}{2 b^{2}}-\frac {3 d^{8} \left (a^{2} d^{2}+4 a b c d +10 b^{2} c^{2}\right ) x^{8}}{4 b^{3}}-\frac {6 d^{7} \left (a^{3} d^{3}+4 a^{2} b c \,d^{2}+10 a \,b^{2} c^{2} d +20 b^{3} c^{3}\right ) x^{7}}{7 b^{4}}-\frac {3 d^{6} \left (a^{4} d^{4}+4 a^{3} b c \,d^{3}+10 a^{2} b^{2} c^{2} d^{2}+20 a \,b^{3} c^{3} d +35 b^{4} c^{4}\right ) x^{6}}{4 b^{5}}-\frac {d^{5} \left (a^{5} d^{5}+4 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}+20 a^{2} b^{3} c^{3} d^{2}+35 a \,b^{4} c^{4} d +56 b^{5} c^{5}\right ) x^{5}}{2 b^{6}}-\frac {d^{4} \left (a^{6} d^{6}+4 a^{5} b c \,d^{5}+10 a^{4} b^{2} c^{2} d^{4}+20 a^{3} b^{3} c^{3} d^{3}+35 a^{2} b^{4} c^{4} d^{2}+56 a \,b^{5} c^{5} d +84 b^{6} c^{6}\right ) x^{4}}{4 b^{7}}-\frac {d^{3} \left (a^{7} d^{7}+4 a^{6} b c \,d^{6}+10 a^{5} b^{2} c^{2} d^{5}+20 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}+56 a^{2} b^{5} c^{5} d^{2}+84 a \,b^{6} c^{6} d +120 b^{7} c^{7}\right ) x^{3}}{11 b^{8}}-\frac {d^{2} \left (a^{8} d^{8}+4 a^{7} b c \,d^{7}+10 a^{6} b^{2} c^{2} d^{6}+20 a^{5} b^{3} c^{3} d^{5}+35 a^{4} b^{4} c^{4} d^{4}+56 a^{3} b^{5} c^{5} d^{3}+84 a^{2} b^{6} c^{6} d^{2}+120 a \,b^{7} c^{7} d +165 b^{8} c^{8}\right ) x^{2}}{44 b^{9}}-\frac {d \left (a^{9} d^{9}+4 a^{8} b c \,d^{8}+10 a^{7} b^{2} c^{2} d^{7}+20 a^{6} b^{3} c^{3} d^{6}+35 a^{5} b^{4} c^{4} d^{5}+56 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}+120 a^{2} b^{7} c^{7} d^{2}+165 a \,b^{8} c^{8} d +220 b^{9} c^{9}\right ) x}{286 b^{10}}-\frac {a^{10} d^{10}+4 a^{9} b c \,d^{9}+10 a^{8} b^{2} c^{2} d^{8}+20 a^{7} b^{3} c^{3} d^{7}+35 a^{6} b^{4} c^{4} d^{6}+56 a^{5} b^{5} c^{5} d^{5}+84 a^{4} b^{6} c^{6} d^{4}+120 a^{3} b^{7} c^{7} d^{3}+165 a^{2} b^{8} c^{8} d^{2}+220 a \,b^{9} c^{9} d +286 b^{10} c^{10}}{4004 b^{11}}}{\left (b x +a \right )^{14}}\)	\(831\)
default	\(\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 )}{11 b^{11} \left (b x +a \right )^{11}}+\frac {10 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}\right )}{13 b^{11} \left (b x +a \right )^{13}}+\frac {28 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 )^{9}}-\frac {15 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{11} \left (b x +a \right )^{6}}-\frac {105 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 )}{4 b^{11} \left (b x +a \right )^{8}}-\frac {d^{10}}{4 b^{11} \left (b x +a \right )^{4}}-\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}}{14 b^{11} \left (b x +a \right )^{14}}-\frac {15 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 )}{4 b^{11} \left (b x +a \right )^{12}}+\frac {120 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 )}{7 b^{11} \left (b x +a \right )^{7}}-\frac {21 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 )}{b^{11} \left (b x +a \right )^{10}}+\frac {2 d^{9} \left (a d -b c \right )}{b^{11} \left (b x +a \right )^{5}}\)	\(867\)
norman	\(\frac {\frac {-a^{10} b^{3} d^{10}-4 a^{9} b^{4} c \,d^{9}-10 a^{8} b^{5} c^{2} d^{8}-20 a^{7} b^{6} c^{3} d^{7}-35 a^{6} b^{7} c^{4} d^{6}-56 a^{5} b^{8} c^{5} d^{5}-84 a^{4} b^{9} c^{6} d^{4}-120 a^{3} b^{10} c^{7} d^{3}-165 a^{2} c^{8} d^{2} b^{11}-220 a \,b^{12} c^{9} d -286 b^{13} c^{10}}{4004 b^{14}}+\frac {\left (-a^{9} b^{3} d^{10}-4 a^{8} b^{4} c \,d^{9}-10 a^{7} b^{5} c^{2} d^{8}-20 a^{6} b^{6} c^{3} d^{7}-35 a^{5} b^{7} c^{4} d^{6}-56 a^{4} b^{8} c^{5} d^{5}-84 a^{3} b^{9} c^{6} d^{4}-120 a^{2} b^{10} c^{7} d^{3}-165 a \,b^{11} c^{8} d^{2}-220 b^{12} c^{9} d \right ) x}{286 b^{13}}+\frac {\left (-a^{8} b^{3} d^{10}-4 a^{7} b^{4} c \,d^{9}-10 a^{6} b^{5} c^{2} d^{8}-20 a^{5} b^{6} c^{3} d^{7}-35 a^{4} b^{7} c^{4} d^{6}-56 a^{3} b^{8} c^{5} d^{5}-84 a^{2} b^{9} c^{6} d^{4}-120 a \,b^{10} c^{7} d^{3}-165 b^{11} c^{8} d^{2}\right ) x^{2}}{44 b^{12}}+\frac {\left (-a^{7} b^{3} d^{10}-4 a^{6} b^{4} c \,d^{9}-10 a^{5} b^{5} c^{2} d^{8}-20 a^{4} b^{6} c^{3} d^{7}-35 a^{3} b^{7} c^{4} d^{6}-56 a^{2} b^{8} c^{5} d^{5}-84 a \,b^{9} c^{6} d^{4}-120 b^{10} c^{7} d^{3}\right ) x^{3}}{11 b^{11}}+\frac {\left (-a^{6} b^{3} d^{10}-4 a^{5} b^{4} c \,d^{9}-10 a^{4} b^{5} c^{2} d^{8}-20 a^{3} b^{6} c^{3} d^{7}-35 a^{2} b^{7} c^{4} d^{6}-56 a \,b^{8} c^{5} d^{5}-84 b^{9} c^{6} d^{4}\right ) x^{4}}{4 b^{10}}+\frac {\left (-a^{5} b^{3} d^{10}-4 a^{4} b^{4} c \,d^{9}-10 a^{3} b^{5} c^{2} d^{8}-20 a^{2} b^{6} c^{3} d^{7}-35 a \,b^{7} c^{4} d^{6}-56 b^{8} c^{5} d^{5}\right ) x^{5}}{2 b^{9}}+\frac {3 \left (-a^{4} b^{3} d^{10}-4 a^{3} b^{4} c \,d^{9}-10 a^{2} b^{5} c^{2} d^{8}-20 a \,b^{6} c^{3} d^{7}-35 b^{7} c^{4} d^{6}\right ) x^{6}}{4 b^{8}}+\frac {6 \left (-a^{3} b^{3} d^{10}-4 a^{2} b^{4} c \,d^{9}-10 a \,b^{5} c^{2} d^{8}-20 b^{6} c^{3} d^{7}\right ) x^{7}}{7 b^{7}}+\frac {3 \left (-a^{2} b^{3} d^{10}-4 a \,b^{4} c \,d^{9}-10 b^{5} c^{2} d^{8}\right ) x^{8}}{4 b^{6}}+\frac {\left (-a \,b^{3} d^{10}-4 b^{4} c \,d^{9}\right ) x^{9}}{2 b^{5}}-\frac {d^{10} x^{10}}{4 b}}{\left (b x +a \right )^{14}}\)	\(909\)
gosper	\(-\frac {1001 x^{10} d^{10} b^{10}+2002 x^{9} a \,b^{9} d^{10}+8008 x^{9} b^{10} c \,d^{9}+3003 x^{8} a^{2} b^{8} d^{10}+12012 x^{8} a \,b^{9} c \,d^{9}+30030 x^{8} b^{10} c^{2} d^{8}+3432 x^{7} a^{3} b^{7} d^{10}+13728 x^{7} a^{2} b^{8} c \,d^{9}+34320 x^{7} a \,b^{9} c^{2} d^{8}+68640 x^{7} b^{10} c^{3} d^{7}+3003 x^{6} a^{4} b^{6} d^{10}+12012 x^{6} a^{3} b^{7} c \,d^{9}+30030 x^{6} a^{2} b^{8} c^{2} d^{8}+60060 x^{6} a \,b^{9} c^{3} d^{7}+105105 x^{6} b^{10} c^{4} d^{6}+2002 x^{5} a^{5} b^{5} d^{10}+8008 x^{5} a^{4} b^{6} c \,d^{9}+20020 x^{5} a^{3} b^{7} c^{2} d^{8}+40040 x^{5} a^{2} b^{8} c^{3} d^{7}+70070 x^{5} a \,b^{9} c^{4} d^{6}+112112 x^{5} b^{10} c^{5} d^{5}+1001 x^{4} a^{6} b^{4} d^{10}+4004 x^{4} a^{5} b^{5} c \,d^{9}+10010 x^{4} a^{4} b^{6} c^{2} d^{8}+20020 x^{4} a^{3} b^{7} c^{3} d^{7}+35035 x^{4} a^{2} b^{8} c^{4} d^{6}+56056 x^{4} a \,b^{9} c^{5} d^{5}+84084 x^{4} b^{10} c^{6} d^{4}+364 x^{3} a^{7} b^{3} d^{10}+1456 x^{3} a^{6} b^{4} c \,d^{9}+3640 x^{3} a^{5} b^{5} c^{2} d^{8}+7280 x^{3} a^{4} b^{6} c^{3} d^{7}+12740 x^{3} a^{3} b^{7} c^{4} d^{6}+20384 x^{3} a^{2} b^{8} c^{5} d^{5}+30576 x^{3} a \,b^{9} c^{6} d^{4}+43680 x^{3} b^{10} c^{7} d^{3}+91 x^{2} a^{8} b^{2} d^{10}+364 x^{2} a^{7} b^{3} c \,d^{9}+910 x^{2} a^{6} b^{4} c^{2} d^{8}+1820 x^{2} a^{5} b^{5} c^{3} d^{7}+3185 x^{2} a^{4} b^{6} c^{4} d^{6}+5096 x^{2} a^{3} b^{7} c^{5} d^{5}+7644 x^{2} a^{2} b^{8} c^{6} d^{4}+10920 x^{2} a \,b^{9} c^{7} d^{3}+15015 x^{2} b^{10} c^{8} d^{2}+14 x \,a^{9} b \,d^{10}+56 x \,a^{8} b^{2} c \,d^{9}+140 x \,a^{7} b^{3} c^{2} d^{8}+280 x \,a^{6} b^{4} c^{3} d^{7}+490 x \,a^{5} b^{5} c^{4} d^{6}+784 x \,a^{4} b^{6} c^{5} d^{5}+1176 x \,a^{3} b^{7} c^{6} d^{4}+1680 x \,a^{2} b^{8} c^{7} d^{3}+2310 x a \,b^{9} c^{8} d^{2}+3080 x \,b^{10} c^{9} d +a^{10} d^{10}+4 a^{9} b c \,d^{9}+10 a^{8} b^{2} c^{2} d^{8}+20 a^{7} b^{3} c^{3} d^{7}+35 a^{6} b^{4} c^{4} d^{6}+56 a^{5} b^{5} c^{5} d^{5}+84 a^{4} b^{6} c^{6} d^{4}+120 a^{3} b^{7} c^{7} d^{3}+165 a^{2} b^{8} c^{8} d^{2}+220 a \,b^{9} c^{9} d +286 b^{10} c^{10}}{4004 b^{11} \left (b x +a \right )^{14}}\)	\(962\)
parallelrisch	\(\frac {-1001 d^{10} x^{10} b^{13}-2002 a \,b^{12} d^{10} x^{9}-8008 b^{13} c \,d^{9} x^{9}-3003 a^{2} b^{11} d^{10} x^{8}-12012 a \,b^{12} c \,d^{9} x^{8}-30030 b^{13} c^{2} d^{8} x^{8}-3432 a^{3} b^{10} d^{10} x^{7}-13728 a^{2} b^{11} c \,d^{9} x^{7}-34320 a \,b^{12} c^{2} d^{8} x^{7}-68640 b^{13} c^{3} d^{7} x^{7}-3003 a^{4} b^{9} d^{10} x^{6}-12012 a^{3} b^{10} c \,d^{9} x^{6}-30030 a^{2} b^{11} c^{2} d^{8} x^{6}-60060 a \,b^{12} c^{3} d^{7} x^{6}-105105 b^{13} c^{4} d^{6} x^{6}-2002 a^{5} b^{8} d^{10} x^{5}-8008 a^{4} b^{9} c \,d^{9} x^{5}-20020 a^{3} b^{10} c^{2} d^{8} x^{5}-40040 a^{2} b^{11} c^{3} d^{7} x^{5}-70070 a \,b^{12} c^{4} d^{6} x^{5}-112112 b^{13} c^{5} d^{5} x^{5}-1001 a^{6} b^{7} d^{10} x^{4}-4004 a^{5} b^{8} c \,d^{9} x^{4}-10010 a^{4} b^{9} c^{2} d^{8} x^{4}-20020 a^{3} b^{10} c^{3} d^{7} x^{4}-35035 a^{2} b^{11} c^{4} d^{6} x^{4}-56056 a \,b^{12} c^{5} d^{5} x^{4}-84084 b^{13} c^{6} d^{4} x^{4}-364 a^{7} b^{6} d^{10} x^{3}-1456 a^{6} b^{7} c \,d^{9} x^{3}-3640 a^{5} b^{8} c^{2} d^{8} x^{3}-7280 a^{4} b^{9} c^{3} d^{7} x^{3}-12740 a^{3} b^{10} c^{4} d^{6} x^{3}-20384 a^{2} b^{11} c^{5} d^{5} x^{3}-30576 a \,b^{12} c^{6} d^{4} x^{3}-43680 b^{13} c^{7} d^{3} x^{3}-91 a^{8} b^{5} d^{10} x^{2}-364 a^{7} b^{6} c \,d^{9} x^{2}-910 a^{6} b^{7} c^{2} d^{8} x^{2}-1820 a^{5} b^{8} c^{3} d^{7} x^{2}-3185 a^{4} b^{9} c^{4} d^{6} x^{2}-5096 a^{3} b^{10} c^{5} d^{5} x^{2}-7644 a^{2} b^{11} c^{6} d^{4} x^{2}-10920 a \,b^{12} c^{7} d^{3} x^{2}-15015 b^{13} c^{8} d^{2} x^{2}-14 a^{9} b^{4} d^{10} x -56 a^{8} b^{5} c \,d^{9} x -140 a^{7} b^{6} c^{2} d^{8} x -280 a^{6} b^{7} c^{3} d^{7} x -490 a^{5} b^{8} c^{4} d^{6} x -784 a^{4} b^{9} c^{5} d^{5} x -1176 a^{3} b^{10} c^{6} d^{4} x -1680 a^{2} b^{11} c^{7} d^{3} x -2310 a \,b^{12} c^{8} d^{2} x -3080 b^{13} c^{9} d x -a^{10} b^{3} d^{10}-4 a^{9} b^{4} c \,d^{9}-10 a^{8} b^{5} c^{2} d^{8}-20 a^{7} b^{6} c^{3} d^{7}-35 a^{6} b^{7} c^{4} d^{6}-56 a^{5} b^{8} c^{5} d^{5}-84 a^{4} b^{9} c^{6} d^{4}-120 a^{3} b^{10} c^{7} d^{3}-165 a^{2} c^{8} d^{2} b^{11}-220 a \,b^{12} c^{9} d -286 b^{13} c^{10}}{4004 b^{14} \left (b x +a \right )^{14}}\)	\(970\)

[In]

int((d*x+c)^10/(b*x+a)^15,x,method=_RETURNVERBOSE)

[Out]

(-1/4/b*d^10*x^10-1/2/b^2*d^9*(a*d+4*b*c)*x^9-3/4/b^3*d^8*(a^2*d^2+4*a*b*c*d+10*b^2*c^2)*x^8-6/7/b^4*d^7*(a^3*
d^3+4*a^2*b*c*d^2+10*a*b^2*c^2*d+20*b^3*c^3)*x^7-3/4/b^5*d^6*(a^4*d^4+4*a^3*b*c*d^3+10*a^2*b^2*c^2*d^2+20*a*b^
3*c^3*d+35*b^4*c^4)*x^6-1/2/b^6*d^5*(a^5*d^5+4*a^4*b*c*d^4+10*a^3*b^2*c^2*d^3+20*a^2*b^3*c^3*d^2+35*a*b^4*c^4*
d+56*b^5*c^5)*x^5-1/4/b^7*d^4*(a^6*d^6+4*a^5*b*c*d^5+10*a^4*b^2*c^2*d^4+20*a^3*b^3*c^3*d^3+35*a^2*b^4*c^4*d^2+
56*a*b^5*c^5*d+84*b^6*c^6)*x^4-1/11/b^8*d^3*(a^7*d^7+4*a^6*b*c*d^6+10*a^5*b^2*c^2*d^5+20*a^4*b^3*c^3*d^4+35*a^
3*b^4*c^4*d^3+56*a^2*b^5*c^5*d^2+84*a*b^6*c^6*d+120*b^7*c^7)*x^3-1/44/b^9*d^2*(a^8*d^8+4*a^7*b*c*d^7+10*a^6*b^
2*c^2*d^6+20*a^5*b^3*c^3*d^5+35*a^4*b^4*c^4*d^4+56*a^3*b^5*c^5*d^3+84*a^2*b^6*c^6*d^2+120*a*b^7*c^7*d+165*b^8*
c^8)*x^2-1/286/b^10*d*(a^9*d^9+4*a^8*b*c*d^8+10*a^7*b^2*c^2*d^7+20*a^6*b^3*c^3*d^6+35*a^5*b^4*c^4*d^5+56*a^4*b
^5*c^5*d^4+84*a^3*b^6*c^6*d^3+120*a^2*b^7*c^7*d^2+165*a*b^8*c^8*d+220*b^9*c^9)*x-1/4004/b^11*(a^10*d^10+4*a^9*
b*c*d^9+10*a^8*b^2*c^2*d^8+20*a^7*b^3*c^3*d^7+35*a^6*b^4*c^4*d^6+56*a^5*b^5*c^5*d^5+84*a^4*b^6*c^6*d^4+120*a^3
*b^7*c^7*d^3+165*a^2*b^8*c^8*d^2+220*a*b^9*c^9*d+286*b^10*c^10))/(b*x+a)^14

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1008 vs. \(2 (112) = 224\).

Time = 0.23 (sec) , antiderivative size = 1008, normalized size of antiderivative = 8.40 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx=-\frac {1001 \, b^{10} d^{10} x^{10} + 286 \, b^{10} c^{10} + 220 \, a b^{9} c^{9} d + 165 \, a^{2} b^{8} c^{8} d^{2} + 120 \, a^{3} b^{7} c^{7} d^{3} + 84 \, a^{4} b^{6} c^{6} d^{4} + 56 \, a^{5} b^{5} c^{5} d^{5} + 35 \, a^{6} b^{4} c^{4} d^{6} + 20 \, a^{7} b^{3} c^{3} d^{7} + 10 \, a^{8} b^{2} c^{2} d^{8} + 4 \, a^{9} b c d^{9} + a^{10} d^{10} + 2002 \, {\left (4 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 3003 \, {\left (10 \, b^{10} c^{2} d^{8} + 4 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 3432 \, {\left (20 \, b^{10} c^{3} d^{7} + 10 \, a b^{9} c^{2} d^{8} + 4 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 3003 \, {\left (35 \, b^{10} c^{4} d^{6} + 20 \, a b^{9} c^{3} d^{7} + 10 \, a^{2} b^{8} c^{2} d^{8} + 4 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 2002 \, {\left (56 \, b^{10} c^{5} d^{5} + 35 \, a b^{9} c^{4} d^{6} + 20 \, a^{2} b^{8} c^{3} d^{7} + 10 \, a^{3} b^{7} c^{2} d^{8} + 4 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 1001 \, {\left (84 \, b^{10} c^{6} d^{4} + 56 \, a b^{9} c^{5} d^{5} + 35 \, a^{2} b^{8} c^{4} d^{6} + 20 \, a^{3} b^{7} c^{3} d^{7} + 10 \, a^{4} b^{6} c^{2} d^{8} + 4 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 364 \, {\left (120 \, b^{10} c^{7} d^{3} + 84 \, a b^{9} c^{6} d^{4} + 56 \, a^{2} b^{8} c^{5} d^{5} + 35 \, a^{3} b^{7} c^{4} d^{6} + 20 \, a^{4} b^{6} c^{3} d^{7} + 10 \, a^{5} b^{5} c^{2} d^{8} + 4 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 91 \, {\left (165 \, b^{10} c^{8} d^{2} + 120 \, a b^{9} c^{7} d^{3} + 84 \, a^{2} b^{8} c^{6} d^{4} + 56 \, a^{3} b^{7} c^{5} d^{5} + 35 \, a^{4} b^{6} c^{4} d^{6} + 20 \, a^{5} b^{5} c^{3} d^{7} + 10 \, a^{6} b^{4} c^{2} d^{8} + 4 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 14 \, {\left (220 \, b^{10} c^{9} d + 165 \, a b^{9} c^{8} d^{2} + 120 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} + 56 \, a^{4} b^{6} c^{5} d^{5} + 35 \, a^{5} b^{5} c^{4} d^{6} + 20 \, a^{6} b^{4} c^{3} d^{7} + 10 \, a^{7} b^{3} c^{2} d^{8} + 4 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{4004 \, {\left (b^{25} x^{14} + 14 \, a b^{24} x^{13} + 91 \, a^{2} b^{23} x^{12} + 364 \, a^{3} b^{22} x^{11} + 1001 \, a^{4} b^{21} x^{10} + 2002 \, a^{5} b^{20} x^{9} + 3003 \, a^{6} b^{19} x^{8} + 3432 \, a^{7} b^{18} x^{7} + 3003 \, a^{8} b^{17} x^{6} + 2002 \, a^{9} b^{16} x^{5} + 1001 \, a^{10} b^{15} x^{4} + 364 \, a^{11} b^{14} x^{3} + 91 \, a^{12} b^{13} x^{2} + 14 \, a^{13} b^{12} x + a^{14} b^{11}\right )}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^15,x, algorithm="fricas")

[Out]

-1/4004*(1001*b^10*d^10*x^10 + 286*b^10*c^10 + 220*a*b^9*c^9*d + 165*a^2*b^8*c^8*d^2 + 120*a^3*b^7*c^7*d^3 + 8
4*a^4*b^6*c^6*d^4 + 56*a^5*b^5*c^5*d^5 + 35*a^6*b^4*c^4*d^6 + 20*a^7*b^3*c^3*d^7 + 10*a^8*b^2*c^2*d^8 + 4*a^9*
b*c*d^9 + a^10*d^10 + 2002*(4*b^10*c*d^9 + a*b^9*d^10)*x^9 + 3003*(10*b^10*c^2*d^8 + 4*a*b^9*c*d^9 + a^2*b^8*d
^10)*x^8 + 3432*(20*b^10*c^3*d^7 + 10*a*b^9*c^2*d^8 + 4*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 3003*(35*b^10*c^4*
d^6 + 20*a*b^9*c^3*d^7 + 10*a^2*b^8*c^2*d^8 + 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 2002*(56*b^10*c^5*d^5 + 35
*a*b^9*c^4*d^6 + 20*a^2*b^8*c^3*d^7 + 10*a^3*b^7*c^2*d^8 + 4*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 1001*(84*b^10
*c^6*d^4 + 56*a*b^9*c^5*d^5 + 35*a^2*b^8*c^4*d^6 + 20*a^3*b^7*c^3*d^7 + 10*a^4*b^6*c^2*d^8 + 4*a^5*b^5*c*d^9 +
a^6*b^4*d^10)*x^4 + 364*(120*b^10*c^7*d^3 + 84*a*b^9*c^6*d^4 + 56*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6 + 20*a
^4*b^6*c^3*d^7 + 10*a^5*b^5*c^2*d^8 + 4*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 91*(165*b^10*c^8*d^2 + 120*a*b^9*c
^7*d^3 + 84*a^2*b^8*c^6*d^4 + 56*a^3*b^7*c^5*d^5 + 35*a^4*b^6*c^4*d^6 + 20*a^5*b^5*c^3*d^7 + 10*a^6*b^4*c^2*d^
8 + 4*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 14*(220*b^10*c^9*d + 165*a*b^9*c^8*d^2 + 120*a^2*b^8*c^7*d^3 + 84*a^
3*b^7*c^6*d^4 + 56*a^4*b^6*c^5*d^5 + 35*a^5*b^5*c^4*d^6 + 20*a^6*b^4*c^3*d^7 + 10*a^7*b^3*c^2*d^8 + 4*a^8*b^2*
c*d^9 + a^9*b*d^10)*x)/(b^25*x^14 + 14*a*b^24*x^13 + 91*a^2*b^23*x^12 + 364*a^3*b^22*x^11 + 1001*a^4*b^21*x^10
+ 2002*a^5*b^20*x^9 + 3003*a^6*b^19*x^8 + 3432*a^7*b^18*x^7 + 3003*a^8*b^17*x^6 + 2002*a^9*b^16*x^5 + 1001*a^
10*b^15*x^4 + 364*a^11*b^14*x^3 + 91*a^12*b^13*x^2 + 14*a^13*b^12*x + a^14*b^11)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx=\text {Timed out} \]

[In]

integrate((d*x+c)**10/(b*x+a)**15,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1008 vs. \(2 (112) = 224\).

Time = 0.25 (sec) , antiderivative size = 1008, normalized size of antiderivative = 8.40 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx=-\frac {1001 \, b^{10} d^{10} x^{10} + 286 \, b^{10} c^{10} + 220 \, a b^{9} c^{9} d + 165 \, a^{2} b^{8} c^{8} d^{2} + 120 \, a^{3} b^{7} c^{7} d^{3} + 84 \, a^{4} b^{6} c^{6} d^{4} + 56 \, a^{5} b^{5} c^{5} d^{5} + 35 \, a^{6} b^{4} c^{4} d^{6} + 20 \, a^{7} b^{3} c^{3} d^{7} + 10 \, a^{8} b^{2} c^{2} d^{8} + 4 \, a^{9} b c d^{9} + a^{10} d^{10} + 2002 \, {\left (4 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 3003 \, {\left (10 \, b^{10} c^{2} d^{8} + 4 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 3432 \, {\left (20 \, b^{10} c^{3} d^{7} + 10 \, a b^{9} c^{2} d^{8} + 4 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 3003 \, {\left (35 \, b^{10} c^{4} d^{6} + 20 \, a b^{9} c^{3} d^{7} + 10 \, a^{2} b^{8} c^{2} d^{8} + 4 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 2002 \, {\left (56 \, b^{10} c^{5} d^{5} + 35 \, a b^{9} c^{4} d^{6} + 20 \, a^{2} b^{8} c^{3} d^{7} + 10 \, a^{3} b^{7} c^{2} d^{8} + 4 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 1001 \, {\left (84 \, b^{10} c^{6} d^{4} + 56 \, a b^{9} c^{5} d^{5} + 35 \, a^{2} b^{8} c^{4} d^{6} + 20 \, a^{3} b^{7} c^{3} d^{7} + 10 \, a^{4} b^{6} c^{2} d^{8} + 4 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 364 \, {\left (120 \, b^{10} c^{7} d^{3} + 84 \, a b^{9} c^{6} d^{4} + 56 \, a^{2} b^{8} c^{5} d^{5} + 35 \, a^{3} b^{7} c^{4} d^{6} + 20 \, a^{4} b^{6} c^{3} d^{7} + 10 \, a^{5} b^{5} c^{2} d^{8} + 4 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 91 \, {\left (165 \, b^{10} c^{8} d^{2} + 120 \, a b^{9} c^{7} d^{3} + 84 \, a^{2} b^{8} c^{6} d^{4} + 56 \, a^{3} b^{7} c^{5} d^{5} + 35 \, a^{4} b^{6} c^{4} d^{6} + 20 \, a^{5} b^{5} c^{3} d^{7} + 10 \, a^{6} b^{4} c^{2} d^{8} + 4 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 14 \, {\left (220 \, b^{10} c^{9} d + 165 \, a b^{9} c^{8} d^{2} + 120 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} + 56 \, a^{4} b^{6} c^{5} d^{5} + 35 \, a^{5} b^{5} c^{4} d^{6} + 20 \, a^{6} b^{4} c^{3} d^{7} + 10 \, a^{7} b^{3} c^{2} d^{8} + 4 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{4004 \, {\left (b^{25} x^{14} + 14 \, a b^{24} x^{13} + 91 \, a^{2} b^{23} x^{12} + 364 \, a^{3} b^{22} x^{11} + 1001 \, a^{4} b^{21} x^{10} + 2002 \, a^{5} b^{20} x^{9} + 3003 \, a^{6} b^{19} x^{8} + 3432 \, a^{7} b^{18} x^{7} + 3003 \, a^{8} b^{17} x^{6} + 2002 \, a^{9} b^{16} x^{5} + 1001 \, a^{10} b^{15} x^{4} + 364 \, a^{11} b^{14} x^{3} + 91 \, a^{12} b^{13} x^{2} + 14 \, a^{13} b^{12} x + a^{14} b^{11}\right )}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^15,x, algorithm="maxima")

[Out]

-1/4004*(1001*b^10*d^10*x^10 + 286*b^10*c^10 + 220*a*b^9*c^9*d + 165*a^2*b^8*c^8*d^2 + 120*a^3*b^7*c^7*d^3 + 8
4*a^4*b^6*c^6*d^4 + 56*a^5*b^5*c^5*d^5 + 35*a^6*b^4*c^4*d^6 + 20*a^7*b^3*c^3*d^7 + 10*a^8*b^2*c^2*d^8 + 4*a^9*
b*c*d^9 + a^10*d^10 + 2002*(4*b^10*c*d^9 + a*b^9*d^10)*x^9 + 3003*(10*b^10*c^2*d^8 + 4*a*b^9*c*d^9 + a^2*b^8*d
^10)*x^8 + 3432*(20*b^10*c^3*d^7 + 10*a*b^9*c^2*d^8 + 4*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 3003*(35*b^10*c^4*
d^6 + 20*a*b^9*c^3*d^7 + 10*a^2*b^8*c^2*d^8 + 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 2002*(56*b^10*c^5*d^5 + 35
*a*b^9*c^4*d^6 + 20*a^2*b^8*c^3*d^7 + 10*a^3*b^7*c^2*d^8 + 4*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 1001*(84*b^10
*c^6*d^4 + 56*a*b^9*c^5*d^5 + 35*a^2*b^8*c^4*d^6 + 20*a^3*b^7*c^3*d^7 + 10*a^4*b^6*c^2*d^8 + 4*a^5*b^5*c*d^9 +
a^6*b^4*d^10)*x^4 + 364*(120*b^10*c^7*d^3 + 84*a*b^9*c^6*d^4 + 56*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6 + 20*a
^4*b^6*c^3*d^7 + 10*a^5*b^5*c^2*d^8 + 4*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 91*(165*b^10*c^8*d^2 + 120*a*b^9*c
^7*d^3 + 84*a^2*b^8*c^6*d^4 + 56*a^3*b^7*c^5*d^5 + 35*a^4*b^6*c^4*d^6 + 20*a^5*b^5*c^3*d^7 + 10*a^6*b^4*c^2*d^
8 + 4*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 14*(220*b^10*c^9*d + 165*a*b^9*c^8*d^2 + 120*a^2*b^8*c^7*d^3 + 84*a^
3*b^7*c^6*d^4 + 56*a^4*b^6*c^5*d^5 + 35*a^5*b^5*c^4*d^6 + 20*a^6*b^4*c^3*d^7 + 10*a^7*b^3*c^2*d^8 + 4*a^8*b^2*
c*d^9 + a^9*b*d^10)*x)/(b^25*x^14 + 14*a*b^24*x^13 + 91*a^2*b^23*x^12 + 364*a^3*b^22*x^11 + 1001*a^4*b^21*x^10
+ 2002*a^5*b^20*x^9 + 3003*a^6*b^19*x^8 + 3432*a^7*b^18*x^7 + 3003*a^8*b^17*x^6 + 2002*a^9*b^16*x^5 + 1001*a^
10*b^15*x^4 + 364*a^11*b^14*x^3 + 91*a^12*b^13*x^2 + 14*a^13*b^12*x + a^14*b^11)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (112) = 224\).

Time = 0.33 (sec) , antiderivative size = 961, normalized size of antiderivative = 8.01 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx=-\frac {1001 \, b^{10} d^{10} x^{10} + 8008 \, b^{10} c d^{9} x^{9} + 2002 \, a b^{9} d^{10} x^{9} + 30030 \, b^{10} c^{2} d^{8} x^{8} + 12012 \, a b^{9} c d^{9} x^{8} + 3003 \, a^{2} b^{8} d^{10} x^{8} + 68640 \, b^{10} c^{3} d^{7} x^{7} + 34320 \, a b^{9} c^{2} d^{8} x^{7} + 13728 \, a^{2} b^{8} c d^{9} x^{7} + 3432 \, a^{3} b^{7} d^{10} x^{7} + 105105 \, b^{10} c^{4} d^{6} x^{6} + 60060 \, a b^{9} c^{3} d^{7} x^{6} + 30030 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 12012 \, a^{3} b^{7} c d^{9} x^{6} + 3003 \, a^{4} b^{6} d^{10} x^{6} + 112112 \, b^{10} c^{5} d^{5} x^{5} + 70070 \, a b^{9} c^{4} d^{6} x^{5} + 40040 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 20020 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 8008 \, a^{4} b^{6} c d^{9} x^{5} + 2002 \, a^{5} b^{5} d^{10} x^{5} + 84084 \, b^{10} c^{6} d^{4} x^{4} + 56056 \, a b^{9} c^{5} d^{5} x^{4} + 35035 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 20020 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 10010 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 4004 \, a^{5} b^{5} c d^{9} x^{4} + 1001 \, a^{6} b^{4} d^{10} x^{4} + 43680 \, b^{10} c^{7} d^{3} x^{3} + 30576 \, a b^{9} c^{6} d^{4} x^{3} + 20384 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 12740 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 7280 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 3640 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 1456 \, a^{6} b^{4} c d^{9} x^{3} + 364 \, a^{7} b^{3} d^{10} x^{3} + 15015 \, b^{10} c^{8} d^{2} x^{2} + 10920 \, a b^{9} c^{7} d^{3} x^{2} + 7644 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 5096 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 3185 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 1820 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 910 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 364 \, a^{7} b^{3} c d^{9} x^{2} + 91 \, a^{8} b^{2} d^{10} x^{2} + 3080 \, b^{10} c^{9} d x + 2310 \, a b^{9} c^{8} d^{2} x + 1680 \, a^{2} b^{8} c^{7} d^{3} x + 1176 \, a^{3} b^{7} c^{6} d^{4} x + 784 \, a^{4} b^{6} c^{5} d^{5} x + 490 \, a^{5} b^{5} c^{4} d^{6} x + 280 \, a^{6} b^{4} c^{3} d^{7} x + 140 \, a^{7} b^{3} c^{2} d^{8} x + 56 \, a^{8} b^{2} c d^{9} x + 14 \, a^{9} b d^{10} x + 286 \, b^{10} c^{10} + 220 \, a b^{9} c^{9} d + 165 \, a^{2} b^{8} c^{8} d^{2} + 120 \, a^{3} b^{7} c^{7} d^{3} + 84 \, a^{4} b^{6} c^{6} d^{4} + 56 \, a^{5} b^{5} c^{5} d^{5} + 35 \, a^{6} b^{4} c^{4} d^{6} + 20 \, a^{7} b^{3} c^{3} d^{7} + 10 \, a^{8} b^{2} c^{2} d^{8} + 4 \, a^{9} b c d^{9} + a^{10} d^{10}}{4004 \, {\left (b x + a\right )}^{14} b^{11}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^15,x, algorithm="giac")

[Out]

-1/4004*(1001*b^10*d^10*x^10 + 8008*b^10*c*d^9*x^9 + 2002*a*b^9*d^10*x^9 + 30030*b^10*c^2*d^8*x^8 + 12012*a*b^
9*c*d^9*x^8 + 3003*a^2*b^8*d^10*x^8 + 68640*b^10*c^3*d^7*x^7 + 34320*a*b^9*c^2*d^8*x^7 + 13728*a^2*b^8*c*d^9*x
^7 + 3432*a^3*b^7*d^10*x^7 + 105105*b^10*c^4*d^6*x^6 + 60060*a*b^9*c^3*d^7*x^6 + 30030*a^2*b^8*c^2*d^8*x^6 + 1
2012*a^3*b^7*c*d^9*x^6 + 3003*a^4*b^6*d^10*x^6 + 112112*b^10*c^5*d^5*x^5 + 70070*a*b^9*c^4*d^6*x^5 + 40040*a^2
*b^8*c^3*d^7*x^5 + 20020*a^3*b^7*c^2*d^8*x^5 + 8008*a^4*b^6*c*d^9*x^5 + 2002*a^5*b^5*d^10*x^5 + 84084*b^10*c^6
*d^4*x^4 + 56056*a*b^9*c^5*d^5*x^4 + 35035*a^2*b^8*c^4*d^6*x^4 + 20020*a^3*b^7*c^3*d^7*x^4 + 10010*a^4*b^6*c^2
*d^8*x^4 + 4004*a^5*b^5*c*d^9*x^4 + 1001*a^6*b^4*d^10*x^4 + 43680*b^10*c^7*d^3*x^3 + 30576*a*b^9*c^6*d^4*x^3 +
20384*a^2*b^8*c^5*d^5*x^3 + 12740*a^3*b^7*c^4*d^6*x^3 + 7280*a^4*b^6*c^3*d^7*x^3 + 3640*a^5*b^5*c^2*d^8*x^3 +
1456*a^6*b^4*c*d^9*x^3 + 364*a^7*b^3*d^10*x^3 + 15015*b^10*c^8*d^2*x^2 + 10920*a*b^9*c^7*d^3*x^2 + 7644*a^2*b
^8*c^6*d^4*x^2 + 5096*a^3*b^7*c^5*d^5*x^2 + 3185*a^4*b^6*c^4*d^6*x^2 + 1820*a^5*b^5*c^3*d^7*x^2 + 910*a^6*b^4*
c^2*d^8*x^2 + 364*a^7*b^3*c*d^9*x^2 + 91*a^8*b^2*d^10*x^2 + 3080*b^10*c^9*d*x + 2310*a*b^9*c^8*d^2*x + 1680*a^
2*b^8*c^7*d^3*x + 1176*a^3*b^7*c^6*d^4*x + 784*a^4*b^6*c^5*d^5*x + 490*a^5*b^5*c^4*d^6*x + 280*a^6*b^4*c^3*d^7
*x + 140*a^7*b^3*c^2*d^8*x + 56*a^8*b^2*c*d^9*x + 14*a^9*b*d^10*x + 286*b^10*c^10 + 220*a*b^9*c^9*d + 165*a^2*
b^8*c^8*d^2 + 120*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 56*a^5*b^5*c^5*d^5 + 35*a^6*b^4*c^4*d^6 + 20*a^7*b^3*
c^3*d^7 + 10*a^8*b^2*c^2*d^8 + 4*a^9*b*c*d^9 + a^10*d^10)/((b*x + a)^14*b^11)

Mupad [B] (verification not implemented)

Time = 1.80 (sec) , antiderivative size = 1109, normalized size of antiderivative = 9.24 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx=-\frac {a^{10}\,d^{10}+4\,a^9\,b\,c\,d^9+14\,a^9\,b\,d^{10}\,x+10\,a^8\,b^2\,c^2\,d^8+56\,a^8\,b^2\,c\,d^9\,x+91\,a^8\,b^2\,d^{10}\,x^2+20\,a^7\,b^3\,c^3\,d^7+140\,a^7\,b^3\,c^2\,d^8\,x+364\,a^7\,b^3\,c\,d^9\,x^2+364\,a^7\,b^3\,d^{10}\,x^3+35\,a^6\,b^4\,c^4\,d^6+280\,a^6\,b^4\,c^3\,d^7\,x+910\,a^6\,b^4\,c^2\,d^8\,x^2+1456\,a^6\,b^4\,c\,d^9\,x^3+1001\,a^6\,b^4\,d^{10}\,x^4+56\,a^5\,b^5\,c^5\,d^5+490\,a^5\,b^5\,c^4\,d^6\,x+1820\,a^5\,b^5\,c^3\,d^7\,x^2+3640\,a^5\,b^5\,c^2\,d^8\,x^3+4004\,a^5\,b^5\,c\,d^9\,x^4+2002\,a^5\,b^5\,d^{10}\,x^5+84\,a^4\,b^6\,c^6\,d^4+784\,a^4\,b^6\,c^5\,d^5\,x+3185\,a^4\,b^6\,c^4\,d^6\,x^2+7280\,a^4\,b^6\,c^3\,d^7\,x^3+10010\,a^4\,b^6\,c^2\,d^8\,x^4+8008\,a^4\,b^6\,c\,d^9\,x^5+3003\,a^4\,b^6\,d^{10}\,x^6+120\,a^3\,b^7\,c^7\,d^3+1176\,a^3\,b^7\,c^6\,d^4\,x+5096\,a^3\,b^7\,c^5\,d^5\,x^2+12740\,a^3\,b^7\,c^4\,d^6\,x^3+20020\,a^3\,b^7\,c^3\,d^7\,x^4+20020\,a^3\,b^7\,c^2\,d^8\,x^5+12012\,a^3\,b^7\,c\,d^9\,x^6+3432\,a^3\,b^7\,d^{10}\,x^7+165\,a^2\,b^8\,c^8\,d^2+1680\,a^2\,b^8\,c^7\,d^3\,x+7644\,a^2\,b^8\,c^6\,d^4\,x^2+20384\,a^2\,b^8\,c^5\,d^5\,x^3+35035\,a^2\,b^8\,c^4\,d^6\,x^4+40040\,a^2\,b^8\,c^3\,d^7\,x^5+30030\,a^2\,b^8\,c^2\,d^8\,x^6+13728\,a^2\,b^8\,c\,d^9\,x^7+3003\,a^2\,b^8\,d^{10}\,x^8+220\,a\,b^9\,c^9\,d+2310\,a\,b^9\,c^8\,d^2\,x+10920\,a\,b^9\,c^7\,d^3\,x^2+30576\,a\,b^9\,c^6\,d^4\,x^3+56056\,a\,b^9\,c^5\,d^5\,x^4+70070\,a\,b^9\,c^4\,d^6\,x^5+60060\,a\,b^9\,c^3\,d^7\,x^6+34320\,a\,b^9\,c^2\,d^8\,x^7+12012\,a\,b^9\,c\,d^9\,x^8+2002\,a\,b^9\,d^{10}\,x^9+286\,b^{10}\,c^{10}+3080\,b^{10}\,c^9\,d\,x+15015\,b^{10}\,c^8\,d^2\,x^2+43680\,b^{10}\,c^7\,d^3\,x^3+84084\,b^{10}\,c^6\,d^4\,x^4+112112\,b^{10}\,c^5\,d^5\,x^5+105105\,b^{10}\,c^4\,d^6\,x^6+68640\,b^{10}\,c^3\,d^7\,x^7+30030\,b^{10}\,c^2\,d^8\,x^8+8008\,b^{10}\,c\,d^9\,x^9+1001\,b^{10}\,d^{10}\,x^{10}}{4004\,a^{14}\,b^{11}+56056\,a^{13}\,b^{12}\,x+364364\,a^{12}\,b^{13}\,x^2+1457456\,a^{11}\,b^{14}\,x^3+4008004\,a^{10}\,b^{15}\,x^4+8016008\,a^9\,b^{16}\,x^5+12024012\,a^8\,b^{17}\,x^6+13741728\,a^7\,b^{18}\,x^7+12024012\,a^6\,b^{19}\,x^8+8016008\,a^5\,b^{20}\,x^9+4008004\,a^4\,b^{21}\,x^{10}+1457456\,a^3\,b^{22}\,x^{11}+364364\,a^2\,b^{23}\,x^{12}+56056\,a\,b^{24}\,x^{13}+4004\,b^{25}\,x^{14}} \]

[In]

int((c + d*x)^10/(a + b*x)^15,x)

[Out]

-(a^10*d^10 + 286*b^10*c^10 + 1001*b^10*d^10*x^10 + 2002*a*b^9*d^10*x^9 + 8008*b^10*c*d^9*x^9 + 165*a^2*b^8*c^
8*d^2 + 120*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 56*a^5*b^5*c^5*d^5 + 35*a^6*b^4*c^4*d^6 + 20*a^7*b^3*c^3*d^
7 + 10*a^8*b^2*c^2*d^8 + 91*a^8*b^2*d^10*x^2 + 364*a^7*b^3*d^10*x^3 + 1001*a^6*b^4*d^10*x^4 + 2002*a^5*b^5*d^1
0*x^5 + 3003*a^4*b^6*d^10*x^6 + 3432*a^3*b^7*d^10*x^7 + 3003*a^2*b^8*d^10*x^8 + 15015*b^10*c^8*d^2*x^2 + 43680
*b^10*c^7*d^3*x^3 + 84084*b^10*c^6*d^4*x^4 + 112112*b^10*c^5*d^5*x^5 + 105105*b^10*c^4*d^6*x^6 + 68640*b^10*c^
3*d^7*x^7 + 30030*b^10*c^2*d^8*x^8 + 220*a*b^9*c^9*d + 4*a^9*b*c*d^9 + 14*a^9*b*d^10*x + 3080*b^10*c^9*d*x + 7
644*a^2*b^8*c^6*d^4*x^2 + 5096*a^3*b^7*c^5*d^5*x^2 + 3185*a^4*b^6*c^4*d^6*x^2 + 1820*a^5*b^5*c^3*d^7*x^2 + 910
*a^6*b^4*c^2*d^8*x^2 + 20384*a^2*b^8*c^5*d^5*x^3 + 12740*a^3*b^7*c^4*d^6*x^3 + 7280*a^4*b^6*c^3*d^7*x^3 + 3640
*a^5*b^5*c^2*d^8*x^3 + 35035*a^2*b^8*c^4*d^6*x^4 + 20020*a^3*b^7*c^3*d^7*x^4 + 10010*a^4*b^6*c^2*d^8*x^4 + 400
40*a^2*b^8*c^3*d^7*x^5 + 20020*a^3*b^7*c^2*d^8*x^5 + 30030*a^2*b^8*c^2*d^8*x^6 + 2310*a*b^9*c^8*d^2*x + 56*a^8
*b^2*c*d^9*x + 12012*a*b^9*c*d^9*x^8 + 1680*a^2*b^8*c^7*d^3*x + 1176*a^3*b^7*c^6*d^4*x + 784*a^4*b^6*c^5*d^5*x
+ 490*a^5*b^5*c^4*d^6*x + 280*a^6*b^4*c^3*d^7*x + 140*a^7*b^3*c^2*d^8*x + 10920*a*b^9*c^7*d^3*x^2 + 364*a^7*b
^3*c*d^9*x^2 + 30576*a*b^9*c^6*d^4*x^3 + 1456*a^6*b^4*c*d^9*x^3 + 56056*a*b^9*c^5*d^5*x^4 + 4004*a^5*b^5*c*d^9
*x^4 + 70070*a*b^9*c^4*d^6*x^5 + 8008*a^4*b^6*c*d^9*x^5 + 60060*a*b^9*c^3*d^7*x^6 + 12012*a^3*b^7*c*d^9*x^6 +
34320*a*b^9*c^2*d^8*x^7 + 13728*a^2*b^8*c*d^9*x^7)/(4004*a^14*b^11 + 4004*b^25*x^14 + 56056*a^13*b^12*x + 5605
6*a*b^24*x^13 + 364364*a^12*b^13*x^2 + 1457456*a^11*b^14*x^3 + 4008004*a^10*b^15*x^4 + 8016008*a^9*b^16*x^5 +
12024012*a^8*b^17*x^6 + 13741728*a^7*b^18*x^7 + 12024012*a^6*b^19*x^8 + 8016008*a^5*b^20*x^9 + 4008004*a^4*b^2
1*x^10 + 1457456*a^3*b^22*x^11 + 364364*a^2*b^23*x^12)