3.13.25 \(\int \frac {(c+d x)^{10}}{(a+b x)^{20}} \, dx\)

Optimal. Leaf size=273 \[ -\frac {d^9 (b c-a d)}{b^{11} (a+b x)^{10}}-\frac {45 d^8 (b c-a d)^2}{11 b^{11} (a+b x)^{11}}-\frac {10 d^7 (b c-a d)^3}{b^{11} (a+b x)^{12}}-\frac {210 d^6 (b c-a d)^4}{13 b^{11} (a+b x)^{13}}-\frac {18 d^5 (b c-a d)^5}{b^{11} (a+b x)^{14}}-\frac {14 d^4 (b c-a d)^6}{b^{11} (a+b x)^{15}}-\frac {15 d^3 (b c-a d)^7}{2 b^{11} (a+b x)^{16}}-\frac {45 d^2 (b c-a d)^8}{17 b^{11} (a+b x)^{17}}-\frac {5 d (b c-a d)^9}{9 b^{11} (a+b x)^{18}}-\frac {(b c-a d)^{10}}{19 b^{11} (a+b x)^{19}}-\frac {d^{10}}{9 b^{11} (a+b x)^9} \]

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Rubi [A]  time = 0.28, antiderivative size = 273, 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 {d^9 (b c-a d)}{b^{11} (a+b x)^{10}}-\frac {45 d^8 (b c-a d)^2}{11 b^{11} (a+b x)^{11}}-\frac {10 d^7 (b c-a d)^3}{b^{11} (a+b x)^{12}}-\frac {210 d^6 (b c-a d)^4}{13 b^{11} (a+b x)^{13}}-\frac {18 d^5 (b c-a d)^5}{b^{11} (a+b x)^{14}}-\frac {14 d^4 (b c-a d)^6}{b^{11} (a+b x)^{15}}-\frac {15 d^3 (b c-a d)^7}{2 b^{11} (a+b x)^{16}}-\frac {45 d^2 (b c-a d)^8}{17 b^{11} (a+b x)^{17}}-\frac {5 d (b c-a d)^9}{9 b^{11} (a+b x)^{18}}-\frac {(b c-a d)^{10}}{19 b^{11} (a+b x)^{19}}-\frac {d^{10}}{9 b^{11} (a+b x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^20,x]

[Out]

-(b*c - a*d)^10/(19*b^11*(a + b*x)^19) - (5*d*(b*c - a*d)^9)/(9*b^11*(a + b*x)^18) - (45*d^2*(b*c - a*d)^8)/(1
7*b^11*(a + b*x)^17) - (15*d^3*(b*c - a*d)^7)/(2*b^11*(a + b*x)^16) - (14*d^4*(b*c - a*d)^6)/(b^11*(a + b*x)^1
5) - (18*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)^14) - (210*d^6*(b*c - a*d)^4)/(13*b^11*(a + b*x)^13) - (10*d^7*(b*
c - a*d)^3)/(b^11*(a + b*x)^12) - (45*d^8*(b*c - a*d)^2)/(11*b^11*(a + b*x)^11) - (d^9*(b*c - a*d))/(b^11*(a +
b*x)^10) - d^10/(9*b^11*(a + b*x)^9)

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

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Mathematica [B]  time = 0.28, size = 692, normalized size = 2.53 \begin {gather*} -\frac {a^{10} d^{10}+a^9 b d^9 (9 c+19 d x)+9 a^8 b^2 d^8 \left (5 c^2+19 c d x+19 d^2 x^2\right )+3 a^7 b^3 d^7 \left (55 c^3+285 c^2 d x+513 c d^2 x^2+323 d^3 x^3\right )+3 a^6 b^4 d^6 \left (165 c^4+1045 c^3 d x+2565 c^2 d^2 x^2+2907 c d^3 x^3+1292 d^4 x^4\right )+9 a^5 b^5 d^5 \left (143 c^5+1045 c^4 d x+3135 c^3 d^2 x^2+4845 c^2 d^3 x^3+3876 c d^4 x^4+1292 d^5 x^5\right )+3 a^4 b^6 d^4 \left (1001 c^6+8151 c^5 d x+28215 c^4 d^2 x^2+53295 c^3 d^3 x^3+58140 c^2 d^4 x^4+34884 c d^5 x^5+9044 d^6 x^6\right )+3 a^3 b^7 d^3 \left (2145 c^7+19019 c^6 d x+73359 c^5 d^2 x^2+159885 c^4 d^3 x^3+213180 c^3 d^4 x^4+174420 c^2 d^5 x^5+81396 c d^6 x^6+16796 d^7 x^7\right )+9 a^2 b^8 d^2 \left (1430 c^8+13585 c^7 d x+57057 c^6 d^2 x^2+138567 c^5 d^3 x^3+213180 c^4 d^4 x^4+213180 c^3 d^5 x^5+135660 c^2 d^6 x^6+50388 c d^7 x^7+8398 d^8 x^8\right )+a b^9 d \left (24310 c^9+244530 c^8 d x+1100385 c^7 d^2 x^2+2909907 c^6 d^3 x^3+4988412 c^5 d^4 x^4+5755860 c^4 d^5 x^5+4476780 c^3 d^6 x^6+2267460 c^2 d^7 x^7+680238 c d^8 x^8+92378 d^9 x^9\right )+b^{10} \left (43758 c^{10}+461890 c^9 d x+2200770 c^8 d^2 x^2+6235515 c^7 d^3 x^3+11639628 c^6 d^4 x^4+14965236 c^5 d^5 x^5+13430340 c^4 d^6 x^6+8314020 c^3 d^7 x^7+3401190 c^2 d^8 x^8+831402 c d^9 x^9+92378 d^{10} x^{10}\right )}{831402 b^{11} (a+b x)^{19}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^20,x]

[Out]

-1/831402*(a^10*d^10 + a^9*b*d^9*(9*c + 19*d*x) + 9*a^8*b^2*d^8*(5*c^2 + 19*c*d*x + 19*d^2*x^2) + 3*a^7*b^3*d^
7*(55*c^3 + 285*c^2*d*x + 513*c*d^2*x^2 + 323*d^3*x^3) + 3*a^6*b^4*d^6*(165*c^4 + 1045*c^3*d*x + 2565*c^2*d^2*
x^2 + 2907*c*d^3*x^3 + 1292*d^4*x^4) + 9*a^5*b^5*d^5*(143*c^5 + 1045*c^4*d*x + 3135*c^3*d^2*x^2 + 4845*c^2*d^3
*x^3 + 3876*c*d^4*x^4 + 1292*d^5*x^5) + 3*a^4*b^6*d^4*(1001*c^6 + 8151*c^5*d*x + 28215*c^4*d^2*x^2 + 53295*c^3
*d^3*x^3 + 58140*c^2*d^4*x^4 + 34884*c*d^5*x^5 + 9044*d^6*x^6) + 3*a^3*b^7*d^3*(2145*c^7 + 19019*c^6*d*x + 733
59*c^5*d^2*x^2 + 159885*c^4*d^3*x^3 + 213180*c^3*d^4*x^4 + 174420*c^2*d^5*x^5 + 81396*c*d^6*x^6 + 16796*d^7*x^
7) + 9*a^2*b^8*d^2*(1430*c^8 + 13585*c^7*d*x + 57057*c^6*d^2*x^2 + 138567*c^5*d^3*x^3 + 213180*c^4*d^4*x^4 + 2
13180*c^3*d^5*x^5 + 135660*c^2*d^6*x^6 + 50388*c*d^7*x^7 + 8398*d^8*x^8) + a*b^9*d*(24310*c^9 + 244530*c^8*d*x
+ 1100385*c^7*d^2*x^2 + 2909907*c^6*d^3*x^3 + 4988412*c^5*d^4*x^4 + 5755860*c^4*d^5*x^5 + 4476780*c^3*d^6*x^6
+ 2267460*c^2*d^7*x^7 + 680238*c*d^8*x^8 + 92378*d^9*x^9) + b^10*(43758*c^10 + 461890*c^9*d*x + 2200770*c^8*d
^2*x^2 + 6235515*c^7*d^3*x^3 + 11639628*c^6*d^4*x^4 + 14965236*c^5*d^5*x^5 + 13430340*c^4*d^6*x^6 + 8314020*c^
3*d^7*x^7 + 3401190*c^2*d^8*x^8 + 831402*c*d^9*x^9 + 92378*d^10*x^10))/(b^11*(a + b*x)^19)

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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)^{20}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^20,x]

[Out]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^20, x]

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fricas [B]  time = 1.30, size = 1063, normalized size = 3.89 \begin {gather*} -\frac {92378 \, b^{10} d^{10} x^{10} + 43758 \, b^{10} c^{10} + 24310 \, a b^{9} c^{9} d + 12870 \, a^{2} b^{8} c^{8} d^{2} + 6435 \, a^{3} b^{7} c^{7} d^{3} + 3003 \, a^{4} b^{6} c^{6} d^{4} + 1287 \, a^{5} b^{5} c^{5} d^{5} + 495 \, a^{6} b^{4} c^{4} d^{6} + 165 \, a^{7} b^{3} c^{3} d^{7} + 45 \, a^{8} b^{2} c^{2} d^{8} + 9 \, a^{9} b c d^{9} + a^{10} d^{10} + 92378 \, {\left (9 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 75582 \, {\left (45 \, b^{10} c^{2} d^{8} + 9 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 50388 \, {\left (165 \, b^{10} c^{3} d^{7} + 45 \, a b^{9} c^{2} d^{8} + 9 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 27132 \, {\left (495 \, b^{10} c^{4} d^{6} + 165 \, a b^{9} c^{3} d^{7} + 45 \, a^{2} b^{8} c^{2} d^{8} + 9 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 11628 \, {\left (1287 \, b^{10} c^{5} d^{5} + 495 \, a b^{9} c^{4} d^{6} + 165 \, a^{2} b^{8} c^{3} d^{7} + 45 \, a^{3} b^{7} c^{2} d^{8} + 9 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 3876 \, {\left (3003 \, b^{10} c^{6} d^{4} + 1287 \, a b^{9} c^{5} d^{5} + 495 \, a^{2} b^{8} c^{4} d^{6} + 165 \, a^{3} b^{7} c^{3} d^{7} + 45 \, a^{4} b^{6} c^{2} d^{8} + 9 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 969 \, {\left (6435 \, b^{10} c^{7} d^{3} + 3003 \, a b^{9} c^{6} d^{4} + 1287 \, a^{2} b^{8} c^{5} d^{5} + 495 \, a^{3} b^{7} c^{4} d^{6} + 165 \, a^{4} b^{6} c^{3} d^{7} + 45 \, a^{5} b^{5} c^{2} d^{8} + 9 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 171 \, {\left (12870 \, b^{10} c^{8} d^{2} + 6435 \, a b^{9} c^{7} d^{3} + 3003 \, a^{2} b^{8} c^{6} d^{4} + 1287 \, a^{3} b^{7} c^{5} d^{5} + 495 \, a^{4} b^{6} c^{4} d^{6} + 165 \, a^{5} b^{5} c^{3} d^{7} + 45 \, a^{6} b^{4} c^{2} d^{8} + 9 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 19 \, {\left (24310 \, b^{10} c^{9} d + 12870 \, a b^{9} c^{8} d^{2} + 6435 \, a^{2} b^{8} c^{7} d^{3} + 3003 \, a^{3} b^{7} c^{6} d^{4} + 1287 \, a^{4} b^{6} c^{5} d^{5} + 495 \, a^{5} b^{5} c^{4} d^{6} + 165 \, a^{6} b^{4} c^{3} d^{7} + 45 \, a^{7} b^{3} c^{2} d^{8} + 9 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{831402 \, {\left (b^{30} x^{19} + 19 \, a b^{29} x^{18} + 171 \, a^{2} b^{28} x^{17} + 969 \, a^{3} b^{27} x^{16} + 3876 \, a^{4} b^{26} x^{15} + 11628 \, a^{5} b^{25} x^{14} + 27132 \, a^{6} b^{24} x^{13} + 50388 \, a^{7} b^{23} x^{12} + 75582 \, a^{8} b^{22} x^{11} + 92378 \, a^{9} b^{21} x^{10} + 92378 \, a^{10} b^{20} x^{9} + 75582 \, a^{11} b^{19} x^{8} + 50388 \, a^{12} b^{18} x^{7} + 27132 \, a^{13} b^{17} x^{6} + 11628 \, a^{14} b^{16} x^{5} + 3876 \, a^{15} b^{15} x^{4} + 969 \, a^{16} b^{14} x^{3} + 171 \, a^{17} b^{13} x^{2} + 19 \, a^{18} b^{12} x + a^{19} b^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^20,x, algorithm="fricas")

[Out]

-1/831402*(92378*b^10*d^10*x^10 + 43758*b^10*c^10 + 24310*a*b^9*c^9*d + 12870*a^2*b^8*c^8*d^2 + 6435*a^3*b^7*c
^7*d^3 + 3003*a^4*b^6*c^6*d^4 + 1287*a^5*b^5*c^5*d^5 + 495*a^6*b^4*c^4*d^6 + 165*a^7*b^3*c^3*d^7 + 45*a^8*b^2*
c^2*d^8 + 9*a^9*b*c*d^9 + a^10*d^10 + 92378*(9*b^10*c*d^9 + a*b^9*d^10)*x^9 + 75582*(45*b^10*c^2*d^8 + 9*a*b^9
*c*d^9 + a^2*b^8*d^10)*x^8 + 50388*(165*b^10*c^3*d^7 + 45*a*b^9*c^2*d^8 + 9*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7
+ 27132*(495*b^10*c^4*d^6 + 165*a*b^9*c^3*d^7 + 45*a^2*b^8*c^2*d^8 + 9*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 116
28*(1287*b^10*c^5*d^5 + 495*a*b^9*c^4*d^6 + 165*a^2*b^8*c^3*d^7 + 45*a^3*b^7*c^2*d^8 + 9*a^4*b^6*c*d^9 + a^5*b
^5*d^10)*x^5 + 3876*(3003*b^10*c^6*d^4 + 1287*a*b^9*c^5*d^5 + 495*a^2*b^8*c^4*d^6 + 165*a^3*b^7*c^3*d^7 + 45*a
^4*b^6*c^2*d^8 + 9*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 969*(6435*b^10*c^7*d^3 + 3003*a*b^9*c^6*d^4 + 1287*a^2*
b^8*c^5*d^5 + 495*a^3*b^7*c^4*d^6 + 165*a^4*b^6*c^3*d^7 + 45*a^5*b^5*c^2*d^8 + 9*a^6*b^4*c*d^9 + a^7*b^3*d^10)
*x^3 + 171*(12870*b^10*c^8*d^2 + 6435*a*b^9*c^7*d^3 + 3003*a^2*b^8*c^6*d^4 + 1287*a^3*b^7*c^5*d^5 + 495*a^4*b^
6*c^4*d^6 + 165*a^5*b^5*c^3*d^7 + 45*a^6*b^4*c^2*d^8 + 9*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 19*(24310*b^10*c^
9*d + 12870*a*b^9*c^8*d^2 + 6435*a^2*b^8*c^7*d^3 + 3003*a^3*b^7*c^6*d^4 + 1287*a^4*b^6*c^5*d^5 + 495*a^5*b^5*c
^4*d^6 + 165*a^6*b^4*c^3*d^7 + 45*a^7*b^3*c^2*d^8 + 9*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^30*x^19 + 19*a*b^29*x^
18 + 171*a^2*b^28*x^17 + 969*a^3*b^27*x^16 + 3876*a^4*b^26*x^15 + 11628*a^5*b^25*x^14 + 27132*a^6*b^24*x^13 +
50388*a^7*b^23*x^12 + 75582*a^8*b^22*x^11 + 92378*a^9*b^21*x^10 + 92378*a^10*b^20*x^9 + 75582*a^11*b^19*x^8 +
50388*a^12*b^18*x^7 + 27132*a^13*b^17*x^6 + 11628*a^14*b^16*x^5 + 3876*a^15*b^15*x^4 + 969*a^16*b^14*x^3 + 171
*a^17*b^13*x^2 + 19*a^18*b^12*x + a^19*b^11)

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giac [B]  time = 1.31, size = 961, normalized size = 3.52 \begin {gather*} -\frac {92378 \, b^{10} d^{10} x^{10} + 831402 \, b^{10} c d^{9} x^{9} + 92378 \, a b^{9} d^{10} x^{9} + 3401190 \, b^{10} c^{2} d^{8} x^{8} + 680238 \, a b^{9} c d^{9} x^{8} + 75582 \, a^{2} b^{8} d^{10} x^{8} + 8314020 \, b^{10} c^{3} d^{7} x^{7} + 2267460 \, a b^{9} c^{2} d^{8} x^{7} + 453492 \, a^{2} b^{8} c d^{9} x^{7} + 50388 \, a^{3} b^{7} d^{10} x^{7} + 13430340 \, b^{10} c^{4} d^{6} x^{6} + 4476780 \, a b^{9} c^{3} d^{7} x^{6} + 1220940 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 244188 \, a^{3} b^{7} c d^{9} x^{6} + 27132 \, a^{4} b^{6} d^{10} x^{6} + 14965236 \, b^{10} c^{5} d^{5} x^{5} + 5755860 \, a b^{9} c^{4} d^{6} x^{5} + 1918620 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 523260 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 104652 \, a^{4} b^{6} c d^{9} x^{5} + 11628 \, a^{5} b^{5} d^{10} x^{5} + 11639628 \, b^{10} c^{6} d^{4} x^{4} + 4988412 \, a b^{9} c^{5} d^{5} x^{4} + 1918620 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 639540 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 174420 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 34884 \, a^{5} b^{5} c d^{9} x^{4} + 3876 \, a^{6} b^{4} d^{10} x^{4} + 6235515 \, b^{10} c^{7} d^{3} x^{3} + 2909907 \, a b^{9} c^{6} d^{4} x^{3} + 1247103 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 479655 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 159885 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 43605 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 8721 \, a^{6} b^{4} c d^{9} x^{3} + 969 \, a^{7} b^{3} d^{10} x^{3} + 2200770 \, b^{10} c^{8} d^{2} x^{2} + 1100385 \, a b^{9} c^{7} d^{3} x^{2} + 513513 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 220077 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 84645 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 28215 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 7695 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 1539 \, a^{7} b^{3} c d^{9} x^{2} + 171 \, a^{8} b^{2} d^{10} x^{2} + 461890 \, b^{10} c^{9} d x + 244530 \, a b^{9} c^{8} d^{2} x + 122265 \, a^{2} b^{8} c^{7} d^{3} x + 57057 \, a^{3} b^{7} c^{6} d^{4} x + 24453 \, a^{4} b^{6} c^{5} d^{5} x + 9405 \, a^{5} b^{5} c^{4} d^{6} x + 3135 \, a^{6} b^{4} c^{3} d^{7} x + 855 \, a^{7} b^{3} c^{2} d^{8} x + 171 \, a^{8} b^{2} c d^{9} x + 19 \, a^{9} b d^{10} x + 43758 \, b^{10} c^{10} + 24310 \, a b^{9} c^{9} d + 12870 \, a^{2} b^{8} c^{8} d^{2} + 6435 \, a^{3} b^{7} c^{7} d^{3} + 3003 \, a^{4} b^{6} c^{6} d^{4} + 1287 \, a^{5} b^{5} c^{5} d^{5} + 495 \, a^{6} b^{4} c^{4} d^{6} + 165 \, a^{7} b^{3} c^{3} d^{7} + 45 \, a^{8} b^{2} c^{2} d^{8} + 9 \, a^{9} b c d^{9} + a^{10} d^{10}}{831402 \, {\left (b x + a\right )}^{19} b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^20,x, algorithm="giac")

[Out]

-1/831402*(92378*b^10*d^10*x^10 + 831402*b^10*c*d^9*x^9 + 92378*a*b^9*d^10*x^9 + 3401190*b^10*c^2*d^8*x^8 + 68
0238*a*b^9*c*d^9*x^8 + 75582*a^2*b^8*d^10*x^8 + 8314020*b^10*c^3*d^7*x^7 + 2267460*a*b^9*c^2*d^8*x^7 + 453492*
a^2*b^8*c*d^9*x^7 + 50388*a^3*b^7*d^10*x^7 + 13430340*b^10*c^4*d^6*x^6 + 4476780*a*b^9*c^3*d^7*x^6 + 1220940*a
^2*b^8*c^2*d^8*x^6 + 244188*a^3*b^7*c*d^9*x^6 + 27132*a^4*b^6*d^10*x^6 + 14965236*b^10*c^5*d^5*x^5 + 5755860*a
*b^9*c^4*d^6*x^5 + 1918620*a^2*b^8*c^3*d^7*x^5 + 523260*a^3*b^7*c^2*d^8*x^5 + 104652*a^4*b^6*c*d^9*x^5 + 11628
*a^5*b^5*d^10*x^5 + 11639628*b^10*c^6*d^4*x^4 + 4988412*a*b^9*c^5*d^5*x^4 + 1918620*a^2*b^8*c^4*d^6*x^4 + 6395
40*a^3*b^7*c^3*d^7*x^4 + 174420*a^4*b^6*c^2*d^8*x^4 + 34884*a^5*b^5*c*d^9*x^4 + 3876*a^6*b^4*d^10*x^4 + 623551
5*b^10*c^7*d^3*x^3 + 2909907*a*b^9*c^6*d^4*x^3 + 1247103*a^2*b^8*c^5*d^5*x^3 + 479655*a^3*b^7*c^4*d^6*x^3 + 15
9885*a^4*b^6*c^3*d^7*x^3 + 43605*a^5*b^5*c^2*d^8*x^3 + 8721*a^6*b^4*c*d^9*x^3 + 969*a^7*b^3*d^10*x^3 + 2200770
*b^10*c^8*d^2*x^2 + 1100385*a*b^9*c^7*d^3*x^2 + 513513*a^2*b^8*c^6*d^4*x^2 + 220077*a^3*b^7*c^5*d^5*x^2 + 8464
5*a^4*b^6*c^4*d^6*x^2 + 28215*a^5*b^5*c^3*d^7*x^2 + 7695*a^6*b^4*c^2*d^8*x^2 + 1539*a^7*b^3*c*d^9*x^2 + 171*a^
8*b^2*d^10*x^2 + 461890*b^10*c^9*d*x + 244530*a*b^9*c^8*d^2*x + 122265*a^2*b^8*c^7*d^3*x + 57057*a^3*b^7*c^6*d
^4*x + 24453*a^4*b^6*c^5*d^5*x + 9405*a^5*b^5*c^4*d^6*x + 3135*a^6*b^4*c^3*d^7*x + 855*a^7*b^3*c^2*d^8*x + 171
*a^8*b^2*c*d^9*x + 19*a^9*b*d^10*x + 43758*b^10*c^10 + 24310*a*b^9*c^9*d + 12870*a^2*b^8*c^8*d^2 + 6435*a^3*b^
7*c^7*d^3 + 3003*a^4*b^6*c^6*d^4 + 1287*a^5*b^5*c^5*d^5 + 495*a^6*b^4*c^4*d^6 + 165*a^7*b^3*c^3*d^7 + 45*a^8*b
^2*c^2*d^8 + 9*a^9*b*c*d^9 + a^10*d^10)/((b*x + a)^19*b^11)

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maple [B]  time = 0.01, size = 866, normalized size = 3.17 \begin {gather*} -\frac {d^{10}}{9 \left (b x +a \right )^{9} b^{11}}+\frac {\left (a d -b c \right ) d^{9}}{\left (b x +a \right )^{10} b^{11}}-\frac {45 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{8}}{11 \left (b x +a \right )^{11} b^{11}}+\frac {10 \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}}{\left (b x +a \right )^{12} b^{11}}-\frac {210 \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}}{13 \left (b x +a \right )^{13} b^{11}}+\frac {18 \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}}{\left (b x +a \right )^{14} b^{11}}-\frac {14 \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}}{\left (b x +a \right )^{15} b^{11}}+\frac {15 \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}}{2 \left (b x +a \right )^{16} b^{11}}-\frac {45 \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}}{17 \left (b x +a \right )^{17} b^{11}}+\frac {5 \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}{9 \left (b x +a \right )^{18} 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}}{19 \left (b x +a \right )^{19} b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^20,x)

[Out]

-210/13*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)^13-1/9*d^10/b^11/(b*x
+a)^9-45/17*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)^17-45/11*d^8*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^11/(b*x
+a)^11-14*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)^15+10*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)^12+5/9*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)^18-1/19*(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)^19+d^9*(a*d-b*c)/b^11/(b*x+a)^10+18*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)^14+15/2*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)^16

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maxima [B]  time = 2.45, size = 1063, normalized size = 3.89 \begin {gather*} -\frac {92378 \, b^{10} d^{10} x^{10} + 43758 \, b^{10} c^{10} + 24310 \, a b^{9} c^{9} d + 12870 \, a^{2} b^{8} c^{8} d^{2} + 6435 \, a^{3} b^{7} c^{7} d^{3} + 3003 \, a^{4} b^{6} c^{6} d^{4} + 1287 \, a^{5} b^{5} c^{5} d^{5} + 495 \, a^{6} b^{4} c^{4} d^{6} + 165 \, a^{7} b^{3} c^{3} d^{7} + 45 \, a^{8} b^{2} c^{2} d^{8} + 9 \, a^{9} b c d^{9} + a^{10} d^{10} + 92378 \, {\left (9 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 75582 \, {\left (45 \, b^{10} c^{2} d^{8} + 9 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 50388 \, {\left (165 \, b^{10} c^{3} d^{7} + 45 \, a b^{9} c^{2} d^{8} + 9 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 27132 \, {\left (495 \, b^{10} c^{4} d^{6} + 165 \, a b^{9} c^{3} d^{7} + 45 \, a^{2} b^{8} c^{2} d^{8} + 9 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 11628 \, {\left (1287 \, b^{10} c^{5} d^{5} + 495 \, a b^{9} c^{4} d^{6} + 165 \, a^{2} b^{8} c^{3} d^{7} + 45 \, a^{3} b^{7} c^{2} d^{8} + 9 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 3876 \, {\left (3003 \, b^{10} c^{6} d^{4} + 1287 \, a b^{9} c^{5} d^{5} + 495 \, a^{2} b^{8} c^{4} d^{6} + 165 \, a^{3} b^{7} c^{3} d^{7} + 45 \, a^{4} b^{6} c^{2} d^{8} + 9 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 969 \, {\left (6435 \, b^{10} c^{7} d^{3} + 3003 \, a b^{9} c^{6} d^{4} + 1287 \, a^{2} b^{8} c^{5} d^{5} + 495 \, a^{3} b^{7} c^{4} d^{6} + 165 \, a^{4} b^{6} c^{3} d^{7} + 45 \, a^{5} b^{5} c^{2} d^{8} + 9 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 171 \, {\left (12870 \, b^{10} c^{8} d^{2} + 6435 \, a b^{9} c^{7} d^{3} + 3003 \, a^{2} b^{8} c^{6} d^{4} + 1287 \, a^{3} b^{7} c^{5} d^{5} + 495 \, a^{4} b^{6} c^{4} d^{6} + 165 \, a^{5} b^{5} c^{3} d^{7} + 45 \, a^{6} b^{4} c^{2} d^{8} + 9 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 19 \, {\left (24310 \, b^{10} c^{9} d + 12870 \, a b^{9} c^{8} d^{2} + 6435 \, a^{2} b^{8} c^{7} d^{3} + 3003 \, a^{3} b^{7} c^{6} d^{4} + 1287 \, a^{4} b^{6} c^{5} d^{5} + 495 \, a^{5} b^{5} c^{4} d^{6} + 165 \, a^{6} b^{4} c^{3} d^{7} + 45 \, a^{7} b^{3} c^{2} d^{8} + 9 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{831402 \, {\left (b^{30} x^{19} + 19 \, a b^{29} x^{18} + 171 \, a^{2} b^{28} x^{17} + 969 \, a^{3} b^{27} x^{16} + 3876 \, a^{4} b^{26} x^{15} + 11628 \, a^{5} b^{25} x^{14} + 27132 \, a^{6} b^{24} x^{13} + 50388 \, a^{7} b^{23} x^{12} + 75582 \, a^{8} b^{22} x^{11} + 92378 \, a^{9} b^{21} x^{10} + 92378 \, a^{10} b^{20} x^{9} + 75582 \, a^{11} b^{19} x^{8} + 50388 \, a^{12} b^{18} x^{7} + 27132 \, a^{13} b^{17} x^{6} + 11628 \, a^{14} b^{16} x^{5} + 3876 \, a^{15} b^{15} x^{4} + 969 \, a^{16} b^{14} x^{3} + 171 \, a^{17} b^{13} x^{2} + 19 \, a^{18} b^{12} x + a^{19} b^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^20,x, algorithm="maxima")

[Out]

-1/831402*(92378*b^10*d^10*x^10 + 43758*b^10*c^10 + 24310*a*b^9*c^9*d + 12870*a^2*b^8*c^8*d^2 + 6435*a^3*b^7*c
^7*d^3 + 3003*a^4*b^6*c^6*d^4 + 1287*a^5*b^5*c^5*d^5 + 495*a^6*b^4*c^4*d^6 + 165*a^7*b^3*c^3*d^7 + 45*a^8*b^2*
c^2*d^8 + 9*a^9*b*c*d^9 + a^10*d^10 + 92378*(9*b^10*c*d^9 + a*b^9*d^10)*x^9 + 75582*(45*b^10*c^2*d^8 + 9*a*b^9
*c*d^9 + a^2*b^8*d^10)*x^8 + 50388*(165*b^10*c^3*d^7 + 45*a*b^9*c^2*d^8 + 9*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7
+ 27132*(495*b^10*c^4*d^6 + 165*a*b^9*c^3*d^7 + 45*a^2*b^8*c^2*d^8 + 9*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 116
28*(1287*b^10*c^5*d^5 + 495*a*b^9*c^4*d^6 + 165*a^2*b^8*c^3*d^7 + 45*a^3*b^7*c^2*d^8 + 9*a^4*b^6*c*d^9 + a^5*b
^5*d^10)*x^5 + 3876*(3003*b^10*c^6*d^4 + 1287*a*b^9*c^5*d^5 + 495*a^2*b^8*c^4*d^6 + 165*a^3*b^7*c^3*d^7 + 45*a
^4*b^6*c^2*d^8 + 9*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 969*(6435*b^10*c^7*d^3 + 3003*a*b^9*c^6*d^4 + 1287*a^2*
b^8*c^5*d^5 + 495*a^3*b^7*c^4*d^6 + 165*a^4*b^6*c^3*d^7 + 45*a^5*b^5*c^2*d^8 + 9*a^6*b^4*c*d^9 + a^7*b^3*d^10)
*x^3 + 171*(12870*b^10*c^8*d^2 + 6435*a*b^9*c^7*d^3 + 3003*a^2*b^8*c^6*d^4 + 1287*a^3*b^7*c^5*d^5 + 495*a^4*b^
6*c^4*d^6 + 165*a^5*b^5*c^3*d^7 + 45*a^6*b^4*c^2*d^8 + 9*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 19*(24310*b^10*c^
9*d + 12870*a*b^9*c^8*d^2 + 6435*a^2*b^8*c^7*d^3 + 3003*a^3*b^7*c^6*d^4 + 1287*a^4*b^6*c^5*d^5 + 495*a^5*b^5*c
^4*d^6 + 165*a^6*b^4*c^3*d^7 + 45*a^7*b^3*c^2*d^8 + 9*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^30*x^19 + 19*a*b^29*x^
18 + 171*a^2*b^28*x^17 + 969*a^3*b^27*x^16 + 3876*a^4*b^26*x^15 + 11628*a^5*b^25*x^14 + 27132*a^6*b^24*x^13 +
50388*a^7*b^23*x^12 + 75582*a^8*b^22*x^11 + 92378*a^9*b^21*x^10 + 92378*a^10*b^20*x^9 + 75582*a^11*b^19*x^8 +
50388*a^12*b^18*x^7 + 27132*a^13*b^17*x^6 + 11628*a^14*b^16*x^5 + 3876*a^15*b^15*x^4 + 969*a^16*b^14*x^3 + 171
*a^17*b^13*x^2 + 19*a^18*b^12*x + a^19*b^11)

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mupad [B]  time = 25.72, size = 1164, normalized size = 4.26 \begin {gather*} -\frac {a^{10}\,d^{10}+9\,a^9\,b\,c\,d^9+19\,a^9\,b\,d^{10}\,x+45\,a^8\,b^2\,c^2\,d^8+171\,a^8\,b^2\,c\,d^9\,x+171\,a^8\,b^2\,d^{10}\,x^2+165\,a^7\,b^3\,c^3\,d^7+855\,a^7\,b^3\,c^2\,d^8\,x+1539\,a^7\,b^3\,c\,d^9\,x^2+969\,a^7\,b^3\,d^{10}\,x^3+495\,a^6\,b^4\,c^4\,d^6+3135\,a^6\,b^4\,c^3\,d^7\,x+7695\,a^6\,b^4\,c^2\,d^8\,x^2+8721\,a^6\,b^4\,c\,d^9\,x^3+3876\,a^6\,b^4\,d^{10}\,x^4+1287\,a^5\,b^5\,c^5\,d^5+9405\,a^5\,b^5\,c^4\,d^6\,x+28215\,a^5\,b^5\,c^3\,d^7\,x^2+43605\,a^5\,b^5\,c^2\,d^8\,x^3+34884\,a^5\,b^5\,c\,d^9\,x^4+11628\,a^5\,b^5\,d^{10}\,x^5+3003\,a^4\,b^6\,c^6\,d^4+24453\,a^4\,b^6\,c^5\,d^5\,x+84645\,a^4\,b^6\,c^4\,d^6\,x^2+159885\,a^4\,b^6\,c^3\,d^7\,x^3+174420\,a^4\,b^6\,c^2\,d^8\,x^4+104652\,a^4\,b^6\,c\,d^9\,x^5+27132\,a^4\,b^6\,d^{10}\,x^6+6435\,a^3\,b^7\,c^7\,d^3+57057\,a^3\,b^7\,c^6\,d^4\,x+220077\,a^3\,b^7\,c^5\,d^5\,x^2+479655\,a^3\,b^7\,c^4\,d^6\,x^3+639540\,a^3\,b^7\,c^3\,d^7\,x^4+523260\,a^3\,b^7\,c^2\,d^8\,x^5+244188\,a^3\,b^7\,c\,d^9\,x^6+50388\,a^3\,b^7\,d^{10}\,x^7+12870\,a^2\,b^8\,c^8\,d^2+122265\,a^2\,b^8\,c^7\,d^3\,x+513513\,a^2\,b^8\,c^6\,d^4\,x^2+1247103\,a^2\,b^8\,c^5\,d^5\,x^3+1918620\,a^2\,b^8\,c^4\,d^6\,x^4+1918620\,a^2\,b^8\,c^3\,d^7\,x^5+1220940\,a^2\,b^8\,c^2\,d^8\,x^6+453492\,a^2\,b^8\,c\,d^9\,x^7+75582\,a^2\,b^8\,d^{10}\,x^8+24310\,a\,b^9\,c^9\,d+244530\,a\,b^9\,c^8\,d^2\,x+1100385\,a\,b^9\,c^7\,d^3\,x^2+2909907\,a\,b^9\,c^6\,d^4\,x^3+4988412\,a\,b^9\,c^5\,d^5\,x^4+5755860\,a\,b^9\,c^4\,d^6\,x^5+4476780\,a\,b^9\,c^3\,d^7\,x^6+2267460\,a\,b^9\,c^2\,d^8\,x^7+680238\,a\,b^9\,c\,d^9\,x^8+92378\,a\,b^9\,d^{10}\,x^9+43758\,b^{10}\,c^{10}+461890\,b^{10}\,c^9\,d\,x+2200770\,b^{10}\,c^8\,d^2\,x^2+6235515\,b^{10}\,c^7\,d^3\,x^3+11639628\,b^{10}\,c^6\,d^4\,x^4+14965236\,b^{10}\,c^5\,d^5\,x^5+13430340\,b^{10}\,c^4\,d^6\,x^6+8314020\,b^{10}\,c^3\,d^7\,x^7+3401190\,b^{10}\,c^2\,d^8\,x^8+831402\,b^{10}\,c\,d^9\,x^9+92378\,b^{10}\,d^{10}\,x^{10}}{831402\,a^{19}\,b^{11}[phone]\,a^{18}\,b^{12}\,x+142169742\,a^{17}\,b^{13}\,x^2+805628538\,a^{16}\,b^{14}\,x^3+3222514152\,a^{15}\,b^{15}\,x^4+9667542456\,a^{14}\,b^{16}\,x^5+22557599064\,a^{13}\,b^{17}\,x^6+41892683976\,a^{12}\,b^{18}\,x^7+62839025964\,a^{11}\,b^{19}\,x^8+76803253956\,a^{10}\,b^{20}\,x^9+76803253956\,a^9\,b^{21}\,x^{10}[phone]\,a^8\,b^{22}\,x^{11}[phone]\,a^7\,b^{23}\,x^{12}[phone]\,a^6\,b^{24}\,x^{13}[phone]\,a^5\,b^{25}\,x^{14}[phone]\,a^4\,b^{26}\,x^{15}[phone]\,a^3\,b^{27}\,x^{16}[phone]\,a^2\,b^{28}\,x^{17}[phone]\,a\,b^{29}\,x^{18}+831402\,b^{30}\,x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^20,x)

[Out]

-(a^10*d^10 + 43758*b^10*c^10 + 92378*b^10*d^10*x^10 + 92378*a*b^9*d^10*x^9 + 831402*b^10*c*d^9*x^9 + 12870*a^
2*b^8*c^8*d^2 + 6435*a^3*b^7*c^7*d^3 + 3003*a^4*b^6*c^6*d^4 + 1287*a^5*b^5*c^5*d^5 + 495*a^6*b^4*c^4*d^6 + 165
*a^7*b^3*c^3*d^7 + 45*a^8*b^2*c^2*d^8 + 171*a^8*b^2*d^10*x^2 + 969*a^7*b^3*d^10*x^3 + 3876*a^6*b^4*d^10*x^4 +
11628*a^5*b^5*d^10*x^5 + 27132*a^4*b^6*d^10*x^6 + 50388*a^3*b^7*d^10*x^7 + 75582*a^2*b^8*d^10*x^8 + 2200770*b^
10*c^8*d^2*x^2 + 6235515*b^10*c^7*d^3*x^3 + 11639628*b^10*c^6*d^4*x^4 + 14965236*b^10*c^5*d^5*x^5 + 13430340*b
^10*c^4*d^6*x^6 + 8314020*b^10*c^3*d^7*x^7 + 3401190*b^10*c^2*d^8*x^8 + 24310*a*b^9*c^9*d + 9*a^9*b*c*d^9 + 19
*a^9*b*d^10*x + 461890*b^10*c^9*d*x + 513513*a^2*b^8*c^6*d^4*x^2 + 220077*a^3*b^7*c^5*d^5*x^2 + 84645*a^4*b^6*
c^4*d^6*x^2 + 28215*a^5*b^5*c^3*d^7*x^2 + 7695*a^6*b^4*c^2*d^8*x^2 + 1247103*a^2*b^8*c^5*d^5*x^3 + 479655*a^3*
b^7*c^4*d^6*x^3 + 159885*a^4*b^6*c^3*d^7*x^3 + 43605*a^5*b^5*c^2*d^8*x^3 + 1918620*a^2*b^8*c^4*d^6*x^4 + 63954
0*a^3*b^7*c^3*d^7*x^4 + 174420*a^4*b^6*c^2*d^8*x^4 + 1918620*a^2*b^8*c^3*d^7*x^5 + 523260*a^3*b^7*c^2*d^8*x^5
+ 1220940*a^2*b^8*c^2*d^8*x^6 + 244530*a*b^9*c^8*d^2*x + 171*a^8*b^2*c*d^9*x + 680238*a*b^9*c*d^9*x^8 + 122265
*a^2*b^8*c^7*d^3*x + 57057*a^3*b^7*c^6*d^4*x + 24453*a^4*b^6*c^5*d^5*x + 9405*a^5*b^5*c^4*d^6*x + 3135*a^6*b^4
*c^3*d^7*x + 855*a^7*b^3*c^2*d^8*x + 1100385*a*b^9*c^7*d^3*x^2 + 1539*a^7*b^3*c*d^9*x^2 + 2909907*a*b^9*c^6*d^
4*x^3 + 8721*a^6*b^4*c*d^9*x^3 + 4988412*a*b^9*c^5*d^5*x^4 + 34884*a^5*b^5*c*d^9*x^4 + 5755860*a*b^9*c^4*d^6*x
^5 + 104652*a^4*b^6*c*d^9*x^5 + 4476780*a*b^9*c^3*d^7*x^6 + 244188*a^3*b^7*c*d^9*x^6 + 2267460*a*b^9*c^2*d^8*x
^7 + 453492*a^2*b^8*c*d^9*x^7)/(831402*a^19*b^11 + 831402*b^30*x^19 + 15796638*a^18*b^12*x + 15796638*a*b^29*x
^18 + 142169742*a^17*b^13*x^2 + 805628538*a^16*b^14*x^3 + 3222514152*a^15*b^15*x^4 + 9667542456*a^14*b^16*x^5
+ 22557599064*a^13*b^17*x^6 + 41892683976*a^12*b^18*x^7 + 62839025964*a^11*b^19*x^8 + 76803253956*a^10*b^20*x^
9 + 76803253956*a^9*b^21*x^10 + 62839025964*a^8*b^22*x^11 + 41892683976*a^7*b^23*x^12 + 22557599064*a^6*b^24*x
^13 + 9667542456*a^5*b^25*x^14 + 3222514152*a^4*b^26*x^15 + 805628538*a^3*b^27*x^16 + 142169742*a^2*b^28*x^17)

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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)**20,x)

[Out]

Timed out

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