\(\int \frac {(c+d x)^{10}}{(a+b x)^3} \, dx\) [1314]

Optimal result

Integrand size = 15, antiderivative size = 262 \[ \int \frac {(c+d x)^{10}}{(a+b x)^3} \, dx=\frac {120 d^3 (b c-a d)^7 x}{b^{10}}-\frac {(b c-a d)^{10}}{2 b^{11} (a+b x)^2}-\frac {10 d (b c-a d)^9}{b^{11} (a+b x)}+\frac {105 d^4 (b c-a d)^6 (a+b x)^2}{b^{11}}+\frac {84 d^5 (b c-a d)^5 (a+b x)^3}{b^{11}}+\frac {105 d^6 (b c-a d)^4 (a+b x)^4}{2 b^{11}}+\frac {24 d^7 (b c-a d)^3 (a+b x)^5}{b^{11}}+\frac {15 d^8 (b c-a d)^2 (a+b x)^6}{2 b^{11}}+\frac {10 d^9 (b c-a d) (a+b x)^7}{7 b^{11}}+\frac {d^{10} (a+b x)^8}{8 b^{11}}+\frac {45 d^2 (b c-a d)^8 \log (a+b x)}{b^{11}} \]

[Out]

120*d^3*(-a*d+b*c)^7*x/b^10-1/2*(-a*d+b*c)^10/b^11/(b*x+a)^2-10*d*(-a*d+b*c)^9/b^11/(b*x+a)+105*d^4*(-a*d+b*c)
^6*(b*x+a)^2/b^11+84*d^5*(-a*d+b*c)^5*(b*x+a)^3/b^11+105/2*d^6*(-a*d+b*c)^4*(b*x+a)^4/b^11+24*d^7*(-a*d+b*c)^3
*(b*x+a)^5/b^11+15/2*d^8*(-a*d+b*c)^2*(b*x+a)^6/b^11+10/7*d^9*(-a*d+b*c)*(b*x+a)^7/b^11+1/8*d^10*(b*x+a)^8/b^1
1+45*d^2*(-a*d+b*c)^8*ln(b*x+a)/b^11

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {(c+d x)^{10}}{(a+b x)^3} \, dx=\frac {10 d^9 (a+b x)^7 (b c-a d)}{7 b^{11}}+\frac {15 d^8 (a+b x)^6 (b c-a d)^2}{2 b^{11}}+\frac {24 d^7 (a+b x)^5 (b c-a d)^3}{b^{11}}+\frac {105 d^6 (a+b x)^4 (b c-a d)^4}{2 b^{11}}+\frac {84 d^5 (a+b x)^3 (b c-a d)^5}{b^{11}}+\frac {105 d^4 (a+b x)^2 (b c-a d)^6}{b^{11}}+\frac {45 d^2 (b c-a d)^8 \log (a+b x)}{b^{11}}-\frac {10 d (b c-a d)^9}{b^{11} (a+b x)}-\frac {(b c-a d)^{10}}{2 b^{11} (a+b x)^2}+\frac {d^{10} (a+b x)^8}{8 b^{11}}+\frac {120 d^3 x (b c-a d)^7}{b^{10}} \]

[In]

Int[(c + d*x)^10/(a + b*x)^3,x]

[Out]

(120*d^3*(b*c - a*d)^7*x)/b^10 - (b*c - a*d)^10/(2*b^11*(a + b*x)^2) - (10*d*(b*c - a*d)^9)/(b^11*(a + b*x)) +
(105*d^4*(b*c - a*d)^6*(a + b*x)^2)/b^11 + (84*d^5*(b*c - a*d)^5*(a + b*x)^3)/b^11 + (105*d^6*(b*c - a*d)^4*(
a + b*x)^4)/(2*b^11) + (24*d^7*(b*c - a*d)^3*(a + b*x)^5)/b^11 + (15*d^8*(b*c - a*d)^2*(a + b*x)^6)/(2*b^11) +
(10*d^9*(b*c - a*d)*(a + b*x)^7)/(7*b^11) + (d^10*(a + b*x)^8)/(8*b^11) + (45*d^2*(b*c - a*d)^8*Log[a + b*x])
/b^11

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {120 d^3 (b c-a d)^7}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^3}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^2}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)}+\frac {210 d^4 (b c-a d)^6 (a+b x)}{b^{10}}+\frac {252 d^5 (b c-a d)^5 (a+b x)^2}{b^{10}}+\frac {210 d^6 (b c-a d)^4 (a+b x)^3}{b^{10}}+\frac {120 d^7 (b c-a d)^3 (a+b x)^4}{b^{10}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^5}{b^{10}}+\frac {10 d^9 (b c-a d) (a+b x)^6}{b^{10}}+\frac {d^{10} (a+b x)^7}{b^{10}}\right ) \, dx \\ & = \frac {120 d^3 (b c-a d)^7 x}{b^{10}}-\frac {(b c-a d)^{10}}{2 b^{11} (a+b x)^2}-\frac {10 d (b c-a d)^9}{b^{11} (a+b x)}+\frac {105 d^4 (b c-a d)^6 (a+b x)^2}{b^{11}}+\frac {84 d^5 (b c-a d)^5 (a+b x)^3}{b^{11}}+\frac {105 d^6 (b c-a d)^4 (a+b x)^4}{2 b^{11}}+\frac {24 d^7 (b c-a d)^3 (a+b x)^5}{b^{11}}+\frac {15 d^8 (b c-a d)^2 (a+b x)^6}{2 b^{11}}+\frac {10 d^9 (b c-a d) (a+b x)^7}{7 b^{11}}+\frac {d^{10} (a+b x)^8}{8 b^{11}}+\frac {45 d^2 (b c-a d)^8 \log (a+b x)}{b^{11}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(708\) vs. \(2(262)=524\).

Time = 0.14 (sec) , antiderivative size = 708, normalized size of antiderivative = 2.70 \[ \int \frac {(c+d x)^{10}}{(a+b x)^3} \, dx=\frac {532 a^{10} d^{10}-56 a^9 b d^9 (85 c+26 d x)+28 a^8 b^2 d^8 \left (675 c^2+380 c d x-116 d^2 x^2\right )-280 a^7 b^3 d^7 \left (156 c^3+117 c^2 d x-91 c d^2 x^2+3 d^3 x^3\right )+210 a^6 b^4 d^6 \left (308 c^4+256 c^3 d x-414 c^2 d^2 x^2+32 c d^3 x^3+d^4 x^4\right )-84 a^5 b^5 d^5 \left (756 c^5+560 c^4 d x-2000 c^3 d^2 x^2+280 c^2 d^3 x^3+20 c d^4 x^4+d^5 x^5\right )+42 a^4 b^6 d^4 \left (980 c^6+336 c^5 d x-4760 c^4 d^2 x^2+1120 c^3 d^3 x^3+140 c^2 d^4 x^4+16 c d^5 x^5+d^6 x^6\right )-24 a^3 b^7 d^3 \left (700 c^7-490 c^6 d x-6174 c^5 d^2 x^2+2450 c^4 d^3 x^3+490 c^3 d^4 x^4+98 c^2 d^5 x^5+14 c d^6 x^6+d^7 x^7\right )+3 a^2 b^8 d^2 \left (1260 c^8-4480 c^7 d x-21560 c^6 d^2 x^2+15680 c^5 d^3 x^3+4900 c^4 d^4 x^4+1568 c^3 d^5 x^5+392 c^2 d^6 x^6+64 c d^7 x^7+5 d^8 x^8\right )-2 a b^9 d \left (140 c^9-2520 c^8 d x-6720 c^7 d^2 x^2+11760 c^6 d^3 x^3+5880 c^5 d^4 x^4+2940 c^4 d^5 x^5+1176 c^3 d^6 x^6+336 c^2 d^7 x^7+60 c d^8 x^8+5 d^9 x^9\right )+b^{10} \left (-28 c^{10}-560 c^9 d x+6720 c^7 d^3 x^3+5880 c^6 d^4 x^4+4704 c^5 d^5 x^5+2940 c^4 d^6 x^6+1344 c^3 d^7 x^7+420 c^2 d^8 x^8+80 c d^9 x^9+7 d^{10} x^{10}\right )+2520 d^2 (b c-a d)^8 (a+b x)^2 \log (a+b x)}{56 b^{11} (a+b x)^2} \]

[In]

Integrate[(c + d*x)^10/(a + b*x)^3,x]

[Out]

(532*a^10*d^10 - 56*a^9*b*d^9*(85*c + 26*d*x) + 28*a^8*b^2*d^8*(675*c^2 + 380*c*d*x - 116*d^2*x^2) - 280*a^7*b
^3*d^7*(156*c^3 + 117*c^2*d*x - 91*c*d^2*x^2 + 3*d^3*x^3) + 210*a^6*b^4*d^6*(308*c^4 + 256*c^3*d*x - 414*c^2*d
^2*x^2 + 32*c*d^3*x^3 + d^4*x^4) - 84*a^5*b^5*d^5*(756*c^5 + 560*c^4*d*x - 2000*c^3*d^2*x^2 + 280*c^2*d^3*x^3
+ 20*c*d^4*x^4 + d^5*x^5) + 42*a^4*b^6*d^4*(980*c^6 + 336*c^5*d*x - 4760*c^4*d^2*x^2 + 1120*c^3*d^3*x^3 + 140*
c^2*d^4*x^4 + 16*c*d^5*x^5 + d^6*x^6) - 24*a^3*b^7*d^3*(700*c^7 - 490*c^6*d*x - 6174*c^5*d^2*x^2 + 2450*c^4*d^
3*x^3 + 490*c^3*d^4*x^4 + 98*c^2*d^5*x^5 + 14*c*d^6*x^6 + d^7*x^7) + 3*a^2*b^8*d^2*(1260*c^8 - 4480*c^7*d*x -
21560*c^6*d^2*x^2 + 15680*c^5*d^3*x^3 + 4900*c^4*d^4*x^4 + 1568*c^3*d^5*x^5 + 392*c^2*d^6*x^6 + 64*c*d^7*x^7 +
5*d^8*x^8) - 2*a*b^9*d*(140*c^9 - 2520*c^8*d*x - 6720*c^7*d^2*x^2 + 11760*c^6*d^3*x^3 + 5880*c^5*d^4*x^4 + 29
40*c^4*d^5*x^5 + 1176*c^3*d^6*x^6 + 336*c^2*d^7*x^7 + 60*c*d^8*x^8 + 5*d^9*x^9) + b^10*(-28*c^10 - 560*c^9*d*x
+ 6720*c^7*d^3*x^3 + 5880*c^6*d^4*x^4 + 4704*c^5*d^5*x^5 + 2940*c^4*d^6*x^6 + 1344*c^3*d^7*x^7 + 420*c^2*d^8*
x^8 + 80*c*d^9*x^9 + 7*d^10*x^10) + 2520*d^2*(b*c - a*d)^8*(a + b*x)^2*Log[a + b*x])/(56*b^11*(a + b*x)^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(837\) vs. \(2(252)=504\).

Time = 0.22 (sec) , antiderivative size = 838, normalized size of antiderivative = 3.20

method	result	size
norman	\(\frac {\frac {135 a^{10} d^{10}-1080 a^{9} b c \,d^{9}+3780 a^{8} b^{2} c^{2} d^{8}-7560 a^{7} b^{3} c^{3} d^{7}+9450 a^{6} b^{4} c^{4} d^{6}-7560 a^{5} b^{5} c^{5} d^{5}+3780 a^{4} b^{6} c^{6} d^{4}-1080 a^{3} b^{7} c^{7} d^{3}+135 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d -b^{10} c^{10}}{2 b^{11}}+\frac {d^{10} x^{10}}{8 b}+\frac {2 \left (45 a^{9} d^{10}-360 a^{8} b c \,d^{9}+1260 a^{7} b^{2} c^{2} d^{8}-2520 a^{6} b^{3} c^{3} d^{7}+3150 a^{5} b^{4} c^{4} d^{6}-2520 a^{4} b^{5} c^{5} d^{5}+1260 a^{3} b^{6} c^{6} d^{4}-360 a^{2} b^{7} c^{7} d^{3}+45 a \,b^{8} c^{8} d^{2}-5 b^{9} c^{9} d \right ) x}{b^{10}}-\frac {15 d^{3} \left (a^{7} d^{7}-8 a^{6} b c \,d^{6}+28 a^{5} b^{2} c^{2} d^{5}-56 a^{4} b^{3} c^{3} d^{4}+70 a^{3} b^{4} c^{4} d^{3}-56 a^{2} b^{5} c^{5} d^{2}+28 a \,b^{6} c^{6} d -8 b^{7} c^{7}\right ) x^{3}}{b^{8}}+\frac {15 d^{4} \left (a^{6} d^{6}-8 a^{5} b c \,d^{5}+28 a^{4} b^{2} c^{2} d^{4}-56 a^{3} b^{3} c^{3} d^{3}+70 a^{2} b^{4} c^{4} d^{2}-56 a \,b^{5} c^{5} d +28 b^{6} c^{6}\right ) x^{4}}{4 b^{7}}-\frac {3 d^{5} \left (a^{5} d^{5}-8 a^{4} b c \,d^{4}+28 a^{3} b^{2} c^{2} d^{3}-56 a^{2} b^{3} c^{3} d^{2}+70 a \,b^{4} c^{4} d -56 b^{5} c^{5}\right ) x^{5}}{2 b^{6}}+\frac {3 d^{6} \left (a^{4} d^{4}-8 a^{3} b c \,d^{3}+28 a^{2} b^{2} c^{2} d^{2}-56 a \,b^{3} c^{3} d +70 b^{4} c^{4}\right ) x^{6}}{4 b^{5}}-\frac {3 d^{7} \left (a^{3} d^{3}-8 a^{2} b c \,d^{2}+28 a \,b^{2} c^{2} d -56 b^{3} c^{3}\right ) x^{7}}{7 b^{4}}+\frac {15 d^{8} \left (a^{2} d^{2}-8 a b c d +28 b^{2} c^{2}\right ) x^{8}}{56 b^{3}}-\frac {5 d^{9} \left (a d -8 b c \right ) x^{9}}{28 b^{2}}}{\left (b x +a \right )^{2}}+\frac {45 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right ) \ln \left (b x +a \right )}{b^{11}}\)	\(838\)
default	\(-\frac {d^{3} \left (-\frac {1}{8} x^{8} d^{7} b^{7}-120 b^{7} c^{7} x +36 a^{7} d^{7} x +630 a \,b^{6} c^{6} d x +105 x^{2} a^{5} b^{2} c \,d^{6}-\frac {675}{2} x^{2} a^{4} b^{3} c^{2} d^{5}+600 x^{2} a^{3} b^{4} c^{3} d^{4}-630 x^{2} a^{2} b^{5} c^{4} d^{3}+378 x^{2} a \,b^{6} c^{5} d^{2}-280 a^{6} b c \,d^{6} x +945 a^{5} b^{2} c^{2} d^{5} x -1800 a^{4} b^{3} c^{3} d^{4} x +2100 a^{3} b^{4} c^{4} d^{3} x -1512 a^{2} b^{5} c^{5} d^{2} x +25 x^{4} a^{3} b^{4} c \,d^{6}-\frac {135}{2} x^{4} a^{2} b^{5} c^{2} d^{5}+90 x^{4} a \,b^{6} c^{3} d^{4}-50 x^{3} a^{4} b^{3} c \,d^{6}+150 x^{3} a^{3} b^{4} c^{2} d^{5}-240 x^{3} a^{2} b^{5} c^{3} d^{4}+210 x^{3} a \,b^{6} c^{4} d^{3}+5 x^{6} a \,b^{6} c \,d^{6}-12 x^{5} a^{2} b^{5} c \,d^{6}+27 x^{5} a \,b^{6} c^{2} d^{5}-84 x^{3} b^{7} c^{5} d^{2}-14 x^{2} a^{6} b \,d^{7}-105 x^{2} b^{7} c^{6} d +\frac {3}{7} x^{7} a \,b^{6} d^{7}-\frac {10}{7} x^{7} b^{7} c \,d^{6}-x^{6} a^{2} b^{5} d^{7}-\frac {15}{2} x^{6} b^{7} c^{2} d^{5}+2 x^{5} a^{3} b^{4} d^{7}-24 x^{5} b^{7} c^{3} d^{4}-\frac {15}{4} x^{4} a^{4} b^{3} d^{7}-\frac {105}{2} x^{4} b^{7} c^{4} d^{3}+7 x^{3} a^{5} b^{2} d^{7}\right )}{b^{10}}+\frac {45 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right ) \ln \left (b x +a \right )}{b^{11}}-\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}}{2 b^{11} \left (b x +a \right )^{2}}+\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 )}{b^{11} \left (b x +a \right )}\)	\(914\)
risch	\(\frac {675 d^{8} x^{2} a^{4} c^{2}}{2 b^{7}}-\frac {600 d^{7} x^{2} a^{3} c^{3}}{b^{6}}+\frac {630 d^{6} x^{2} a^{2} c^{4}}{b^{5}}-\frac {378 d^{5} x^{2} a \,c^{5}}{b^{4}}+\frac {280 d^{9} a^{6} c x}{b^{9}}-\frac {945 d^{8} a^{5} c^{2} x}{b^{8}}+\frac {1800 d^{7} a^{4} c^{3} x}{b^{7}}-\frac {2100 d^{6} a^{3} c^{4} x}{b^{6}}+\frac {1512 d^{5} a^{2} c^{5} x}{b^{5}}-\frac {25 d^{9} x^{4} a^{3} c}{b^{6}}+\frac {135 d^{8} x^{4} a^{2} c^{2}}{2 b^{5}}-\frac {90 d^{7} x^{4} a \,c^{3}}{b^{4}}+\frac {50 d^{9} x^{3} a^{4} c}{b^{7}}-\frac {150 d^{8} x^{3} a^{3} c^{2}}{b^{6}}+\frac {240 d^{7} x^{3} a^{2} c^{3}}{b^{5}}-\frac {210 d^{6} x^{3} a \,c^{4}}{b^{4}}-\frac {5 d^{9} x^{6} a c}{b^{4}}+\frac {12 d^{9} x^{5} a^{2} c}{b^{5}}-\frac {27 d^{8} x^{5} a \,c^{2}}{b^{4}}-\frac {360 d^{9} \ln \left (b x +a \right ) a^{7} c}{b^{10}}+\frac {1260 d^{8} \ln \left (b x +a \right ) a^{6} c^{2}}{b^{9}}-\frac {2520 d^{7} \ln \left (b x +a \right ) a^{5} c^{3}}{b^{8}}+\frac {3150 d^{6} \ln \left (b x +a \right ) a^{4} c^{4}}{b^{7}}-\frac {2520 d^{5} \ln \left (b x +a \right ) a^{3} c^{5}}{b^{6}}+\frac {1260 d^{4} \ln \left (b x +a \right ) a^{2} c^{6}}{b^{5}}-\frac {360 d^{3} \ln \left (b x +a \right ) a \,c^{7}}{b^{4}}+\frac {120 d^{3} c^{7} x}{b^{3}}-\frac {36 d^{10} a^{7} x}{b^{10}}+\frac {84 d^{5} x^{3} c^{5}}{b^{3}}+\frac {14 d^{10} x^{2} a^{6}}{b^{9}}+\frac {105 d^{4} x^{2} c^{6}}{b^{3}}-\frac {3 d^{10} x^{7} a}{7 b^{4}}+\frac {10 d^{9} x^{7} c}{7 b^{3}}+\frac {d^{10} x^{6} a^{2}}{b^{5}}+\frac {15 d^{8} x^{6} c^{2}}{2 b^{3}}-\frac {2 d^{10} x^{5} a^{3}}{b^{6}}+\frac {24 d^{7} x^{5} c^{3}}{b^{3}}+\frac {15 d^{10} x^{4} a^{4}}{4 b^{7}}+\frac {105 d^{6} x^{4} c^{4}}{2 b^{3}}-\frac {7 d^{10} x^{3} a^{5}}{b^{8}}+\frac {45 d^{10} \ln \left (b x +a \right ) a^{8}}{b^{11}}+\frac {45 d^{2} \ln \left (b x +a \right ) c^{8}}{b^{3}}-\frac {630 d^{4} a \,c^{6} x}{b^{4}}-\frac {105 d^{9} x^{2} a^{5} c}{b^{8}}+\frac {d^{10} x^{8}}{8 b^{3}}+\frac {\left (10 a^{9} d^{10}-90 a^{8} b c \,d^{9}+360 a^{7} b^{2} c^{2} d^{8}-840 a^{6} b^{3} c^{3} d^{7}+1260 a^{5} b^{4} c^{4} d^{6}-1260 a^{4} b^{5} c^{5} d^{5}+840 a^{3} b^{6} c^{6} d^{4}-360 a^{2} b^{7} c^{7} d^{3}+90 a \,b^{8} c^{8} d^{2}-10 b^{9} c^{9} d \right ) x +\frac {19 a^{10} d^{10}-170 a^{9} b c \,d^{9}+675 a^{8} b^{2} c^{2} d^{8}-1560 a^{7} b^{3} c^{3} d^{7}+2310 a^{6} b^{4} c^{4} d^{6}-2268 a^{5} b^{5} c^{5} d^{5}+1470 a^{4} b^{6} c^{6} d^{4}-600 a^{3} b^{7} c^{7} d^{3}+135 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d -b^{10} c^{10}}{2 b}}{b^{10} \left (b x +a \right )^{2}}\)	\(969\)
parallelrisch	\(\text {Expression too large to display}\)	\(1367\)

[In]

int((d*x+c)^10/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

(1/2*(135*a^10*d^10-1080*a^9*b*c*d^9+3780*a^8*b^2*c^2*d^8-7560*a^7*b^3*c^3*d^7+9450*a^6*b^4*c^4*d^6-7560*a^5*b
^5*c^5*d^5+3780*a^4*b^6*c^6*d^4-1080*a^3*b^7*c^7*d^3+135*a^2*b^8*c^8*d^2-10*a*b^9*c^9*d-b^10*c^10)/b^11+1/8/b*
d^10*x^10+2*(45*a^9*d^10-360*a^8*b*c*d^9+1260*a^7*b^2*c^2*d^8-2520*a^6*b^3*c^3*d^7+3150*a^5*b^4*c^4*d^6-2520*a
^4*b^5*c^5*d^5+1260*a^3*b^6*c^6*d^4-360*a^2*b^7*c^7*d^3+45*a*b^8*c^8*d^2-5*b^9*c^9*d)/b^10*x-15*d^3*(a^7*d^7-8
*a^6*b*c*d^6+28*a^5*b^2*c^2*d^5-56*a^4*b^3*c^3*d^4+70*a^3*b^4*c^4*d^3-56*a^2*b^5*c^5*d^2+28*a*b^6*c^6*d-8*b^7*
c^7)/b^8*x^3+15/4*d^4*(a^6*d^6-8*a^5*b*c*d^5+28*a^4*b^2*c^2*d^4-56*a^3*b^3*c^3*d^3+70*a^2*b^4*c^4*d^2-56*a*b^5
*c^5*d+28*b^6*c^6)/b^7*x^4-3/2*d^5*(a^5*d^5-8*a^4*b*c*d^4+28*a^3*b^2*c^2*d^3-56*a^2*b^3*c^3*d^2+70*a*b^4*c^4*d
-56*b^5*c^5)/b^6*x^5+3/4*d^6*(a^4*d^4-8*a^3*b*c*d^3+28*a^2*b^2*c^2*d^2-56*a*b^3*c^3*d+70*b^4*c^4)/b^5*x^6-3/7*
d^7*(a^3*d^3-8*a^2*b*c*d^2+28*a*b^2*c^2*d-56*b^3*c^3)/b^4*x^7+15/56*d^8*(a^2*d^2-8*a*b*c*d+28*b^2*c^2)/b^3*x^8
-5/28*d^9*(a*d-8*b*c)/b^2*x^9)/(b*x+a)^2+45/b^11*d^2*(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)*ln(b*x+a)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1233 vs. \(2 (252) = 504\).

Time = 0.24 (sec) , antiderivative size = 1233, normalized size of antiderivative = 4.71 \[ \int \frac {(c+d x)^{10}}{(a+b x)^3} \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^10/(b*x+a)^3,x, algorithm="fricas")

[Out]

1/56*(7*b^10*d^10*x^10 - 28*b^10*c^10 - 280*a*b^9*c^9*d + 3780*a^2*b^8*c^8*d^2 - 16800*a^3*b^7*c^7*d^3 + 41160
*a^4*b^6*c^6*d^4 - 63504*a^5*b^5*c^5*d^5 + 64680*a^6*b^4*c^4*d^6 - 43680*a^7*b^3*c^3*d^7 + 18900*a^8*b^2*c^2*d
^8 - 4760*a^9*b*c*d^9 + 532*a^10*d^10 + 10*(8*b^10*c*d^9 - a*b^9*d^10)*x^9 + 15*(28*b^10*c^2*d^8 - 8*a*b^9*c*d
^9 + a^2*b^8*d^10)*x^8 + 24*(56*b^10*c^3*d^7 - 28*a*b^9*c^2*d^8 + 8*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 42*(70
*b^10*c^4*d^6 - 56*a*b^9*c^3*d^7 + 28*a^2*b^8*c^2*d^8 - 8*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 84*(56*b^10*c^5*
d^5 - 70*a*b^9*c^4*d^6 + 56*a^2*b^8*c^3*d^7 - 28*a^3*b^7*c^2*d^8 + 8*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 + 210*(
28*b^10*c^6*d^4 - 56*a*b^9*c^5*d^5 + 70*a^2*b^8*c^4*d^6 - 56*a^3*b^7*c^3*d^7 + 28*a^4*b^6*c^2*d^8 - 8*a^5*b^5*
c*d^9 + a^6*b^4*d^10)*x^4 + 840*(8*b^10*c^7*d^3 - 28*a*b^9*c^6*d^4 + 56*a^2*b^8*c^5*d^5 - 70*a^3*b^7*c^4*d^6 +
56*a^4*b^6*c^3*d^7 - 28*a^5*b^5*c^2*d^8 + 8*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 28*(480*a*b^9*c^7*d^3 - 2310*
a^2*b^8*c^6*d^4 + 5292*a^3*b^7*c^5*d^5 - 7140*a^4*b^6*c^4*d^6 + 6000*a^5*b^5*c^3*d^7 - 3105*a^6*b^4*c^2*d^8 +
910*a^7*b^3*c*d^9 - 116*a^8*b^2*d^10)*x^2 - 56*(10*b^10*c^9*d - 90*a*b^9*c^8*d^2 + 240*a^2*b^8*c^7*d^3 - 210*a
^3*b^7*c^6*d^4 - 252*a^4*b^6*c^5*d^5 + 840*a^5*b^5*c^4*d^6 - 960*a^6*b^4*c^3*d^7 + 585*a^7*b^3*c^2*d^8 - 190*a
^8*b^2*c*d^9 + 26*a^9*b*d^10)*x + 2520*(a^2*b^8*c^8*d^2 - 8*a^3*b^7*c^7*d^3 + 28*a^4*b^6*c^6*d^4 - 56*a^5*b^5*
c^5*d^5 + 70*a^6*b^4*c^4*d^6 - 56*a^7*b^3*c^3*d^7 + 28*a^8*b^2*c^2*d^8 - 8*a^9*b*c*d^9 + a^10*d^10 + (b^10*c^8
*d^2 - 8*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 - 56*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 - 56*a^5*b^5*c^3*d^7 + 2
8*a^6*b^4*c^2*d^8 - 8*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 2*(a*b^9*c^8*d^2 - 8*a^2*b^8*c^7*d^3 + 28*a^3*b^7*c^
6*d^4 - 56*a^4*b^6*c^5*d^5 + 70*a^5*b^5*c^4*d^6 - 56*a^6*b^4*c^3*d^7 + 28*a^7*b^3*c^2*d^8 - 8*a^8*b^2*c*d^9 +
a^9*b*d^10)*x)*log(b*x + a))/(b^13*x^2 + 2*a*b^12*x + a^2*b^11)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 843 vs. \(2 (243) = 486\).

Time = 6.42 (sec) , antiderivative size = 843, normalized size of antiderivative = 3.22 \[ \int \frac {(c+d x)^{10}}{(a+b x)^3} \, dx=x^{7} \left (- \frac {3 a d^{10}}{7 b^{4}} + \frac {10 c d^{9}}{7 b^{3}}\right ) + x^{6} \left (\frac {a^{2} d^{10}}{b^{5}} - \frac {5 a c d^{9}}{b^{4}} + \frac {15 c^{2} d^{8}}{2 b^{3}}\right ) + x^{5} \left (- \frac {2 a^{3} d^{10}}{b^{6}} + \frac {12 a^{2} c d^{9}}{b^{5}} - \frac {27 a c^{2} d^{8}}{b^{4}} + \frac {24 c^{3} d^{7}}{b^{3}}\right ) + x^{4} \cdot \left (\frac {15 a^{4} d^{10}}{4 b^{7}} - \frac {25 a^{3} c d^{9}}{b^{6}} + \frac {135 a^{2} c^{2} d^{8}}{2 b^{5}} - \frac {90 a c^{3} d^{7}}{b^{4}} + \frac {105 c^{4} d^{6}}{2 b^{3}}\right ) + x^{3} \left (- \frac {7 a^{5} d^{10}}{b^{8}} + \frac {50 a^{4} c d^{9}}{b^{7}} - \frac {150 a^{3} c^{2} d^{8}}{b^{6}} + \frac {240 a^{2} c^{3} d^{7}}{b^{5}} - \frac {210 a c^{4} d^{6}}{b^{4}} + \frac {84 c^{5} d^{5}}{b^{3}}\right ) + x^{2} \cdot \left (\frac {14 a^{6} d^{10}}{b^{9}} - \frac {105 a^{5} c d^{9}}{b^{8}} + \frac {675 a^{4} c^{2} d^{8}}{2 b^{7}} - \frac {600 a^{3} c^{3} d^{7}}{b^{6}} + \frac {630 a^{2} c^{4} d^{6}}{b^{5}} - \frac {378 a c^{5} d^{5}}{b^{4}} + \frac {105 c^{6} d^{4}}{b^{3}}\right ) + x \left (- \frac {36 a^{7} d^{10}}{b^{10}} + \frac {280 a^{6} c d^{9}}{b^{9}} - \frac {945 a^{5} c^{2} d^{8}}{b^{8}} + \frac {1800 a^{4} c^{3} d^{7}}{b^{7}} - \frac {2100 a^{3} c^{4} d^{6}}{b^{6}} + \frac {1512 a^{2} c^{5} d^{5}}{b^{5}} - \frac {630 a c^{6} d^{4}}{b^{4}} + \frac {120 c^{7} d^{3}}{b^{3}}\right ) + \frac {19 a^{10} d^{10} - 170 a^{9} b c d^{9} + 675 a^{8} b^{2} c^{2} d^{8} - 1560 a^{7} b^{3} c^{3} d^{7} + 2310 a^{6} b^{4} c^{4} d^{6} - 2268 a^{5} b^{5} c^{5} d^{5} + 1470 a^{4} b^{6} c^{6} d^{4} - 600 a^{3} b^{7} c^{7} d^{3} + 135 a^{2} b^{8} c^{8} d^{2} - 10 a b^{9} c^{9} d - b^{10} c^{10} + x \left (20 a^{9} b d^{10} - 180 a^{8} b^{2} c d^{9} + 720 a^{7} b^{3} c^{2} d^{8} - 1680 a^{6} b^{4} c^{3} d^{7} + 2520 a^{5} b^{5} c^{4} d^{6} - 2520 a^{4} b^{6} c^{5} d^{5} + 1680 a^{3} b^{7} c^{6} d^{4} - 720 a^{2} b^{8} c^{7} d^{3} + 180 a b^{9} c^{8} d^{2} - 20 b^{10} c^{9} d\right )}{2 a^{2} b^{11} + 4 a b^{12} x + 2 b^{13} x^{2}} + \frac {d^{10} x^{8}}{8 b^{3}} + \frac {45 d^{2} \left (a d - b c\right )^{8} \log {\left (a + b x \right )}}{b^{11}} \]

[In]

integrate((d*x+c)**10/(b*x+a)**3,x)

[Out]

x**7*(-3*a*d**10/(7*b**4) + 10*c*d**9/(7*b**3)) + x**6*(a**2*d**10/b**5 - 5*a*c*d**9/b**4 + 15*c**2*d**8/(2*b*
*3)) + x**5*(-2*a**3*d**10/b**6 + 12*a**2*c*d**9/b**5 - 27*a*c**2*d**8/b**4 + 24*c**3*d**7/b**3) + x**4*(15*a*
*4*d**10/(4*b**7) - 25*a**3*c*d**9/b**6 + 135*a**2*c**2*d**8/(2*b**5) - 90*a*c**3*d**7/b**4 + 105*c**4*d**6/(2
*b**3)) + x**3*(-7*a**5*d**10/b**8 + 50*a**4*c*d**9/b**7 - 150*a**3*c**2*d**8/b**6 + 240*a**2*c**3*d**7/b**5 -
210*a*c**4*d**6/b**4 + 84*c**5*d**5/b**3) + x**2*(14*a**6*d**10/b**9 - 105*a**5*c*d**9/b**8 + 675*a**4*c**2*d
**8/(2*b**7) - 600*a**3*c**3*d**7/b**6 + 630*a**2*c**4*d**6/b**5 - 378*a*c**5*d**5/b**4 + 105*c**6*d**4/b**3)
+ x*(-36*a**7*d**10/b**10 + 280*a**6*c*d**9/b**9 - 945*a**5*c**2*d**8/b**8 + 1800*a**4*c**3*d**7/b**7 - 2100*a
**3*c**4*d**6/b**6 + 1512*a**2*c**5*d**5/b**5 - 630*a*c**6*d**4/b**4 + 120*c**7*d**3/b**3) + (19*a**10*d**10 -
170*a**9*b*c*d**9 + 675*a**8*b**2*c**2*d**8 - 1560*a**7*b**3*c**3*d**7 + 2310*a**6*b**4*c**4*d**6 - 2268*a**5
*b**5*c**5*d**5 + 1470*a**4*b**6*c**6*d**4 - 600*a**3*b**7*c**7*d**3 + 135*a**2*b**8*c**8*d**2 - 10*a*b**9*c**
9*d - b**10*c**10 + x*(20*a**9*b*d**10 - 180*a**8*b**2*c*d**9 + 720*a**7*b**3*c**2*d**8 - 1680*a**6*b**4*c**3*
d**7 + 2520*a**5*b**5*c**4*d**6 - 2520*a**4*b**6*c**5*d**5 + 1680*a**3*b**7*c**6*d**4 - 720*a**2*b**8*c**7*d**
3 + 180*a*b**9*c**8*d**2 - 20*b**10*c**9*d))/(2*a**2*b**11 + 4*a*b**12*x + 2*b**13*x**2) + d**10*x**8/(8*b**3)
+ 45*d**2*(a*d - b*c)**8*log(a + b*x)/b**11

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 881 vs. \(2 (252) = 504\).

Time = 0.22 (sec) , antiderivative size = 881, normalized size of antiderivative = 3.36 \[ \int \frac {(c+d x)^{10}}{(a+b x)^3} \, dx=-\frac {b^{10} c^{10} + 10 \, a b^{9} c^{9} d - 135 \, a^{2} b^{8} c^{8} d^{2} + 600 \, a^{3} b^{7} c^{7} d^{3} - 1470 \, a^{4} b^{6} c^{6} d^{4} + 2268 \, a^{5} b^{5} c^{5} d^{5} - 2310 \, a^{6} b^{4} c^{4} d^{6} + 1560 \, a^{7} b^{3} c^{3} d^{7} - 675 \, a^{8} b^{2} c^{2} d^{8} + 170 \, a^{9} b c d^{9} - 19 \, a^{10} d^{10} + 20 \, {\left (b^{10} c^{9} d - 9 \, a b^{9} c^{8} d^{2} + 36 \, a^{2} b^{8} c^{7} d^{3} - 84 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} - 126 \, a^{5} b^{5} c^{4} d^{6} + 84 \, a^{6} b^{4} c^{3} d^{7} - 36 \, a^{7} b^{3} c^{2} d^{8} + 9 \, a^{8} b^{2} c d^{9} - a^{9} b d^{10}\right )} x}{2 \, {\left (b^{13} x^{2} + 2 \, a b^{12} x + a^{2} b^{11}\right )}} + \frac {7 \, b^{7} d^{10} x^{8} + 8 \, {\left (10 \, b^{7} c d^{9} - 3 \, a b^{6} d^{10}\right )} x^{7} + 28 \, {\left (15 \, b^{7} c^{2} d^{8} - 10 \, a b^{6} c d^{9} + 2 \, a^{2} b^{5} d^{10}\right )} x^{6} + 56 \, {\left (24 \, b^{7} c^{3} d^{7} - 27 \, a b^{6} c^{2} d^{8} + 12 \, a^{2} b^{5} c d^{9} - 2 \, a^{3} b^{4} d^{10}\right )} x^{5} + 70 \, {\left (42 \, b^{7} c^{4} d^{6} - 72 \, a b^{6} c^{3} d^{7} + 54 \, a^{2} b^{5} c^{2} d^{8} - 20 \, a^{3} b^{4} c d^{9} + 3 \, a^{4} b^{3} d^{10}\right )} x^{4} + 56 \, {\left (84 \, b^{7} c^{5} d^{5} - 210 \, a b^{6} c^{4} d^{6} + 240 \, a^{2} b^{5} c^{3} d^{7} - 150 \, a^{3} b^{4} c^{2} d^{8} + 50 \, a^{4} b^{3} c d^{9} - 7 \, a^{5} b^{2} d^{10}\right )} x^{3} + 28 \, {\left (210 \, b^{7} c^{6} d^{4} - 756 \, a b^{6} c^{5} d^{5} + 1260 \, a^{2} b^{5} c^{4} d^{6} - 1200 \, a^{3} b^{4} c^{3} d^{7} + 675 \, a^{4} b^{3} c^{2} d^{8} - 210 \, a^{5} b^{2} c d^{9} + 28 \, a^{6} b d^{10}\right )} x^{2} + 56 \, {\left (120 \, b^{7} c^{7} d^{3} - 630 \, a b^{6} c^{6} d^{4} + 1512 \, a^{2} b^{5} c^{5} d^{5} - 2100 \, a^{3} b^{4} c^{4} d^{6} + 1800 \, a^{4} b^{3} c^{3} d^{7} - 945 \, a^{5} b^{2} c^{2} d^{8} + 280 \, a^{6} b c d^{9} - 36 \, a^{7} d^{10}\right )} x}{56 \, b^{10}} + \frac {45 \, {\left (b^{8} c^{8} d^{2} - 8 \, a b^{7} c^{7} d^{3} + 28 \, a^{2} b^{6} c^{6} d^{4} - 56 \, a^{3} b^{5} c^{5} d^{5} + 70 \, a^{4} b^{4} c^{4} d^{6} - 56 \, a^{5} b^{3} c^{3} d^{7} + 28 \, a^{6} b^{2} c^{2} d^{8} - 8 \, a^{7} b c d^{9} + a^{8} d^{10}\right )} \log \left (b x + a\right )}{b^{11}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2*(b^10*c^10 + 10*a*b^9*c^9*d - 135*a^2*b^8*c^8*d^2 + 600*a^3*b^7*c^7*d^3 - 1470*a^4*b^6*c^6*d^4 + 2268*a^5
*b^5*c^5*d^5 - 2310*a^6*b^4*c^4*d^6 + 1560*a^7*b^3*c^3*d^7 - 675*a^8*b^2*c^2*d^8 + 170*a^9*b*c*d^9 - 19*a^10*d
^10 + 20*(b^10*c^9*d - 9*a*b^9*c^8*d^2 + 36*a^2*b^8*c^7*d^3 - 84*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 - 126*a
^5*b^5*c^4*d^6 + 84*a^6*b^4*c^3*d^7 - 36*a^7*b^3*c^2*d^8 + 9*a^8*b^2*c*d^9 - a^9*b*d^10)*x)/(b^13*x^2 + 2*a*b^
12*x + a^2*b^11) + 1/56*(7*b^7*d^10*x^8 + 8*(10*b^7*c*d^9 - 3*a*b^6*d^10)*x^7 + 28*(15*b^7*c^2*d^8 - 10*a*b^6*
c*d^9 + 2*a^2*b^5*d^10)*x^6 + 56*(24*b^7*c^3*d^7 - 27*a*b^6*c^2*d^8 + 12*a^2*b^5*c*d^9 - 2*a^3*b^4*d^10)*x^5 +
70*(42*b^7*c^4*d^6 - 72*a*b^6*c^3*d^7 + 54*a^2*b^5*c^2*d^8 - 20*a^3*b^4*c*d^9 + 3*a^4*b^3*d^10)*x^4 + 56*(84*
b^7*c^5*d^5 - 210*a*b^6*c^4*d^6 + 240*a^2*b^5*c^3*d^7 - 150*a^3*b^4*c^2*d^8 + 50*a^4*b^3*c*d^9 - 7*a^5*b^2*d^1
0)*x^3 + 28*(210*b^7*c^6*d^4 - 756*a*b^6*c^5*d^5 + 1260*a^2*b^5*c^4*d^6 - 1200*a^3*b^4*c^3*d^7 + 675*a^4*b^3*c
^2*d^8 - 210*a^5*b^2*c*d^9 + 28*a^6*b*d^10)*x^2 + 56*(120*b^7*c^7*d^3 - 630*a*b^6*c^6*d^4 + 1512*a^2*b^5*c^5*d
^5 - 2100*a^3*b^4*c^4*d^6 + 1800*a^4*b^3*c^3*d^7 - 945*a^5*b^2*c^2*d^8 + 280*a^6*b*c*d^9 - 36*a^7*d^10)*x)/b^1
0 + 45*(b^8*c^8*d^2 - 8*a*b^7*c^7*d^3 + 28*a^2*b^6*c^6*d^4 - 56*a^3*b^5*c^5*d^5 + 70*a^4*b^4*c^4*d^6 - 56*a^5*
b^3*c^3*d^7 + 28*a^6*b^2*c^2*d^8 - 8*a^7*b*c*d^9 + a^8*d^10)*log(b*x + a)/b^11

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 924 vs. \(2 (252) = 504\).

Time = 0.40 (sec) , antiderivative size = 924, normalized size of antiderivative = 3.53 \[ \int \frac {(c+d x)^{10}}{(a+b x)^3} \, dx=\frac {45 \, {\left (b^{8} c^{8} d^{2} - 8 \, a b^{7} c^{7} d^{3} + 28 \, a^{2} b^{6} c^{6} d^{4} - 56 \, a^{3} b^{5} c^{5} d^{5} + 70 \, a^{4} b^{4} c^{4} d^{6} - 56 \, a^{5} b^{3} c^{3} d^{7} + 28 \, a^{6} b^{2} c^{2} d^{8} - 8 \, a^{7} b c d^{9} + a^{8} d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {b^{10} c^{10} + 10 \, a b^{9} c^{9} d - 135 \, a^{2} b^{8} c^{8} d^{2} + 600 \, a^{3} b^{7} c^{7} d^{3} - 1470 \, a^{4} b^{6} c^{6} d^{4} + 2268 \, a^{5} b^{5} c^{5} d^{5} - 2310 \, a^{6} b^{4} c^{4} d^{6} + 1560 \, a^{7} b^{3} c^{3} d^{7} - 675 \, a^{8} b^{2} c^{2} d^{8} + 170 \, a^{9} b c d^{9} - 19 \, a^{10} d^{10} + 20 \, {\left (b^{10} c^{9} d - 9 \, a b^{9} c^{8} d^{2} + 36 \, a^{2} b^{8} c^{7} d^{3} - 84 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} - 126 \, a^{5} b^{5} c^{4} d^{6} + 84 \, a^{6} b^{4} c^{3} d^{7} - 36 \, a^{7} b^{3} c^{2} d^{8} + 9 \, a^{8} b^{2} c d^{9} - a^{9} b d^{10}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{11}} + \frac {7 \, b^{21} d^{10} x^{8} + 80 \, b^{21} c d^{9} x^{7} - 24 \, a b^{20} d^{10} x^{7} + 420 \, b^{21} c^{2} d^{8} x^{6} - 280 \, a b^{20} c d^{9} x^{6} + 56 \, a^{2} b^{19} d^{10} x^{6} + 1344 \, b^{21} c^{3} d^{7} x^{5} - 1512 \, a b^{20} c^{2} d^{8} x^{5} + 672 \, a^{2} b^{19} c d^{9} x^{5} - 112 \, a^{3} b^{18} d^{10} x^{5} + 2940 \, b^{21} c^{4} d^{6} x^{4} - 5040 \, a b^{20} c^{3} d^{7} x^{4} + 3780 \, a^{2} b^{19} c^{2} d^{8} x^{4} - 1400 \, a^{3} b^{18} c d^{9} x^{4} + 210 \, a^{4} b^{17} d^{10} x^{4} + 4704 \, b^{21} c^{5} d^{5} x^{3} - 11760 \, a b^{20} c^{4} d^{6} x^{3} + 13440 \, a^{2} b^{19} c^{3} d^{7} x^{3} - 8400 \, a^{3} b^{18} c^{2} d^{8} x^{3} + 2800 \, a^{4} b^{17} c d^{9} x^{3} - 392 \, a^{5} b^{16} d^{10} x^{3} + 5880 \, b^{21} c^{6} d^{4} x^{2} - 21168 \, a b^{20} c^{5} d^{5} x^{2} + 35280 \, a^{2} b^{19} c^{4} d^{6} x^{2} - 33600 \, a^{3} b^{18} c^{3} d^{7} x^{2} + 18900 \, a^{4} b^{17} c^{2} d^{8} x^{2} - 5880 \, a^{5} b^{16} c d^{9} x^{2} + 784 \, a^{6} b^{15} d^{10} x^{2} + 6720 \, b^{21} c^{7} d^{3} x - 35280 \, a b^{20} c^{6} d^{4} x + 84672 \, a^{2} b^{19} c^{5} d^{5} x - 117600 \, a^{3} b^{18} c^{4} d^{6} x + 100800 \, a^{4} b^{17} c^{3} d^{7} x - 52920 \, a^{5} b^{16} c^{2} d^{8} x + 15680 \, a^{6} b^{15} c d^{9} x - 2016 \, a^{7} b^{14} d^{10} x}{56 \, b^{24}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^3,x, algorithm="giac")

[Out]

45*(b^8*c^8*d^2 - 8*a*b^7*c^7*d^3 + 28*a^2*b^6*c^6*d^4 - 56*a^3*b^5*c^5*d^5 + 70*a^4*b^4*c^4*d^6 - 56*a^5*b^3*
c^3*d^7 + 28*a^6*b^2*c^2*d^8 - 8*a^7*b*c*d^9 + a^8*d^10)*log(abs(b*x + a))/b^11 - 1/2*(b^10*c^10 + 10*a*b^9*c^
9*d - 135*a^2*b^8*c^8*d^2 + 600*a^3*b^7*c^7*d^3 - 1470*a^4*b^6*c^6*d^4 + 2268*a^5*b^5*c^5*d^5 - 2310*a^6*b^4*c
^4*d^6 + 1560*a^7*b^3*c^3*d^7 - 675*a^8*b^2*c^2*d^8 + 170*a^9*b*c*d^9 - 19*a^10*d^10 + 20*(b^10*c^9*d - 9*a*b^
9*c^8*d^2 + 36*a^2*b^8*c^7*d^3 - 84*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 - 126*a^5*b^5*c^4*d^6 + 84*a^6*b^4*c
^3*d^7 - 36*a^7*b^3*c^2*d^8 + 9*a^8*b^2*c*d^9 - a^9*b*d^10)*x)/((b*x + a)^2*b^11) + 1/56*(7*b^21*d^10*x^8 + 80
*b^21*c*d^9*x^7 - 24*a*b^20*d^10*x^7 + 420*b^21*c^2*d^8*x^6 - 280*a*b^20*c*d^9*x^6 + 56*a^2*b^19*d^10*x^6 + 13
44*b^21*c^3*d^7*x^5 - 1512*a*b^20*c^2*d^8*x^5 + 672*a^2*b^19*c*d^9*x^5 - 112*a^3*b^18*d^10*x^5 + 2940*b^21*c^4
*d^6*x^4 - 5040*a*b^20*c^3*d^7*x^4 + 3780*a^2*b^19*c^2*d^8*x^4 - 1400*a^3*b^18*c*d^9*x^4 + 210*a^4*b^17*d^10*x
^4 + 4704*b^21*c^5*d^5*x^3 - 11760*a*b^20*c^4*d^6*x^3 + 13440*a^2*b^19*c^3*d^7*x^3 - 8400*a^3*b^18*c^2*d^8*x^3
+ 2800*a^4*b^17*c*d^9*x^3 - 392*a^5*b^16*d^10*x^3 + 5880*b^21*c^6*d^4*x^2 - 21168*a*b^20*c^5*d^5*x^2 + 35280*
a^2*b^19*c^4*d^6*x^2 - 33600*a^3*b^18*c^3*d^7*x^2 + 18900*a^4*b^17*c^2*d^8*x^2 - 5880*a^5*b^16*c*d^9*x^2 + 784
*a^6*b^15*d^10*x^2 + 6720*b^21*c^7*d^3*x - 35280*a*b^20*c^6*d^4*x + 84672*a^2*b^19*c^5*d^5*x - 117600*a^3*b^18
*c^4*d^6*x + 100800*a^4*b^17*c^3*d^7*x - 52920*a^5*b^16*c^2*d^8*x + 15680*a^6*b^15*c*d^9*x - 2016*a^7*b^14*d^1
0*x)/b^24

Mupad [B] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 3299, normalized size of antiderivative = 12.59 \[ \int \frac {(c+d x)^{10}}{(a+b x)^3} \, dx=\text {Too large to display} \]

[In]

int((c + d*x)^10/(a + b*x)^3,x)

[Out]

x^3*((84*c^5*d^5)/b^3 - (a*((3*a*((3*a*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2
*d^8)/b^3))/b + (a^3*d^10)/b^6 - (120*c^3*d^7)/b^3 - (3*a^2*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b^2))/b + (210*
c^4*d^6)/b^3 + (a^3*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b^3 - (3*a^2*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b
- (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b^2))/b + (a^2*((3*a*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*
a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b + (a^3*d^10)/b^6 - (120*c^3*d^7)/b^3 - (3*a^2*((3*a*d^10)/b^4 - (10*c*d^9
)/b^3))/b^2))/b^2 - (a^3*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/(3
*b^3)) - x^7*((3*a*d^10)/(7*b^4) - (10*c*d^9)/(7*b^3)) - ((b^10*c^10 - 19*a^10*d^10 - 135*a^2*b^8*c^8*d^2 + 60
0*a^3*b^7*c^7*d^3 - 1470*a^4*b^6*c^6*d^4 + 2268*a^5*b^5*c^5*d^5 - 2310*a^6*b^4*c^4*d^6 + 1560*a^7*b^3*c^3*d^7
- 675*a^8*b^2*c^2*d^8 + 10*a*b^9*c^9*d + 170*a^9*b*c*d^9)/(2*b) - x*(10*a^9*d^10 - 10*b^9*c^9*d + 90*a*b^8*c^8
*d^2 - 360*a^2*b^7*c^7*d^3 + 840*a^3*b^6*c^6*d^4 - 1260*a^4*b^5*c^5*d^5 + 1260*a^5*b^4*c^4*d^6 - 840*a^6*b^3*c
^3*d^7 + 360*a^7*b^2*c^2*d^8 - 90*a^8*b*c*d^9))/(a^2*b^10 + b^12*x^2 + 2*a*b^11*x) - x^5*((3*a*((3*a*((3*a*d^1
0)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/(5*b) + (a^3*d^10)/(5*b^6) - (24*c^3*d^7)/
b^3 - (3*a^2*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/(5*b^2)) + x^6*((a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/(2*b) -
(a^2*d^10)/(2*b^5) + (15*c^2*d^8)/(2*b^3)) + x^4*((3*a*((3*a*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a
^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b + (a^3*d^10)/b^6 - (120*c^3*d^7)/b^3 - (3*a^2*((3*a*d^10)/b^4 - (10*c*d^9)
/b^3))/b^2))/(4*b) + (105*c^4*d^6)/(2*b^3) + (a^3*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/(4*b^3) - (3*a^2*((3*a*((
3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/(4*b^2)) - x^2*((3*a^2*((3*a*((3*a*
((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b + (a^3*d^10)/b^6 - (120*c
^3*d^7)/b^3 - (3*a^2*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b^2))/b + (210*c^4*d^6)/b^3 + (a^3*((3*a*d^10)/b^4 - (
10*c*d^9)/b^3))/b^3 - (3*a^2*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3)
)/b^2))/(2*b^2) + (3*a*((252*c^5*d^5)/b^3 - (3*a*((3*a*((3*a*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a
^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b + (a^3*d^10)/b^6 - (120*c^3*d^7)/b^3 - (3*a^2*((3*a*d^10)/b^4 - (10*c*d^9)
/b^3))/b^2))/b + (210*c^4*d^6)/b^3 + (a^3*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b^3 - (3*a^2*((3*a*((3*a*d^10)/b^
4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b^2))/b + (3*a^2*((3*a*((3*a*((3*a*d^10)/b^4 -
(10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b + (a^3*d^10)/b^6 - (120*c^3*d^7)/b^3 - (3*a^2*((3
*a*d^10)/b^4 - (10*c*d^9)/b^3))/b^2))/b^2 - (a^3*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5
+ (45*c^2*d^8)/b^3))/b^3))/(2*b) - (105*c^6*d^4)/b^3 - (a^3*((3*a*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b
- (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b + (a^3*d^10)/b^6 - (120*c^3*d^7)/b^3 - (3*a^2*((3*a*d^10)/b^4 - (10*
c*d^9)/b^3))/b^2))/(2*b^3)) + x*((3*a*((3*a^2*((3*a*((3*a*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*
d^10)/b^5 + (45*c^2*d^8)/b^3))/b + (a^3*d^10)/b^6 - (120*c^3*d^7)/b^3 - (3*a^2*((3*a*d^10)/b^4 - (10*c*d^9)/b^
3))/b^2))/b + (210*c^4*d^6)/b^3 + (a^3*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b^3 - (3*a^2*((3*a*((3*a*d^10)/b^4 -
(10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b^2))/b^2 + (3*a*((252*c^5*d^5)/b^3 - (3*a*((3*a*(
(3*a*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b + (a^3*d^10)/b^6 - (
120*c^3*d^7)/b^3 - (3*a^2*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b^2))/b + (210*c^4*d^6)/b^3 + (a^3*((3*a*d^10)/b^
4 - (10*c*d^9)/b^3))/b^3 - (3*a^2*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)
/b^3))/b^2))/b + (3*a^2*((3*a*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3
))/b + (a^3*d^10)/b^6 - (120*c^3*d^7)/b^3 - (3*a^2*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b^2))/b^2 - (a^3*((3*a*(
(3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b^3))/b - (210*c^6*d^4)/b^3 - (a^3
*((3*a*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b + (a^3*d^10)/b^6 -
(120*c^3*d^7)/b^3 - (3*a^2*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b^2))/b^3))/b - (a^3*((3*a*((3*a*((3*a*((3*a*d^
10)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b + (a^3*d^10)/b^6 - (120*c^3*d^7)/b^3 -
(3*a^2*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b^2))/b + (210*c^4*d^6)/b^3 + (a^3*((3*a*d^10)/b^4 - (10*c*d^9)/b^3)
)/b^3 - (3*a^2*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b^2))/b^3 +
(120*c^7*d^3)/b^3 - (3*a^2*((252*c^5*d^5)/b^3 - (3*a*((3*a*((3*a*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b -
(3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b + (a^3*d^10)/b^6 - (120*c^3*d^7)/b^3 - (3*a^2*((3*a*d^10)/b^4 - (10*c*
d^9)/b^3))/b^2))/b + (210*c^4*d^6)/b^3 + (a^3*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b^3 - (3*a^2*((3*a*((3*a*d^10
)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b^2))/b + (3*a^2*((3*a*((3*a*((3*a*d^10)/b^
4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)/b^5 + (45*c^2*d^8)/b^3))/b + (a^3*d^10)/b^6 - (120*c^3*d^7)/b^3 - (3*a^2
*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b^2))/b^2 - (a^3*((3*a*((3*a*d^10)/b^4 - (10*c*d^9)/b^3))/b - (3*a^2*d^10)
/b^5 + (45*c^2*d^8)/b^3))/b^3))/b^2) + (log(a + b*x)*(45*a^8*d^10 + 45*b^8*c^8*d^2 - 360*a*b^7*c^7*d^3 + 1260*
a^2*b^6*c^6*d^4 - 2520*a^3*b^5*c^5*d^5 + 3150*a^4*b^4*c^4*d^6 - 2520*a^5*b^3*c^3*d^7 + 1260*a^6*b^2*c^2*d^8 -
360*a^7*b*c*d^9))/b^11 + (d^10*x^8)/(8*b^3)