\(\int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx\) [1323]

Optimal result

Integrand size = 15, antiderivative size = 28 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx=-\frac {(c+d x)^{11}}{11 (b c-a d) (a+b x)^{11}} \]

[Out]

-1/11*(d*x+c)^11/(-a*d+b*c)/(b*x+a)^11

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx=-\frac {(c+d x)^{11}}{11 (a+b x)^{11} (b c-a d)} \]

[In]

Int[(c + d*x)^10/(a + b*x)^12,x]

[Out]

-1/11*(c + d*x)^11/((b*c - a*d)*(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^{11}}{11 (b c-a d) (a+b x)^{11}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(665\) vs. \(2(28)=56\).

Time = 0.17 (sec) , antiderivative size = 665, normalized size of antiderivative = 23.75 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx=-\frac {a^{10} d^{10}+a^9 b d^9 (c+11 d x)+a^8 b^2 d^8 \left (c^2+11 c d x+55 d^2 x^2\right )+a^7 b^3 d^7 \left (c^3+11 c^2 d x+55 c d^2 x^2+165 d^3 x^3\right )+a^6 b^4 d^6 \left (c^4+11 c^3 d x+55 c^2 d^2 x^2+165 c d^3 x^3+330 d^4 x^4\right )+a^5 b^5 d^5 \left (c^5+11 c^4 d x+55 c^3 d^2 x^2+165 c^2 d^3 x^3+330 c d^4 x^4+462 d^5 x^5\right )+a^4 b^6 d^4 \left (c^6+11 c^5 d x+55 c^4 d^2 x^2+165 c^3 d^3 x^3+330 c^2 d^4 x^4+462 c d^5 x^5+462 d^6 x^6\right )+a^3 b^7 d^3 \left (c^7+11 c^6 d x+55 c^5 d^2 x^2+165 c^4 d^3 x^3+330 c^3 d^4 x^4+462 c^2 d^5 x^5+462 c d^6 x^6+330 d^7 x^7\right )+a^2 b^8 d^2 \left (c^8+11 c^7 d x+55 c^6 d^2 x^2+165 c^5 d^3 x^3+330 c^4 d^4 x^4+462 c^3 d^5 x^5+462 c^2 d^6 x^6+330 c d^7 x^7+165 d^8 x^8\right )+a b^9 d \left (c^9+11 c^8 d x+55 c^7 d^2 x^2+165 c^6 d^3 x^3+330 c^5 d^4 x^4+462 c^4 d^5 x^5+462 c^3 d^6 x^6+330 c^2 d^7 x^7+165 c d^8 x^8+55 d^9 x^9\right )+b^{10} \left (c^{10}+11 c^9 d x+55 c^8 d^2 x^2+165 c^7 d^3 x^3+330 c^6 d^4 x^4+462 c^5 d^5 x^5+462 c^4 d^6 x^6+330 c^3 d^7 x^7+165 c^2 d^8 x^8+55 c d^9 x^9+11 d^{10} x^{10}\right )}{11 b^{11} (a+b x)^{11}} \]

[In]

Integrate[(c + d*x)^10/(a + b*x)^12,x]

[Out]

-1/11*(a^10*d^10 + a^9*b*d^9*(c + 11*d*x) + a^8*b^2*d^8*(c^2 + 11*c*d*x + 55*d^2*x^2) + a^7*b^3*d^7*(c^3 + 11*
c^2*d*x + 55*c*d^2*x^2 + 165*d^3*x^3) + a^6*b^4*d^6*(c^4 + 11*c^3*d*x + 55*c^2*d^2*x^2 + 165*c*d^3*x^3 + 330*d
^4*x^4) + a^5*b^5*d^5*(c^5 + 11*c^4*d*x + 55*c^3*d^2*x^2 + 165*c^2*d^3*x^3 + 330*c*d^4*x^4 + 462*d^5*x^5) + a^
4*b^6*d^4*(c^6 + 11*c^5*d*x + 55*c^4*d^2*x^2 + 165*c^3*d^3*x^3 + 330*c^2*d^4*x^4 + 462*c*d^5*x^5 + 462*d^6*x^6
) + a^3*b^7*d^3*(c^7 + 11*c^6*d*x + 55*c^5*d^2*x^2 + 165*c^4*d^3*x^3 + 330*c^3*d^4*x^4 + 462*c^2*d^5*x^5 + 462
*c*d^6*x^6 + 330*d^7*x^7) + a^2*b^8*d^2*(c^8 + 11*c^7*d*x + 55*c^6*d^2*x^2 + 165*c^5*d^3*x^3 + 330*c^4*d^4*x^4
+ 462*c^3*d^5*x^5 + 462*c^2*d^6*x^6 + 330*c*d^7*x^7 + 165*d^8*x^8) + a*b^9*d*(c^9 + 11*c^8*d*x + 55*c^7*d^2*x
^2 + 165*c^6*d^3*x^3 + 330*c^5*d^4*x^4 + 462*c^4*d^5*x^5 + 462*c^3*d^6*x^6 + 330*c^2*d^7*x^7 + 165*c*d^8*x^8 +
55*d^9*x^9) + b^10*(c^10 + 11*c^9*d*x + 55*c^8*d^2*x^2 + 165*c^7*d^3*x^3 + 330*c^6*d^4*x^4 + 462*c^5*d^5*x^5
+ 462*c^4*d^6*x^6 + 330*c^3*d^7*x^7 + 165*c^2*d^8*x^8 + 55*c*d^9*x^9 + 11*d^10*x^10))/(b^11*(a + b*x)^11)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(775\) vs. \(2(26)=52\).

Time = 0.24 (sec) , antiderivative size = 776, normalized size of antiderivative = 27.71

method	result	size
risch	\(\frac {-\frac {d^{10} x^{10}}{b}-\frac {5 d^{9} \left (a d +b c \right ) x^{9}}{b^{2}}-\frac {15 d^{8} \left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) x^{8}}{b^{3}}-\frac {30 d^{7} \left (a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{7}}{b^{4}}-\frac {42 d^{6} \left (a^{4} d^{4}+a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}+a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x^{6}}{b^{5}}-\frac {42 d^{5} \left (a^{5} d^{5}+a^{4} b c \,d^{4}+a^{3} b^{2} c^{2} d^{3}+a^{2} b^{3} c^{3} d^{2}+a \,b^{4} c^{4} d +b^{5} c^{5}\right ) x^{5}}{b^{6}}-\frac {30 d^{4} \left (a^{6} d^{6}+a^{5} b c \,d^{5}+a^{4} b^{2} c^{2} d^{4}+a^{3} b^{3} c^{3} d^{3}+a^{2} b^{4} c^{4} d^{2}+a \,b^{5} c^{5} d +b^{6} c^{6}\right ) x^{4}}{b^{7}}-\frac {15 d^{3} \left (a^{7} d^{7}+a^{6} b c \,d^{6}+a^{5} b^{2} c^{2} d^{5}+a^{4} b^{3} c^{3} d^{4}+a^{3} b^{4} c^{4} d^{3}+a^{2} b^{5} c^{5} d^{2}+a \,b^{6} c^{6} d +b^{7} c^{7}\right ) x^{3}}{b^{8}}-\frac {5 d^{2} \left (a^{8} d^{8}+a^{7} b c \,d^{7}+a^{6} b^{2} c^{2} d^{6}+a^{5} b^{3} c^{3} d^{5}+a^{4} b^{4} c^{4} d^{4}+a^{3} b^{5} c^{5} d^{3}+a^{2} b^{6} c^{6} d^{2}+a \,b^{7} c^{7} d +b^{8} c^{8}\right ) x^{2}}{b^{9}}-\frac {d \left (a^{9} d^{9}+a^{8} b c \,d^{8}+a^{7} b^{2} c^{2} d^{7}+a^{6} b^{3} c^{3} d^{6}+a^{5} b^{4} c^{4} d^{5}+a^{4} b^{5} c^{5} d^{4}+a^{3} b^{6} c^{6} d^{3}+a^{2} b^{7} c^{7} d^{2}+a \,b^{8} c^{8} d +b^{9} c^{9}\right ) x}{b^{10}}-\frac {a^{10} d^{10}+a^{9} b c \,d^{9}+a^{8} b^{2} c^{2} d^{8}+a^{7} b^{3} c^{3} d^{7}+a^{6} b^{4} c^{4} d^{6}+a^{5} b^{5} c^{5} d^{5}+a^{4} b^{6} c^{6} d^{4}+a^{3} b^{7} c^{7} d^{3}+a^{2} b^{8} c^{8} d^{2}+a \,b^{9} c^{9} d +b^{10} c^{10}}{11 b^{11}}}{\left (b x +a \right )^{11}}\)	\(776\)
norman	\(\frac {-\frac {d^{10} x^{10}}{b}+\frac {5 \left (-a \,d^{10}-b c \,d^{9}\right ) x^{9}}{b^{2}}+\frac {15 \left (-a^{2} d^{10}-a b c \,d^{9}-b^{2} c^{2} d^{8}\right ) x^{8}}{b^{3}}+\frac {30 \left (-a^{3} d^{10}-a^{2} b c \,d^{9}-a \,b^{2} c^{2} d^{8}-b^{3} c^{3} d^{7}\right ) x^{7}}{b^{4}}+\frac {42 \left (-a^{4} d^{10}-a^{3} b c \,d^{9}-a^{2} b^{2} c^{2} d^{8}-a \,b^{3} c^{3} d^{7}-b^{4} c^{4} d^{6}\right ) x^{6}}{b^{5}}+\frac {42 \left (-a^{5} d^{10}-a^{4} b c \,d^{9}-a^{3} b^{2} c^{2} d^{8}-a^{2} b^{3} c^{3} d^{7}-a \,b^{4} c^{4} d^{6}-b^{5} c^{5} d^{5}\right ) x^{5}}{b^{6}}+\frac {30 \left (-a^{6} d^{10}-a^{5} b c \,d^{9}-a^{4} b^{2} c^{2} d^{8}-a^{3} b^{3} c^{3} d^{7}-a^{2} b^{4} c^{4} d^{6}-a \,b^{5} c^{5} d^{5}-b^{6} c^{6} d^{4}\right ) x^{4}}{b^{7}}+\frac {15 \left (-a^{7} d^{10}-a^{6} b c \,d^{9}-a^{5} b^{2} c^{2} d^{8}-a^{4} b^{3} c^{3} d^{7}-a^{3} b^{4} c^{4} d^{6}-a^{2} b^{5} c^{5} d^{5}-a \,b^{6} c^{6} d^{4}-b^{7} c^{7} d^{3}\right ) x^{3}}{b^{8}}+\frac {5 \left (-a^{8} d^{10}-a^{7} b c \,d^{9}-a^{6} b^{2} c^{2} d^{8}-a^{5} b^{3} c^{3} d^{7}-a^{4} b^{4} c^{4} d^{6}-a^{3} b^{5} c^{5} d^{5}-a^{2} b^{6} c^{6} d^{4}-a \,b^{7} c^{7} d^{3}-b^{8} c^{8} d^{2}\right ) x^{2}}{b^{9}}+\frac {\left (-a^{9} d^{10}-a^{8} b c \,d^{9}-a^{7} b^{2} c^{2} d^{8}-a^{6} b^{3} c^{3} d^{7}-a^{5} b^{4} c^{4} d^{6}-a^{4} b^{5} c^{5} d^{5}-a^{3} b^{6} c^{6} d^{4}-a^{2} b^{7} c^{7} d^{3}-a \,b^{8} c^{8} d^{2}-b^{9} c^{9} d \right ) x}{b^{10}}+\frac {-a^{10} d^{10}-a^{9} b c \,d^{9}-a^{8} b^{2} c^{2} d^{8}-a^{7} b^{3} c^{3} d^{7}-a^{6} b^{4} c^{4} d^{6}-a^{5} b^{5} c^{5} d^{5}-a^{4} b^{6} c^{6} d^{4}-a^{3} b^{7} c^{7} d^{3}-a^{2} b^{8} c^{8} d^{2}-a \,b^{9} c^{9} d -b^{10} c^{10}}{11 b^{11}}}{\left (b x +a \right )^{11}}\)	\(858\)
default	\(-\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}}{11 b^{11} \left (b x +a \right )^{11}}-\frac {15 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{11} \left (b x +a \right )^{3}}-\frac {5 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 )^{9}}+\frac {42 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 )^{6}}+\frac {15 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{b^{11} \left (b x +a \right )^{8}}+\frac {30 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 )^{4}}-\frac {30 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 )^{7}}+\frac {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 )}{b^{11} \left (b x +a \right )^{10}}+\frac {5 d^{9} \left (a d -b c \right )}{b^{11} \left (b x +a \right )^{2}}-\frac {42 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 )^{5}}-\frac {d^{10}}{b^{11} \left (b x +a \right )}\)	\(866\)
gosper	\(-\frac {11 x^{10} d^{10} b^{10}+55 x^{9} a \,b^{9} d^{10}+55 x^{9} b^{10} c \,d^{9}+165 x^{8} a^{2} b^{8} d^{10}+165 x^{8} a \,b^{9} c \,d^{9}+165 x^{8} b^{10} c^{2} d^{8}+330 x^{7} a^{3} b^{7} d^{10}+330 x^{7} a^{2} b^{8} c \,d^{9}+330 x^{7} a \,b^{9} c^{2} d^{8}+330 x^{7} b^{10} c^{3} d^{7}+462 x^{6} a^{4} b^{6} d^{10}+462 x^{6} a^{3} b^{7} c \,d^{9}+462 x^{6} a^{2} b^{8} c^{2} d^{8}+462 x^{6} a \,b^{9} c^{3} d^{7}+462 x^{6} b^{10} c^{4} d^{6}+462 x^{5} a^{5} b^{5} d^{10}+462 x^{5} a^{4} b^{6} c \,d^{9}+462 x^{5} a^{3} b^{7} c^{2} d^{8}+462 x^{5} a^{2} b^{8} c^{3} d^{7}+462 x^{5} a \,b^{9} c^{4} d^{6}+462 x^{5} b^{10} c^{5} d^{5}+330 x^{4} a^{6} b^{4} d^{10}+330 x^{4} a^{5} b^{5} c \,d^{9}+330 x^{4} a^{4} b^{6} c^{2} d^{8}+330 x^{4} a^{3} b^{7} c^{3} d^{7}+330 x^{4} a^{2} b^{8} c^{4} d^{6}+330 x^{4} a \,b^{9} c^{5} d^{5}+330 x^{4} b^{10} c^{6} d^{4}+165 x^{3} a^{7} b^{3} d^{10}+165 x^{3} a^{6} b^{4} c \,d^{9}+165 x^{3} a^{5} b^{5} c^{2} d^{8}+165 x^{3} a^{4} b^{6} c^{3} d^{7}+165 x^{3} a^{3} b^{7} c^{4} d^{6}+165 x^{3} a^{2} b^{8} c^{5} d^{5}+165 x^{3} a \,b^{9} c^{6} d^{4}+165 x^{3} b^{10} c^{7} d^{3}+55 x^{2} a^{8} b^{2} d^{10}+55 x^{2} a^{7} b^{3} c \,d^{9}+55 x^{2} a^{6} b^{4} c^{2} d^{8}+55 x^{2} a^{5} b^{5} c^{3} d^{7}+55 x^{2} a^{4} b^{6} c^{4} d^{6}+55 x^{2} a^{3} b^{7} c^{5} d^{5}+55 x^{2} a^{2} b^{8} c^{6} d^{4}+55 x^{2} a \,b^{9} c^{7} d^{3}+55 x^{2} b^{10} c^{8} d^{2}+11 x \,a^{9} b \,d^{10}+11 x \,a^{8} b^{2} c \,d^{9}+11 x \,a^{7} b^{3} c^{2} d^{8}+11 x \,a^{6} b^{4} c^{3} d^{7}+11 x \,a^{5} b^{5} c^{4} d^{6}+11 x \,a^{4} b^{6} c^{5} d^{5}+11 x \,a^{3} b^{7} c^{6} d^{4}+11 x \,a^{2} b^{8} c^{7} d^{3}+11 x a \,b^{9} c^{8} d^{2}+11 x \,b^{10} c^{9} d +a^{10} d^{10}+a^{9} b c \,d^{9}+a^{8} b^{2} c^{2} d^{8}+a^{7} b^{3} c^{3} d^{7}+a^{6} b^{4} c^{4} d^{6}+a^{5} b^{5} c^{5} d^{5}+a^{4} b^{6} c^{6} d^{4}+a^{3} b^{7} c^{7} d^{3}+a^{2} b^{8} c^{8} d^{2}+a \,b^{9} c^{9} d +b^{10} c^{10}}{11 \left (b x +a \right )^{11} b^{11}}\)	\(952\)
parallelrisch	\(\frac {-11 x^{10} d^{10} b^{10}-55 x^{9} a \,b^{9} d^{10}-55 x^{9} b^{10} c \,d^{9}-165 x^{8} a^{2} b^{8} d^{10}-165 x^{8} a \,b^{9} c \,d^{9}-165 x^{8} b^{10} c^{2} d^{8}-330 x^{7} a^{3} b^{7} d^{10}-330 x^{7} a^{2} b^{8} c \,d^{9}-330 x^{7} a \,b^{9} c^{2} d^{8}-330 x^{7} b^{10} c^{3} d^{7}-462 x^{6} a^{4} b^{6} d^{10}-462 x^{6} a^{3} b^{7} c \,d^{9}-462 x^{6} a^{2} b^{8} c^{2} d^{8}-462 x^{6} a \,b^{9} c^{3} d^{7}-462 x^{6} b^{10} c^{4} d^{6}-462 x^{5} a^{5} b^{5} d^{10}-462 x^{5} a^{4} b^{6} c \,d^{9}-462 x^{5} a^{3} b^{7} c^{2} d^{8}-462 x^{5} a^{2} b^{8} c^{3} d^{7}-462 x^{5} a \,b^{9} c^{4} d^{6}-462 x^{5} b^{10} c^{5} d^{5}-330 x^{4} a^{6} b^{4} d^{10}-330 x^{4} a^{5} b^{5} c \,d^{9}-330 x^{4} a^{4} b^{6} c^{2} d^{8}-330 x^{4} a^{3} b^{7} c^{3} d^{7}-330 x^{4} a^{2} b^{8} c^{4} d^{6}-330 x^{4} a \,b^{9} c^{5} d^{5}-330 x^{4} b^{10} c^{6} d^{4}-165 x^{3} a^{7} b^{3} d^{10}-165 x^{3} a^{6} b^{4} c \,d^{9}-165 x^{3} a^{5} b^{5} c^{2} d^{8}-165 x^{3} a^{4} b^{6} c^{3} d^{7}-165 x^{3} a^{3} b^{7} c^{4} d^{6}-165 x^{3} a^{2} b^{8} c^{5} d^{5}-165 x^{3} a \,b^{9} c^{6} d^{4}-165 x^{3} b^{10} c^{7} d^{3}-55 x^{2} a^{8} b^{2} d^{10}-55 x^{2} a^{7} b^{3} c \,d^{9}-55 x^{2} a^{6} b^{4} c^{2} d^{8}-55 x^{2} a^{5} b^{5} c^{3} d^{7}-55 x^{2} a^{4} b^{6} c^{4} d^{6}-55 x^{2} a^{3} b^{7} c^{5} d^{5}-55 x^{2} a^{2} b^{8} c^{6} d^{4}-55 x^{2} a \,b^{9} c^{7} d^{3}-55 x^{2} b^{10} c^{8} d^{2}-11 x \,a^{9} b \,d^{10}-11 x \,a^{8} b^{2} c \,d^{9}-11 x \,a^{7} b^{3} c^{2} d^{8}-11 x \,a^{6} b^{4} c^{3} d^{7}-11 x \,a^{5} b^{5} c^{4} d^{6}-11 x \,a^{4} b^{6} c^{5} d^{5}-11 x \,a^{3} b^{7} c^{6} d^{4}-11 x \,a^{2} b^{8} c^{7} d^{3}-11 x a \,b^{9} c^{8} d^{2}-11 x \,b^{10} c^{9} d -a^{10} d^{10}-a^{9} b c \,d^{9}-a^{8} b^{2} c^{2} d^{8}-a^{7} b^{3} c^{3} d^{7}-a^{6} b^{4} c^{4} d^{6}-a^{5} b^{5} c^{5} d^{5}-a^{4} b^{6} c^{6} d^{4}-a^{3} b^{7} c^{7} d^{3}-a^{2} b^{8} c^{8} d^{2}-a \,b^{9} c^{9} d -b^{10} c^{10}}{11 b^{11} \left (b x +a \right )^{11}}\)	\(963\)

[In]

int((d*x+c)^10/(b*x+a)^12,x,method=_RETURNVERBOSE)

[Out]

(-1/b*d^10*x^10-5*d^9*(a*d+b*c)/b^2*x^9-15*d^8*(a^2*d^2+a*b*c*d+b^2*c^2)/b^3*x^8-30*d^7*(a^3*d^3+a^2*b*c*d^2+a
*b^2*c^2*d+b^3*c^3)/b^4*x^7-42*d^6*(a^4*d^4+a^3*b*c*d^3+a^2*b^2*c^2*d^2+a*b^3*c^3*d+b^4*c^4)/b^5*x^6-42*d^5*(a
^5*d^5+a^4*b*c*d^4+a^3*b^2*c^2*d^3+a^2*b^3*c^3*d^2+a*b^4*c^4*d+b^5*c^5)/b^6*x^5-30*d^4*(a^6*d^6+a^5*b*c*d^5+a^
4*b^2*c^2*d^4+a^3*b^3*c^3*d^3+a^2*b^4*c^4*d^2+a*b^5*c^5*d+b^6*c^6)/b^7*x^4-15*d^3*(a^7*d^7+a^6*b*c*d^6+a^5*b^2
*c^2*d^5+a^4*b^3*c^3*d^4+a^3*b^4*c^4*d^3+a^2*b^5*c^5*d^2+a*b^6*c^6*d+b^7*c^7)/b^8*x^3-5*d^2*(a^8*d^8+a^7*b*c*d
^7+a^6*b^2*c^2*d^6+a^5*b^3*c^3*d^5+a^4*b^4*c^4*d^4+a^3*b^5*c^5*d^3+a^2*b^6*c^6*d^2+a*b^7*c^7*d+b^8*c^8)/b^9*x^
2-d*(a^9*d^9+a^8*b*c*d^8+a^7*b^2*c^2*d^7+a^6*b^3*c^3*d^6+a^5*b^4*c^4*d^5+a^4*b^5*c^5*d^4+a^3*b^6*c^6*d^3+a^2*b
^7*c^7*d^2+a*b^8*c^8*d+b^9*c^9)/b^10*x-1/11*(a^10*d^10+a^9*b*c*d^9+a^8*b^2*c^2*d^8+a^7*b^3*c^3*d^7+a^6*b^4*c^4
*d^6+a^5*b^5*c^5*d^5+a^4*b^6*c^6*d^4+a^3*b^7*c^7*d^3+a^2*b^8*c^8*d^2+a*b^9*c^9*d+b^10*c^10)/b^11)/(b*x+a)^11

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 920 vs. \(2 (26) = 52\).

Time = 0.23 (sec) , antiderivative size = 920, normalized size of antiderivative = 32.86 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx=-\frac {11 \, b^{10} d^{10} x^{10} + b^{10} c^{10} + a b^{9} c^{9} d + a^{2} b^{8} c^{8} d^{2} + a^{3} b^{7} c^{7} d^{3} + a^{4} b^{6} c^{6} d^{4} + a^{5} b^{5} c^{5} d^{5} + a^{6} b^{4} c^{4} d^{6} + a^{7} b^{3} c^{3} d^{7} + a^{8} b^{2} c^{2} d^{8} + a^{9} b c d^{9} + a^{10} d^{10} + 55 \, {\left (b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 165 \, {\left (b^{10} c^{2} d^{8} + a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 330 \, {\left (b^{10} c^{3} d^{7} + a b^{9} c^{2} d^{8} + a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 462 \, {\left (b^{10} c^{4} d^{6} + a b^{9} c^{3} d^{7} + a^{2} b^{8} c^{2} d^{8} + a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 462 \, {\left (b^{10} c^{5} d^{5} + a b^{9} c^{4} d^{6} + a^{2} b^{8} c^{3} d^{7} + a^{3} b^{7} c^{2} d^{8} + a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 330 \, {\left (b^{10} c^{6} d^{4} + a b^{9} c^{5} d^{5} + a^{2} b^{8} c^{4} d^{6} + a^{3} b^{7} c^{3} d^{7} + a^{4} b^{6} c^{2} d^{8} + a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 165 \, {\left (b^{10} c^{7} d^{3} + a b^{9} c^{6} d^{4} + a^{2} b^{8} c^{5} d^{5} + a^{3} b^{7} c^{4} d^{6} + a^{4} b^{6} c^{3} d^{7} + a^{5} b^{5} c^{2} d^{8} + a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 55 \, {\left (b^{10} c^{8} d^{2} + a b^{9} c^{7} d^{3} + a^{2} b^{8} c^{6} d^{4} + a^{3} b^{7} c^{5} d^{5} + a^{4} b^{6} c^{4} d^{6} + a^{5} b^{5} c^{3} d^{7} + a^{6} b^{4} c^{2} d^{8} + a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 11 \, {\left (b^{10} c^{9} d + a b^{9} c^{8} d^{2} + a^{2} b^{8} c^{7} d^{3} + a^{3} b^{7} c^{6} d^{4} + a^{4} b^{6} c^{5} d^{5} + a^{5} b^{5} c^{4} d^{6} + a^{6} b^{4} c^{3} d^{7} + a^{7} b^{3} c^{2} d^{8} + a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{11 \, {\left (b^{22} x^{11} + 11 \, a b^{21} x^{10} + 55 \, a^{2} b^{20} x^{9} + 165 \, a^{3} b^{19} x^{8} + 330 \, a^{4} b^{18} x^{7} + 462 \, a^{5} b^{17} x^{6} + 462 \, a^{6} b^{16} x^{5} + 330 \, a^{7} b^{15} x^{4} + 165 \, a^{8} b^{14} x^{3} + 55 \, a^{9} b^{13} x^{2} + 11 \, a^{10} b^{12} x + a^{11} b^{11}\right )}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^12,x, algorithm="fricas")

[Out]

-1/11*(11*b^10*d^10*x^10 + b^10*c^10 + a*b^9*c^9*d + a^2*b^8*c^8*d^2 + a^3*b^7*c^7*d^3 + a^4*b^6*c^6*d^4 + a^5
*b^5*c^5*d^5 + a^6*b^4*c^4*d^6 + a^7*b^3*c^3*d^7 + a^8*b^2*c^2*d^8 + a^9*b*c*d^9 + a^10*d^10 + 55*(b^10*c*d^9
+ a*b^9*d^10)*x^9 + 165*(b^10*c^2*d^8 + a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 330*(b^10*c^3*d^7 + a*b^9*c^2*d^8 +
a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 462*(b^10*c^4*d^6 + a*b^9*c^3*d^7 + a^2*b^8*c^2*d^8 + a^3*b^7*c*d^9 + a^4*
b^6*d^10)*x^6 + 462*(b^10*c^5*d^5 + a*b^9*c^4*d^6 + a^2*b^8*c^3*d^7 + a^3*b^7*c^2*d^8 + a^4*b^6*c*d^9 + a^5*b^
5*d^10)*x^5 + 330*(b^10*c^6*d^4 + a*b^9*c^5*d^5 + a^2*b^8*c^4*d^6 + a^3*b^7*c^3*d^7 + a^4*b^6*c^2*d^8 + a^5*b^
5*c*d^9 + a^6*b^4*d^10)*x^4 + 165*(b^10*c^7*d^3 + a*b^9*c^6*d^4 + a^2*b^8*c^5*d^5 + a^3*b^7*c^4*d^6 + a^4*b^6*
c^3*d^7 + a^5*b^5*c^2*d^8 + a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 55*(b^10*c^8*d^2 + a*b^9*c^7*d^3 + a^2*b^8*c^6
*d^4 + a^3*b^7*c^5*d^5 + a^4*b^6*c^4*d^6 + a^5*b^5*c^3*d^7 + a^6*b^4*c^2*d^8 + a^7*b^3*c*d^9 + a^8*b^2*d^10)*x
^2 + 11*(b^10*c^9*d + a*b^9*c^8*d^2 + a^2*b^8*c^7*d^3 + a^3*b^7*c^6*d^4 + a^4*b^6*c^5*d^5 + a^5*b^5*c^4*d^6 +
a^6*b^4*c^3*d^7 + a^7*b^3*c^2*d^8 + a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^22*x^11 + 11*a*b^21*x^10 + 55*a^2*b^20*x
^9 + 165*a^3*b^19*x^8 + 330*a^4*b^18*x^7 + 462*a^5*b^17*x^6 + 462*a^6*b^16*x^5 + 330*a^7*b^15*x^4 + 165*a^8*b^
14*x^3 + 55*a^9*b^13*x^2 + 11*a^10*b^12*x + a^11*b^11)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx=\text {Timed out} \]

[In]

integrate((d*x+c)**10/(b*x+a)**12,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 920 vs. \(2 (26) = 52\).

Time = 0.24 (sec) , antiderivative size = 920, normalized size of antiderivative = 32.86 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx=-\frac {11 \, b^{10} d^{10} x^{10} + b^{10} c^{10} + a b^{9} c^{9} d + a^{2} b^{8} c^{8} d^{2} + a^{3} b^{7} c^{7} d^{3} + a^{4} b^{6} c^{6} d^{4} + a^{5} b^{5} c^{5} d^{5} + a^{6} b^{4} c^{4} d^{6} + a^{7} b^{3} c^{3} d^{7} + a^{8} b^{2} c^{2} d^{8} + a^{9} b c d^{9} + a^{10} d^{10} + 55 \, {\left (b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 165 \, {\left (b^{10} c^{2} d^{8} + a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 330 \, {\left (b^{10} c^{3} d^{7} + a b^{9} c^{2} d^{8} + a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 462 \, {\left (b^{10} c^{4} d^{6} + a b^{9} c^{3} d^{7} + a^{2} b^{8} c^{2} d^{8} + a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 462 \, {\left (b^{10} c^{5} d^{5} + a b^{9} c^{4} d^{6} + a^{2} b^{8} c^{3} d^{7} + a^{3} b^{7} c^{2} d^{8} + a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 330 \, {\left (b^{10} c^{6} d^{4} + a b^{9} c^{5} d^{5} + a^{2} b^{8} c^{4} d^{6} + a^{3} b^{7} c^{3} d^{7} + a^{4} b^{6} c^{2} d^{8} + a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 165 \, {\left (b^{10} c^{7} d^{3} + a b^{9} c^{6} d^{4} + a^{2} b^{8} c^{5} d^{5} + a^{3} b^{7} c^{4} d^{6} + a^{4} b^{6} c^{3} d^{7} + a^{5} b^{5} c^{2} d^{8} + a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 55 \, {\left (b^{10} c^{8} d^{2} + a b^{9} c^{7} d^{3} + a^{2} b^{8} c^{6} d^{4} + a^{3} b^{7} c^{5} d^{5} + a^{4} b^{6} c^{4} d^{6} + a^{5} b^{5} c^{3} d^{7} + a^{6} b^{4} c^{2} d^{8} + a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 11 \, {\left (b^{10} c^{9} d + a b^{9} c^{8} d^{2} + a^{2} b^{8} c^{7} d^{3} + a^{3} b^{7} c^{6} d^{4} + a^{4} b^{6} c^{5} d^{5} + a^{5} b^{5} c^{4} d^{6} + a^{6} b^{4} c^{3} d^{7} + a^{7} b^{3} c^{2} d^{8} + a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{11 \, {\left (b^{22} x^{11} + 11 \, a b^{21} x^{10} + 55 \, a^{2} b^{20} x^{9} + 165 \, a^{3} b^{19} x^{8} + 330 \, a^{4} b^{18} x^{7} + 462 \, a^{5} b^{17} x^{6} + 462 \, a^{6} b^{16} x^{5} + 330 \, a^{7} b^{15} x^{4} + 165 \, a^{8} b^{14} x^{3} + 55 \, a^{9} b^{13} x^{2} + 11 \, a^{10} b^{12} x + a^{11} b^{11}\right )}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^12,x, algorithm="maxima")

[Out]

-1/11*(11*b^10*d^10*x^10 + b^10*c^10 + a*b^9*c^9*d + a^2*b^8*c^8*d^2 + a^3*b^7*c^7*d^3 + a^4*b^6*c^6*d^4 + a^5
*b^5*c^5*d^5 + a^6*b^4*c^4*d^6 + a^7*b^3*c^3*d^7 + a^8*b^2*c^2*d^8 + a^9*b*c*d^9 + a^10*d^10 + 55*(b^10*c*d^9
+ a*b^9*d^10)*x^9 + 165*(b^10*c^2*d^8 + a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 330*(b^10*c^3*d^7 + a*b^9*c^2*d^8 +
a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 462*(b^10*c^4*d^6 + a*b^9*c^3*d^7 + a^2*b^8*c^2*d^8 + a^3*b^7*c*d^9 + a^4*
b^6*d^10)*x^6 + 462*(b^10*c^5*d^5 + a*b^9*c^4*d^6 + a^2*b^8*c^3*d^7 + a^3*b^7*c^2*d^8 + a^4*b^6*c*d^9 + a^5*b^
5*d^10)*x^5 + 330*(b^10*c^6*d^4 + a*b^9*c^5*d^5 + a^2*b^8*c^4*d^6 + a^3*b^7*c^3*d^7 + a^4*b^6*c^2*d^8 + a^5*b^
5*c*d^9 + a^6*b^4*d^10)*x^4 + 165*(b^10*c^7*d^3 + a*b^9*c^6*d^4 + a^2*b^8*c^5*d^5 + a^3*b^7*c^4*d^6 + a^4*b^6*
c^3*d^7 + a^5*b^5*c^2*d^8 + a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 55*(b^10*c^8*d^2 + a*b^9*c^7*d^3 + a^2*b^8*c^6
*d^4 + a^3*b^7*c^5*d^5 + a^4*b^6*c^4*d^6 + a^5*b^5*c^3*d^7 + a^6*b^4*c^2*d^8 + a^7*b^3*c*d^9 + a^8*b^2*d^10)*x
^2 + 11*(b^10*c^9*d + a*b^9*c^8*d^2 + a^2*b^8*c^7*d^3 + a^3*b^7*c^6*d^4 + a^4*b^6*c^5*d^5 + a^5*b^5*c^4*d^6 +
a^6*b^4*c^3*d^7 + a^7*b^3*c^2*d^8 + a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^22*x^11 + 11*a*b^21*x^10 + 55*a^2*b^20*x
^9 + 165*a^3*b^19*x^8 + 330*a^4*b^18*x^7 + 462*a^5*b^17*x^6 + 462*a^6*b^16*x^5 + 330*a^7*b^15*x^4 + 165*a^8*b^
14*x^3 + 55*a^9*b^13*x^2 + 11*a^10*b^12*x + a^11*b^11)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 951 vs. \(2 (26) = 52\).

Time = 0.32 (sec) , antiderivative size = 951, normalized size of antiderivative = 33.96 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx=-\frac {11 \, b^{10} d^{10} x^{10} + 55 \, b^{10} c d^{9} x^{9} + 55 \, a b^{9} d^{10} x^{9} + 165 \, b^{10} c^{2} d^{8} x^{8} + 165 \, a b^{9} c d^{9} x^{8} + 165 \, a^{2} b^{8} d^{10} x^{8} + 330 \, b^{10} c^{3} d^{7} x^{7} + 330 \, a b^{9} c^{2} d^{8} x^{7} + 330 \, a^{2} b^{8} c d^{9} x^{7} + 330 \, a^{3} b^{7} d^{10} x^{7} + 462 \, b^{10} c^{4} d^{6} x^{6} + 462 \, a b^{9} c^{3} d^{7} x^{6} + 462 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 462 \, a^{3} b^{7} c d^{9} x^{6} + 462 \, a^{4} b^{6} d^{10} x^{6} + 462 \, b^{10} c^{5} d^{5} x^{5} + 462 \, a b^{9} c^{4} d^{6} x^{5} + 462 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 462 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 462 \, a^{4} b^{6} c d^{9} x^{5} + 462 \, a^{5} b^{5} d^{10} x^{5} + 330 \, b^{10} c^{6} d^{4} x^{4} + 330 \, a b^{9} c^{5} d^{5} x^{4} + 330 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 330 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 330 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 330 \, a^{5} b^{5} c d^{9} x^{4} + 330 \, a^{6} b^{4} d^{10} x^{4} + 165 \, b^{10} c^{7} d^{3} x^{3} + 165 \, a b^{9} c^{6} d^{4} x^{3} + 165 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 165 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 165 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 165 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 165 \, a^{6} b^{4} c d^{9} x^{3} + 165 \, a^{7} b^{3} d^{10} x^{3} + 55 \, b^{10} c^{8} d^{2} x^{2} + 55 \, a b^{9} c^{7} d^{3} x^{2} + 55 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 55 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 55 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 55 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 55 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 55 \, a^{7} b^{3} c d^{9} x^{2} + 55 \, a^{8} b^{2} d^{10} x^{2} + 11 \, b^{10} c^{9} d x + 11 \, a b^{9} c^{8} d^{2} x + 11 \, a^{2} b^{8} c^{7} d^{3} x + 11 \, a^{3} b^{7} c^{6} d^{4} x + 11 \, a^{4} b^{6} c^{5} d^{5} x + 11 \, a^{5} b^{5} c^{4} d^{6} x + 11 \, a^{6} b^{4} c^{3} d^{7} x + 11 \, a^{7} b^{3} c^{2} d^{8} x + 11 \, a^{8} b^{2} c d^{9} x + 11 \, a^{9} b d^{10} x + b^{10} c^{10} + a b^{9} c^{9} d + a^{2} b^{8} c^{8} d^{2} + a^{3} b^{7} c^{7} d^{3} + a^{4} b^{6} c^{6} d^{4} + a^{5} b^{5} c^{5} d^{5} + a^{6} b^{4} c^{4} d^{6} + a^{7} b^{3} c^{3} d^{7} + a^{8} b^{2} c^{2} d^{8} + a^{9} b c d^{9} + a^{10} d^{10}}{11 \, {\left (b x + a\right )}^{11} b^{11}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^12,x, algorithm="giac")

[Out]

-1/11*(11*b^10*d^10*x^10 + 55*b^10*c*d^9*x^9 + 55*a*b^9*d^10*x^9 + 165*b^10*c^2*d^8*x^8 + 165*a*b^9*c*d^9*x^8
+ 165*a^2*b^8*d^10*x^8 + 330*b^10*c^3*d^7*x^7 + 330*a*b^9*c^2*d^8*x^7 + 330*a^2*b^8*c*d^9*x^7 + 330*a^3*b^7*d^
10*x^7 + 462*b^10*c^4*d^6*x^6 + 462*a*b^9*c^3*d^7*x^6 + 462*a^2*b^8*c^2*d^8*x^6 + 462*a^3*b^7*c*d^9*x^6 + 462*
a^4*b^6*d^10*x^6 + 462*b^10*c^5*d^5*x^5 + 462*a*b^9*c^4*d^6*x^5 + 462*a^2*b^8*c^3*d^7*x^5 + 462*a^3*b^7*c^2*d^
8*x^5 + 462*a^4*b^6*c*d^9*x^5 + 462*a^5*b^5*d^10*x^5 + 330*b^10*c^6*d^4*x^4 + 330*a*b^9*c^5*d^5*x^4 + 330*a^2*
b^8*c^4*d^6*x^4 + 330*a^3*b^7*c^3*d^7*x^4 + 330*a^4*b^6*c^2*d^8*x^4 + 330*a^5*b^5*c*d^9*x^4 + 330*a^6*b^4*d^10
*x^4 + 165*b^10*c^7*d^3*x^3 + 165*a*b^9*c^6*d^4*x^3 + 165*a^2*b^8*c^5*d^5*x^3 + 165*a^3*b^7*c^4*d^6*x^3 + 165*
a^4*b^6*c^3*d^7*x^3 + 165*a^5*b^5*c^2*d^8*x^3 + 165*a^6*b^4*c*d^9*x^3 + 165*a^7*b^3*d^10*x^3 + 55*b^10*c^8*d^2
*x^2 + 55*a*b^9*c^7*d^3*x^2 + 55*a^2*b^8*c^6*d^4*x^2 + 55*a^3*b^7*c^5*d^5*x^2 + 55*a^4*b^6*c^4*d^6*x^2 + 55*a^
5*b^5*c^3*d^7*x^2 + 55*a^6*b^4*c^2*d^8*x^2 + 55*a^7*b^3*c*d^9*x^2 + 55*a^8*b^2*d^10*x^2 + 11*b^10*c^9*d*x + 11
*a*b^9*c^8*d^2*x + 11*a^2*b^8*c^7*d^3*x + 11*a^3*b^7*c^6*d^4*x + 11*a^4*b^6*c^5*d^5*x + 11*a^5*b^5*c^4*d^6*x +
11*a^6*b^4*c^3*d^7*x + 11*a^7*b^3*c^2*d^8*x + 11*a^8*b^2*c*d^9*x + 11*a^9*b*d^10*x + b^10*c^10 + a*b^9*c^9*d
+ a^2*b^8*c^8*d^2 + a^3*b^7*c^7*d^3 + a^4*b^6*c^6*d^4 + a^5*b^5*c^5*d^5 + a^6*b^4*c^4*d^6 + a^7*b^3*c^3*d^7 +
a^8*b^2*c^2*d^8 + a^9*b*c*d^9 + a^10*d^10)/((b*x + a)^11*b^11)

Mupad [B] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 1066, normalized size of antiderivative = 38.07 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx=-\frac {a^{10}\,d^{10}+a^9\,b\,c\,d^9+11\,a^9\,b\,d^{10}\,x+a^8\,b^2\,c^2\,d^8+11\,a^8\,b^2\,c\,d^9\,x+55\,a^8\,b^2\,d^{10}\,x^2+a^7\,b^3\,c^3\,d^7+11\,a^7\,b^3\,c^2\,d^8\,x+55\,a^7\,b^3\,c\,d^9\,x^2+165\,a^7\,b^3\,d^{10}\,x^3+a^6\,b^4\,c^4\,d^6+11\,a^6\,b^4\,c^3\,d^7\,x+55\,a^6\,b^4\,c^2\,d^8\,x^2+165\,a^6\,b^4\,c\,d^9\,x^3+330\,a^6\,b^4\,d^{10}\,x^4+a^5\,b^5\,c^5\,d^5+11\,a^5\,b^5\,c^4\,d^6\,x+55\,a^5\,b^5\,c^3\,d^7\,x^2+165\,a^5\,b^5\,c^2\,d^8\,x^3+330\,a^5\,b^5\,c\,d^9\,x^4+462\,a^5\,b^5\,d^{10}\,x^5+a^4\,b^6\,c^6\,d^4+11\,a^4\,b^6\,c^5\,d^5\,x+55\,a^4\,b^6\,c^4\,d^6\,x^2+165\,a^4\,b^6\,c^3\,d^7\,x^3+330\,a^4\,b^6\,c^2\,d^8\,x^4+462\,a^4\,b^6\,c\,d^9\,x^5+462\,a^4\,b^6\,d^{10}\,x^6+a^3\,b^7\,c^7\,d^3+11\,a^3\,b^7\,c^6\,d^4\,x+55\,a^3\,b^7\,c^5\,d^5\,x^2+165\,a^3\,b^7\,c^4\,d^6\,x^3+330\,a^3\,b^7\,c^3\,d^7\,x^4+462\,a^3\,b^7\,c^2\,d^8\,x^5+462\,a^3\,b^7\,c\,d^9\,x^6+330\,a^3\,b^7\,d^{10}\,x^7+a^2\,b^8\,c^8\,d^2+11\,a^2\,b^8\,c^7\,d^3\,x+55\,a^2\,b^8\,c^6\,d^4\,x^2+165\,a^2\,b^8\,c^5\,d^5\,x^3+330\,a^2\,b^8\,c^4\,d^6\,x^4+462\,a^2\,b^8\,c^3\,d^7\,x^5+462\,a^2\,b^8\,c^2\,d^8\,x^6+330\,a^2\,b^8\,c\,d^9\,x^7+165\,a^2\,b^8\,d^{10}\,x^8+a\,b^9\,c^9\,d+11\,a\,b^9\,c^8\,d^2\,x+55\,a\,b^9\,c^7\,d^3\,x^2+165\,a\,b^9\,c^6\,d^4\,x^3+330\,a\,b^9\,c^5\,d^5\,x^4+462\,a\,b^9\,c^4\,d^6\,x^5+462\,a\,b^9\,c^3\,d^7\,x^6+330\,a\,b^9\,c^2\,d^8\,x^7+165\,a\,b^9\,c\,d^9\,x^8+55\,a\,b^9\,d^{10}\,x^9+b^{10}\,c^{10}+11\,b^{10}\,c^9\,d\,x+55\,b^{10}\,c^8\,d^2\,x^2+165\,b^{10}\,c^7\,d^3\,x^3+330\,b^{10}\,c^6\,d^4\,x^4+462\,b^{10}\,c^5\,d^5\,x^5+462\,b^{10}\,c^4\,d^6\,x^6+330\,b^{10}\,c^3\,d^7\,x^7+165\,b^{10}\,c^2\,d^8\,x^8+55\,b^{10}\,c\,d^9\,x^9+11\,b^{10}\,d^{10}\,x^{10}}{11\,a^{11}\,b^{11}+121\,a^{10}\,b^{12}\,x+605\,a^9\,b^{13}\,x^2+1815\,a^8\,b^{14}\,x^3+3630\,a^7\,b^{15}\,x^4+5082\,a^6\,b^{16}\,x^5+5082\,a^5\,b^{17}\,x^6+3630\,a^4\,b^{18}\,x^7+1815\,a^3\,b^{19}\,x^8+605\,a^2\,b^{20}\,x^9+121\,a\,b^{21}\,x^{10}+11\,b^{22}\,x^{11}} \]

[In]

int((c + d*x)^10/(a + b*x)^12,x)

[Out]

-(a^10*d^10 + b^10*c^10 + 11*b^10*d^10*x^10 + 55*a*b^9*d^10*x^9 + 55*b^10*c*d^9*x^9 + a^2*b^8*c^8*d^2 + a^3*b^
7*c^7*d^3 + a^4*b^6*c^6*d^4 + a^5*b^5*c^5*d^5 + a^6*b^4*c^4*d^6 + a^7*b^3*c^3*d^7 + a^8*b^2*c^2*d^8 + 55*a^8*b
^2*d^10*x^2 + 165*a^7*b^3*d^10*x^3 + 330*a^6*b^4*d^10*x^4 + 462*a^5*b^5*d^10*x^5 + 462*a^4*b^6*d^10*x^6 + 330*
a^3*b^7*d^10*x^7 + 165*a^2*b^8*d^10*x^8 + 55*b^10*c^8*d^2*x^2 + 165*b^10*c^7*d^3*x^3 + 330*b^10*c^6*d^4*x^4 +
462*b^10*c^5*d^5*x^5 + 462*b^10*c^4*d^6*x^6 + 330*b^10*c^3*d^7*x^7 + 165*b^10*c^2*d^8*x^8 + a*b^9*c^9*d + a^9*
b*c*d^9 + 11*a^9*b*d^10*x + 11*b^10*c^9*d*x + 55*a^2*b^8*c^6*d^4*x^2 + 55*a^3*b^7*c^5*d^5*x^2 + 55*a^4*b^6*c^4
*d^6*x^2 + 55*a^5*b^5*c^3*d^7*x^2 + 55*a^6*b^4*c^2*d^8*x^2 + 165*a^2*b^8*c^5*d^5*x^3 + 165*a^3*b^7*c^4*d^6*x^3
+ 165*a^4*b^6*c^3*d^7*x^3 + 165*a^5*b^5*c^2*d^8*x^3 + 330*a^2*b^8*c^4*d^6*x^4 + 330*a^3*b^7*c^3*d^7*x^4 + 330
*a^4*b^6*c^2*d^8*x^4 + 462*a^2*b^8*c^3*d^7*x^5 + 462*a^3*b^7*c^2*d^8*x^5 + 462*a^2*b^8*c^2*d^8*x^6 + 11*a*b^9*
c^8*d^2*x + 11*a^8*b^2*c*d^9*x + 165*a*b^9*c*d^9*x^8 + 11*a^2*b^8*c^7*d^3*x + 11*a^3*b^7*c^6*d^4*x + 11*a^4*b^
6*c^5*d^5*x + 11*a^5*b^5*c^4*d^6*x + 11*a^6*b^4*c^3*d^7*x + 11*a^7*b^3*c^2*d^8*x + 55*a*b^9*c^7*d^3*x^2 + 55*a
^7*b^3*c*d^9*x^2 + 165*a*b^9*c^6*d^4*x^3 + 165*a^6*b^4*c*d^9*x^3 + 330*a*b^9*c^5*d^5*x^4 + 330*a^5*b^5*c*d^9*x
^4 + 462*a*b^9*c^4*d^6*x^5 + 462*a^4*b^6*c*d^9*x^5 + 462*a*b^9*c^3*d^7*x^6 + 462*a^3*b^7*c*d^9*x^6 + 330*a*b^9
*c^2*d^8*x^7 + 330*a^2*b^8*c*d^9*x^7)/(11*a^11*b^11 + 11*b^22*x^11 + 121*a^10*b^12*x + 121*a*b^21*x^10 + 605*a
^9*b^13*x^2 + 1815*a^8*b^14*x^3 + 3630*a^7*b^15*x^4 + 5082*a^6*b^16*x^5 + 5082*a^5*b^17*x^6 + 3630*a^4*b^18*x^
7 + 1815*a^3*b^19*x^8 + 605*a^2*b^20*x^9)