3.13.26 \(\int \frac {(c+d x)^{10}}{(a+b x)^{21}} \, dx\)

Optimal. Leaf size=279 \[ -\frac {10 d^9 (b c-a d)}{11 b^{11} (a+b x)^{11}}-\frac {15 d^8 (b c-a d)^2}{4 b^{11} (a+b x)^{12}}-\frac {120 d^7 (b c-a d)^3}{13 b^{11} (a+b x)^{13}}-\frac {15 d^6 (b c-a d)^4}{b^{11} (a+b x)^{14}}-\frac {84 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^{15}}-\frac {105 d^4 (b c-a d)^6}{8 b^{11} (a+b x)^{16}}-\frac {120 d^3 (b c-a d)^7}{17 b^{11} (a+b x)^{17}}-\frac {5 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^{18}}-\frac {10 d (b c-a d)^9}{19 b^{11} (a+b x)^{19}}-\frac {(b c-a d)^{10}}{20 b^{11} (a+b x)^{20}}-\frac {d^{10}}{10 b^{11} (a+b x)^{10}} \]

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Rubi [A]  time = 0.27, antiderivative size = 279, 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)}{11 b^{11} (a+b x)^{11}}-\frac {15 d^8 (b c-a d)^2}{4 b^{11} (a+b x)^{12}}-\frac {120 d^7 (b c-a d)^3}{13 b^{11} (a+b x)^{13}}-\frac {15 d^6 (b c-a d)^4}{b^{11} (a+b x)^{14}}-\frac {84 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^{15}}-\frac {105 d^4 (b c-a d)^6}{8 b^{11} (a+b x)^{16}}-\frac {120 d^3 (b c-a d)^7}{17 b^{11} (a+b x)^{17}}-\frac {5 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^{18}}-\frac {10 d (b c-a d)^9}{19 b^{11} (a+b x)^{19}}-\frac {(b c-a d)^{10}}{20 b^{11} (a+b x)^{20}}-\frac {d^{10}}{10 b^{11} (a+b x)^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^21,x]

[Out]

-(b*c - a*d)^10/(20*b^11*(a + b*x)^20) - (10*d*(b*c - a*d)^9)/(19*b^11*(a + b*x)^19) - (5*d^2*(b*c - a*d)^8)/(
2*b^11*(a + b*x)^18) - (120*d^3*(b*c - a*d)^7)/(17*b^11*(a + b*x)^17) - (105*d^4*(b*c - a*d)^6)/(8*b^11*(a + b
*x)^16) - (84*d^5*(b*c - a*d)^5)/(5*b^11*(a + b*x)^15) - (15*d^6*(b*c - a*d)^4)/(b^11*(a + b*x)^14) - (120*d^7
*(b*c - a*d)^3)/(13*b^11*(a + b*x)^13) - (15*d^8*(b*c - a*d)^2)/(4*b^11*(a + b*x)^12) - (10*d^9*(b*c - a*d))/(
11*b^11*(a + b*x)^11) - d^10/(10*b^11*(a + b*x)^10)

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)^{21}} \, dx &=\int \left (\frac {(b c-a d)^{10}}{b^{10} (a+b x)^{21}}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^{20}}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^{19}}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^{18}}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^{17}}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^{16}}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)^{15}}+\frac {120 d^7 (b c-a d)^3}{b^{10} (a+b x)^{14}}+\frac {45 d^8 (b c-a d)^2}{b^{10} (a+b x)^{13}}+\frac {10 d^9 (b c-a d)}{b^{10} (a+b x)^{12}}+\frac {d^{10}}{b^{10} (a+b x)^{11}}\right ) \, dx\\ &=-\frac {(b c-a d)^{10}}{20 b^{11} (a+b x)^{20}}-\frac {10 d (b c-a d)^9}{19 b^{11} (a+b x)^{19}}-\frac {5 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^{18}}-\frac {120 d^3 (b c-a d)^7}{17 b^{11} (a+b x)^{17}}-\frac {105 d^4 (b c-a d)^6}{8 b^{11} (a+b x)^{16}}-\frac {84 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^{15}}-\frac {15 d^6 (b c-a d)^4}{b^{11} (a+b x)^{14}}-\frac {120 d^7 (b c-a d)^3}{13 b^{11} (a+b x)^{13}}-\frac {15 d^8 (b c-a d)^2}{4 b^{11} (a+b x)^{12}}-\frac {10 d^9 (b c-a d)}{11 b^{11} (a+b x)^{11}}-\frac {d^{10}}{10 b^{11} (a+b x)^{10}}\\ \end {align*}

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Mathematica [B]  time = 0.29, size = 692, normalized size = 2.48 \begin {gather*} -\frac {a^{10} d^{10}+10 a^9 b d^9 (c+2 d x)+5 a^8 b^2 d^8 \left (11 c^2+40 c d x+38 d^2 x^2\right )+20 a^7 b^3 d^7 \left (11 c^3+55 c^2 d x+95 c d^2 x^2+57 d^3 x^3\right )+5 a^6 b^4 d^6 \left (143 c^4+880 c^3 d x+2090 c^2 d^2 x^2+2280 c d^3 x^3+969 d^4 x^4\right )+2 a^5 b^5 d^5 \left (1001 c^5+7150 c^4 d x+20900 c^3 d^2 x^2+31350 c^2 d^3 x^3+24225 c d^4 x^4+7752 d^5 x^5\right )+5 a^4 b^6 d^4 \left (1001 c^6+8008 c^5 d x+27170 c^4 d^2 x^2+50160 c^3 d^3 x^3+53295 c^2 d^4 x^4+31008 c d^5 x^5+7752 d^6 x^6\right )+20 a^3 b^7 d^3 \left (572 c^7+5005 c^6 d x+19019 c^5 d^2 x^2+40755 c^4 d^3 x^3+53295 c^3 d^4 x^4+42636 c^2 d^5 x^5+19380 c d^6 x^6+3876 d^7 x^7\right )+5 a^2 b^8 d^2 \left (4862 c^8+45760 c^7 d x+190190 c^6 d^2 x^2+456456 c^5 d^3 x^3+692835 c^4 d^4 x^4+682176 c^3 d^5 x^5+426360 c^2 d^6 x^6+155040 c d^7 x^7+25194 d^8 x^8\right )+10 a b^9 d \left (4862 c^9+48620 c^8 d x+217360 c^7 d^2 x^2+570570 c^6 d^3 x^3+969969 c^5 d^4 x^4+1108536 c^4 d^5 x^5+852720 c^3 d^6 x^6+426360 c^2 d^7 x^7+125970 c d^8 x^8+16796 d^9 x^9\right )+b^{10} \left (92378 c^{10}+972400 c^9 d x+4618900 c^8 d^2 x^2+13041600 c^7 d^3 x^3+24249225 c^6 d^4 x^4+31039008 c^5 d^5 x^5+27713400 c^4 d^6 x^6+17054400 c^3 d^7 x^7+6928350 c^2 d^8 x^8+1679600 c d^9 x^9+184756 d^{10} x^{10}\right )}{1847560 b^{11} (a+b x)^{20}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^21,x]

[Out]

-1/1847560*(a^10*d^10 + 10*a^9*b*d^9*(c + 2*d*x) + 5*a^8*b^2*d^8*(11*c^2 + 40*c*d*x + 38*d^2*x^2) + 20*a^7*b^3
*d^7*(11*c^3 + 55*c^2*d*x + 95*c*d^2*x^2 + 57*d^3*x^3) + 5*a^6*b^4*d^6*(143*c^4 + 880*c^3*d*x + 2090*c^2*d^2*x
^2 + 2280*c*d^3*x^3 + 969*d^4*x^4) + 2*a^5*b^5*d^5*(1001*c^5 + 7150*c^4*d*x + 20900*c^3*d^2*x^2 + 31350*c^2*d^
3*x^3 + 24225*c*d^4*x^4 + 7752*d^5*x^5) + 5*a^4*b^6*d^4*(1001*c^6 + 8008*c^5*d*x + 27170*c^4*d^2*x^2 + 50160*c
^3*d^3*x^3 + 53295*c^2*d^4*x^4 + 31008*c*d^5*x^5 + 7752*d^6*x^6) + 20*a^3*b^7*d^3*(572*c^7 + 5005*c^6*d*x + 19
019*c^5*d^2*x^2 + 40755*c^4*d^3*x^3 + 53295*c^3*d^4*x^4 + 42636*c^2*d^5*x^5 + 19380*c*d^6*x^6 + 3876*d^7*x^7)
+ 5*a^2*b^8*d^2*(4862*c^8 + 45760*c^7*d*x + 190190*c^6*d^2*x^2 + 456456*c^5*d^3*x^3 + 692835*c^4*d^4*x^4 + 682
176*c^3*d^5*x^5 + 426360*c^2*d^6*x^6 + 155040*c*d^7*x^7 + 25194*d^8*x^8) + 10*a*b^9*d*(4862*c^9 + 48620*c^8*d*
x + 217360*c^7*d^2*x^2 + 570570*c^6*d^3*x^3 + 969969*c^5*d^4*x^4 + 1108536*c^4*d^5*x^5 + 852720*c^3*d^6*x^6 +
426360*c^2*d^7*x^7 + 125970*c*d^8*x^8 + 16796*d^9*x^9) + b^10*(92378*c^10 + 972400*c^9*d*x + 4618900*c^8*d^2*x
^2 + 13041600*c^7*d^3*x^3 + 24249225*c^6*d^4*x^4 + 31039008*c^5*d^5*x^5 + 27713400*c^4*d^6*x^6 + 17054400*c^3*
d^7*x^7 + 6928350*c^2*d^8*x^8 + 1679600*c*d^9*x^9 + 184756*d^10*x^10))/(b^11*(a + b*x)^20)

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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)^{21}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^21,x]

[Out]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^21, x]

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fricas [B]  time = 1.23, size = 1074, normalized size = 3.85 \begin {gather*} -\frac {184756 \, b^{10} d^{10} x^{10} + 92378 \, b^{10} c^{10} + 48620 \, a b^{9} c^{9} d + 24310 \, a^{2} b^{8} c^{8} d^{2} + 11440 \, a^{3} b^{7} c^{7} d^{3} + 5005 \, a^{4} b^{6} c^{6} d^{4} + 2002 \, a^{5} b^{5} c^{5} d^{5} + 715 \, a^{6} b^{4} c^{4} d^{6} + 220 \, a^{7} b^{3} c^{3} d^{7} + 55 \, a^{8} b^{2} c^{2} d^{8} + 10 \, a^{9} b c d^{9} + a^{10} d^{10} + 167960 \, {\left (10 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 125970 \, {\left (55 \, b^{10} c^{2} d^{8} + 10 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 77520 \, {\left (220 \, b^{10} c^{3} d^{7} + 55 \, a b^{9} c^{2} d^{8} + 10 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 38760 \, {\left (715 \, b^{10} c^{4} d^{6} + 220 \, a b^{9} c^{3} d^{7} + 55 \, a^{2} b^{8} c^{2} d^{8} + 10 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 15504 \, {\left (2002 \, b^{10} c^{5} d^{5} + 715 \, a b^{9} c^{4} d^{6} + 220 \, a^{2} b^{8} c^{3} d^{7} + 55 \, a^{3} b^{7} c^{2} d^{8} + 10 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 4845 \, {\left (5005 \, b^{10} c^{6} d^{4} + 2002 \, a b^{9} c^{5} d^{5} + 715 \, a^{2} b^{8} c^{4} d^{6} + 220 \, a^{3} b^{7} c^{3} d^{7} + 55 \, a^{4} b^{6} c^{2} d^{8} + 10 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 1140 \, {\left (11440 \, b^{10} c^{7} d^{3} + 5005 \, a b^{9} c^{6} d^{4} + 2002 \, a^{2} b^{8} c^{5} d^{5} + 715 \, a^{3} b^{7} c^{4} d^{6} + 220 \, a^{4} b^{6} c^{3} d^{7} + 55 \, a^{5} b^{5} c^{2} d^{8} + 10 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 190 \, {\left (24310 \, b^{10} c^{8} d^{2} + 11440 \, a b^{9} c^{7} d^{3} + 5005 \, a^{2} b^{8} c^{6} d^{4} + 2002 \, a^{3} b^{7} c^{5} d^{5} + 715 \, a^{4} b^{6} c^{4} d^{6} + 220 \, a^{5} b^{5} c^{3} d^{7} + 55 \, a^{6} b^{4} c^{2} d^{8} + 10 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 20 \, {\left (48620 \, b^{10} c^{9} d + 24310 \, a b^{9} c^{8} d^{2} + 11440 \, a^{2} b^{8} c^{7} d^{3} + 5005 \, a^{3} b^{7} c^{6} d^{4} + 2002 \, a^{4} b^{6} c^{5} d^{5} + 715 \, a^{5} b^{5} c^{4} d^{6} + 220 \, a^{6} b^{4} c^{3} d^{7} + 55 \, a^{7} b^{3} c^{2} d^{8} + 10 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{1847560 \, {\left (b^{31} x^{20} + 20 \, a b^{30} x^{19} + 190 \, a^{2} b^{29} x^{18} + 1140 \, a^{3} b^{28} x^{17} + 4845 \, a^{4} b^{27} x^{16} + 15504 \, a^{5} b^{26} x^{15} + 38760 \, a^{6} b^{25} x^{14} + 77520 \, a^{7} b^{24} x^{13} + 125970 \, a^{8} b^{23} x^{12} + 167960 \, a^{9} b^{22} x^{11} + 184756 \, a^{10} b^{21} x^{10} + 167960 \, a^{11} b^{20} x^{9} + 125970 \, a^{12} b^{19} x^{8} + 77520 \, a^{13} b^{18} x^{7} + 38760 \, a^{14} b^{17} x^{6} + 15504 \, a^{15} b^{16} x^{5} + 4845 \, a^{16} b^{15} x^{4} + 1140 \, a^{17} b^{14} x^{3} + 190 \, a^{18} b^{13} x^{2} + 20 \, a^{19} b^{12} x + a^{20} b^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^21,x, algorithm="fricas")

[Out]

-1/1847560*(184756*b^10*d^10*x^10 + 92378*b^10*c^10 + 48620*a*b^9*c^9*d + 24310*a^2*b^8*c^8*d^2 + 11440*a^3*b^
7*c^7*d^3 + 5005*a^4*b^6*c^6*d^4 + 2002*a^5*b^5*c^5*d^5 + 715*a^6*b^4*c^4*d^6 + 220*a^7*b^3*c^3*d^7 + 55*a^8*b
^2*c^2*d^8 + 10*a^9*b*c*d^9 + a^10*d^10 + 167960*(10*b^10*c*d^9 + a*b^9*d^10)*x^9 + 125970*(55*b^10*c^2*d^8 +
10*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 77520*(220*b^10*c^3*d^7 + 55*a*b^9*c^2*d^8 + 10*a^2*b^8*c*d^9 + a^3*b^7*d
^10)*x^7 + 38760*(715*b^10*c^4*d^6 + 220*a*b^9*c^3*d^7 + 55*a^2*b^8*c^2*d^8 + 10*a^3*b^7*c*d^9 + a^4*b^6*d^10)
*x^6 + 15504*(2002*b^10*c^5*d^5 + 715*a*b^9*c^4*d^6 + 220*a^2*b^8*c^3*d^7 + 55*a^3*b^7*c^2*d^8 + 10*a^4*b^6*c*
d^9 + a^5*b^5*d^10)*x^5 + 4845*(5005*b^10*c^6*d^4 + 2002*a*b^9*c^5*d^5 + 715*a^2*b^8*c^4*d^6 + 220*a^3*b^7*c^3
*d^7 + 55*a^4*b^6*c^2*d^8 + 10*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 1140*(11440*b^10*c^7*d^3 + 5005*a*b^9*c^6*d
^4 + 2002*a^2*b^8*c^5*d^5 + 715*a^3*b^7*c^4*d^6 + 220*a^4*b^6*c^3*d^7 + 55*a^5*b^5*c^2*d^8 + 10*a^6*b^4*c*d^9
+ a^7*b^3*d^10)*x^3 + 190*(24310*b^10*c^8*d^2 + 11440*a*b^9*c^7*d^3 + 5005*a^2*b^8*c^6*d^4 + 2002*a^3*b^7*c^5*
d^5 + 715*a^4*b^6*c^4*d^6 + 220*a^5*b^5*c^3*d^7 + 55*a^6*b^4*c^2*d^8 + 10*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 +
20*(48620*b^10*c^9*d + 24310*a*b^9*c^8*d^2 + 11440*a^2*b^8*c^7*d^3 + 5005*a^3*b^7*c^6*d^4 + 2002*a^4*b^6*c^5*d
^5 + 715*a^5*b^5*c^4*d^6 + 220*a^6*b^4*c^3*d^7 + 55*a^7*b^3*c^2*d^8 + 10*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^31*
x^20 + 20*a*b^30*x^19 + 190*a^2*b^29*x^18 + 1140*a^3*b^28*x^17 + 4845*a^4*b^27*x^16 + 15504*a^5*b^26*x^15 + 38
760*a^6*b^25*x^14 + 77520*a^7*b^24*x^13 + 125970*a^8*b^23*x^12 + 167960*a^9*b^22*x^11 + 184756*a^10*b^21*x^10
+ 167960*a^11*b^20*x^9 + 125970*a^12*b^19*x^8 + 77520*a^13*b^18*x^7 + 38760*a^14*b^17*x^6 + 15504*a^15*b^16*x^
5 + 4845*a^16*b^15*x^4 + 1140*a^17*b^14*x^3 + 190*a^18*b^13*x^2 + 20*a^19*b^12*x + a^20*b^11)

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giac [B]  time = 1.31, size = 961, normalized size = 3.44 \begin {gather*} -\frac {184756 \, b^{10} d^{10} x^{10} + 1679600 \, b^{10} c d^{9} x^{9} + 167960 \, a b^{9} d^{10} x^{9} + 6928350 \, b^{10} c^{2} d^{8} x^{8} + 1259700 \, a b^{9} c d^{9} x^{8} + 125970 \, a^{2} b^{8} d^{10} x^{8} + 17054400 \, b^{10} c^{3} d^{7} x^{7} + 4263600 \, a b^{9} c^{2} d^{8} x^{7} + 775200 \, a^{2} b^{8} c d^{9} x^{7} + 77520 \, a^{3} b^{7} d^{10} x^{7} + 27713400 \, b^{10} c^{4} d^{6} x^{6} + 8527200 \, a b^{9} c^{3} d^{7} x^{6} + 2131800 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 387600 \, a^{3} b^{7} c d^{9} x^{6} + 38760 \, a^{4} b^{6} d^{10} x^{6} + 31039008 \, b^{10} c^{5} d^{5} x^{5} + 11085360 \, a b^{9} c^{4} d^{6} x^{5} + 3410880 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 852720 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 155040 \, a^{4} b^{6} c d^{9} x^{5} + 15504 \, a^{5} b^{5} d^{10} x^{5} + 24249225 \, b^{10} c^{6} d^{4} x^{4} + 9699690 \, a b^{9} c^{5} d^{5} x^{4} + 3464175 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 1065900 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 266475 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 48450 \, a^{5} b^{5} c d^{9} x^{4} + 4845 \, a^{6} b^{4} d^{10} x^{4} + 13041600 \, b^{10} c^{7} d^{3} x^{3} + 5705700 \, a b^{9} c^{6} d^{4} x^{3} + 2282280 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 815100 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 250800 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 62700 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 11400 \, a^{6} b^{4} c d^{9} x^{3} + 1140 \, a^{7} b^{3} d^{10} x^{3} + 4618900 \, b^{10} c^{8} d^{2} x^{2} + 2173600 \, a b^{9} c^{7} d^{3} x^{2} + 950950 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 380380 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 135850 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 41800 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 10450 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 1900 \, a^{7} b^{3} c d^{9} x^{2} + 190 \, a^{8} b^{2} d^{10} x^{2} + 972400 \, b^{10} c^{9} d x + 486200 \, a b^{9} c^{8} d^{2} x + 228800 \, a^{2} b^{8} c^{7} d^{3} x + 100100 \, a^{3} b^{7} c^{6} d^{4} x + 40040 \, a^{4} b^{6} c^{5} d^{5} x + 14300 \, a^{5} b^{5} c^{4} d^{6} x + 4400 \, a^{6} b^{4} c^{3} d^{7} x + 1100 \, a^{7} b^{3} c^{2} d^{8} x + 200 \, a^{8} b^{2} c d^{9} x + 20 \, a^{9} b d^{10} x + 92378 \, b^{10} c^{10} + 48620 \, a b^{9} c^{9} d + 24310 \, a^{2} b^{8} c^{8} d^{2} + 11440 \, a^{3} b^{7} c^{7} d^{3} + 5005 \, a^{4} b^{6} c^{6} d^{4} + 2002 \, a^{5} b^{5} c^{5} d^{5} + 715 \, a^{6} b^{4} c^{4} d^{6} + 220 \, a^{7} b^{3} c^{3} d^{7} + 55 \, a^{8} b^{2} c^{2} d^{8} + 10 \, a^{9} b c d^{9} + a^{10} d^{10}}{1847560 \, {\left (b x + a\right )}^{20} b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^21,x, algorithm="giac")

[Out]

-1/1847560*(184756*b^10*d^10*x^10 + 1679600*b^10*c*d^9*x^9 + 167960*a*b^9*d^10*x^9 + 6928350*b^10*c^2*d^8*x^8
+ 1259700*a*b^9*c*d^9*x^8 + 125970*a^2*b^8*d^10*x^8 + 17054400*b^10*c^3*d^7*x^7 + 4263600*a*b^9*c^2*d^8*x^7 +
775200*a^2*b^8*c*d^9*x^7 + 77520*a^3*b^7*d^10*x^7 + 27713400*b^10*c^4*d^6*x^6 + 8527200*a*b^9*c^3*d^7*x^6 + 21
31800*a^2*b^8*c^2*d^8*x^6 + 387600*a^3*b^7*c*d^9*x^6 + 38760*a^4*b^6*d^10*x^6 + 31039008*b^10*c^5*d^5*x^5 + 11
085360*a*b^9*c^4*d^6*x^5 + 3410880*a^2*b^8*c^3*d^7*x^5 + 852720*a^3*b^7*c^2*d^8*x^5 + 155040*a^4*b^6*c*d^9*x^5
+ 15504*a^5*b^5*d^10*x^5 + 24249225*b^10*c^6*d^4*x^4 + 9699690*a*b^9*c^5*d^5*x^4 + 3464175*a^2*b^8*c^4*d^6*x^
4 + 1065900*a^3*b^7*c^3*d^7*x^4 + 266475*a^4*b^6*c^2*d^8*x^4 + 48450*a^5*b^5*c*d^9*x^4 + 4845*a^6*b^4*d^10*x^4
+ 13041600*b^10*c^7*d^3*x^3 + 5705700*a*b^9*c^6*d^4*x^3 + 2282280*a^2*b^8*c^5*d^5*x^3 + 815100*a^3*b^7*c^4*d^
6*x^3 + 250800*a^4*b^6*c^3*d^7*x^3 + 62700*a^5*b^5*c^2*d^8*x^3 + 11400*a^6*b^4*c*d^9*x^3 + 1140*a^7*b^3*d^10*x
^3 + 4618900*b^10*c^8*d^2*x^2 + 2173600*a*b^9*c^7*d^3*x^2 + 950950*a^2*b^8*c^6*d^4*x^2 + 380380*a^3*b^7*c^5*d^
5*x^2 + 135850*a^4*b^6*c^4*d^6*x^2 + 41800*a^5*b^5*c^3*d^7*x^2 + 10450*a^6*b^4*c^2*d^8*x^2 + 1900*a^7*b^3*c*d^
9*x^2 + 190*a^8*b^2*d^10*x^2 + 972400*b^10*c^9*d*x + 486200*a*b^9*c^8*d^2*x + 228800*a^2*b^8*c^7*d^3*x + 10010
0*a^3*b^7*c^6*d^4*x + 40040*a^4*b^6*c^5*d^5*x + 14300*a^5*b^5*c^4*d^6*x + 4400*a^6*b^4*c^3*d^7*x + 1100*a^7*b^
3*c^2*d^8*x + 200*a^8*b^2*c*d^9*x + 20*a^9*b*d^10*x + 92378*b^10*c^10 + 48620*a*b^9*c^9*d + 24310*a^2*b^8*c^8*
d^2 + 11440*a^3*b^7*c^7*d^3 + 5005*a^4*b^6*c^6*d^4 + 2002*a^5*b^5*c^5*d^5 + 715*a^6*b^4*c^4*d^6 + 220*a^7*b^3*
c^3*d^7 + 55*a^8*b^2*c^2*d^8 + 10*a^9*b*c*d^9 + a^10*d^10)/((b*x + a)^20*b^11)

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maple [B]  time = 0.01, size = 867, normalized size = 3.11 \begin {gather*} -\frac {d^{10}}{10 \left (b x +a \right )^{10} b^{11}}+\frac {10 \left (a d -b c \right ) d^{9}}{11 \left (b x +a \right )^{11} b^{11}}-\frac {15 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{8}}{4 \left (b x +a \right )^{12} b^{11}}+\frac {120 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d^{7}}{13 \left (b x +a \right )^{13} b^{11}}-\frac {15 \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 ) d^{6}}{\left (b x +a \right )^{14} b^{11}}+\frac {84 \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 ) d^{5}}{5 \left (b x +a \right )^{15} b^{11}}-\frac {105 \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 ) d^{4}}{8 \left (b x +a \right )^{16} b^{11}}+\frac {120 \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 ) d^{3}}{17 \left (b x +a \right )^{17} b^{11}}-\frac {5 \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 ) d^{2}}{2 \left (b x +a \right )^{18} b^{11}}+\frac {10 \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 ) d}{19 \left (b x +a \right )^{19} b^{11}}-\frac {d^{10} a^{10}-10 c \,d^{9} a^{9} b +45 c^{2} d^{8} a^{8} b^{2}-120 a^{7} c^{3} d^{7} b^{3}+210 c^{4} d^{6} a^{6} b^{4}-252 a^{5} c^{5} d^{5} b^{5}+210 a^{4} c^{6} d^{4} b^{6}-120 a^{3} c^{7} d^{3} b^{7}+45 a^{2} c^{8} d^{2} b^{8}-10 a \,b^{9} c^{9} d +c^{10} b^{10}}{20 \left (b x +a \right )^{20} b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^21,x)

[Out]

120/13*d^7*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^11/(b*x+a)^13-1/20*(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)/b^11/(b*x+a)^20+120/17*d^3*(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)/b^11/(b*x+a)^17+10/11*d
^9*(a*d-b*c)/b^11/(b*x+a)^11+84/5*d^5*(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)/b^11/(b*x+a)^15-15/4*d^8*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^11/(b*x+a)^12-5/2*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)/b^11/(b*x+a)^18+10/19*d*(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)/b^11/(b*x+a)^19-1/10*d^1
0/b^11/(b*x+a)^10-15*d^6*(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)/b^11/(b*x+a)^14-105/8
*d^4*(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)/b^
11/(b*x+a)^16

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maxima [B]  time = 2.51, size = 1074, normalized size = 3.85 \begin {gather*} -\frac {184756 \, b^{10} d^{10} x^{10} + 92378 \, b^{10} c^{10} + 48620 \, a b^{9} c^{9} d + 24310 \, a^{2} b^{8} c^{8} d^{2} + 11440 \, a^{3} b^{7} c^{7} d^{3} + 5005 \, a^{4} b^{6} c^{6} d^{4} + 2002 \, a^{5} b^{5} c^{5} d^{5} + 715 \, a^{6} b^{4} c^{4} d^{6} + 220 \, a^{7} b^{3} c^{3} d^{7} + 55 \, a^{8} b^{2} c^{2} d^{8} + 10 \, a^{9} b c d^{9} + a^{10} d^{10} + 167960 \, {\left (10 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 125970 \, {\left (55 \, b^{10} c^{2} d^{8} + 10 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 77520 \, {\left (220 \, b^{10} c^{3} d^{7} + 55 \, a b^{9} c^{2} d^{8} + 10 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 38760 \, {\left (715 \, b^{10} c^{4} d^{6} + 220 \, a b^{9} c^{3} d^{7} + 55 \, a^{2} b^{8} c^{2} d^{8} + 10 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 15504 \, {\left (2002 \, b^{10} c^{5} d^{5} + 715 \, a b^{9} c^{4} d^{6} + 220 \, a^{2} b^{8} c^{3} d^{7} + 55 \, a^{3} b^{7} c^{2} d^{8} + 10 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 4845 \, {\left (5005 \, b^{10} c^{6} d^{4} + 2002 \, a b^{9} c^{5} d^{5} + 715 \, a^{2} b^{8} c^{4} d^{6} + 220 \, a^{3} b^{7} c^{3} d^{7} + 55 \, a^{4} b^{6} c^{2} d^{8} + 10 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 1140 \, {\left (11440 \, b^{10} c^{7} d^{3} + 5005 \, a b^{9} c^{6} d^{4} + 2002 \, a^{2} b^{8} c^{5} d^{5} + 715 \, a^{3} b^{7} c^{4} d^{6} + 220 \, a^{4} b^{6} c^{3} d^{7} + 55 \, a^{5} b^{5} c^{2} d^{8} + 10 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 190 \, {\left (24310 \, b^{10} c^{8} d^{2} + 11440 \, a b^{9} c^{7} d^{3} + 5005 \, a^{2} b^{8} c^{6} d^{4} + 2002 \, a^{3} b^{7} c^{5} d^{5} + 715 \, a^{4} b^{6} c^{4} d^{6} + 220 \, a^{5} b^{5} c^{3} d^{7} + 55 \, a^{6} b^{4} c^{2} d^{8} + 10 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 20 \, {\left (48620 \, b^{10} c^{9} d + 24310 \, a b^{9} c^{8} d^{2} + 11440 \, a^{2} b^{8} c^{7} d^{3} + 5005 \, a^{3} b^{7} c^{6} d^{4} + 2002 \, a^{4} b^{6} c^{5} d^{5} + 715 \, a^{5} b^{5} c^{4} d^{6} + 220 \, a^{6} b^{4} c^{3} d^{7} + 55 \, a^{7} b^{3} c^{2} d^{8} + 10 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{1847560 \, {\left (b^{31} x^{20} + 20 \, a b^{30} x^{19} + 190 \, a^{2} b^{29} x^{18} + 1140 \, a^{3} b^{28} x^{17} + 4845 \, a^{4} b^{27} x^{16} + 15504 \, a^{5} b^{26} x^{15} + 38760 \, a^{6} b^{25} x^{14} + 77520 \, a^{7} b^{24} x^{13} + 125970 \, a^{8} b^{23} x^{12} + 167960 \, a^{9} b^{22} x^{11} + 184756 \, a^{10} b^{21} x^{10} + 167960 \, a^{11} b^{20} x^{9} + 125970 \, a^{12} b^{19} x^{8} + 77520 \, a^{13} b^{18} x^{7} + 38760 \, a^{14} b^{17} x^{6} + 15504 \, a^{15} b^{16} x^{5} + 4845 \, a^{16} b^{15} x^{4} + 1140 \, a^{17} b^{14} x^{3} + 190 \, a^{18} b^{13} x^{2} + 20 \, a^{19} b^{12} x + a^{20} b^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^21,x, algorithm="maxima")

[Out]

-1/1847560*(184756*b^10*d^10*x^10 + 92378*b^10*c^10 + 48620*a*b^9*c^9*d + 24310*a^2*b^8*c^8*d^2 + 11440*a^3*b^
7*c^7*d^3 + 5005*a^4*b^6*c^6*d^4 + 2002*a^5*b^5*c^5*d^5 + 715*a^6*b^4*c^4*d^6 + 220*a^7*b^3*c^3*d^7 + 55*a^8*b
^2*c^2*d^8 + 10*a^9*b*c*d^9 + a^10*d^10 + 167960*(10*b^10*c*d^9 + a*b^9*d^10)*x^9 + 125970*(55*b^10*c^2*d^8 +
10*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 77520*(220*b^10*c^3*d^7 + 55*a*b^9*c^2*d^8 + 10*a^2*b^8*c*d^9 + a^3*b^7*d
^10)*x^7 + 38760*(715*b^10*c^4*d^6 + 220*a*b^9*c^3*d^7 + 55*a^2*b^8*c^2*d^8 + 10*a^3*b^7*c*d^9 + a^4*b^6*d^10)
*x^6 + 15504*(2002*b^10*c^5*d^5 + 715*a*b^9*c^4*d^6 + 220*a^2*b^8*c^3*d^7 + 55*a^3*b^7*c^2*d^8 + 10*a^4*b^6*c*
d^9 + a^5*b^5*d^10)*x^5 + 4845*(5005*b^10*c^6*d^4 + 2002*a*b^9*c^5*d^5 + 715*a^2*b^8*c^4*d^6 + 220*a^3*b^7*c^3
*d^7 + 55*a^4*b^6*c^2*d^8 + 10*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 1140*(11440*b^10*c^7*d^3 + 5005*a*b^9*c^6*d
^4 + 2002*a^2*b^8*c^5*d^5 + 715*a^3*b^7*c^4*d^6 + 220*a^4*b^6*c^3*d^7 + 55*a^5*b^5*c^2*d^8 + 10*a^6*b^4*c*d^9
+ a^7*b^3*d^10)*x^3 + 190*(24310*b^10*c^8*d^2 + 11440*a*b^9*c^7*d^3 + 5005*a^2*b^8*c^6*d^4 + 2002*a^3*b^7*c^5*
d^5 + 715*a^4*b^6*c^4*d^6 + 220*a^5*b^5*c^3*d^7 + 55*a^6*b^4*c^2*d^8 + 10*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 +
20*(48620*b^10*c^9*d + 24310*a*b^9*c^8*d^2 + 11440*a^2*b^8*c^7*d^3 + 5005*a^3*b^7*c^6*d^4 + 2002*a^4*b^6*c^5*d
^5 + 715*a^5*b^5*c^4*d^6 + 220*a^6*b^4*c^3*d^7 + 55*a^7*b^3*c^2*d^8 + 10*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^31*
x^20 + 20*a*b^30*x^19 + 190*a^2*b^29*x^18 + 1140*a^3*b^28*x^17 + 4845*a^4*b^27*x^16 + 15504*a^5*b^26*x^15 + 38
760*a^6*b^25*x^14 + 77520*a^7*b^24*x^13 + 125970*a^8*b^23*x^12 + 167960*a^9*b^22*x^11 + 184756*a^10*b^21*x^10
+ 167960*a^11*b^20*x^9 + 125970*a^12*b^19*x^8 + 77520*a^13*b^18*x^7 + 38760*a^14*b^17*x^6 + 15504*a^15*b^16*x^
5 + 4845*a^16*b^15*x^4 + 1140*a^17*b^14*x^3 + 190*a^18*b^13*x^2 + 20*a^19*b^12*x + a^20*b^11)

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mupad [B]  time = 0.80, size = 1175, normalized size = 4.21 \begin {gather*} -\frac {a^{10}\,d^{10}+10\,a^9\,b\,c\,d^9+20\,a^9\,b\,d^{10}\,x+55\,a^8\,b^2\,c^2\,d^8+200\,a^8\,b^2\,c\,d^9\,x+190\,a^8\,b^2\,d^{10}\,x^2+220\,a^7\,b^3\,c^3\,d^7+1100\,a^7\,b^3\,c^2\,d^8\,x+1900\,a^7\,b^3\,c\,d^9\,x^2+1140\,a^7\,b^3\,d^{10}\,x^3+715\,a^6\,b^4\,c^4\,d^6+4400\,a^6\,b^4\,c^3\,d^7\,x+10450\,a^6\,b^4\,c^2\,d^8\,x^2+11400\,a^6\,b^4\,c\,d^9\,x^3+4845\,a^6\,b^4\,d^{10}\,x^4+2002\,a^5\,b^5\,c^5\,d^5+14300\,a^5\,b^5\,c^4\,d^6\,x+41800\,a^5\,b^5\,c^3\,d^7\,x^2+62700\,a^5\,b^5\,c^2\,d^8\,x^3+48450\,a^5\,b^5\,c\,d^9\,x^4+15504\,a^5\,b^5\,d^{10}\,x^5+5005\,a^4\,b^6\,c^6\,d^4+40040\,a^4\,b^6\,c^5\,d^5\,x+135850\,a^4\,b^6\,c^4\,d^6\,x^2+250800\,a^4\,b^6\,c^3\,d^7\,x^3+266475\,a^4\,b^6\,c^2\,d^8\,x^4+155040\,a^4\,b^6\,c\,d^9\,x^5+38760\,a^4\,b^6\,d^{10}\,x^6+11440\,a^3\,b^7\,c^7\,d^3+100100\,a^3\,b^7\,c^6\,d^4\,x+380380\,a^3\,b^7\,c^5\,d^5\,x^2+815100\,a^3\,b^7\,c^4\,d^6\,x^3+1065900\,a^3\,b^7\,c^3\,d^7\,x^4+852720\,a^3\,b^7\,c^2\,d^8\,x^5+387600\,a^3\,b^7\,c\,d^9\,x^6+77520\,a^3\,b^7\,d^{10}\,x^7+24310\,a^2\,b^8\,c^8\,d^2+228800\,a^2\,b^8\,c^7\,d^3\,x+950950\,a^2\,b^8\,c^6\,d^4\,x^2+2282280\,a^2\,b^8\,c^5\,d^5\,x^3+3464175\,a^2\,b^8\,c^4\,d^6\,x^4+3410880\,a^2\,b^8\,c^3\,d^7\,x^5+2131800\,a^2\,b^8\,c^2\,d^8\,x^6+775200\,a^2\,b^8\,c\,d^9\,x^7+125970\,a^2\,b^8\,d^{10}\,x^8+48620\,a\,b^9\,c^9\,d+486200\,a\,b^9\,c^8\,d^2\,x+2173600\,a\,b^9\,c^7\,d^3\,x^2+5705700\,a\,b^9\,c^6\,d^4\,x^3+9699690\,a\,b^9\,c^5\,d^5\,x^4+11085360\,a\,b^9\,c^4\,d^6\,x^5+8527200\,a\,b^9\,c^3\,d^7\,x^6+4263600\,a\,b^9\,c^2\,d^8\,x^7+1259700\,a\,b^9\,c\,d^9\,x^8+167960\,a\,b^9\,d^{10}\,x^9+92378\,b^{10}\,c^{10}+972400\,b^{10}\,c^9\,d\,x+4618900\,b^{10}\,c^8\,d^2\,x^2+13041600\,b^{10}\,c^7\,d^3\,x^3+24249225\,b^{10}\,c^6\,d^4\,x^4+31039008\,b^{10}\,c^5\,d^5\,x^5+27713400\,b^{10}\,c^4\,d^6\,x^6+17054400\,b^{10}\,c^3\,d^7\,x^7+6928350\,b^{10}\,c^2\,d^8\,x^8+1679600\,b^{10}\,c\,d^9\,x^9+184756\,b^{10}\,d^{10}\,x^{10}}{1847560\,a^{20}\,b^{11}[phone]\,a^{19}\,b^{12}\,x+351036400\,a^{18}\,b^{13}\,x^2+2106218400\,a^{17}\,b^{14}\,x^3+8951428200\,a^{16}\,b^{15}\,x^4+28644570240\,a^{15}\,b^{16}\,x^5+71611425600\,a^{14}\,b^{17}\,x^6+143222851200\,a^{13}\,b^{18}\,x^7+232737133200\,a^{12}\,b^{19}\,x^8+310316177600\,a^{11}\,b^{20}\,x^9+341347795360\,a^{10}\,b^{21}\,x^{10}[phone]\,a^9\,b^{22}\,x^{11}[phone]\,a^8\,b^{23}\,x^{12}[phone]\,a^7\,b^{24}\,x^{13}[phone]\,a^6\,b^{25}\,x^{14}[phone]\,a^5\,b^{26}\,x^{15}[phone]\,a^4\,b^{27}\,x^{16}[phone]\,a^3\,b^{28}\,x^{17}[phone]\,a^2\,b^{29}\,x^{18}[phone]\,a\,b^{30}\,x^{19}+1847560\,b^{31}\,x^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^21,x)

[Out]

-(a^10*d^10 + 92378*b^10*c^10 + 184756*b^10*d^10*x^10 + 167960*a*b^9*d^10*x^9 + 1679600*b^10*c*d^9*x^9 + 24310
*a^2*b^8*c^8*d^2 + 11440*a^3*b^7*c^7*d^3 + 5005*a^4*b^6*c^6*d^4 + 2002*a^5*b^5*c^5*d^5 + 715*a^6*b^4*c^4*d^6 +
220*a^7*b^3*c^3*d^7 + 55*a^8*b^2*c^2*d^8 + 190*a^8*b^2*d^10*x^2 + 1140*a^7*b^3*d^10*x^3 + 4845*a^6*b^4*d^10*x
^4 + 15504*a^5*b^5*d^10*x^5 + 38760*a^4*b^6*d^10*x^6 + 77520*a^3*b^7*d^10*x^7 + 125970*a^2*b^8*d^10*x^8 + 4618
900*b^10*c^8*d^2*x^2 + 13041600*b^10*c^7*d^3*x^3 + 24249225*b^10*c^6*d^4*x^4 + 31039008*b^10*c^5*d^5*x^5 + 277
13400*b^10*c^4*d^6*x^6 + 17054400*b^10*c^3*d^7*x^7 + 6928350*b^10*c^2*d^8*x^8 + 48620*a*b^9*c^9*d + 10*a^9*b*c
*d^9 + 20*a^9*b*d^10*x + 972400*b^10*c^9*d*x + 950950*a^2*b^8*c^6*d^4*x^2 + 380380*a^3*b^7*c^5*d^5*x^2 + 13585
0*a^4*b^6*c^4*d^6*x^2 + 41800*a^5*b^5*c^3*d^7*x^2 + 10450*a^6*b^4*c^2*d^8*x^2 + 2282280*a^2*b^8*c^5*d^5*x^3 +
815100*a^3*b^7*c^4*d^6*x^3 + 250800*a^4*b^6*c^3*d^7*x^3 + 62700*a^5*b^5*c^2*d^8*x^3 + 3464175*a^2*b^8*c^4*d^6*
x^4 + 1065900*a^3*b^7*c^3*d^7*x^4 + 266475*a^4*b^6*c^2*d^8*x^4 + 3410880*a^2*b^8*c^3*d^7*x^5 + 852720*a^3*b^7*
c^2*d^8*x^5 + 2131800*a^2*b^8*c^2*d^8*x^6 + 486200*a*b^9*c^8*d^2*x + 200*a^8*b^2*c*d^9*x + 1259700*a*b^9*c*d^9
*x^8 + 228800*a^2*b^8*c^7*d^3*x + 100100*a^3*b^7*c^6*d^4*x + 40040*a^4*b^6*c^5*d^5*x + 14300*a^5*b^5*c^4*d^6*x
+ 4400*a^6*b^4*c^3*d^7*x + 1100*a^7*b^3*c^2*d^8*x + 2173600*a*b^9*c^7*d^3*x^2 + 1900*a^7*b^3*c*d^9*x^2 + 5705
700*a*b^9*c^6*d^4*x^3 + 11400*a^6*b^4*c*d^9*x^3 + 9699690*a*b^9*c^5*d^5*x^4 + 48450*a^5*b^5*c*d^9*x^4 + 110853
60*a*b^9*c^4*d^6*x^5 + 155040*a^4*b^6*c*d^9*x^5 + 8527200*a*b^9*c^3*d^7*x^6 + 387600*a^3*b^7*c*d^9*x^6 + 42636
00*a*b^9*c^2*d^8*x^7 + 775200*a^2*b^8*c*d^9*x^7)/(1847560*a^20*b^11 + 1847560*b^31*x^20 + 36951200*a^19*b^12*x
+ 36951200*a*b^30*x^19 + 351036400*a^18*b^13*x^2 + 2106218400*a^17*b^14*x^3 + 8951428200*a^16*b^15*x^4 + 2864
4570240*a^15*b^16*x^5 + 71611425600*a^14*b^17*x^6 + 143222851200*a^13*b^18*x^7 + 232737133200*a^12*b^19*x^8 +
310316177600*a^11*b^20*x^9 + 341347795360*a^10*b^21*x^10 + 310316177600*a^9*b^22*x^11 + 232737133200*a^8*b^23*
x^12 + 143222851200*a^7*b^24*x^13 + 71611425600*a^6*b^25*x^14 + 28644570240*a^5*b^26*x^15 + 8951428200*a^4*b^2
7*x^16 + 2106218400*a^3*b^28*x^17 + 351036400*a^2*b^29*x^18)

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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)**21,x)

[Out]

Timed out

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