3.13.16 \(\int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx\)

Optimal. Leaf size=271 \[ -\frac {10 d^9 (b c-a d)}{b^{11} (a+b x)}-\frac {45 d^8 (b c-a d)^2}{2 b^{11} (a+b x)^2}-\frac {40 d^7 (b c-a d)^3}{b^{11} (a+b x)^3}-\frac {105 d^6 (b c-a d)^4}{2 b^{11} (a+b x)^4}-\frac {252 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^5}-\frac {35 d^4 (b c-a d)^6}{b^{11} (a+b x)^6}-\frac {120 d^3 (b c-a d)^7}{7 b^{11} (a+b x)^7}-\frac {45 d^2 (b c-a d)^8}{8 b^{11} (a+b x)^8}-\frac {10 d (b c-a d)^9}{9 b^{11} (a+b x)^9}-\frac {(b c-a d)^{10}}{10 b^{11} (a+b x)^{10}}+\frac {d^{10} \log (a+b x)}{b^{11}} \]

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Rubi [A]  time = 0.29, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \begin {gather*} -\frac {10 d^9 (b c-a d)}{b^{11} (a+b x)}-\frac {45 d^8 (b c-a d)^2}{2 b^{11} (a+b x)^2}-\frac {40 d^7 (b c-a d)^3}{b^{11} (a+b x)^3}-\frac {105 d^6 (b c-a d)^4}{2 b^{11} (a+b x)^4}-\frac {252 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^5}-\frac {35 d^4 (b c-a d)^6}{b^{11} (a+b x)^6}-\frac {120 d^3 (b c-a d)^7}{7 b^{11} (a+b x)^7}-\frac {45 d^2 (b c-a d)^8}{8 b^{11} (a+b x)^8}-\frac {10 d (b c-a d)^9}{9 b^{11} (a+b x)^9}-\frac {(b c-a d)^{10}}{10 b^{11} (a+b x)^{10}}+\frac {d^{10} \log (a+b x)}{b^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^11,x]

[Out]

-(b*c - a*d)^10/(10*b^11*(a + b*x)^10) - (10*d*(b*c - a*d)^9)/(9*b^11*(a + b*x)^9) - (45*d^2*(b*c - a*d)^8)/(8
*b^11*(a + b*x)^8) - (120*d^3*(b*c - a*d)^7)/(7*b^11*(a + b*x)^7) - (35*d^4*(b*c - a*d)^6)/(b^11*(a + b*x)^6)
- (252*d^5*(b*c - a*d)^5)/(5*b^11*(a + b*x)^5) - (105*d^6*(b*c - a*d)^4)/(2*b^11*(a + b*x)^4) - (40*d^7*(b*c -
a*d)^3)/(b^11*(a + b*x)^3) - (45*d^8*(b*c - a*d)^2)/(2*b^11*(a + b*x)^2) - (10*d^9*(b*c - a*d))/(b^11*(a + b*
x)) + (d^10*Log[a + b*x])/b^11

Rule 43

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*} \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx &=\int \left (\frac {(b c-a d)^{10}}{b^{10} (a+b x)^{11}}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^{10}}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^9}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^8}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^7}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^6}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)^5}+\frac {120 d^7 (b c-a d)^3}{b^{10} (a+b x)^4}+\frac {45 d^8 (b c-a d)^2}{b^{10} (a+b x)^3}+\frac {10 d^9 (b c-a d)}{b^{10} (a+b x)^2}+\frac {d^{10}}{b^{10} (a+b x)}\right ) \, dx\\ &=-\frac {(b c-a d)^{10}}{10 b^{11} (a+b x)^{10}}-\frac {10 d (b c-a d)^9}{9 b^{11} (a+b x)^9}-\frac {45 d^2 (b c-a d)^8}{8 b^{11} (a+b x)^8}-\frac {120 d^3 (b c-a d)^7}{7 b^{11} (a+b x)^7}-\frac {35 d^4 (b c-a d)^6}{b^{11} (a+b x)^6}-\frac {252 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^5}-\frac {105 d^6 (b c-a d)^4}{2 b^{11} (a+b x)^4}-\frac {40 d^7 (b c-a d)^3}{b^{11} (a+b x)^3}-\frac {45 d^8 (b c-a d)^2}{2 b^{11} (a+b x)^2}-\frac {10 d^9 (b c-a d)}{b^{11} (a+b x)}+\frac {d^{10} \log (a+b x)}{b^{11}}\\ \end {align*}

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Mathematica [B]  time = 0.36, size = 591, normalized size = 2.18 \begin {gather*} \frac {d^{10} \log (a+b x)}{b^{11}}-\frac {(b c-a d) \left (7381 a^9 d^9+a^8 b d^8 (4861 c+71290 d x)+a^7 b^2 d^7 \left (3601 c^2+46090 c d x+308205 d^2 x^2\right )+a^6 b^3 d^6 \left (2761 c^3+33490 c^2 d x+194805 c d^2 x^2+784080 d^3 x^3\right )+a^5 b^4 d^5 \left (2131 c^4+25090 c^3 d x+138105 c^2 d^2 x^2+481680 c d^3 x^3+1296540 d^4 x^4\right )+a^4 b^5 d^4 \left (1627 c^5+18790 c^4 d x+100305 c^3 d^2 x^2+330480 c^2 d^3 x^3+767340 c d^4 x^4+1450008 d^5 x^5\right )+a^3 b^6 d^3 \left (1207 c^6+13750 c^5 d x+71955 c^4 d^2 x^2+229680 c^3 d^3 x^3+502740 c^2 d^4 x^4+814968 c d^5 x^5+1102500 d^6 x^6\right )+a^2 b^7 d^2 \left (847 c^7+9550 c^6 d x+49275 c^5 d^2 x^2+154080 c^4 d^3 x^3+326340 c^3 d^4 x^4+497448 c^2 d^5 x^5+573300 c d^6 x^6+554400 d^7 x^7\right )+a b^8 d \left (532 c^8+5950 c^7 d x+30375 c^6 d^2 x^2+93600 c^5 d^3 x^3+194040 c^4 d^4 x^4+285768 c^3 d^5 x^5+308700 c^2 d^6 x^6+252000 c d^7 x^7+170100 d^8 x^8\right )+b^9 \left (252 c^9+2800 c^8 d x+14175 c^7 d^2 x^2+43200 c^6 d^3 x^3+88200 c^5 d^4 x^4+127008 c^4 d^5 x^5+132300 c^3 d^6 x^6+100800 c^2 d^7 x^7+56700 c d^8 x^8+25200 d^9 x^9\right )\right )}{2520 b^{11} (a+b x)^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^11,x]

[Out]

-1/2520*((b*c - a*d)*(7381*a^9*d^9 + a^8*b*d^8*(4861*c + 71290*d*x) + a^7*b^2*d^7*(3601*c^2 + 46090*c*d*x + 30
8205*d^2*x^2) + a^6*b^3*d^6*(2761*c^3 + 33490*c^2*d*x + 194805*c*d^2*x^2 + 784080*d^3*x^3) + a^5*b^4*d^5*(2131
*c^4 + 25090*c^3*d*x + 138105*c^2*d^2*x^2 + 481680*c*d^3*x^3 + 1296540*d^4*x^4) + a^4*b^5*d^4*(1627*c^5 + 1879
0*c^4*d*x + 100305*c^3*d^2*x^2 + 330480*c^2*d^3*x^3 + 767340*c*d^4*x^4 + 1450008*d^5*x^5) + a^3*b^6*d^3*(1207*
c^6 + 13750*c^5*d*x + 71955*c^4*d^2*x^2 + 229680*c^3*d^3*x^3 + 502740*c^2*d^4*x^4 + 814968*c*d^5*x^5 + 1102500
*d^6*x^6) + a^2*b^7*d^2*(847*c^7 + 9550*c^6*d*x + 49275*c^5*d^2*x^2 + 154080*c^4*d^3*x^3 + 326340*c^3*d^4*x^4
+ 497448*c^2*d^5*x^5 + 573300*c*d^6*x^6 + 554400*d^7*x^7) + a*b^8*d*(532*c^8 + 5950*c^7*d*x + 30375*c^6*d^2*x^
2 + 93600*c^5*d^3*x^3 + 194040*c^4*d^4*x^4 + 285768*c^3*d^5*x^5 + 308700*c^2*d^6*x^6 + 252000*c*d^7*x^7 + 1701
00*d^8*x^8) + b^9*(252*c^9 + 2800*c^8*d*x + 14175*c^7*d^2*x^2 + 43200*c^6*d^3*x^3 + 88200*c^5*d^4*x^4 + 127008
*c^4*d^5*x^5 + 132300*c^3*d^6*x^6 + 100800*c^2*d^7*x^7 + 56700*c*d^8*x^8 + 25200*d^9*x^9)))/(b^11*(a + b*x)^10
) + (d^10*Log[a + b*x])/b^11

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^11,x]

[Out]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^11, x]

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fricas [B]  time = 1.22, size = 1107, normalized size = 4.08 \begin {gather*} -\frac {252 \, b^{10} c^{10} + 280 \, a b^{9} c^{9} d + 315 \, a^{2} b^{8} c^{8} d^{2} + 360 \, a^{3} b^{7} c^{7} d^{3} + 420 \, a^{4} b^{6} c^{6} d^{4} + 504 \, a^{5} b^{5} c^{5} d^{5} + 630 \, a^{6} b^{4} c^{4} d^{6} + 840 \, a^{7} b^{3} c^{3} d^{7} + 1260 \, a^{8} b^{2} c^{2} d^{8} + 2520 \, a^{9} b c d^{9} - 7381 \, a^{10} d^{10} + 25200 \, {\left (b^{10} c d^{9} - a b^{9} d^{10}\right )} x^{9} + 56700 \, {\left (b^{10} c^{2} d^{8} + 2 \, a b^{9} c d^{9} - 3 \, a^{2} b^{8} d^{10}\right )} x^{8} + 50400 \, {\left (2 \, b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} + 6 \, a^{2} b^{8} c d^{9} - 11 \, a^{3} b^{7} d^{10}\right )} x^{7} + 44100 \, {\left (3 \, b^{10} c^{4} d^{6} + 4 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} + 12 \, a^{3} b^{7} c d^{9} - 25 \, a^{4} b^{6} d^{10}\right )} x^{6} + 10584 \, {\left (12 \, b^{10} c^{5} d^{5} + 15 \, a b^{9} c^{4} d^{6} + 20 \, a^{2} b^{8} c^{3} d^{7} + 30 \, a^{3} b^{7} c^{2} d^{8} + 60 \, a^{4} b^{6} c d^{9} - 137 \, a^{5} b^{5} d^{10}\right )} x^{5} + 8820 \, {\left (10 \, b^{10} c^{6} d^{4} + 12 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} + 20 \, a^{3} b^{7} c^{3} d^{7} + 30 \, a^{4} b^{6} c^{2} d^{8} + 60 \, a^{5} b^{5} c d^{9} - 147 \, a^{6} b^{4} d^{10}\right )} x^{4} + 720 \, {\left (60 \, b^{10} c^{7} d^{3} + 70 \, a b^{9} c^{6} d^{4} + 84 \, a^{2} b^{8} c^{5} d^{5} + 105 \, a^{3} b^{7} c^{4} d^{6} + 140 \, a^{4} b^{6} c^{3} d^{7} + 210 \, a^{5} b^{5} c^{2} d^{8} + 420 \, a^{6} b^{4} c d^{9} - 1089 \, a^{7} b^{3} d^{10}\right )} x^{3} + 135 \, {\left (105 \, b^{10} c^{8} d^{2} + 120 \, a b^{9} c^{7} d^{3} + 140 \, a^{2} b^{8} c^{6} d^{4} + 168 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} + 280 \, a^{5} b^{5} c^{3} d^{7} + 420 \, a^{6} b^{4} c^{2} d^{8} + 840 \, a^{7} b^{3} c d^{9} - 2283 \, a^{8} b^{2} d^{10}\right )} x^{2} + 10 \, {\left (280 \, b^{10} c^{9} d + 315 \, a b^{9} c^{8} d^{2} + 360 \, a^{2} b^{8} c^{7} d^{3} + 420 \, a^{3} b^{7} c^{6} d^{4} + 504 \, a^{4} b^{6} c^{5} d^{5} + 630 \, a^{5} b^{5} c^{4} d^{6} + 840 \, a^{6} b^{4} c^{3} d^{7} + 1260 \, a^{7} b^{3} c^{2} d^{8} + 2520 \, a^{8} b^{2} c d^{9} - 7129 \, a^{9} b d^{10}\right )} x - 2520 \, {\left (b^{10} d^{10} x^{10} + 10 \, a b^{9} d^{10} x^{9} + 45 \, a^{2} b^{8} d^{10} x^{8} + 120 \, a^{3} b^{7} d^{10} x^{7} + 210 \, a^{4} b^{6} d^{10} x^{6} + 252 \, a^{5} b^{5} d^{10} x^{5} + 210 \, a^{6} b^{4} d^{10} x^{4} + 120 \, a^{7} b^{3} d^{10} x^{3} + 45 \, a^{8} b^{2} d^{10} x^{2} + 10 \, a^{9} b d^{10} x + a^{10} d^{10}\right )} \log \left (b x + a\right )}{2520 \, {\left (b^{21} x^{10} + 10 \, a b^{20} x^{9} + 45 \, a^{2} b^{19} x^{8} + 120 \, a^{3} b^{18} x^{7} + 210 \, a^{4} b^{17} x^{6} + 252 \, a^{5} b^{16} x^{5} + 210 \, a^{6} b^{15} x^{4} + 120 \, a^{7} b^{14} x^{3} + 45 \, a^{8} b^{13} x^{2} + 10 \, a^{9} b^{12} x + a^{10} b^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^11,x, algorithm="fricas")

[Out]

-1/2520*(252*b^10*c^10 + 280*a*b^9*c^9*d + 315*a^2*b^8*c^8*d^2 + 360*a^3*b^7*c^7*d^3 + 420*a^4*b^6*c^6*d^4 + 5
04*a^5*b^5*c^5*d^5 + 630*a^6*b^4*c^4*d^6 + 840*a^7*b^3*c^3*d^7 + 1260*a^8*b^2*c^2*d^8 + 2520*a^9*b*c*d^9 - 738
1*a^10*d^10 + 25200*(b^10*c*d^9 - a*b^9*d^10)*x^9 + 56700*(b^10*c^2*d^8 + 2*a*b^9*c*d^9 - 3*a^2*b^8*d^10)*x^8
+ 50400*(2*b^10*c^3*d^7 + 3*a*b^9*c^2*d^8 + 6*a^2*b^8*c*d^9 - 11*a^3*b^7*d^10)*x^7 + 44100*(3*b^10*c^4*d^6 + 4
*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 + 12*a^3*b^7*c*d^9 - 25*a^4*b^6*d^10)*x^6 + 10584*(12*b^10*c^5*d^5 + 15*a*b
^9*c^4*d^6 + 20*a^2*b^8*c^3*d^7 + 30*a^3*b^7*c^2*d^8 + 60*a^4*b^6*c*d^9 - 137*a^5*b^5*d^10)*x^5 + 8820*(10*b^1
0*c^6*d^4 + 12*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 20*a^3*b^7*c^3*d^7 + 30*a^4*b^6*c^2*d^8 + 60*a^5*b^5*c*d^9
- 147*a^6*b^4*d^10)*x^4 + 720*(60*b^10*c^7*d^3 + 70*a*b^9*c^6*d^4 + 84*a^2*b^8*c^5*d^5 + 105*a^3*b^7*c^4*d^6
+ 140*a^4*b^6*c^3*d^7 + 210*a^5*b^5*c^2*d^8 + 420*a^6*b^4*c*d^9 - 1089*a^7*b^3*d^10)*x^3 + 135*(105*b^10*c^8*d
^2 + 120*a*b^9*c^7*d^3 + 140*a^2*b^8*c^6*d^4 + 168*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 + 280*a^5*b^5*c^3*d^7
+ 420*a^6*b^4*c^2*d^8 + 840*a^7*b^3*c*d^9 - 2283*a^8*b^2*d^10)*x^2 + 10*(280*b^10*c^9*d + 315*a*b^9*c^8*d^2 +
360*a^2*b^8*c^7*d^3 + 420*a^3*b^7*c^6*d^4 + 504*a^4*b^6*c^5*d^5 + 630*a^5*b^5*c^4*d^6 + 840*a^6*b^4*c^3*d^7 +
1260*a^7*b^3*c^2*d^8 + 2520*a^8*b^2*c*d^9 - 7129*a^9*b*d^10)*x - 2520*(b^10*d^10*x^10 + 10*a*b^9*d^10*x^9 + 4
5*a^2*b^8*d^10*x^8 + 120*a^3*b^7*d^10*x^7 + 210*a^4*b^6*d^10*x^6 + 252*a^5*b^5*d^10*x^5 + 210*a^6*b^4*d^10*x^4
+ 120*a^7*b^3*d^10*x^3 + 45*a^8*b^2*d^10*x^2 + 10*a^9*b*d^10*x + a^10*d^10)*log(b*x + a))/(b^21*x^10 + 10*a*b
^20*x^9 + 45*a^2*b^19*x^8 + 120*a^3*b^18*x^7 + 210*a^4*b^17*x^6 + 252*a^5*b^16*x^5 + 210*a^6*b^15*x^4 + 120*a^
7*b^14*x^3 + 45*a^8*b^13*x^2 + 10*a^9*b^12*x + a^10*b^11)

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giac [B]  time = 1.36, size = 874, normalized size = 3.23 \begin {gather*} \frac {d^{10} \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {25200 \, {\left (b^{9} c d^{9} - a b^{8} d^{10}\right )} x^{9} + 56700 \, {\left (b^{9} c^{2} d^{8} + 2 \, a b^{8} c d^{9} - 3 \, a^{2} b^{7} d^{10}\right )} x^{8} + 50400 \, {\left (2 \, b^{9} c^{3} d^{7} + 3 \, a b^{8} c^{2} d^{8} + 6 \, a^{2} b^{7} c d^{9} - 11 \, a^{3} b^{6} d^{10}\right )} x^{7} + 44100 \, {\left (3 \, b^{9} c^{4} d^{6} + 4 \, a b^{8} c^{3} d^{7} + 6 \, a^{2} b^{7} c^{2} d^{8} + 12 \, a^{3} b^{6} c d^{9} - 25 \, a^{4} b^{5} d^{10}\right )} x^{6} + 10584 \, {\left (12 \, b^{9} c^{5} d^{5} + 15 \, a b^{8} c^{4} d^{6} + 20 \, a^{2} b^{7} c^{3} d^{7} + 30 \, a^{3} b^{6} c^{2} d^{8} + 60 \, a^{4} b^{5} c d^{9} - 137 \, a^{5} b^{4} d^{10}\right )} x^{5} + 8820 \, {\left (10 \, b^{9} c^{6} d^{4} + 12 \, a b^{8} c^{5} d^{5} + 15 \, a^{2} b^{7} c^{4} d^{6} + 20 \, a^{3} b^{6} c^{3} d^{7} + 30 \, a^{4} b^{5} c^{2} d^{8} + 60 \, a^{5} b^{4} c d^{9} - 147 \, a^{6} b^{3} d^{10}\right )} x^{4} + 720 \, {\left (60 \, b^{9} c^{7} d^{3} + 70 \, a b^{8} c^{6} d^{4} + 84 \, a^{2} b^{7} c^{5} d^{5} + 105 \, a^{3} b^{6} c^{4} d^{6} + 140 \, a^{4} b^{5} c^{3} d^{7} + 210 \, a^{5} b^{4} c^{2} d^{8} + 420 \, a^{6} b^{3} c d^{9} - 1089 \, a^{7} b^{2} d^{10}\right )} x^{3} + 135 \, {\left (105 \, b^{9} c^{8} d^{2} + 120 \, a b^{8} c^{7} d^{3} + 140 \, a^{2} b^{7} c^{6} d^{4} + 168 \, a^{3} b^{6} c^{5} d^{5} + 210 \, a^{4} b^{5} c^{4} d^{6} + 280 \, a^{5} b^{4} c^{3} d^{7} + 420 \, a^{6} b^{3} c^{2} d^{8} + 840 \, a^{7} b^{2} c d^{9} - 2283 \, a^{8} b d^{10}\right )} x^{2} + 10 \, {\left (280 \, b^{9} c^{9} d + 315 \, a b^{8} c^{8} d^{2} + 360 \, a^{2} b^{7} c^{7} d^{3} + 420 \, a^{3} b^{6} c^{6} d^{4} + 504 \, a^{4} b^{5} c^{5} d^{5} + 630 \, a^{5} b^{4} c^{4} d^{6} + 840 \, a^{6} b^{3} c^{3} d^{7} + 1260 \, a^{7} b^{2} c^{2} d^{8} + 2520 \, a^{8} b c d^{9} - 7129 \, a^{9} d^{10}\right )} x + \frac {252 \, b^{10} c^{10} + 280 \, a b^{9} c^{9} d + 315 \, a^{2} b^{8} c^{8} d^{2} + 360 \, a^{3} b^{7} c^{7} d^{3} + 420 \, a^{4} b^{6} c^{6} d^{4} + 504 \, a^{5} b^{5} c^{5} d^{5} + 630 \, a^{6} b^{4} c^{4} d^{6} + 840 \, a^{7} b^{3} c^{3} d^{7} + 1260 \, a^{8} b^{2} c^{2} d^{8} + 2520 \, a^{9} b c d^{9} - 7381 \, a^{10} d^{10}}{b}}{2520 \, {\left (b x + a\right )}^{10} b^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^11,x, algorithm="giac")

[Out]

d^10*log(abs(b*x + a))/b^11 - 1/2520*(25200*(b^9*c*d^9 - a*b^8*d^10)*x^9 + 56700*(b^9*c^2*d^8 + 2*a*b^8*c*d^9
- 3*a^2*b^7*d^10)*x^8 + 50400*(2*b^9*c^3*d^7 + 3*a*b^8*c^2*d^8 + 6*a^2*b^7*c*d^9 - 11*a^3*b^6*d^10)*x^7 + 4410
0*(3*b^9*c^4*d^6 + 4*a*b^8*c^3*d^7 + 6*a^2*b^7*c^2*d^8 + 12*a^3*b^6*c*d^9 - 25*a^4*b^5*d^10)*x^6 + 10584*(12*b
^9*c^5*d^5 + 15*a*b^8*c^4*d^6 + 20*a^2*b^7*c^3*d^7 + 30*a^3*b^6*c^2*d^8 + 60*a^4*b^5*c*d^9 - 137*a^5*b^4*d^10)
*x^5 + 8820*(10*b^9*c^6*d^4 + 12*a*b^8*c^5*d^5 + 15*a^2*b^7*c^4*d^6 + 20*a^3*b^6*c^3*d^7 + 30*a^4*b^5*c^2*d^8
+ 60*a^5*b^4*c*d^9 - 147*a^6*b^3*d^10)*x^4 + 720*(60*b^9*c^7*d^3 + 70*a*b^8*c^6*d^4 + 84*a^2*b^7*c^5*d^5 + 105
*a^3*b^6*c^4*d^6 + 140*a^4*b^5*c^3*d^7 + 210*a^5*b^4*c^2*d^8 + 420*a^6*b^3*c*d^9 - 1089*a^7*b^2*d^10)*x^3 + 13
5*(105*b^9*c^8*d^2 + 120*a*b^8*c^7*d^3 + 140*a^2*b^7*c^6*d^4 + 168*a^3*b^6*c^5*d^5 + 210*a^4*b^5*c^4*d^6 + 280
*a^5*b^4*c^3*d^7 + 420*a^6*b^3*c^2*d^8 + 840*a^7*b^2*c*d^9 - 2283*a^8*b*d^10)*x^2 + 10*(280*b^9*c^9*d + 315*a*
b^8*c^8*d^2 + 360*a^2*b^7*c^7*d^3 + 420*a^3*b^6*c^6*d^4 + 504*a^4*b^5*c^5*d^5 + 630*a^5*b^4*c^4*d^6 + 840*a^6*
b^3*c^3*d^7 + 1260*a^7*b^2*c^2*d^8 + 2520*a^8*b*c*d^9 - 7129*a^9*d^10)*x + (252*b^10*c^10 + 280*a*b^9*c^9*d +
315*a^2*b^8*c^8*d^2 + 360*a^3*b^7*c^7*d^3 + 420*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^5*d^5 + 630*a^6*b^4*c^4*d^6 +
840*a^7*b^3*c^3*d^7 + 1260*a^8*b^2*c^2*d^8 + 2520*a^9*b*c*d^9 - 7381*a^10*d^10)/b)/((b*x + a)^10*b^10)

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maple [B]  time = 0.01, size = 1271, normalized size = 4.69

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^11,x)

[Out]

-35*d^10/b^11/(b*x+a)^6*a^6-35*d^4/b^5/(b*x+a)^6*c^6-1/10/b^11/(b*x+a)^10*a^10*d^10-120/7*d^3/b^4/(b*x+a)^7*c^
7+10/9*d^10/b^11/(b*x+a)^9*a^9-10/9*d/b^2/(b*x+a)^9*c^9+252/5*d^10/b^11/(b*x+a)^5*a^5-252/5*d^5/b^6/(b*x+a)^5*
c^5-45/2*d^10/b^11/(b*x+a)^2*a^2-45/2*d^8/b^9/(b*x+a)^2*c^2-105/2*d^10/b^11/(b*x+a)^4*a^4-105/2*d^6/b^7/(b*x+a
)^4*c^4+10/b^11*d^10/(b*x+a)*a-10/b^10*d^9/(b*x+a)*c-45/8*d^10/b^11/(b*x+a)^8*a^8-45/8*d^2/b^3/(b*x+a)^8*c^8+4
0*d^10/b^11/(b*x+a)^3*a^3-40*d^7/b^8/(b*x+a)^3*c^3+120/7*d^10/b^11/(b*x+a)^7*a^7-315/2*d^4/b^5/(b*x+a)^8*a^2*c
^6+45*d^3/b^4/(b*x+a)^8*a*c^7-120*d^9/b^10/(b*x+a)^3*a^2*c+120*d^8/b^9/(b*x+a)^3*a*c^2-120*d^9/b^10/(b*x+a)^7*
a^6*c+360*d^8/b^9/(b*x+a)^7*a^5*c^2-600*d^7/b^8/(b*x+a)^7*a^4*c^3+600*d^6/b^7/(b*x+a)^7*a^3*c^4-360*d^5/b^6/(b
*x+a)^7*a^2*c^5+120*d^4/b^5/(b*x+a)^7*a*c^6-10*d^9/b^10/(b*x+a)^9*a^8*c+40*d^8/b^9/(b*x+a)^9*a^7*c^2-280/3*d^7
/b^8/(b*x+a)^9*a^6*c^3+140*d^6/b^7/(b*x+a)^9*a^5*c^4-140*d^5/b^6/(b*x+a)^9*a^4*c^5+280/3*d^4/b^5/(b*x+a)^9*a^3
*c^6-40*d^3/b^4/(b*x+a)^9*a^2*c^7+10*d^2/b^3/(b*x+a)^9*a*c^8-252*d^9/b^10/(b*x+a)^5*a^4*c+504*d^8/b^9/(b*x+a)^
5*a^3*c^2-504*d^7/b^8/(b*x+a)^5*a^2*c^3+252*d^6/b^7/(b*x+a)^5*a*c^4+45*d^9/b^10/(b*x+a)^2*a*c+210*d^9/b^10/(b*
x+a)^4*a^3*c-315*d^8/b^9/(b*x+a)^4*a^2*c^2+210*d^7/b^8/(b*x+a)^4*a*c^3+210*d^9/b^10/(b*x+a)^6*a^5*c-525*d^8/b^
9/(b*x+a)^6*a^4*c^2+700*d^7/b^8/(b*x+a)^6*a^3*c^3-525*d^6/b^7/(b*x+a)^6*a^2*c^4+210*d^5/b^6/(b*x+a)^6*a*c^5+1/
b^10/(b*x+a)^10*a^9*c*d^9-9/2/b^9/(b*x+a)^10*a^8*c^2*d^8+12/b^8/(b*x+a)^10*a^7*c^3*d^7-21/b^7/(b*x+a)^10*a^6*c
^4*d^6+126/5/b^6/(b*x+a)^10*a^5*c^5*d^5-21/b^5/(b*x+a)^10*a^4*c^6*d^4+12/b^4/(b*x+a)^10*a^3*c^7*d^3-9/2/b^3/(b
*x+a)^10*a^2*c^8*d^2+1/b^2/(b*x+a)^10*a*c^9*d+45*d^9/b^10/(b*x+a)^8*a^7*c-315/2*d^8/b^9/(b*x+a)^8*a^6*c^2+315*
d^5/b^6/(b*x+a)^8*a^3*c^5+d^10*ln(b*x+a)/b^11-1/10/b/(b*x+a)^10*c^10+315*d^7/b^8/(b*x+a)^8*a^5*c^3-1575/4*d^6/
b^7/(b*x+a)^8*a^4*c^4

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maxima [B]  time = 2.01, size = 975, normalized size = 3.60 \begin {gather*} -\frac {252 \, b^{10} c^{10} + 280 \, a b^{9} c^{9} d + 315 \, a^{2} b^{8} c^{8} d^{2} + 360 \, a^{3} b^{7} c^{7} d^{3} + 420 \, a^{4} b^{6} c^{6} d^{4} + 504 \, a^{5} b^{5} c^{5} d^{5} + 630 \, a^{6} b^{4} c^{4} d^{6} + 840 \, a^{7} b^{3} c^{3} d^{7} + 1260 \, a^{8} b^{2} c^{2} d^{8} + 2520 \, a^{9} b c d^{9} - 7381 \, a^{10} d^{10} + 25200 \, {\left (b^{10} c d^{9} - a b^{9} d^{10}\right )} x^{9} + 56700 \, {\left (b^{10} c^{2} d^{8} + 2 \, a b^{9} c d^{9} - 3 \, a^{2} b^{8} d^{10}\right )} x^{8} + 50400 \, {\left (2 \, b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} + 6 \, a^{2} b^{8} c d^{9} - 11 \, a^{3} b^{7} d^{10}\right )} x^{7} + 44100 \, {\left (3 \, b^{10} c^{4} d^{6} + 4 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} + 12 \, a^{3} b^{7} c d^{9} - 25 \, a^{4} b^{6} d^{10}\right )} x^{6} + 10584 \, {\left (12 \, b^{10} c^{5} d^{5} + 15 \, a b^{9} c^{4} d^{6} + 20 \, a^{2} b^{8} c^{3} d^{7} + 30 \, a^{3} b^{7} c^{2} d^{8} + 60 \, a^{4} b^{6} c d^{9} - 137 \, a^{5} b^{5} d^{10}\right )} x^{5} + 8820 \, {\left (10 \, b^{10} c^{6} d^{4} + 12 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} + 20 \, a^{3} b^{7} c^{3} d^{7} + 30 \, a^{4} b^{6} c^{2} d^{8} + 60 \, a^{5} b^{5} c d^{9} - 147 \, a^{6} b^{4} d^{10}\right )} x^{4} + 720 \, {\left (60 \, b^{10} c^{7} d^{3} + 70 \, a b^{9} c^{6} d^{4} + 84 \, a^{2} b^{8} c^{5} d^{5} + 105 \, a^{3} b^{7} c^{4} d^{6} + 140 \, a^{4} b^{6} c^{3} d^{7} + 210 \, a^{5} b^{5} c^{2} d^{8} + 420 \, a^{6} b^{4} c d^{9} - 1089 \, a^{7} b^{3} d^{10}\right )} x^{3} + 135 \, {\left (105 \, b^{10} c^{8} d^{2} + 120 \, a b^{9} c^{7} d^{3} + 140 \, a^{2} b^{8} c^{6} d^{4} + 168 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} + 280 \, a^{5} b^{5} c^{3} d^{7} + 420 \, a^{6} b^{4} c^{2} d^{8} + 840 \, a^{7} b^{3} c d^{9} - 2283 \, a^{8} b^{2} d^{10}\right )} x^{2} + 10 \, {\left (280 \, b^{10} c^{9} d + 315 \, a b^{9} c^{8} d^{2} + 360 \, a^{2} b^{8} c^{7} d^{3} + 420 \, a^{3} b^{7} c^{6} d^{4} + 504 \, a^{4} b^{6} c^{5} d^{5} + 630 \, a^{5} b^{5} c^{4} d^{6} + 840 \, a^{6} b^{4} c^{3} d^{7} + 1260 \, a^{7} b^{3} c^{2} d^{8} + 2520 \, a^{8} b^{2} c d^{9} - 7129 \, a^{9} b d^{10}\right )} x}{2520 \, {\left (b^{21} x^{10} + 10 \, a b^{20} x^{9} + 45 \, a^{2} b^{19} x^{8} + 120 \, a^{3} b^{18} x^{7} + 210 \, a^{4} b^{17} x^{6} + 252 \, a^{5} b^{16} x^{5} + 210 \, a^{6} b^{15} x^{4} + 120 \, a^{7} b^{14} x^{3} + 45 \, a^{8} b^{13} x^{2} + 10 \, a^{9} b^{12} x + a^{10} b^{11}\right )}} + \frac {d^{10} \log \left (b x + a\right )}{b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^11,x, algorithm="maxima")

[Out]

-1/2520*(252*b^10*c^10 + 280*a*b^9*c^9*d + 315*a^2*b^8*c^8*d^2 + 360*a^3*b^7*c^7*d^3 + 420*a^4*b^6*c^6*d^4 + 5
04*a^5*b^5*c^5*d^5 + 630*a^6*b^4*c^4*d^6 + 840*a^7*b^3*c^3*d^7 + 1260*a^8*b^2*c^2*d^8 + 2520*a^9*b*c*d^9 - 738
1*a^10*d^10 + 25200*(b^10*c*d^9 - a*b^9*d^10)*x^9 + 56700*(b^10*c^2*d^8 + 2*a*b^9*c*d^9 - 3*a^2*b^8*d^10)*x^8
+ 50400*(2*b^10*c^3*d^7 + 3*a*b^9*c^2*d^8 + 6*a^2*b^8*c*d^9 - 11*a^3*b^7*d^10)*x^7 + 44100*(3*b^10*c^4*d^6 + 4
*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 + 12*a^3*b^7*c*d^9 - 25*a^4*b^6*d^10)*x^6 + 10584*(12*b^10*c^5*d^5 + 15*a*b
^9*c^4*d^6 + 20*a^2*b^8*c^3*d^7 + 30*a^3*b^7*c^2*d^8 + 60*a^4*b^6*c*d^9 - 137*a^5*b^5*d^10)*x^5 + 8820*(10*b^1
0*c^6*d^4 + 12*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 20*a^3*b^7*c^3*d^7 + 30*a^4*b^6*c^2*d^8 + 60*a^5*b^5*c*d^9
- 147*a^6*b^4*d^10)*x^4 + 720*(60*b^10*c^7*d^3 + 70*a*b^9*c^6*d^4 + 84*a^2*b^8*c^5*d^5 + 105*a^3*b^7*c^4*d^6
+ 140*a^4*b^6*c^3*d^7 + 210*a^5*b^5*c^2*d^8 + 420*a^6*b^4*c*d^9 - 1089*a^7*b^3*d^10)*x^3 + 135*(105*b^10*c^8*d
^2 + 120*a*b^9*c^7*d^3 + 140*a^2*b^8*c^6*d^4 + 168*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 + 280*a^5*b^5*c^3*d^7
+ 420*a^6*b^4*c^2*d^8 + 840*a^7*b^3*c*d^9 - 2283*a^8*b^2*d^10)*x^2 + 10*(280*b^10*c^9*d + 315*a*b^9*c^8*d^2 +
360*a^2*b^8*c^7*d^3 + 420*a^3*b^7*c^6*d^4 + 504*a^4*b^6*c^5*d^5 + 630*a^5*b^5*c^4*d^6 + 840*a^6*b^4*c^3*d^7 +
1260*a^7*b^3*c^2*d^8 + 2520*a^8*b^2*c*d^9 - 7129*a^9*b*d^10)*x)/(b^21*x^10 + 10*a*b^20*x^9 + 45*a^2*b^19*x^8
+ 120*a^3*b^18*x^7 + 210*a^4*b^17*x^6 + 252*a^5*b^16*x^5 + 210*a^6*b^15*x^4 + 120*a^7*b^14*x^3 + 45*a^8*b^13*x
^2 + 10*a^9*b^12*x + a^10*b^11) + d^10*log(b*x + a)/b^11

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mupad [B]  time = 0.56, size = 866, normalized size = 3.20 \begin {gather*} \frac {d^{10}\,\ln \left (a+b\,x\right )}{b^{11}}-\frac {x^4\,\left (-\frac {1029\,a^6\,b^4\,d^{10}}{2}+210\,a^5\,b^5\,c\,d^9+105\,a^4\,b^6\,c^2\,d^8+70\,a^3\,b^7\,c^3\,d^7+\frac {105\,a^2\,b^8\,c^4\,d^6}{2}+42\,a\,b^9\,c^5\,d^5+35\,b^{10}\,c^6\,d^4\right )-x^9\,\left (10\,a\,b^9\,d^{10}-10\,b^{10}\,c\,d^9\right )+x\,\left (-\frac {7129\,a^9\,b\,d^{10}}{252}+10\,a^8\,b^2\,c\,d^9+5\,a^7\,b^3\,c^2\,d^8+\frac {10\,a^6\,b^4\,c^3\,d^7}{3}+\frac {5\,a^5\,b^5\,c^4\,d^6}{2}+2\,a^4\,b^6\,c^5\,d^5+\frac {5\,a^3\,b^7\,c^6\,d^4}{3}+\frac {10\,a^2\,b^8\,c^7\,d^3}{7}+\frac {5\,a\,b^9\,c^8\,d^2}{4}+\frac {10\,b^{10}\,c^9\,d}{9}\right )+x^6\,\left (-\frac {875\,a^4\,b^6\,d^{10}}{2}+210\,a^3\,b^7\,c\,d^9+105\,a^2\,b^8\,c^2\,d^8+70\,a\,b^9\,c^3\,d^7+\frac {105\,b^{10}\,c^4\,d^6}{2}\right )+x^8\,\left (-\frac {135\,a^2\,b^8\,d^{10}}{2}+45\,a\,b^9\,c\,d^9+\frac {45\,b^{10}\,c^2\,d^8}{2}\right )+x^3\,\left (-\frac {2178\,a^7\,b^3\,d^{10}}{7}+120\,a^6\,b^4\,c\,d^9+60\,a^5\,b^5\,c^2\,d^8+40\,a^4\,b^6\,c^3\,d^7+30\,a^3\,b^7\,c^4\,d^6+24\,a^2\,b^8\,c^5\,d^5+20\,a\,b^9\,c^6\,d^4+\frac {120\,b^{10}\,c^7\,d^3}{7}\right )+x^5\,\left (-\frac {2877\,a^5\,b^5\,d^{10}}{5}+252\,a^4\,b^6\,c\,d^9+126\,a^3\,b^7\,c^2\,d^8+84\,a^2\,b^8\,c^3\,d^7+63\,a\,b^9\,c^4\,d^6+\frac {252\,b^{10}\,c^5\,d^5}{5}\right )-\frac {7381\,a^{10}\,d^{10}}{2520}+\frac {b^{10}\,c^{10}}{10}+x^7\,\left (-220\,a^3\,b^7\,d^{10}+120\,a^2\,b^8\,c\,d^9+60\,a\,b^9\,c^2\,d^8+40\,b^{10}\,c^3\,d^7\right )+x^2\,\left (-\frac {6849\,a^8\,b^2\,d^{10}}{56}+45\,a^7\,b^3\,c\,d^9+\frac {45\,a^6\,b^4\,c^2\,d^8}{2}+15\,a^5\,b^5\,c^3\,d^7+\frac {45\,a^4\,b^6\,c^4\,d^6}{4}+9\,a^3\,b^7\,c^5\,d^5+\frac {15\,a^2\,b^8\,c^6\,d^4}{2}+\frac {45\,a\,b^9\,c^7\,d^3}{7}+\frac {45\,b^{10}\,c^8\,d^2}{8}\right )+\frac {a^2\,b^8\,c^8\,d^2}{8}+\frac {a^3\,b^7\,c^7\,d^3}{7}+\frac {a^4\,b^6\,c^6\,d^4}{6}+\frac {a^5\,b^5\,c^5\,d^5}{5}+\frac {a^6\,b^4\,c^4\,d^6}{4}+\frac {a^7\,b^3\,c^3\,d^7}{3}+\frac {a^8\,b^2\,c^2\,d^8}{2}+\frac {a\,b^9\,c^9\,d}{9}+a^9\,b\,c\,d^9}{b^{11}\,{\left (a+b\,x\right )}^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^11,x)

[Out]

(d^10*log(a + b*x))/b^11 - (x^4*(35*b^10*c^6*d^4 - (1029*a^6*b^4*d^10)/2 + 42*a*b^9*c^5*d^5 + 210*a^5*b^5*c*d^
9 + (105*a^2*b^8*c^4*d^6)/2 + 70*a^3*b^7*c^3*d^7 + 105*a^4*b^6*c^2*d^8) - x^9*(10*a*b^9*d^10 - 10*b^10*c*d^9)
+ x*((10*b^10*c^9*d)/9 - (7129*a^9*b*d^10)/252 + (5*a*b^9*c^8*d^2)/4 + 10*a^8*b^2*c*d^9 + (10*a^2*b^8*c^7*d^3)
/7 + (5*a^3*b^7*c^6*d^4)/3 + 2*a^4*b^6*c^5*d^5 + (5*a^5*b^5*c^4*d^6)/2 + (10*a^6*b^4*c^3*d^7)/3 + 5*a^7*b^3*c^
2*d^8) + x^6*((105*b^10*c^4*d^6)/2 - (875*a^4*b^6*d^10)/2 + 70*a*b^9*c^3*d^7 + 210*a^3*b^7*c*d^9 + 105*a^2*b^8
*c^2*d^8) + x^8*((45*b^10*c^2*d^8)/2 - (135*a^2*b^8*d^10)/2 + 45*a*b^9*c*d^9) + x^3*((120*b^10*c^7*d^3)/7 - (2
178*a^7*b^3*d^10)/7 + 20*a*b^9*c^6*d^4 + 120*a^6*b^4*c*d^9 + 24*a^2*b^8*c^5*d^5 + 30*a^3*b^7*c^4*d^6 + 40*a^4*
b^6*c^3*d^7 + 60*a^5*b^5*c^2*d^8) + x^5*((252*b^10*c^5*d^5)/5 - (2877*a^5*b^5*d^10)/5 + 63*a*b^9*c^4*d^6 + 252
*a^4*b^6*c*d^9 + 84*a^2*b^8*c^3*d^7 + 126*a^3*b^7*c^2*d^8) - (7381*a^10*d^10)/2520 + (b^10*c^10)/10 + x^7*(40*
b^10*c^3*d^7 - 220*a^3*b^7*d^10 + 60*a*b^9*c^2*d^8 + 120*a^2*b^8*c*d^9) + x^2*((45*b^10*c^8*d^2)/8 - (6849*a^8
*b^2*d^10)/56 + (45*a*b^9*c^7*d^3)/7 + 45*a^7*b^3*c*d^9 + (15*a^2*b^8*c^6*d^4)/2 + 9*a^3*b^7*c^5*d^5 + (45*a^4
*b^6*c^4*d^6)/4 + 15*a^5*b^5*c^3*d^7 + (45*a^6*b^4*c^2*d^8)/2) + (a^2*b^8*c^8*d^2)/8 + (a^3*b^7*c^7*d^3)/7 + (
a^4*b^6*c^6*d^4)/6 + (a^5*b^5*c^5*d^5)/5 + (a^6*b^4*c^4*d^6)/4 + (a^7*b^3*c^3*d^7)/3 + (a^8*b^2*c^2*d^8)/2 + (
a*b^9*c^9*d)/9 + a^9*b*c*d^9)/(b^11*(a + b*x)^10)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**10/(b*x+a)**11,x)

[Out]

Timed out

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