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

Optimal. Leaf size=241 \[ \frac {(b c-a d)^{10} \log (a+b x)}{b^{11}}+\frac {d x (b c-a d)^9}{b^{10}}+\frac {(c+d x)^2 (b c-a d)^8}{2 b^9}+\frac {(c+d x)^3 (b c-a d)^7}{3 b^8}+\frac {(c+d x)^4 (b c-a d)^6}{4 b^7}+\frac {(c+d x)^5 (b c-a d)^5}{5 b^6}+\frac {(c+d x)^6 (b c-a d)^4}{6 b^5}+\frac {(c+d x)^7 (b c-a d)^3}{7 b^4}+\frac {(c+d x)^8 (b c-a d)^2}{8 b^3}+\frac {(c+d x)^9 (b c-a d)}{9 b^2}+\frac {(c+d x)^{10}}{10 b} \]

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Rubi [A]  time = 0.10, antiderivative size = 241, 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 x (b c-a d)^9}{b^{10}}+\frac {(c+d x)^2 (b c-a d)^8}{2 b^9}+\frac {(c+d x)^3 (b c-a d)^7}{3 b^8}+\frac {(c+d x)^4 (b c-a d)^6}{4 b^7}+\frac {(c+d x)^5 (b c-a d)^5}{5 b^6}+\frac {(c+d x)^6 (b c-a d)^4}{6 b^5}+\frac {(c+d x)^7 (b c-a d)^3}{7 b^4}+\frac {(c+d x)^8 (b c-a d)^2}{8 b^3}+\frac {(c+d x)^9 (b c-a d)}{9 b^2}+\frac {(b c-a d)^{10} \log (a+b x)}{b^{11}}+\frac {(c+d x)^{10}}{10 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x),x]

[Out]

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

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Mathematica [B]  time = 0.34, size = 591, normalized size = 2.45 \begin {gather*} \frac {d x \left (-2520 a^9 d^9+1260 a^8 b d^8 (20 c+d x)-840 a^7 b^2 d^7 \left (135 c^2+15 c d x+d^2 x^2\right )+210 a^6 b^3 d^6 \left (1440 c^3+270 c^2 d x+40 c d^2 x^2+3 d^3 x^3\right )-252 a^5 b^4 d^5 \left (2100 c^4+600 c^3 d x+150 c^2 d^2 x^2+25 c d^3 x^3+2 d^4 x^4\right )+210 a^4 b^5 d^4 \left (3024 c^5+1260 c^4 d x+480 c^3 d^2 x^2+135 c^2 d^3 x^3+24 c d^4 x^4+2 d^5 x^5\right )-120 a^3 b^6 d^3 \left (4410 c^6+2646 c^5 d x+1470 c^4 d^2 x^2+630 c^3 d^3 x^3+189 c^2 d^4 x^4+35 c d^5 x^5+3 d^6 x^6\right )+45 a^2 b^7 d^2 \left (6720 c^7+5880 c^6 d x+4704 c^5 d^2 x^2+2940 c^4 d^3 x^3+1344 c^3 d^4 x^4+420 c^2 d^5 x^5+80 c d^6 x^6+7 d^7 x^7\right )-10 a b^8 d \left (11340 c^8+15120 c^7 d x+17640 c^6 d^2 x^2+15876 c^5 d^3 x^3+10584 c^4 d^4 x^4+5040 c^3 d^5 x^5+1620 c^2 d^6 x^6+315 c d^7 x^7+28 d^8 x^8\right )+b^9 \left (25200 c^9+56700 c^8 d x+100800 c^7 d^2 x^2+132300 c^6 d^3 x^3+127008 c^5 d^4 x^4+88200 c^4 d^5 x^5+43200 c^3 d^6 x^6+14175 c^2 d^7 x^7+2800 c d^8 x^8+252 d^9 x^9\right )\right )}{2520 b^{10}}+\frac {(b c-a d)^{10} \log (a+b x)}{b^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x),x]

[Out]

(d*x*(-2520*a^9*d^9 + 1260*a^8*b*d^8*(20*c + d*x) - 840*a^7*b^2*d^7*(135*c^2 + 15*c*d*x + d^2*x^2) + 210*a^6*b
^3*d^6*(1440*c^3 + 270*c^2*d*x + 40*c*d^2*x^2 + 3*d^3*x^3) - 252*a^5*b^4*d^5*(2100*c^4 + 600*c^3*d*x + 150*c^2
*d^2*x^2 + 25*c*d^3*x^3 + 2*d^4*x^4) + 210*a^4*b^5*d^4*(3024*c^5 + 1260*c^4*d*x + 480*c^3*d^2*x^2 + 135*c^2*d^
3*x^3 + 24*c*d^4*x^4 + 2*d^5*x^5) - 120*a^3*b^6*d^3*(4410*c^6 + 2646*c^5*d*x + 1470*c^4*d^2*x^2 + 630*c^3*d^3*
x^3 + 189*c^2*d^4*x^4 + 35*c*d^5*x^5 + 3*d^6*x^6) + 45*a^2*b^7*d^2*(6720*c^7 + 5880*c^6*d*x + 4704*c^5*d^2*x^2
+ 2940*c^4*d^3*x^3 + 1344*c^3*d^4*x^4 + 420*c^2*d^5*x^5 + 80*c*d^6*x^6 + 7*d^7*x^7) - 10*a*b^8*d*(11340*c^8 +
15120*c^7*d*x + 17640*c^6*d^2*x^2 + 15876*c^5*d^3*x^3 + 10584*c^4*d^4*x^4 + 5040*c^3*d^5*x^5 + 1620*c^2*d^6*x
^6 + 315*c*d^7*x^7 + 28*d^8*x^8) + b^9*(25200*c^9 + 56700*c^8*d*x + 100800*c^7*d^2*x^2 + 132300*c^6*d^3*x^3 +
127008*c^5*d^4*x^4 + 88200*c^4*d^5*x^5 + 43200*c^3*d^6*x^6 + 14175*c^2*d^7*x^7 + 2800*c*d^8*x^8 + 252*d^9*x^9)
))/(2520*b^10) + ((b*c - a*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} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.28, size = 868, normalized size = 3.60 \begin {gather*} \frac {252 \, b^{10} d^{10} x^{10} + 280 \, {\left (10 \, b^{10} c d^{9} - a b^{9} d^{10}\right )} x^{9} + 315 \, {\left (45 \, b^{10} c^{2} d^{8} - 10 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 360 \, {\left (120 \, b^{10} c^{3} d^{7} - 45 \, a b^{9} c^{2} d^{8} + 10 \, a^{2} b^{8} c d^{9} - a^{3} b^{7} d^{10}\right )} x^{7} + 420 \, {\left (210 \, b^{10} c^{4} d^{6} - 120 \, a b^{9} c^{3} d^{7} + 45 \, 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} + 504 \, {\left (252 \, b^{10} c^{5} d^{5} - 210 \, a b^{9} c^{4} d^{6} + 120 \, a^{2} b^{8} c^{3} d^{7} - 45 \, 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} + 630 \, {\left (210 \, b^{10} c^{6} d^{4} - 252 \, a b^{9} c^{5} d^{5} + 210 \, a^{2} b^{8} c^{4} d^{6} - 120 \, a^{3} b^{7} c^{3} d^{7} + 45 \, 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} + 840 \, {\left (120 \, b^{10} c^{7} d^{3} - 210 \, a b^{9} c^{6} d^{4} + 252 \, a^{2} b^{8} c^{5} d^{5} - 210 \, a^{3} b^{7} c^{4} d^{6} + 120 \, a^{4} b^{6} c^{3} d^{7} - 45 \, 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} + 1260 \, {\left (45 \, b^{10} c^{8} d^{2} - 120 \, a b^{9} c^{7} d^{3} + 210 \, a^{2} b^{8} c^{6} d^{4} - 252 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} - 120 \, a^{5} b^{5} c^{3} d^{7} + 45 \, 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} + 2520 \, {\left (10 \, b^{10} c^{9} d - 45 \, a b^{9} c^{8} d^{2} + 120 \, 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} - 210 \, a^{5} b^{5} c^{4} d^{6} + 120 \, a^{6} b^{4} c^{3} d^{7} - 45 \, a^{7} b^{3} c^{2} d^{8} + 10 \, a^{8} b^{2} c d^{9} - a^{9} b d^{10}\right )} x + 2520 \, {\left (b^{10} c^{10} - 10 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} - 120 \, a^{3} b^{7} c^{7} d^{3} + 210 \, a^{4} b^{6} c^{6} d^{4} - 252 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} - 120 \, a^{7} b^{3} c^{3} d^{7} + 45 \, a^{8} b^{2} c^{2} d^{8} - 10 \, a^{9} b c d^{9} + a^{10} d^{10}\right )} \log \left (b x + a\right )}{2520 \, b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*(252*b^10*d^10*x^10 + 280*(10*b^10*c*d^9 - a*b^9*d^10)*x^9 + 315*(45*b^10*c^2*d^8 - 10*a*b^9*c*d^9 + a^
2*b^8*d^10)*x^8 + 360*(120*b^10*c^3*d^7 - 45*a*b^9*c^2*d^8 + 10*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 420*(210*b
^10*c^4*d^6 - 120*a*b^9*c^3*d^7 + 45*a^2*b^8*c^2*d^8 - 10*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 504*(252*b^10*c^
5*d^5 - 210*a*b^9*c^4*d^6 + 120*a^2*b^8*c^3*d^7 - 45*a^3*b^7*c^2*d^8 + 10*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 +
630*(210*b^10*c^6*d^4 - 252*a*b^9*c^5*d^5 + 210*a^2*b^8*c^4*d^6 - 120*a^3*b^7*c^3*d^7 + 45*a^4*b^6*c^2*d^8 - 1
0*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 840*(120*b^10*c^7*d^3 - 210*a*b^9*c^6*d^4 + 252*a^2*b^8*c^5*d^5 - 210*a^
3*b^7*c^4*d^6 + 120*a^4*b^6*c^3*d^7 - 45*a^5*b^5*c^2*d^8 + 10*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 1260*(45*b^1
0*c^8*d^2 - 120*a*b^9*c^7*d^3 + 210*a^2*b^8*c^6*d^4 - 252*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 - 120*a^5*b^5*
c^3*d^7 + 45*a^6*b^4*c^2*d^8 - 10*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 2520*(10*b^10*c^9*d - 45*a*b^9*c^8*d^2 +
120*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 - 210*a^5*b^5*c^4*d^6 + 120*a^6*b^4*c^3*d^7 -
45*a^7*b^3*c^2*d^8 + 10*a^8*b^2*c*d^9 - a^9*b*d^10)*x + 2520*(b^10*c^10 - 10*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2
- 120*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c^6*d^4 - 252*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 - 120*a^7*b^3*c^3*d^7
+ 45*a^8*b^2*c^2*d^8 - 10*a^9*b*c*d^9 + a^10*d^10)*log(b*x + a))/b^11

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giac [B]  time = 1.35, size = 961, normalized size = 3.99 \begin {gather*} \frac {252 \, b^{9} d^{10} x^{10} + 2800 \, b^{9} c d^{9} x^{9} - 280 \, a b^{8} d^{10} x^{9} + 14175 \, b^{9} c^{2} d^{8} x^{8} - 3150 \, a b^{8} c d^{9} x^{8} + 315 \, a^{2} b^{7} d^{10} x^{8} + 43200 \, b^{9} c^{3} d^{7} x^{7} - 16200 \, a b^{8} c^{2} d^{8} x^{7} + 3600 \, a^{2} b^{7} c d^{9} x^{7} - 360 \, a^{3} b^{6} d^{10} x^{7} + 88200 \, b^{9} c^{4} d^{6} x^{6} - 50400 \, a b^{8} c^{3} d^{7} x^{6} + 18900 \, a^{2} b^{7} c^{2} d^{8} x^{6} - 4200 \, a^{3} b^{6} c d^{9} x^{6} + 420 \, a^{4} b^{5} d^{10} x^{6} + 127008 \, b^{9} c^{5} d^{5} x^{5} - 105840 \, a b^{8} c^{4} d^{6} x^{5} + 60480 \, a^{2} b^{7} c^{3} d^{7} x^{5} - 22680 \, a^{3} b^{6} c^{2} d^{8} x^{5} + 5040 \, a^{4} b^{5} c d^{9} x^{5} - 504 \, a^{5} b^{4} d^{10} x^{5} + 132300 \, b^{9} c^{6} d^{4} x^{4} - 158760 \, a b^{8} c^{5} d^{5} x^{4} + 132300 \, a^{2} b^{7} c^{4} d^{6} x^{4} - 75600 \, a^{3} b^{6} c^{3} d^{7} x^{4} + 28350 \, a^{4} b^{5} c^{2} d^{8} x^{4} - 6300 \, a^{5} b^{4} c d^{9} x^{4} + 630 \, a^{6} b^{3} d^{10} x^{4} + 100800 \, b^{9} c^{7} d^{3} x^{3} - 176400 \, a b^{8} c^{6} d^{4} x^{3} + 211680 \, a^{2} b^{7} c^{5} d^{5} x^{3} - 176400 \, a^{3} b^{6} c^{4} d^{6} x^{3} + 100800 \, a^{4} b^{5} c^{3} d^{7} x^{3} - 37800 \, a^{5} b^{4} c^{2} d^{8} x^{3} + 8400 \, a^{6} b^{3} c d^{9} x^{3} - 840 \, a^{7} b^{2} d^{10} x^{3} + 56700 \, b^{9} c^{8} d^{2} x^{2} - 151200 \, a b^{8} c^{7} d^{3} x^{2} + 264600 \, a^{2} b^{7} c^{6} d^{4} x^{2} - 317520 \, a^{3} b^{6} c^{5} d^{5} x^{2} + 264600 \, a^{4} b^{5} c^{4} d^{6} x^{2} - 151200 \, a^{5} b^{4} c^{3} d^{7} x^{2} + 56700 \, a^{6} b^{3} c^{2} d^{8} x^{2} - 12600 \, a^{7} b^{2} c d^{9} x^{2} + 1260 \, a^{8} b d^{10} x^{2} + 25200 \, b^{9} c^{9} d x - 113400 \, a b^{8} c^{8} d^{2} x + 302400 \, a^{2} b^{7} c^{7} d^{3} x - 529200 \, a^{3} b^{6} c^{6} d^{4} x + 635040 \, a^{4} b^{5} c^{5} d^{5} x - 529200 \, a^{5} b^{4} c^{4} d^{6} x + 302400 \, a^{6} b^{3} c^{3} d^{7} x - 113400 \, a^{7} b^{2} c^{2} d^{8} x + 25200 \, a^{8} b c d^{9} x - 2520 \, a^{9} d^{10} x}{2520 \, b^{10}} + \frac {{\left (b^{10} c^{10} - 10 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} - 120 \, a^{3} b^{7} c^{7} d^{3} + 210 \, a^{4} b^{6} c^{6} d^{4} - 252 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} - 120 \, a^{7} b^{3} c^{3} d^{7} + 45 \, a^{8} b^{2} c^{2} d^{8} - 10 \, a^{9} b c d^{9} + a^{10} d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*(252*b^9*d^10*x^10 + 2800*b^9*c*d^9*x^9 - 280*a*b^8*d^10*x^9 + 14175*b^9*c^2*d^8*x^8 - 3150*a*b^8*c*d^9
*x^8 + 315*a^2*b^7*d^10*x^8 + 43200*b^9*c^3*d^7*x^7 - 16200*a*b^8*c^2*d^8*x^7 + 3600*a^2*b^7*c*d^9*x^7 - 360*a
^3*b^6*d^10*x^7 + 88200*b^9*c^4*d^6*x^6 - 50400*a*b^8*c^3*d^7*x^6 + 18900*a^2*b^7*c^2*d^8*x^6 - 4200*a^3*b^6*c
*d^9*x^6 + 420*a^4*b^5*d^10*x^6 + 127008*b^9*c^5*d^5*x^5 - 105840*a*b^8*c^4*d^6*x^5 + 60480*a^2*b^7*c^3*d^7*x^
5 - 22680*a^3*b^6*c^2*d^8*x^5 + 5040*a^4*b^5*c*d^9*x^5 - 504*a^5*b^4*d^10*x^5 + 132300*b^9*c^6*d^4*x^4 - 15876
0*a*b^8*c^5*d^5*x^4 + 132300*a^2*b^7*c^4*d^6*x^4 - 75600*a^3*b^6*c^3*d^7*x^4 + 28350*a^4*b^5*c^2*d^8*x^4 - 630
0*a^5*b^4*c*d^9*x^4 + 630*a^6*b^3*d^10*x^4 + 100800*b^9*c^7*d^3*x^3 - 176400*a*b^8*c^6*d^4*x^3 + 211680*a^2*b^
7*c^5*d^5*x^3 - 176400*a^3*b^6*c^4*d^6*x^3 + 100800*a^4*b^5*c^3*d^7*x^3 - 37800*a^5*b^4*c^2*d^8*x^3 + 8400*a^6
*b^3*c*d^9*x^3 - 840*a^7*b^2*d^10*x^3 + 56700*b^9*c^8*d^2*x^2 - 151200*a*b^8*c^7*d^3*x^2 + 264600*a^2*b^7*c^6*
d^4*x^2 - 317520*a^3*b^6*c^5*d^5*x^2 + 264600*a^4*b^5*c^4*d^6*x^2 - 151200*a^5*b^4*c^3*d^7*x^2 + 56700*a^6*b^3
*c^2*d^8*x^2 - 12600*a^7*b^2*c*d^9*x^2 + 1260*a^8*b*d^10*x^2 + 25200*b^9*c^9*d*x - 113400*a*b^8*c^8*d^2*x + 30
2400*a^2*b^7*c^7*d^3*x - 529200*a^3*b^6*c^6*d^4*x + 635040*a^4*b^5*c^5*d^5*x - 529200*a^5*b^4*c^4*d^6*x + 3024
00*a^6*b^3*c^3*d^7*x - 113400*a^7*b^2*c^2*d^8*x + 25200*a^8*b*c*d^9*x - 2520*a^9*d^10*x)/b^10 + (b^10*c^10 - 1
0*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 - 120*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c^6*d^4 - 252*a^5*b^5*c^5*d^5 + 210*a^6
*b^4*c^4*d^6 - 120*a^7*b^3*c^3*d^7 + 45*a^8*b^2*c^2*d^8 - 10*a^9*b*c*d^9 + a^10*d^10)*log(abs(b*x + a))/b^11

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maple [B]  time = 0.01, size = 1022, normalized size = 4.24 \begin {gather*} \frac {d^{10} x^{10}}{10 b}-\frac {a \,d^{10} x^{9}}{9 b^{2}}+\frac {10 c \,d^{9} x^{9}}{9 b}+\frac {a^{2} d^{10} x^{8}}{8 b^{3}}-\frac {5 a c \,d^{9} x^{8}}{4 b^{2}}+\frac {45 c^{2} d^{8} x^{8}}{8 b}-\frac {a^{3} d^{10} x^{7}}{7 b^{4}}+\frac {10 a^{2} c \,d^{9} x^{7}}{7 b^{3}}-\frac {45 a \,c^{2} d^{8} x^{7}}{7 b^{2}}+\frac {120 c^{3} d^{7} x^{7}}{7 b}+\frac {a^{4} d^{10} x^{6}}{6 b^{5}}-\frac {5 a^{3} c \,d^{9} x^{6}}{3 b^{4}}+\frac {15 a^{2} c^{2} d^{8} x^{6}}{2 b^{3}}-\frac {20 a \,c^{3} d^{7} x^{6}}{b^{2}}+\frac {35 c^{4} d^{6} x^{6}}{b}-\frac {a^{5} d^{10} x^{5}}{5 b^{6}}+\frac {2 a^{4} c \,d^{9} x^{5}}{b^{5}}-\frac {9 a^{3} c^{2} d^{8} x^{5}}{b^{4}}+\frac {24 a^{2} c^{3} d^{7} x^{5}}{b^{3}}-\frac {42 a \,c^{4} d^{6} x^{5}}{b^{2}}+\frac {252 c^{5} d^{5} x^{5}}{5 b}+\frac {a^{6} d^{10} x^{4}}{4 b^{7}}-\frac {5 a^{5} c \,d^{9} x^{4}}{2 b^{6}}+\frac {45 a^{4} c^{2} d^{8} x^{4}}{4 b^{5}}-\frac {30 a^{3} c^{3} d^{7} x^{4}}{b^{4}}+\frac {105 a^{2} c^{4} d^{6} x^{4}}{2 b^{3}}-\frac {63 a \,c^{5} d^{5} x^{4}}{b^{2}}+\frac {105 c^{6} d^{4} x^{4}}{2 b}-\frac {a^{7} d^{10} x^{3}}{3 b^{8}}+\frac {10 a^{6} c \,d^{9} x^{3}}{3 b^{7}}-\frac {15 a^{5} c^{2} d^{8} x^{3}}{b^{6}}+\frac {40 a^{4} c^{3} d^{7} x^{3}}{b^{5}}-\frac {70 a^{3} c^{4} d^{6} x^{3}}{b^{4}}+\frac {84 a^{2} c^{5} d^{5} x^{3}}{b^{3}}-\frac {70 a \,c^{6} d^{4} x^{3}}{b^{2}}+\frac {40 c^{7} d^{3} x^{3}}{b}+\frac {a^{8} d^{10} x^{2}}{2 b^{9}}-\frac {5 a^{7} c \,d^{9} x^{2}}{b^{8}}+\frac {45 a^{6} c^{2} d^{8} x^{2}}{2 b^{7}}-\frac {60 a^{5} c^{3} d^{7} x^{2}}{b^{6}}+\frac {105 a^{4} c^{4} d^{6} x^{2}}{b^{5}}-\frac {126 a^{3} c^{5} d^{5} x^{2}}{b^{4}}+\frac {105 a^{2} c^{6} d^{4} x^{2}}{b^{3}}-\frac {60 a \,c^{7} d^{3} x^{2}}{b^{2}}+\frac {45 c^{8} d^{2} x^{2}}{2 b}+\frac {a^{10} d^{10} \ln \left (b x +a \right )}{b^{11}}-\frac {10 a^{9} c \,d^{9} \ln \left (b x +a \right )}{b^{10}}-\frac {a^{9} d^{10} x}{b^{10}}+\frac {45 a^{8} c^{2} d^{8} \ln \left (b x +a \right )}{b^{9}}+\frac {10 a^{8} c \,d^{9} x}{b^{9}}-\frac {120 a^{7} c^{3} d^{7} \ln \left (b x +a \right )}{b^{8}}-\frac {45 a^{7} c^{2} d^{8} x}{b^{8}}+\frac {210 a^{6} c^{4} d^{6} \ln \left (b x +a \right )}{b^{7}}+\frac {120 a^{6} c^{3} d^{7} x}{b^{7}}-\frac {252 a^{5} c^{5} d^{5} \ln \left (b x +a \right )}{b^{6}}-\frac {210 a^{5} c^{4} d^{6} x}{b^{6}}+\frac {210 a^{4} c^{6} d^{4} \ln \left (b x +a \right )}{b^{5}}+\frac {252 a^{4} c^{5} d^{5} x}{b^{5}}-\frac {120 a^{3} c^{7} d^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {210 a^{3} c^{6} d^{4} x}{b^{4}}+\frac {45 a^{2} c^{8} d^{2} \ln \left (b x +a \right )}{b^{3}}+\frac {120 a^{2} c^{7} d^{3} x}{b^{3}}-\frac {10 a \,c^{9} d \ln \left (b x +a \right )}{b^{2}}-\frac {45 a \,c^{8} d^{2} x}{b^{2}}+\frac {c^{10} \ln \left (b x +a \right )}{b}+\frac {10 c^{9} d x}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

105/2*d^4/b*x^4*c^6-1/3*d^10/b^8*x^3*a^7+40*d^3/b*x^3*c^7+1/2*d^10/b^9*x^2*a^8+45/2*d^2/b*x^2*c^8+1/8*d^10/b^3
*x^8*a^2+45/8*d^8/b*x^8*c^2-1/7*d^10/b^4*x^7*a^3+120/7*d^7/b*x^7*c^3+1/6*d^10/b^5*x^6*a^4+35*d^6/b*x^6*c^4-1/5
*d^10/b^6*x^5*a^5+252/5*d^5/b*x^5*c^5+1/4*d^10/b^7*x^4*a^6-1/9*d^10/b^2*x^9*a+10/9*d^9/b*x^9*c+1/b^11*ln(b*x+a
)*a^10*d^10+10*d/b*c^9*x-d^10/b^10*a^9*x-42*d^6/b^2*x^5*a*c^4-5/2*d^9/b^6*x^4*a^5*c+10/3*d^9/b^7*x^3*a^6*c-15*
d^8/b^6*x^3*a^5*c^2+24*d^7/b^3*x^5*a^2*c^3+40*d^7/b^5*x^3*a^4*c^3-30*d^7/b^4*x^4*a^3*c^3+105/2*d^6/b^3*x^4*a^2
*c^4-63*d^5/b^2*x^4*a*c^5-5/4*d^9/b^2*x^8*a*c+2*d^9/b^5*x^5*a^4*c-9*d^8/b^4*x^5*a^3*c^2-5/3*d^9/b^4*x^6*a^3*c+
15/2*d^8/b^3*x^6*a^2*c^2-20*d^7/b^2*x^6*a*c^3+10/7*d^9/b^3*x^7*a^2*c-45/7*d^8/b^2*x^7*a*c^2-120/b^8*ln(b*x+a)*
a^7*c^3*d^7+210/b^7*ln(b*x+a)*a^6*c^4*d^6-252/b^6*ln(b*x+a)*a^5*c^5*d^5+210/b^5*ln(b*x+a)*a^4*c^6*d^4-120/b^4*
ln(b*x+a)*a^3*c^7*d^3+45/b^3*ln(b*x+a)*a^2*c^8*d^2-10/b^2*ln(b*x+a)*a*c^9*d-10/b^10*ln(b*x+a)*a^9*c*d^9+45/b^9
*ln(b*x+a)*a^8*c^2*d^8+1/10*d^10/b*x^10+1/b*ln(b*x+a)*c^10+45/4*d^8/b^5*x^4*a^4*c^2+120*d^3/b^3*a^2*c^7*x-45*d
^2/b^2*a*c^8*x-70*d^4/b^2*x^3*a*c^6-5*d^9/b^8*x^2*a^7*c+45/2*d^8/b^7*x^2*a^6*c^2-60*d^7/b^6*x^2*a^5*c^3+105*d^
6/b^5*x^2*a^4*c^4-126*d^5/b^4*x^2*a^3*c^5+105*d^4/b^3*x^2*a^2*c^6-60*d^3/b^2*x^2*a*c^7+10*d^9/b^9*a^8*c*x-45*d
^8/b^8*a^7*c^2*x+120*d^7/b^7*a^6*c^3*x-210*d^6/b^6*a^5*c^4*x-70*d^6/b^4*x^3*a^3*c^4+84*d^5/b^3*x^3*a^2*c^5+252
*d^5/b^5*a^4*c^5*x-210*d^4/b^4*a^3*c^6*x

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maxima [B]  time = 1.52, size = 866, normalized size = 3.59 \begin {gather*} \frac {252 \, b^{9} d^{10} x^{10} + 280 \, {\left (10 \, b^{9} c d^{9} - a b^{8} d^{10}\right )} x^{9} + 315 \, {\left (45 \, b^{9} c^{2} d^{8} - 10 \, a b^{8} c d^{9} + a^{2} b^{7} d^{10}\right )} x^{8} + 360 \, {\left (120 \, b^{9} c^{3} d^{7} - 45 \, a b^{8} c^{2} d^{8} + 10 \, a^{2} b^{7} c d^{9} - a^{3} b^{6} d^{10}\right )} x^{7} + 420 \, {\left (210 \, b^{9} c^{4} d^{6} - 120 \, a b^{8} c^{3} d^{7} + 45 \, a^{2} b^{7} c^{2} d^{8} - 10 \, a^{3} b^{6} c d^{9} + a^{4} b^{5} d^{10}\right )} x^{6} + 504 \, {\left (252 \, b^{9} c^{5} d^{5} - 210 \, a b^{8} c^{4} d^{6} + 120 \, a^{2} b^{7} c^{3} d^{7} - 45 \, a^{3} b^{6} c^{2} d^{8} + 10 \, a^{4} b^{5} c d^{9} - a^{5} b^{4} d^{10}\right )} x^{5} + 630 \, {\left (210 \, b^{9} c^{6} d^{4} - 252 \, a b^{8} c^{5} d^{5} + 210 \, a^{2} b^{7} c^{4} d^{6} - 120 \, a^{3} b^{6} c^{3} d^{7} + 45 \, a^{4} b^{5} c^{2} d^{8} - 10 \, a^{5} b^{4} c d^{9} + a^{6} b^{3} d^{10}\right )} x^{4} + 840 \, {\left (120 \, b^{9} c^{7} d^{3} - 210 \, a b^{8} c^{6} d^{4} + 252 \, a^{2} b^{7} c^{5} d^{5} - 210 \, a^{3} b^{6} c^{4} d^{6} + 120 \, a^{4} b^{5} c^{3} d^{7} - 45 \, a^{5} b^{4} c^{2} d^{8} + 10 \, a^{6} b^{3} c d^{9} - a^{7} b^{2} d^{10}\right )} x^{3} + 1260 \, {\left (45 \, b^{9} c^{8} d^{2} - 120 \, a b^{8} c^{7} d^{3} + 210 \, a^{2} b^{7} c^{6} d^{4} - 252 \, a^{3} b^{6} c^{5} d^{5} + 210 \, a^{4} b^{5} c^{4} d^{6} - 120 \, a^{5} b^{4} c^{3} d^{7} + 45 \, a^{6} b^{3} c^{2} d^{8} - 10 \, a^{7} b^{2} c d^{9} + a^{8} b d^{10}\right )} x^{2} + 2520 \, {\left (10 \, b^{9} c^{9} d - 45 \, a b^{8} c^{8} d^{2} + 120 \, a^{2} b^{7} c^{7} d^{3} - 210 \, a^{3} b^{6} c^{6} d^{4} + 252 \, a^{4} b^{5} c^{5} d^{5} - 210 \, a^{5} b^{4} c^{4} d^{6} + 120 \, a^{6} b^{3} c^{3} d^{7} - 45 \, a^{7} b^{2} c^{2} d^{8} + 10 \, a^{8} b c d^{9} - a^{9} d^{10}\right )} x}{2520 \, b^{10}} + \frac {{\left (b^{10} c^{10} - 10 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} - 120 \, a^{3} b^{7} c^{7} d^{3} + 210 \, a^{4} b^{6} c^{6} d^{4} - 252 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} - 120 \, a^{7} b^{3} c^{3} d^{7} + 45 \, a^{8} b^{2} c^{2} d^{8} - 10 \, a^{9} b c d^{9} + a^{10} d^{10}\right )} \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),x, algorithm="maxima")

[Out]

1/2520*(252*b^9*d^10*x^10 + 280*(10*b^9*c*d^9 - a*b^8*d^10)*x^9 + 315*(45*b^9*c^2*d^8 - 10*a*b^8*c*d^9 + a^2*b
^7*d^10)*x^8 + 360*(120*b^9*c^3*d^7 - 45*a*b^8*c^2*d^8 + 10*a^2*b^7*c*d^9 - a^3*b^6*d^10)*x^7 + 420*(210*b^9*c
^4*d^6 - 120*a*b^8*c^3*d^7 + 45*a^2*b^7*c^2*d^8 - 10*a^3*b^6*c*d^9 + a^4*b^5*d^10)*x^6 + 504*(252*b^9*c^5*d^5
- 210*a*b^8*c^4*d^6 + 120*a^2*b^7*c^3*d^7 - 45*a^3*b^6*c^2*d^8 + 10*a^4*b^5*c*d^9 - a^5*b^4*d^10)*x^5 + 630*(2
10*b^9*c^6*d^4 - 252*a*b^8*c^5*d^5 + 210*a^2*b^7*c^4*d^6 - 120*a^3*b^6*c^3*d^7 + 45*a^4*b^5*c^2*d^8 - 10*a^5*b
^4*c*d^9 + a^6*b^3*d^10)*x^4 + 840*(120*b^9*c^7*d^3 - 210*a*b^8*c^6*d^4 + 252*a^2*b^7*c^5*d^5 - 210*a^3*b^6*c^
4*d^6 + 120*a^4*b^5*c^3*d^7 - 45*a^5*b^4*c^2*d^8 + 10*a^6*b^3*c*d^9 - a^7*b^2*d^10)*x^3 + 1260*(45*b^9*c^8*d^2
- 120*a*b^8*c^7*d^3 + 210*a^2*b^7*c^6*d^4 - 252*a^3*b^6*c^5*d^5 + 210*a^4*b^5*c^4*d^6 - 120*a^5*b^4*c^3*d^7 +
45*a^6*b^3*c^2*d^8 - 10*a^7*b^2*c*d^9 + a^8*b*d^10)*x^2 + 2520*(10*b^9*c^9*d - 45*a*b^8*c^8*d^2 + 120*a^2*b^7
*c^7*d^3 - 210*a^3*b^6*c^6*d^4 + 252*a^4*b^5*c^5*d^5 - 210*a^5*b^4*c^4*d^6 + 120*a^6*b^3*c^3*d^7 - 45*a^7*b^2*
c^2*d^8 + 10*a^8*b*c*d^9 - a^9*d^10)*x)/b^10 + (b^10*c^10 - 10*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 - 120*a^3*b^7*
c^7*d^3 + 210*a^4*b^6*c^6*d^4 - 252*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 - 120*a^7*b^3*c^3*d^7 + 45*a^8*b^2*c
^2*d^8 - 10*a^9*b*c*d^9 + a^10*d^10)*log(b*x + a)/b^11

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mupad [B]  time = 0.13, size = 979, normalized size = 4.06 \begin {gather*} x^7\,\left (\frac {120\,c^3\,d^7}{7\,b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{b}+\frac {45\,c^2\,d^8}{b}\right )}{7\,b}\right )-x^9\,\left (\frac {a\,d^{10}}{9\,b^2}-\frac {10\,c\,d^9}{9\,b}\right )+x^5\,\left (\frac {a\,\left (\frac {a\,\left (\frac {120\,c^3\,d^7}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{b}+\frac {45\,c^2\,d^8}{b}\right )}{b}\right )}{b}-\frac {210\,c^4\,d^6}{b}\right )}{5\,b}+\frac {252\,c^5\,d^5}{5\,b}\right )+x^3\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {120\,c^3\,d^7}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{b}+\frac {45\,c^2\,d^8}{b}\right )}{b}\right )}{b}-\frac {210\,c^4\,d^6}{b}\right )}{b}+\frac {252\,c^5\,d^5}{b}\right )}{b}-\frac {210\,c^6\,d^4}{b}\right )}{3\,b}+\frac {40\,c^7\,d^3}{b}\right )+x\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {120\,c^3\,d^7}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{b}+\frac {45\,c^2\,d^8}{b}\right )}{b}\right )}{b}-\frac {210\,c^4\,d^6}{b}\right )}{b}+\frac {252\,c^5\,d^5}{b}\right )}{b}-\frac {210\,c^6\,d^4}{b}\right )}{b}+\frac {120\,c^7\,d^3}{b}\right )}{b}-\frac {45\,c^8\,d^2}{b}\right )}{b}+\frac {10\,c^9\,d}{b}\right )+x^8\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{8\,b}+\frac {45\,c^2\,d^8}{8\,b}\right )-x^6\,\left (\frac {a\,\left (\frac {120\,c^3\,d^7}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{b}+\frac {45\,c^2\,d^8}{b}\right )}{b}\right )}{6\,b}-\frac {35\,c^4\,d^6}{b}\right )-x^4\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {120\,c^3\,d^7}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{b}+\frac {45\,c^2\,d^8}{b}\right )}{b}\right )}{b}-\frac {210\,c^4\,d^6}{b}\right )}{b}+\frac {252\,c^5\,d^5}{b}\right )}{4\,b}-\frac {105\,c^6\,d^4}{2\,b}\right )-x^2\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {120\,c^3\,d^7}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^{10}}{b^2}-\frac {10\,c\,d^9}{b}\right )}{b}+\frac {45\,c^2\,d^8}{b}\right )}{b}\right )}{b}-\frac {210\,c^4\,d^6}{b}\right )}{b}+\frac {252\,c^5\,d^5}{b}\right )}{b}-\frac {210\,c^6\,d^4}{b}\right )}{b}+\frac {120\,c^7\,d^3}{b}\right )}{2\,b}-\frac {45\,c^8\,d^2}{2\,b}\right )+\frac {d^{10}\,x^{10}}{10\,b}+\frac {\ln \left (a+b\,x\right )\,\left (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}\right )}{b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x),x)

[Out]

x^7*((120*c^3*d^7)/(7*b) - (a*((a*((a*d^10)/b^2 - (10*c*d^9)/b))/b + (45*c^2*d^8)/b))/(7*b)) - x^9*((a*d^10)/(
9*b^2) - (10*c*d^9)/(9*b)) + x^5*((a*((a*((120*c^3*d^7)/b - (a*((a*((a*d^10)/b^2 - (10*c*d^9)/b))/b + (45*c^2*
d^8)/b))/b))/b - (210*c^4*d^6)/b))/(5*b) + (252*c^5*d^5)/(5*b)) + x^3*((a*((a*((a*((a*((120*c^3*d^7)/b - (a*((
a*((a*d^10)/b^2 - (10*c*d^9)/b))/b + (45*c^2*d^8)/b))/b))/b - (210*c^4*d^6)/b))/b + (252*c^5*d^5)/b))/b - (210
*c^6*d^4)/b))/(3*b) + (40*c^7*d^3)/b) + x*((a*((a*((a*((a*((a*((a*((120*c^3*d^7)/b - (a*((a*((a*d^10)/b^2 - (1
0*c*d^9)/b))/b + (45*c^2*d^8)/b))/b))/b - (210*c^4*d^6)/b))/b + (252*c^5*d^5)/b))/b - (210*c^6*d^4)/b))/b + (1
20*c^7*d^3)/b))/b - (45*c^8*d^2)/b))/b + (10*c^9*d)/b) + x^8*((a*((a*d^10)/b^2 - (10*c*d^9)/b))/(8*b) + (45*c^
2*d^8)/(8*b)) - x^6*((a*((120*c^3*d^7)/b - (a*((a*((a*d^10)/b^2 - (10*c*d^9)/b))/b + (45*c^2*d^8)/b))/b))/(6*b
) - (35*c^4*d^6)/b) - x^4*((a*((a*((a*((120*c^3*d^7)/b - (a*((a*((a*d^10)/b^2 - (10*c*d^9)/b))/b + (45*c^2*d^8
)/b))/b))/b - (210*c^4*d^6)/b))/b + (252*c^5*d^5)/b))/(4*b) - (105*c^6*d^4)/(2*b)) - x^2*((a*((a*((a*((a*((a*(
(120*c^3*d^7)/b - (a*((a*((a*d^10)/b^2 - (10*c*d^9)/b))/b + (45*c^2*d^8)/b))/b))/b - (210*c^4*d^6)/b))/b + (25
2*c^5*d^5)/b))/b - (210*c^6*d^4)/b))/b + (120*c^7*d^3)/b))/(2*b) - (45*c^8*d^2)/(2*b)) + (d^10*x^10)/(10*b) +
(log(a + b*x)*(a^10*d^10 + b^10*c^10 + 45*a^2*b^8*c^8*d^2 - 120*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c^6*d^4 - 252*a^
5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 - 120*a^7*b^3*c^3*d^7 + 45*a^8*b^2*c^2*d^8 - 10*a*b^9*c^9*d - 10*a^9*b*c*d
^9))/b^11

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sympy [B]  time = 1.42, size = 799, normalized size = 3.32 \begin {gather*} x^{9} \left (- \frac {a d^{10}}{9 b^{2}} + \frac {10 c d^{9}}{9 b}\right ) + x^{8} \left (\frac {a^{2} d^{10}}{8 b^{3}} - \frac {5 a c d^{9}}{4 b^{2}} + \frac {45 c^{2} d^{8}}{8 b}\right ) + x^{7} \left (- \frac {a^{3} d^{10}}{7 b^{4}} + \frac {10 a^{2} c d^{9}}{7 b^{3}} - \frac {45 a c^{2} d^{8}}{7 b^{2}} + \frac {120 c^{3} d^{7}}{7 b}\right ) + x^{6} \left (\frac {a^{4} d^{10}}{6 b^{5}} - \frac {5 a^{3} c d^{9}}{3 b^{4}} + \frac {15 a^{2} c^{2} d^{8}}{2 b^{3}} - \frac {20 a c^{3} d^{7}}{b^{2}} + \frac {35 c^{4} d^{6}}{b}\right ) + x^{5} \left (- \frac {a^{5} d^{10}}{5 b^{6}} + \frac {2 a^{4} c d^{9}}{b^{5}} - \frac {9 a^{3} c^{2} d^{8}}{b^{4}} + \frac {24 a^{2} c^{3} d^{7}}{b^{3}} - \frac {42 a c^{4} d^{6}}{b^{2}} + \frac {252 c^{5} d^{5}}{5 b}\right ) + x^{4} \left (\frac {a^{6} d^{10}}{4 b^{7}} - \frac {5 a^{5} c d^{9}}{2 b^{6}} + \frac {45 a^{4} c^{2} d^{8}}{4 b^{5}} - \frac {30 a^{3} c^{3} d^{7}}{b^{4}} + \frac {105 a^{2} c^{4} d^{6}}{2 b^{3}} - \frac {63 a c^{5} d^{5}}{b^{2}} + \frac {105 c^{6} d^{4}}{2 b}\right ) + x^{3} \left (- \frac {a^{7} d^{10}}{3 b^{8}} + \frac {10 a^{6} c d^{9}}{3 b^{7}} - \frac {15 a^{5} c^{2} d^{8}}{b^{6}} + \frac {40 a^{4} c^{3} d^{7}}{b^{5}} - \frac {70 a^{3} c^{4} d^{6}}{b^{4}} + \frac {84 a^{2} c^{5} d^{5}}{b^{3}} - \frac {70 a c^{6} d^{4}}{b^{2}} + \frac {40 c^{7} d^{3}}{b}\right ) + x^{2} \left (\frac {a^{8} d^{10}}{2 b^{9}} - \frac {5 a^{7} c d^{9}}{b^{8}} + \frac {45 a^{6} c^{2} d^{8}}{2 b^{7}} - \frac {60 a^{5} c^{3} d^{7}}{b^{6}} + \frac {105 a^{4} c^{4} d^{6}}{b^{5}} - \frac {126 a^{3} c^{5} d^{5}}{b^{4}} + \frac {105 a^{2} c^{6} d^{4}}{b^{3}} - \frac {60 a c^{7} d^{3}}{b^{2}} + \frac {45 c^{8} d^{2}}{2 b}\right ) + x \left (- \frac {a^{9} d^{10}}{b^{10}} + \frac {10 a^{8} c d^{9}}{b^{9}} - \frac {45 a^{7} c^{2} d^{8}}{b^{8}} + \frac {120 a^{6} c^{3} d^{7}}{b^{7}} - \frac {210 a^{5} c^{4} d^{6}}{b^{6}} + \frac {252 a^{4} c^{5} d^{5}}{b^{5}} - \frac {210 a^{3} c^{6} d^{4}}{b^{4}} + \frac {120 a^{2} c^{7} d^{3}}{b^{3}} - \frac {45 a c^{8} d^{2}}{b^{2}} + \frac {10 c^{9} d}{b}\right ) + \frac {d^{10} x^{10}}{10 b} + \frac {\left (a d - b c\right )^{10} \log {\left (a + b x \right )}}{b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**9*(-a*d**10/(9*b**2) + 10*c*d**9/(9*b)) + x**8*(a**2*d**10/(8*b**3) - 5*a*c*d**9/(4*b**2) + 45*c**2*d**8/(8
*b)) + x**7*(-a**3*d**10/(7*b**4) + 10*a**2*c*d**9/(7*b**3) - 45*a*c**2*d**8/(7*b**2) + 120*c**3*d**7/(7*b)) +
x**6*(a**4*d**10/(6*b**5) - 5*a**3*c*d**9/(3*b**4) + 15*a**2*c**2*d**8/(2*b**3) - 20*a*c**3*d**7/b**2 + 35*c*
*4*d**6/b) + x**5*(-a**5*d**10/(5*b**6) + 2*a**4*c*d**9/b**5 - 9*a**3*c**2*d**8/b**4 + 24*a**2*c**3*d**7/b**3
- 42*a*c**4*d**6/b**2 + 252*c**5*d**5/(5*b)) + x**4*(a**6*d**10/(4*b**7) - 5*a**5*c*d**9/(2*b**6) + 45*a**4*c*
*2*d**8/(4*b**5) - 30*a**3*c**3*d**7/b**4 + 105*a**2*c**4*d**6/(2*b**3) - 63*a*c**5*d**5/b**2 + 105*c**6*d**4/
(2*b)) + x**3*(-a**7*d**10/(3*b**8) + 10*a**6*c*d**9/(3*b**7) - 15*a**5*c**2*d**8/b**6 + 40*a**4*c**3*d**7/b**
5 - 70*a**3*c**4*d**6/b**4 + 84*a**2*c**5*d**5/b**3 - 70*a*c**6*d**4/b**2 + 40*c**7*d**3/b) + x**2*(a**8*d**10
/(2*b**9) - 5*a**7*c*d**9/b**8 + 45*a**6*c**2*d**8/(2*b**7) - 60*a**5*c**3*d**7/b**6 + 105*a**4*c**4*d**6/b**5
- 126*a**3*c**5*d**5/b**4 + 105*a**2*c**6*d**4/b**3 - 60*a*c**7*d**3/b**2 + 45*c**8*d**2/(2*b)) + x*(-a**9*d*
*10/b**10 + 10*a**8*c*d**9/b**9 - 45*a**7*c**2*d**8/b**8 + 120*a**6*c**3*d**7/b**7 - 210*a**5*c**4*d**6/b**6 +
252*a**4*c**5*d**5/b**5 - 210*a**3*c**6*d**4/b**4 + 120*a**2*c**7*d**3/b**3 - 45*a*c**8*d**2/b**2 + 10*c**9*d
/b) + d**10*x**10/(10*b) + (a*d - b*c)**10*log(a + b*x)/b**11

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