3.14.13 \(\int \frac {(c+d x)^{10}}{(a+b x)^2} \, dx\) [1313]

Optimal. Leaf size=258 \[ \frac {45 d^2 (b c-a d)^8 x}{b^{10}}-\frac {(b c-a d)^{10}}{b^{11} (a+b x)}+\frac {60 d^3 (b c-a d)^7 (a+b x)^2}{b^{11}}+\frac {70 d^4 (b c-a d)^6 (a+b x)^3}{b^{11}}+\frac {63 d^5 (b c-a d)^5 (a+b x)^4}{b^{11}}+\frac {42 d^6 (b c-a d)^4 (a+b x)^5}{b^{11}}+\frac {20 d^7 (b c-a d)^3 (a+b x)^6}{b^{11}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^7}{7 b^{11}}+\frac {5 d^9 (b c-a d) (a+b x)^8}{4 b^{11}}+\frac {d^{10} (a+b x)^9}{9 b^{11}}+\frac {10 d (b c-a d)^9 \log (a+b x)}{b^{11}} \]

[Out]

45*d^2*(-a*d+b*c)^8*x/b^10-(-a*d+b*c)^10/b^11/(b*x+a)+60*d^3*(-a*d+b*c)^7*(b*x+a)^2/b^11+70*d^4*(-a*d+b*c)^6*(
b*x+a)^3/b^11+63*d^5*(-a*d+b*c)^5*(b*x+a)^4/b^11+42*d^6*(-a*d+b*c)^4*(b*x+a)^5/b^11+20*d^7*(-a*d+b*c)^3*(b*x+a
)^6/b^11+45/7*d^8*(-a*d+b*c)^2*(b*x+a)^7/b^11+5/4*d^9*(-a*d+b*c)*(b*x+a)^8/b^11+1/9*d^10*(b*x+a)^9/b^11+10*d*(
-a*d+b*c)^9*ln(b*x+a)/b^11

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Rubi [A]
time = 0.32, antiderivative size = 258, 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 = {45} \begin {gather*} \frac {5 d^9 (a+b x)^8 (b c-a d)}{4 b^{11}}+\frac {45 d^8 (a+b x)^7 (b c-a d)^2}{7 b^{11}}+\frac {20 d^7 (a+b x)^6 (b c-a d)^3}{b^{11}}+\frac {42 d^6 (a+b x)^5 (b c-a d)^4}{b^{11}}+\frac {63 d^5 (a+b x)^4 (b c-a d)^5}{b^{11}}+\frac {70 d^4 (a+b x)^3 (b c-a d)^6}{b^{11}}+\frac {60 d^3 (a+b x)^2 (b c-a d)^7}{b^{11}}-\frac {(b c-a d)^{10}}{b^{11} (a+b x)}+\frac {10 d (b c-a d)^9 \log (a+b x)}{b^{11}}+\frac {d^{10} (a+b x)^9}{9 b^{11}}+\frac {45 d^2 x (b c-a d)^8}{b^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^2,x]

[Out]

(45*d^2*(b*c - a*d)^8*x)/b^10 - (b*c - a*d)^10/(b^11*(a + b*x)) + (60*d^3*(b*c - a*d)^7*(a + b*x)^2)/b^11 + (7
0*d^4*(b*c - a*d)^6*(a + b*x)^3)/b^11 + (63*d^5*(b*c - a*d)^5*(a + b*x)^4)/b^11 + (42*d^6*(b*c - a*d)^4*(a + b
*x)^5)/b^11 + (20*d^7*(b*c - a*d)^3*(a + b*x)^6)/b^11 + (45*d^8*(b*c - a*d)^2*(a + b*x)^7)/(7*b^11) + (5*d^9*(
b*c - a*d)*(a + b*x)^8)/(4*b^11) + (d^10*(a + b*x)^9)/(9*b^11) + (10*d*(b*c - a*d)^9*Log[a + b*x])/b^11

Rule 45

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

Rubi steps

\begin {align*} \int \frac {(c+d x)^{10}}{(a+b x)^2} \, dx &=\int \left (\frac {45 d^2 (b c-a d)^8}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^2}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)}+\frac {120 d^3 (b c-a d)^7 (a+b x)}{b^{10}}+\frac {210 d^4 (b c-a d)^6 (a+b x)^2}{b^{10}}+\frac {252 d^5 (b c-a d)^5 (a+b x)^3}{b^{10}}+\frac {210 d^6 (b c-a d)^4 (a+b x)^4}{b^{10}}+\frac {120 d^7 (b c-a d)^3 (a+b x)^5}{b^{10}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^6}{b^{10}}+\frac {10 d^9 (b c-a d) (a+b x)^7}{b^{10}}+\frac {d^{10} (a+b x)^8}{b^{10}}\right ) \, dx\\ &=\frac {45 d^2 (b c-a d)^8 x}{b^{10}}-\frac {(b c-a d)^{10}}{b^{11} (a+b x)}+\frac {60 d^3 (b c-a d)^7 (a+b x)^2}{b^{11}}+\frac {70 d^4 (b c-a d)^6 (a+b x)^3}{b^{11}}+\frac {63 d^5 (b c-a d)^5 (a+b x)^4}{b^{11}}+\frac {42 d^6 (b c-a d)^4 (a+b x)^5}{b^{11}}+\frac {20 d^7 (b c-a d)^3 (a+b x)^6}{b^{11}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^7}{7 b^{11}}+\frac {5 d^9 (b c-a d) (a+b x)^8}{4 b^{11}}+\frac {d^{10} (a+b x)^9}{9 b^{11}}+\frac {10 d (b c-a d)^9 \log (a+b x)}{b^{11}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(708\) vs. \(2(258)=516\).
time = 0.14, size = 708, normalized size = 2.74 \begin {gather*} \frac {-252 a^{10} d^{10}+252 a^9 b d^9 (10 c+9 d x)+1260 a^8 b^2 d^8 \left (-9 c^2-16 c d x+d^2 x^2\right )-420 a^7 b^3 d^7 \left (-72 c^3-189 c^2 d x+27 c d^2 x^2+d^3 x^3\right )+210 a^6 b^4 d^6 \left (-252 c^4-864 c^3 d x+216 c^2 d^2 x^2+18 c d^3 x^3+d^4 x^4\right )-126 a^5 b^5 d^5 \left (-504 c^5-2100 c^4 d x+840 c^3 d^2 x^2+120 c^2 d^3 x^3+15 c d^4 x^4+d^5 x^5\right )+42 a^4 b^6 d^4 \left (-1260 c^6-6048 c^5 d x+3780 c^4 d^2 x^2+840 c^3 d^3 x^3+180 c^2 d^4 x^4+27 c d^5 x^5+2 d^6 x^6\right )-12 a^3 b^7 d^3 \left (-2520 c^7-13230 c^6 d x+13230 c^5 d^2 x^2+4410 c^4 d^3 x^3+1470 c^3 d^4 x^4+378 c^2 d^5 x^5+63 c d^6 x^6+5 d^7 x^7\right )+9 a^2 b^8 d^2 \left (-1260 c^8-6720 c^7 d x+11760 c^6 d^2 x^2+5880 c^5 d^3 x^3+2940 c^4 d^4 x^4+1176 c^3 d^5 x^5+336 c^2 d^6 x^6+60 c d^7 x^7+5 d^8 x^8\right )-a b^9 d \left (-2520 c^9-11340 c^8 d x+45360 c^7 d^2 x^2+35280 c^6 d^3 x^3+26460 c^5 d^4 x^4+15876 c^4 d^5 x^5+7056 c^3 d^6 x^6+2160 c^2 d^7 x^7+405 c d^8 x^8+35 d^9 x^9\right )+b^{10} \left (-252 c^{10}+11340 c^8 d^2 x^2+15120 c^7 d^3 x^3+17640 c^6 d^4 x^4+15876 c^5 d^5 x^5+10584 c^4 d^6 x^6+5040 c^3 d^7 x^7+1620 c^2 d^8 x^8+315 c d^9 x^9+28 d^{10} x^{10}\right )-2520 d (-b c+a d)^9 (a+b x) \log (a+b x)}{252 b^{11} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^2,x]

[Out]

(-252*a^10*d^10 + 252*a^9*b*d^9*(10*c + 9*d*x) + 1260*a^8*b^2*d^8*(-9*c^2 - 16*c*d*x + d^2*x^2) - 420*a^7*b^3*
d^7*(-72*c^3 - 189*c^2*d*x + 27*c*d^2*x^2 + d^3*x^3) + 210*a^6*b^4*d^6*(-252*c^4 - 864*c^3*d*x + 216*c^2*d^2*x
^2 + 18*c*d^3*x^3 + d^4*x^4) - 126*a^5*b^5*d^5*(-504*c^5 - 2100*c^4*d*x + 840*c^3*d^2*x^2 + 120*c^2*d^3*x^3 +
15*c*d^4*x^4 + d^5*x^5) + 42*a^4*b^6*d^4*(-1260*c^6 - 6048*c^5*d*x + 3780*c^4*d^2*x^2 + 840*c^3*d^3*x^3 + 180*
c^2*d^4*x^4 + 27*c*d^5*x^5 + 2*d^6*x^6) - 12*a^3*b^7*d^3*(-2520*c^7 - 13230*c^6*d*x + 13230*c^5*d^2*x^2 + 4410
*c^4*d^3*x^3 + 1470*c^3*d^4*x^4 + 378*c^2*d^5*x^5 + 63*c*d^6*x^6 + 5*d^7*x^7) + 9*a^2*b^8*d^2*(-1260*c^8 - 672
0*c^7*d*x + 11760*c^6*d^2*x^2 + 5880*c^5*d^3*x^3 + 2940*c^4*d^4*x^4 + 1176*c^3*d^5*x^5 + 336*c^2*d^6*x^6 + 60*
c*d^7*x^7 + 5*d^8*x^8) - a*b^9*d*(-2520*c^9 - 11340*c^8*d*x + 45360*c^7*d^2*x^2 + 35280*c^6*d^3*x^3 + 26460*c^
5*d^4*x^4 + 15876*c^4*d^5*x^5 + 7056*c^3*d^6*x^6 + 2160*c^2*d^7*x^7 + 405*c*d^8*x^8 + 35*d^9*x^9) + b^10*(-252
*c^10 + 11340*c^8*d^2*x^2 + 15120*c^7*d^3*x^3 + 17640*c^6*d^4*x^4 + 15876*c^5*d^5*x^5 + 10584*c^4*d^6*x^6 + 50
40*c^3*d^7*x^7 + 1620*c^2*d^8*x^8 + 315*c*d^9*x^9 + 28*d^10*x^10) - 2520*d*(-(b*c) + a*d)^9*(a + b*x)*Log[a +
b*x])/(252*b^11*(a + b*x))

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(782\) vs. \(2(258)=516\).
time = 9.26, size = 770, normalized size = 2.98 \begin {gather*} \frac {-10 d \text {Log}\left [a+b x\right ] \left (a+b x\right ) \left (a d-b c\right )^9-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 d^2 x \left (a+b x\right ) \left (9 a^8 d^8-80 a^7 b c d^7+315 a^6 b^2 c^2 d^6-720 a^5 b^3 c^3 d^5+1050 a^4 b^4 c^4 d^4-1008 a^3 b^5 c^5 d^3+630 a^2 b^6 c^6 d^2-240 a b^7 c^7 d+45 b^8 c^8\right )-b^2 d^3 x^2 \left (a+b x\right ) \left (4 a^7 d^7-35 a^6 b c d^6+135 a^5 b^2 c^2 d^5-300 a^4 b^3 c^3 d^4+420 a^3 b^4 c^4 d^3-378 a^2 b^5 c^5 d^2+210 a b^6 c^6 d-60 b^7 c^7\right )+\frac {b^3 d^4 x^3 \left (a+b x\right ) \left (7 a^6 d^6-60 a^5 b c d^5+225 a^4 b^2 c^2 d^4-480 a^3 b^3 c^3 d^3+630 a^2 b^4 c^4 d^2-504 a b^5 c^5 d+210 b^6 c^6\right )}{3}-\frac {b^4 d^5 x^4 \left (a+b x\right ) \left (3 a^5 d^5-25 a^4 b c d^4+90 a^3 b^2 c^2 d^3-180 a^2 b^3 c^3 d^2+210 a b^4 c^4 d-126 b^5 c^5\right )}{2}+b^5 d^6 x^5 \left (a+b x\right ) \left (a^4 d^4-8 a^3 b c d^3+27 a^2 b^2 c^2 d^2-48 a b^3 c^3 d+42 b^4 c^4\right )-\frac {b^6 d^7 x^6 \left (a+b x\right ) \left (2 a^3 d^3-15 a^2 b c d^2+45 a b^2 c^2 d-60 b^3 c^3\right )}{3}+\frac {b^7 d^8 x^7 \left (a+b x\right ) \left (3 a^2 d^2-20 a b c d+45 b^2 c^2\right )}{7}-\frac {b^8 d^9 x^8 \left (a+b x\right ) \left (a d-5 b c\right )}{4}+\frac {b^9 d^{10} x^9 \left (a+b x\right )}{9}}{b^{11} \left (a+b x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(c + d*x)^10/(a + b*x)^2,x]')

[Out]

(-10 d Log[a + b x] (a + b x) (a d - b c) ^ 9 - 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 d ^ 2 x (a + b x) (9 a ^ 8 d ^ 8 - 80 a ^ 7 b c d ^ 7 + 315 a ^ 6 b ^ 2 c ^ 2 d ^ 6 - 720 a ^ 5 b ^ 3 c ^ 3
d ^ 5 + 1050 a ^ 4 b ^ 4 c ^ 4 d ^ 4 - 1008 a ^ 3 b ^ 5 c ^ 5 d ^ 3 + 630 a ^ 2 b ^ 6 c ^ 6 d ^ 2 - 240 a b ^
7 c ^ 7 d + 45 b ^ 8 c ^ 8) - b ^ 2 d ^ 3 x ^ 2 (a + b x) (4 a ^ 7 d ^ 7 - 35 a ^ 6 b c d ^ 6 + 135 a ^ 5 b ^
2 c ^ 2 d ^ 5 - 300 a ^ 4 b ^ 3 c ^ 3 d ^ 4 + 420 a ^ 3 b ^ 4 c ^ 4 d ^ 3 - 378 a ^ 2 b ^ 5 c ^ 5 d ^ 2 + 210
a b ^ 6 c ^ 6 d - 60 b ^ 7 c ^ 7) + b ^ 3 d ^ 4 x ^ 3 (a + b x) (7 a ^ 6 d ^ 6 - 60 a ^ 5 b c d ^ 5 + 225 a ^
4 b ^ 2 c ^ 2 d ^ 4 - 480 a ^ 3 b ^ 3 c ^ 3 d ^ 3 + 630 a ^ 2 b ^ 4 c ^ 4 d ^ 2 - 504 a b ^ 5 c ^ 5 d + 210 b
^ 6 c ^ 6) / 3 - b ^ 4 d ^ 5 x ^ 4 (a + b x) (3 a ^ 5 d ^ 5 - 25 a ^ 4 b c d ^ 4 + 90 a ^ 3 b ^ 2 c ^ 2 d ^ 3
- 180 a ^ 2 b ^ 3 c ^ 3 d ^ 2 + 210 a b ^ 4 c ^ 4 d - 126 b ^ 5 c ^ 5) / 2 + b ^ 5 d ^ 6 x ^ 5 (a + b x) (a ^
4 d ^ 4 - 8 a ^ 3 b c d ^ 3 + 27 a ^ 2 b ^ 2 c ^ 2 d ^ 2 - 48 a b ^ 3 c ^ 3 d + 42 b ^ 4 c ^ 4) - b ^ 6 d ^ 7
x ^ 6 (a + b x) (2 a ^ 3 d ^ 3 - 15 a ^ 2 b c d ^ 2 + 45 a b ^ 2 c ^ 2 d - 60 b ^ 3 c ^ 3) / 3 + b ^ 7 d ^ 8 x
^ 7 (a + b x) (3 a ^ 2 d ^ 2 - 20 a b c d + 45 b ^ 2 c ^ 2) / 7 - b ^ 8 d ^ 9 x ^ 8 (a + b x) (a d - 5 b c) /
4 + b ^ 9 d ^ 10 x ^ 9 (a + b x) / 9) / (b ^ 11 (a + b x))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(932\) vs. \(2(252)=504\).
time = 0.15, size = 933, normalized size = 3.62

method	result	size
norman	\(\frac {\frac {\left (10 a^{10} d^{10}-90 a^{9} b c \,d^{9}+360 a^{8} b^{2} c^{2} d^{8}-840 a^{7} b^{3} c^{3} d^{7}+1260 a^{6} b^{4} c^{4} d^{6}-1260 a^{5} b^{5} c^{5} d^{5}+840 a^{4} b^{6} c^{6} d^{4}-360 a^{3} b^{7} c^{7} d^{3}+90 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}\right ) x}{b^{10} a}+\frac {d^{10} x^{10}}{9 b}+\frac {5 d^{2} \left (a^{8} d^{8}-9 a^{7} b c \,d^{7}+36 a^{6} b^{2} c^{2} d^{6}-84 a^{5} b^{3} c^{3} d^{5}+126 a^{4} b^{4} c^{4} d^{4}-126 a^{3} b^{5} c^{5} d^{3}+84 a^{2} b^{6} c^{6} d^{2}-36 a \,b^{7} c^{7} d +9 b^{8} c^{8}\right ) x^{2}}{b^{9}}-\frac {5 d^{3} \left (a^{7} d^{7}-9 a^{6} b c \,d^{6}+36 a^{5} b^{2} c^{2} d^{5}-84 a^{4} b^{3} c^{3} d^{4}+126 a^{3} b^{4} c^{4} d^{3}-126 a^{2} b^{5} c^{5} d^{2}+84 a \,b^{6} c^{6} d -36 b^{7} c^{7}\right ) x^{3}}{3 b^{8}}+\frac {5 d^{4} \left (a^{6} d^{6}-9 a^{5} b c \,d^{5}+36 a^{4} b^{2} c^{2} d^{4}-84 a^{3} b^{3} c^{3} d^{3}+126 a^{2} b^{4} c^{4} d^{2}-126 a \,b^{5} c^{5} d +84 b^{6} c^{6}\right ) x^{4}}{6 b^{7}}-\frac {d^{5} \left (a^{5} d^{5}-9 a^{4} b c \,d^{4}+36 a^{3} b^{2} c^{2} d^{3}-84 a^{2} b^{3} c^{3} d^{2}+126 a \,b^{4} c^{4} d -126 b^{5} c^{5}\right ) x^{5}}{2 b^{6}}+\frac {d^{6} \left (a^{4} d^{4}-9 a^{3} b c \,d^{3}+36 a^{2} b^{2} c^{2} d^{2}-84 a \,b^{3} c^{3} d +126 b^{4} c^{4}\right ) x^{6}}{3 b^{5}}-\frac {5 d^{7} \left (a^{3} d^{3}-9 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -84 b^{3} c^{3}\right ) x^{7}}{21 b^{4}}+\frac {5 d^{8} \left (a^{2} d^{2}-9 a b c d +36 b^{2} c^{2}\right ) x^{8}}{28 b^{3}}-\frac {5 d^{9} \left (a d -9 b c \right ) x^{9}}{36 b^{2}}}{b x +a}-\frac {10 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}\right ) \ln \left (b x +a \right )}{b^{11}}\)	\(840\)
default	\(\frac {d^{2} \left (45 b^{8} c^{8} x +9 a^{8} d^{8} x +\frac {1}{9} d^{8} x^{9} b^{8}-\frac {20}{7} a \,b^{7} c \,d^{7} x^{7}+5 a^{2} b^{6} c \,d^{7} x^{6}-15 a \,b^{7} c^{2} d^{6} x^{6}-8 a^{3} b^{5} c \,d^{7} x^{5}+27 a^{2} b^{6} c^{2} d^{6} x^{5}-48 a \,b^{7} c^{3} d^{5} x^{5}+\frac {25}{2} a^{4} b^{4} c \,d^{7} x^{4}-45 a^{3} b^{5} c^{2} d^{6} x^{4}+90 a^{2} b^{6} c^{3} d^{5} x^{4}-105 a \,b^{7} c^{4} d^{4} x^{4}-20 a^{5} b^{3} c \,d^{7} x^{3}+75 a^{4} b^{4} c^{2} d^{6} x^{3}-160 a^{3} b^{5} c^{3} d^{5} x^{3}+210 a^{2} b^{6} c^{4} d^{4} x^{3}-168 a \,b^{7} c^{5} d^{3} x^{3}+35 a^{6} b^{2} c \,d^{7} x^{2}-135 a^{5} b^{3} c^{2} d^{6} x^{2}+300 a^{4} b^{4} c^{3} d^{5} x^{2}-420 a^{3} b^{5} c^{4} d^{4} x^{2}+378 a^{2} b^{6} c^{5} d^{3} x^{2}-210 a \,b^{7} c^{6} d^{2} x^{2}-80 a^{7} b c \,d^{7} x +315 a^{6} b^{2} c^{2} d^{6} x -720 a^{5} b^{3} c^{3} d^{5} x +1050 a^{4} b^{4} c^{4} d^{4} x -1008 a^{3} b^{5} c^{5} d^{3} x +630 a^{2} b^{6} c^{6} d^{2} x -240 a \,b^{7} c^{7} d x +a^{4} b^{4} d^{8} x^{5}-\frac {1}{4} a \,b^{7} d^{8} x^{8}+\frac {5}{4} b^{8} c \,d^{7} x^{8}+\frac {3}{7} a^{2} b^{6} d^{8} x^{7}+\frac {45}{7} b^{8} c^{2} d^{6} x^{7}-\frac {2}{3} a^{3} b^{5} d^{8} x^{6}+20 b^{8} c^{3} d^{5} x^{6}+42 b^{8} c^{4} d^{4} x^{5}-\frac {3}{2} a^{5} b^{3} d^{8} x^{4}+63 b^{8} c^{5} d^{3} x^{4}+\frac {7}{3} a^{6} b^{2} d^{8} x^{3}+70 b^{8} c^{6} d^{2} x^{3}-4 a^{7} b \,d^{8} x^{2}+60 b^{8} c^{7} d \,x^{2}\right )}{b^{10}}-\frac {a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}}{b^{11} \left (b x +a \right )}-\frac {10 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}\right ) \ln \left (b x +a \right )}{b^{11}}\)	\(933\)
risch	\(\frac {10 a^{9} c \,d^{9}}{b^{10} \left (b x +a \right )}-\frac {45 a^{8} c^{2} d^{8}}{b^{9} \left (b x +a \right )}+\frac {120 a^{7} c^{3} d^{7}}{b^{8} \left (b x +a \right )}-\frac {210 a^{6} c^{4} d^{6}}{b^{7} \left (b x +a \right )}+\frac {252 a^{5} c^{5} d^{5}}{b^{6} \left (b x +a \right )}-\frac {210 a^{4} c^{6} d^{4}}{b^{5} \left (b x +a \right )}+\frac {120 a^{3} c^{7} d^{3}}{b^{4} \left (b x +a \right )}-\frac {45 a^{2} c^{8} d^{2}}{b^{3} \left (b x +a \right )}+\frac {10 a \,c^{9} d}{b^{2} \left (b x +a \right )}+\frac {d^{10} x^{9}}{9 b^{2}}-\frac {c^{10}}{b \left (b x +a \right )}+\frac {90 d^{9} \ln \left (b x +a \right ) a^{8} c}{b^{10}}-\frac {360 d^{8} \ln \left (b x +a \right ) a^{7} c^{2}}{b^{9}}+\frac {840 d^{7} \ln \left (b x +a \right ) a^{6} c^{3}}{b^{8}}-\frac {1260 d^{6} \ln \left (b x +a \right ) a^{5} c^{4}}{b^{7}}+\frac {1260 d^{5} \ln \left (b x +a \right ) a^{4} c^{5}}{b^{6}}-\frac {840 d^{4} \ln \left (b x +a \right ) a^{3} c^{6}}{b^{5}}+\frac {360 d^{3} \ln \left (b x +a \right ) a^{2} c^{7}}{b^{4}}-\frac {90 d^{2} \ln \left (b x +a \right ) a \,c^{8}}{b^{3}}+\frac {45 d^{8} c^{2} x^{7}}{7 b^{2}}-\frac {2 d^{10} a^{3} x^{6}}{3 b^{5}}+\frac {20 d^{7} c^{3} x^{6}}{b^{2}}+\frac {42 d^{6} c^{4} x^{5}}{b^{2}}-\frac {3 d^{10} a^{5} x^{4}}{2 b^{7}}+\frac {63 d^{5} c^{5} x^{4}}{b^{2}}+\frac {7 d^{10} a^{6} x^{3}}{3 b^{8}}+\frac {70 d^{4} c^{6} x^{3}}{b^{2}}-\frac {4 d^{10} a^{7} x^{2}}{b^{9}}+\frac {60 d^{3} c^{7} x^{2}}{b^{2}}-\frac {a^{10} d^{10}}{b^{11} \left (b x +a \right )}-\frac {10 d^{10} \ln \left (b x +a \right ) a^{9}}{b^{11}}+\frac {10 d \ln \left (b x +a \right ) c^{9}}{b^{2}}-\frac {20 d^{9} a c \,x^{7}}{7 b^{3}}+\frac {5 d^{9} a^{2} c \,x^{6}}{b^{4}}-\frac {15 d^{8} a \,c^{2} x^{6}}{b^{3}}-\frac {8 d^{9} a^{3} c \,x^{5}}{b^{5}}+\frac {27 d^{8} a^{2} c^{2} x^{5}}{b^{4}}-\frac {48 d^{7} a \,c^{3} x^{5}}{b^{3}}+\frac {25 d^{9} a^{4} c \,x^{4}}{2 b^{6}}-\frac {45 d^{8} a^{3} c^{2} x^{4}}{b^{5}}+\frac {90 d^{7} a^{2} c^{3} x^{4}}{b^{4}}-\frac {105 d^{6} a \,c^{4} x^{4}}{b^{3}}-\frac {20 d^{9} a^{5} c \,x^{3}}{b^{7}}+\frac {75 d^{8} a^{4} c^{2} x^{3}}{b^{6}}-\frac {160 d^{7} a^{3} c^{3} x^{3}}{b^{5}}+\frac {210 d^{6} a^{2} c^{4} x^{3}}{b^{4}}-\frac {168 d^{5} a \,c^{5} x^{3}}{b^{3}}+\frac {35 d^{9} a^{6} c \,x^{2}}{b^{8}}-\frac {135 d^{8} a^{5} c^{2} x^{2}}{b^{7}}+\frac {300 d^{7} a^{4} c^{3} x^{2}}{b^{6}}-\frac {420 d^{6} a^{3} c^{4} x^{2}}{b^{5}}+\frac {378 d^{5} a^{2} c^{5} x^{2}}{b^{4}}-\frac {210 d^{4} a \,c^{6} x^{2}}{b^{3}}-\frac {80 d^{9} a^{7} c x}{b^{9}}+\frac {315 d^{8} a^{6} c^{2} x}{b^{8}}-\frac {720 d^{7} a^{5} c^{3} x}{b^{7}}+\frac {1050 d^{6} a^{4} c^{4} x}{b^{6}}-\frac {1008 d^{5} a^{3} c^{5} x}{b^{5}}+\frac {630 d^{4} a^{2} c^{6} x}{b^{4}}-\frac {240 d^{3} a \,c^{7} x}{b^{3}}+\frac {45 d^{2} c^{8} x}{b^{2}}+\frac {9 d^{10} a^{8} x}{b^{10}}+\frac {d^{10} a^{4} x^{5}}{b^{6}}-\frac {d^{10} a \,x^{8}}{4 b^{3}}+\frac {5 d^{9} c \,x^{8}}{4 b^{2}}+\frac {3 d^{10} a^{2} x^{7}}{7 b^{4}}\)	\(1066\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

d^2/b^10*(45*b^8*c^8*x+9*a^8*d^8*x+1/9*d^8*x^9*b^8-20/7*a*b^7*c*d^7*x^7+5*a^2*b^6*c*d^7*x^6-15*a*b^7*c^2*d^6*x
^6-8*a^3*b^5*c*d^7*x^5+27*a^2*b^6*c^2*d^6*x^5-48*a*b^7*c^3*d^5*x^5+25/2*a^4*b^4*c*d^7*x^4-45*a^3*b^5*c^2*d^6*x
^4+90*a^2*b^6*c^3*d^5*x^4-105*a*b^7*c^4*d^4*x^4-20*a^5*b^3*c*d^7*x^3+75*a^4*b^4*c^2*d^6*x^3-160*a^3*b^5*c^3*d^
5*x^3+210*a^2*b^6*c^4*d^4*x^3-168*a*b^7*c^5*d^3*x^3+35*a^6*b^2*c*d^7*x^2-135*a^5*b^3*c^2*d^6*x^2+300*a^4*b^4*c
^3*d^5*x^2-420*a^3*b^5*c^4*d^4*x^2+378*a^2*b^6*c^5*d^3*x^2-210*a*b^7*c^6*d^2*x^2-80*a^7*b*c*d^7*x+315*a^6*b^2*
c^2*d^6*x-720*a^5*b^3*c^3*d^5*x+1050*a^4*b^4*c^4*d^4*x-1008*a^3*b^5*c^5*d^3*x+630*a^2*b^6*c^6*d^2*x-240*a*b^7*
c^7*d*x+a^4*b^4*d^8*x^5-1/4*a*b^7*d^8*x^8+5/4*b^8*c*d^7*x^8+3/7*a^2*b^6*d^8*x^7+45/7*b^8*c^2*d^6*x^7-2/3*a^3*b
^5*d^8*x^6+20*b^8*c^3*d^5*x^6+42*b^8*c^4*d^4*x^5-3/2*a^5*b^3*d^8*x^4+63*b^8*c^5*d^3*x^4+7/3*a^6*b^2*d^8*x^3+70
*b^8*c^6*d^2*x^3-4*a^7*b*d^8*x^2+60*b^8*c^7*d*x^2)-(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)-10/b^11*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)*ln(b*x+a)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 874 vs. \(2 (252) = 504\).
time = 0.28, size = 874, normalized size = 3.39 \begin {gather*} -\frac {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}}{b^{12} x + a b^{11}} + \frac {28 \, b^{8} d^{10} x^{9} + 63 \, {\left (5 \, b^{8} c d^{9} - a b^{7} d^{10}\right )} x^{8} + 36 \, {\left (45 \, b^{8} c^{2} d^{8} - 20 \, a b^{7} c d^{9} + 3 \, a^{2} b^{6} d^{10}\right )} x^{7} + 84 \, {\left (60 \, b^{8} c^{3} d^{7} - 45 \, a b^{7} c^{2} d^{8} + 15 \, a^{2} b^{6} c d^{9} - 2 \, a^{3} b^{5} d^{10}\right )} x^{6} + 252 \, {\left (42 \, b^{8} c^{4} d^{6} - 48 \, a b^{7} c^{3} d^{7} + 27 \, a^{2} b^{6} c^{2} d^{8} - 8 \, a^{3} b^{5} c d^{9} + a^{4} b^{4} d^{10}\right )} x^{5} + 126 \, {\left (126 \, b^{8} c^{5} d^{5} - 210 \, a b^{7} c^{4} d^{6} + 180 \, a^{2} b^{6} c^{3} d^{7} - 90 \, a^{3} b^{5} c^{2} d^{8} + 25 \, a^{4} b^{4} c d^{9} - 3 \, a^{5} b^{3} d^{10}\right )} x^{4} + 84 \, {\left (210 \, b^{8} c^{6} d^{4} - 504 \, a b^{7} c^{5} d^{5} + 630 \, a^{2} b^{6} c^{4} d^{6} - 480 \, a^{3} b^{5} c^{3} d^{7} + 225 \, a^{4} b^{4} c^{2} d^{8} - 60 \, a^{5} b^{3} c d^{9} + 7 \, a^{6} b^{2} d^{10}\right )} x^{3} + 252 \, {\left (60 \, b^{8} c^{7} d^{3} - 210 \, a b^{7} c^{6} d^{4} + 378 \, a^{2} b^{6} c^{5} d^{5} - 420 \, a^{3} b^{5} c^{4} d^{6} + 300 \, a^{4} b^{4} c^{3} d^{7} - 135 \, a^{5} b^{3} c^{2} d^{8} + 35 \, a^{6} b^{2} c d^{9} - 4 \, a^{7} b d^{10}\right )} x^{2} + 252 \, {\left (45 \, b^{8} c^{8} d^{2} - 240 \, a b^{7} c^{7} d^{3} + 630 \, a^{2} b^{6} c^{6} d^{4} - 1008 \, a^{3} b^{5} c^{5} d^{5} + 1050 \, a^{4} b^{4} c^{4} d^{6} - 720 \, a^{5} b^{3} c^{3} d^{7} + 315 \, a^{6} b^{2} c^{2} d^{8} - 80 \, a^{7} b c d^{9} + 9 \, a^{8} d^{10}\right )} x}{252 \, b^{10}} + \frac {10 \, {\left (b^{9} c^{9} d - 9 \, a b^{8} c^{8} d^{2} + 36 \, a^{2} b^{7} c^{7} d^{3} - 84 \, a^{3} b^{6} c^{6} d^{4} + 126 \, a^{4} b^{5} c^{5} d^{5} - 126 \, a^{5} b^{4} c^{4} d^{6} + 84 \, a^{6} b^{3} c^{3} d^{7} - 36 \, a^{7} b^{2} c^{2} d^{8} + 9 \, a^{8} b c d^{9} - a^{9} 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)^2,x, algorithm="maxima")

[Out]

-(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)/(b^12*x +
a*b^11) + 1/252*(28*b^8*d^10*x^9 + 63*(5*b^8*c*d^9 - a*b^7*d^10)*x^8 + 36*(45*b^8*c^2*d^8 - 20*a*b^7*c*d^9 +
3*a^2*b^6*d^10)*x^7 + 84*(60*b^8*c^3*d^7 - 45*a*b^7*c^2*d^8 + 15*a^2*b^6*c*d^9 - 2*a^3*b^5*d^10)*x^6 + 252*(42
*b^8*c^4*d^6 - 48*a*b^7*c^3*d^7 + 27*a^2*b^6*c^2*d^8 - 8*a^3*b^5*c*d^9 + a^4*b^4*d^10)*x^5 + 126*(126*b^8*c^5*
d^5 - 210*a*b^7*c^4*d^6 + 180*a^2*b^6*c^3*d^7 - 90*a^3*b^5*c^2*d^8 + 25*a^4*b^4*c*d^9 - 3*a^5*b^3*d^10)*x^4 +
84*(210*b^8*c^6*d^4 - 504*a*b^7*c^5*d^5 + 630*a^2*b^6*c^4*d^6 - 480*a^3*b^5*c^3*d^7 + 225*a^4*b^4*c^2*d^8 - 60
*a^5*b^3*c*d^9 + 7*a^6*b^2*d^10)*x^3 + 252*(60*b^8*c^7*d^3 - 210*a*b^7*c^6*d^4 + 378*a^2*b^6*c^5*d^5 - 420*a^3
*b^5*c^4*d^6 + 300*a^4*b^4*c^3*d^7 - 135*a^5*b^3*c^2*d^8 + 35*a^6*b^2*c*d^9 - 4*a^7*b*d^10)*x^2 + 252*(45*b^8*
c^8*d^2 - 240*a*b^7*c^7*d^3 + 630*a^2*b^6*c^6*d^4 - 1008*a^3*b^5*c^5*d^5 + 1050*a^4*b^4*c^4*d^6 - 720*a^5*b^3*
c^3*d^7 + 315*a^6*b^2*c^2*d^8 - 80*a^7*b*c*d^9 + 9*a^8*d^10)*x)/b^10 + 10*(b^9*c^9*d - 9*a*b^8*c^8*d^2 + 36*a^
2*b^7*c^7*d^3 - 84*a^3*b^6*c^6*d^4 + 126*a^4*b^5*c^5*d^5 - 126*a^5*b^4*c^4*d^6 + 84*a^6*b^3*c^3*d^7 - 36*a^7*b
^2*c^2*d^8 + 9*a^8*b*c*d^9 - a^9*d^10)*log(b*x + a)/b^11

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1124 vs. \(2 (252) = 504\).
time = 0.31, size = 1124, normalized size = 4.36 \begin {gather*} \frac {28 \, b^{10} d^{10} x^{10} - 252 \, b^{10} c^{10} + 2520 \, a b^{9} c^{9} d - 11340 \, a^{2} b^{8} c^{8} d^{2} + 30240 \, a^{3} b^{7} c^{7} d^{3} - 52920 \, a^{4} b^{6} c^{6} d^{4} + 63504 \, a^{5} b^{5} c^{5} d^{5} - 52920 \, a^{6} b^{4} c^{4} d^{6} + 30240 \, a^{7} b^{3} c^{3} d^{7} - 11340 \, a^{8} b^{2} c^{2} d^{8} + 2520 \, a^{9} b c d^{9} - 252 \, a^{10} d^{10} + 35 \, {\left (9 \, b^{10} c d^{9} - a b^{9} d^{10}\right )} x^{9} + 45 \, {\left (36 \, b^{10} c^{2} d^{8} - 9 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 60 \, {\left (84 \, b^{10} c^{3} d^{7} - 36 \, a b^{9} c^{2} d^{8} + 9 \, a^{2} b^{8} c d^{9} - a^{3} b^{7} d^{10}\right )} x^{7} + 84 \, {\left (126 \, b^{10} c^{4} d^{6} - 84 \, a b^{9} c^{3} d^{7} + 36 \, 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} + 126 \, {\left (126 \, b^{10} c^{5} d^{5} - 126 \, a b^{9} c^{4} d^{6} + 84 \, a^{2} b^{8} c^{3} d^{7} - 36 \, 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} + 210 \, {\left (84 \, b^{10} c^{6} d^{4} - 126 \, a b^{9} c^{5} d^{5} + 126 \, a^{2} b^{8} c^{4} d^{6} - 84 \, a^{3} b^{7} c^{3} d^{7} + 36 \, 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} + 420 \, {\left (36 \, b^{10} c^{7} d^{3} - 84 \, a b^{9} c^{6} d^{4} + 126 \, a^{2} b^{8} c^{5} d^{5} - 126 \, a^{3} b^{7} c^{4} d^{6} + 84 \, a^{4} b^{6} c^{3} d^{7} - 36 \, 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} + 1260 \, {\left (9 \, b^{10} c^{8} d^{2} - 36 \, a b^{9} c^{7} d^{3} + 84 \, a^{2} b^{8} c^{6} d^{4} - 126 \, a^{3} b^{7} c^{5} d^{5} + 126 \, a^{4} b^{6} c^{4} d^{6} - 84 \, a^{5} b^{5} c^{3} d^{7} + 36 \, 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} + 252 \, {\left (45 \, a b^{9} c^{8} d^{2} - 240 \, a^{2} b^{8} c^{7} d^{3} + 630 \, a^{3} b^{7} c^{6} d^{4} - 1008 \, a^{4} b^{6} c^{5} d^{5} + 1050 \, a^{5} b^{5} c^{4} d^{6} - 720 \, a^{6} b^{4} c^{3} d^{7} + 315 \, a^{7} b^{3} c^{2} d^{8} - 80 \, a^{8} b^{2} c d^{9} + 9 \, a^{9} b d^{10}\right )} x + 2520 \, {\left (a b^{9} c^{9} d - 9 \, a^{2} b^{8} c^{8} d^{2} + 36 \, a^{3} b^{7} c^{7} d^{3} - 84 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} - 126 \, a^{6} b^{4} c^{4} d^{6} + 84 \, a^{7} b^{3} c^{3} d^{7} - 36 \, a^{8} b^{2} c^{2} d^{8} + 9 \, a^{9} b c d^{9} - a^{10} d^{10} + {\left (b^{10} c^{9} d - 9 \, a b^{9} c^{8} d^{2} + 36 \, a^{2} b^{8} c^{7} d^{3} - 84 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} - 126 \, a^{5} b^{5} c^{4} d^{6} + 84 \, a^{6} b^{4} c^{3} d^{7} - 36 \, a^{7} b^{3} c^{2} d^{8} + 9 \, a^{8} b^{2} c d^{9} - a^{9} b d^{10}\right )} x\right )} \log \left (b x + a\right )}{252 \, {\left (b^{12} x + a b^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/252*(28*b^10*d^10*x^10 - 252*b^10*c^10 + 2520*a*b^9*c^9*d - 11340*a^2*b^8*c^8*d^2 + 30240*a^3*b^7*c^7*d^3 -
52920*a^4*b^6*c^6*d^4 + 63504*a^5*b^5*c^5*d^5 - 52920*a^6*b^4*c^4*d^6 + 30240*a^7*b^3*c^3*d^7 - 11340*a^8*b^2*
c^2*d^8 + 2520*a^9*b*c*d^9 - 252*a^10*d^10 + 35*(9*b^10*c*d^9 - a*b^9*d^10)*x^9 + 45*(36*b^10*c^2*d^8 - 9*a*b^
9*c*d^9 + a^2*b^8*d^10)*x^8 + 60*(84*b^10*c^3*d^7 - 36*a*b^9*c^2*d^8 + 9*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 8
4*(126*b^10*c^4*d^6 - 84*a*b^9*c^3*d^7 + 36*a^2*b^8*c^2*d^8 - 9*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 126*(126*b
^10*c^5*d^5 - 126*a*b^9*c^4*d^6 + 84*a^2*b^8*c^3*d^7 - 36*a^3*b^7*c^2*d^8 + 9*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^
5 + 210*(84*b^10*c^6*d^4 - 126*a*b^9*c^5*d^5 + 126*a^2*b^8*c^4*d^6 - 84*a^3*b^7*c^3*d^7 + 36*a^4*b^6*c^2*d^8 -
9*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 420*(36*b^10*c^7*d^3 - 84*a*b^9*c^6*d^4 + 126*a^2*b^8*c^5*d^5 - 126*a^3
*b^7*c^4*d^6 + 84*a^4*b^6*c^3*d^7 - 36*a^5*b^5*c^2*d^8 + 9*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 1260*(9*b^10*c^
8*d^2 - 36*a*b^9*c^7*d^3 + 84*a^2*b^8*c^6*d^4 - 126*a^3*b^7*c^5*d^5 + 126*a^4*b^6*c^4*d^6 - 84*a^5*b^5*c^3*d^7
+ 36*a^6*b^4*c^2*d^8 - 9*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 252*(45*a*b^9*c^8*d^2 - 240*a^2*b^8*c^7*d^3 + 63
0*a^3*b^7*c^6*d^4 - 1008*a^4*b^6*c^5*d^5 + 1050*a^5*b^5*c^4*d^6 - 720*a^6*b^4*c^3*d^7 + 315*a^7*b^3*c^2*d^8 -
80*a^8*b^2*c*d^9 + 9*a^9*b*d^10)*x + 2520*(a*b^9*c^9*d - 9*a^2*b^8*c^8*d^2 + 36*a^3*b^7*c^7*d^3 - 84*a^4*b^6*c
^6*d^4 + 126*a^5*b^5*c^5*d^5 - 126*a^6*b^4*c^4*d^6 + 84*a^7*b^3*c^3*d^7 - 36*a^8*b^2*c^2*d^8 + 9*a^9*b*c*d^9 -
a^10*d^10 + (b^10*c^9*d - 9*a*b^9*c^8*d^2 + 36*a^2*b^8*c^7*d^3 - 84*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 - 1
26*a^5*b^5*c^4*d^6 + 84*a^6*b^4*c^3*d^7 - 36*a^7*b^3*c^2*d^8 + 9*a^8*b^2*c*d^9 - a^9*b*d^10)*x)*log(b*x + a))/
(b^12*x + a*b^11)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 816 vs. \(2 (240) = 480\).
time = 1.51, size = 816, normalized size = 3.16 \begin {gather*} x^{8} \left (- \frac {a d^{10}}{4 b^{3}} + \frac {5 c d^{9}}{4 b^{2}}\right ) + x^{7} \cdot \left (\frac {3 a^{2} d^{10}}{7 b^{4}} - \frac {20 a c d^{9}}{7 b^{3}} + \frac {45 c^{2} d^{8}}{7 b^{2}}\right ) + x^{6} \left (- \frac {2 a^{3} d^{10}}{3 b^{5}} + \frac {5 a^{2} c d^{9}}{b^{4}} - \frac {15 a c^{2} d^{8}}{b^{3}} + \frac {20 c^{3} d^{7}}{b^{2}}\right ) + x^{5} \left (\frac {a^{4} d^{10}}{b^{6}} - \frac {8 a^{3} c d^{9}}{b^{5}} + \frac {27 a^{2} c^{2} d^{8}}{b^{4}} - \frac {48 a c^{3} d^{7}}{b^{3}} + \frac {42 c^{4} d^{6}}{b^{2}}\right ) + x^{4} \left (- \frac {3 a^{5} d^{10}}{2 b^{7}} + \frac {25 a^{4} c d^{9}}{2 b^{6}} - \frac {45 a^{3} c^{2} d^{8}}{b^{5}} + \frac {90 a^{2} c^{3} d^{7}}{b^{4}} - \frac {105 a c^{4} d^{6}}{b^{3}} + \frac {63 c^{5} d^{5}}{b^{2}}\right ) + x^{3} \cdot \left (\frac {7 a^{6} d^{10}}{3 b^{8}} - \frac {20 a^{5} c d^{9}}{b^{7}} + \frac {75 a^{4} c^{2} d^{8}}{b^{6}} - \frac {160 a^{3} c^{3} d^{7}}{b^{5}} + \frac {210 a^{2} c^{4} d^{6}}{b^{4}} - \frac {168 a c^{5} d^{5}}{b^{3}} + \frac {70 c^{6} d^{4}}{b^{2}}\right ) + x^{2} \left (- \frac {4 a^{7} d^{10}}{b^{9}} + \frac {35 a^{6} c d^{9}}{b^{8}} - \frac {135 a^{5} c^{2} d^{8}}{b^{7}} + \frac {300 a^{4} c^{3} d^{7}}{b^{6}} - \frac {420 a^{3} c^{4} d^{6}}{b^{5}} + \frac {378 a^{2} c^{5} d^{5}}{b^{4}} - \frac {210 a c^{6} d^{4}}{b^{3}} + \frac {60 c^{7} d^{3}}{b^{2}}\right ) + x \left (\frac {9 a^{8} d^{10}}{b^{10}} - \frac {80 a^{7} c d^{9}}{b^{9}} + \frac {315 a^{6} c^{2} d^{8}}{b^{8}} - \frac {720 a^{5} c^{3} d^{7}}{b^{7}} + \frac {1050 a^{4} c^{4} d^{6}}{b^{6}} - \frac {1008 a^{3} c^{5} d^{5}}{b^{5}} + \frac {630 a^{2} c^{6} d^{4}}{b^{4}} - \frac {240 a c^{7} d^{3}}{b^{3}} + \frac {45 c^{8} d^{2}}{b^{2}}\right ) + \frac {- a^{10} d^{10} + 10 a^{9} b c d^{9} - 45 a^{8} b^{2} c^{2} d^{8} + 120 a^{7} b^{3} c^{3} d^{7} - 210 a^{6} b^{4} c^{4} d^{6} + 252 a^{5} b^{5} c^{5} d^{5} - 210 a^{4} b^{6} c^{6} d^{4} + 120 a^{3} b^{7} c^{7} d^{3} - 45 a^{2} b^{8} c^{8} d^{2} + 10 a b^{9} c^{9} d - b^{10} c^{10}}{a b^{11} + b^{12} x} + \frac {d^{10} x^{9}}{9 b^{2}} - \frac {10 d \left (a d - b c\right )^{9} \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)**2,x)

[Out]

x**8*(-a*d**10/(4*b**3) + 5*c*d**9/(4*b**2)) + x**7*(3*a**2*d**10/(7*b**4) - 20*a*c*d**9/(7*b**3) + 45*c**2*d*
*8/(7*b**2)) + x**6*(-2*a**3*d**10/(3*b**5) + 5*a**2*c*d**9/b**4 - 15*a*c**2*d**8/b**3 + 20*c**3*d**7/b**2) +
x**5*(a**4*d**10/b**6 - 8*a**3*c*d**9/b**5 + 27*a**2*c**2*d**8/b**4 - 48*a*c**3*d**7/b**3 + 42*c**4*d**6/b**2)
+ x**4*(-3*a**5*d**10/(2*b**7) + 25*a**4*c*d**9/(2*b**6) - 45*a**3*c**2*d**8/b**5 + 90*a**2*c**3*d**7/b**4 -
105*a*c**4*d**6/b**3 + 63*c**5*d**5/b**2) + x**3*(7*a**6*d**10/(3*b**8) - 20*a**5*c*d**9/b**7 + 75*a**4*c**2*d
**8/b**6 - 160*a**3*c**3*d**7/b**5 + 210*a**2*c**4*d**6/b**4 - 168*a*c**5*d**5/b**3 + 70*c**6*d**4/b**2) + x**
2*(-4*a**7*d**10/b**9 + 35*a**6*c*d**9/b**8 - 135*a**5*c**2*d**8/b**7 + 300*a**4*c**3*d**7/b**6 - 420*a**3*c**
4*d**6/b**5 + 378*a**2*c**5*d**5/b**4 - 210*a*c**6*d**4/b**3 + 60*c**7*d**3/b**2) + x*(9*a**8*d**10/b**10 - 80
*a**7*c*d**9/b**9 + 315*a**6*c**2*d**8/b**8 - 720*a**5*c**3*d**7/b**7 + 1050*a**4*c**4*d**6/b**6 - 1008*a**3*c
**5*d**5/b**5 + 630*a**2*c**6*d**4/b**4 - 240*a*c**7*d**3/b**3 + 45*c**8*d**2/b**2) + (-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**1
0)/(a*b**11 + b**12*x) + d**10*x**9/(9*b**2) - 10*d*(a*d - b*c)**9*log(a + b*x)/b**11

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 948 vs. \(2 (252) = 504\).
time = 0.00, size = 998, normalized size = 3.87 \begin {gather*} \frac {\frac {1}{9} x^{9} d^{10} b^{16}-\frac {1}{4} x^{8} d^{10} b^{15} a+\frac {5}{4} x^{8} d^{9} c b^{16}+\frac {3}{7} x^{7} d^{10} b^{14} a^{2}-\frac {20}{7} x^{7} d^{9} c b^{15} a+\frac {45}{7} x^{7} d^{8} c^{2} b^{16}-\frac {2}{3} x^{6} d^{10} b^{13} a^{3}+5 x^{6} d^{9} c b^{14} a^{2}-15 x^{6} d^{8} c^{2} b^{15} a+20 x^{6} d^{7} c^{3} b^{16}+x^{5} d^{10} b^{12} a^{4}-8 x^{5} d^{9} c b^{13} a^{3}+27 x^{5} d^{8} c^{2} b^{14} a^{2}-48 x^{5} d^{7} c^{3} b^{15} a+42 x^{5} d^{6} c^{4} b^{16}-\frac {3}{2} x^{4} d^{10} b^{11} a^{5}+\frac {25}{2} x^{4} d^{9} c b^{12} a^{4}-45 x^{4} d^{8} c^{2} b^{13} a^{3}+90 x^{4} d^{7} c^{3} b^{14} a^{2}-105 x^{4} d^{6} c^{4} b^{15} a+63 x^{4} d^{5} c^{5} b^{16}+\frac {7}{3} x^{3} d^{10} b^{10} a^{6}-20 x^{3} d^{9} c b^{11} a^{5}+75 x^{3} d^{8} c^{2} b^{12} a^{4}-160 x^{3} d^{7} c^{3} b^{13} a^{3}+210 x^{3} d^{6} c^{4} b^{14} a^{2}-168 x^{3} d^{5} c^{5} b^{15} a+70 x^{3} d^{4} c^{6} b^{16}-4 x^{2} d^{10} b^{9} a^{7}+35 x^{2} d^{9} c b^{10} a^{6}-135 x^{2} d^{8} c^{2} b^{11} a^{5}+300 x^{2} d^{7} c^{3} b^{12} a^{4}-420 x^{2} d^{6} c^{4} b^{13} a^{3}+378 x^{2} d^{5} c^{5} b^{14} a^{2}-210 x^{2} d^{4} c^{6} b^{15} a+60 x^{2} d^{3} c^{7} b^{16}+9 x d^{10} b^{8} a^{8}-80 x d^{9} c b^{9} a^{7}+315 x d^{8} c^{2} b^{10} a^{6}-720 x d^{7} c^{3} b^{11} a^{5}+1050 x d^{6} c^{4} b^{12} a^{4}-1008 x d^{5} c^{5} b^{13} a^{3}+630 x d^{4} c^{6} b^{14} a^{2}-240 x d^{3} c^{7} b^{15} a+45 x d^{2} c^{8} b^{16}}{b^{18}}+\frac {-d^{10} a^{10}+10 d^{9} b a^{9} c-45 d^{8} b^{2} a^{8} c^{2}+120 d^{7} b^{3} a^{7} c^{3}-210 d^{6} b^{4} a^{6} c^{4}+252 d^{5} b^{5} a^{5} c^{5}-210 d^{4} b^{6} a^{4} c^{6}+120 d^{3} b^{7} a^{3} c^{7}-45 d^{2} b^{8} a^{2} c^{8}+10 d b^{9} a c^{9}-b^{10} c^{10}}{b^{11} \left (x b+a\right )}+\frac {\left (-10 d^{10} a^{9}+90 d^{9} c b a^{8}-360 d^{8} c^{2} b^{2} a^{7}+840 d^{7} c^{3} b^{3} a^{6}-1260 d^{6} c^{4} b^{4} a^{5}+1260 d^{5} c^{5} b^{5} a^{4}-840 d^{4} c^{6} b^{6} a^{3}+360 d^{3} c^{7} b^{7} a^{2}-90 d^{2} c^{8} b^{8} a+10 d c^{9} b^{9}\right ) \ln \left |x b+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)^2,x)

[Out]

10*(b^9*c^9*d - 9*a*b^8*c^8*d^2 + 36*a^2*b^7*c^7*d^3 - 84*a^3*b^6*c^6*d^4 + 126*a^4*b^5*c^5*d^5 - 126*a^5*b^4*
c^4*d^6 + 84*a^6*b^3*c^3*d^7 - 36*a^7*b^2*c^2*d^8 + 9*a^8*b*c*d^9 - a^9*d^10)*log(abs(b*x + a))/b^11 - (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)/((b*x + a)*b^11)
+ 1/252*(28*b^16*d^10*x^9 + 315*b^16*c*d^9*x^8 - 63*a*b^15*d^10*x^8 + 1620*b^16*c^2*d^8*x^7 - 720*a*b^15*c*d^9
*x^7 + 108*a^2*b^14*d^10*x^7 + 5040*b^16*c^3*d^7*x^6 - 3780*a*b^15*c^2*d^8*x^6 + 1260*a^2*b^14*c*d^9*x^6 - 168
*a^3*b^13*d^10*x^6 + 10584*b^16*c^4*d^6*x^5 - 12096*a*b^15*c^3*d^7*x^5 + 6804*a^2*b^14*c^2*d^8*x^5 - 2016*a^3*
b^13*c*d^9*x^5 + 252*a^4*b^12*d^10*x^5 + 15876*b^16*c^5*d^5*x^4 - 26460*a*b^15*c^4*d^6*x^4 + 22680*a^2*b^14*c^
3*d^7*x^4 - 11340*a^3*b^13*c^2*d^8*x^4 + 3150*a^4*b^12*c*d^9*x^4 - 378*a^5*b^11*d^10*x^4 + 17640*b^16*c^6*d^4*
x^3 - 42336*a*b^15*c^5*d^5*x^3 + 52920*a^2*b^14*c^4*d^6*x^3 - 40320*a^3*b^13*c^3*d^7*x^3 + 18900*a^4*b^12*c^2*
d^8*x^3 - 5040*a^5*b^11*c*d^9*x^3 + 588*a^6*b^10*d^10*x^3 + 15120*b^16*c^7*d^3*x^2 - 52920*a*b^15*c^6*d^4*x^2
+ 95256*a^2*b^14*c^5*d^5*x^2 - 105840*a^3*b^13*c^4*d^6*x^2 + 75600*a^4*b^12*c^3*d^7*x^2 - 34020*a^5*b^11*c^2*d
^8*x^2 + 8820*a^6*b^10*c*d^9*x^2 - 1008*a^7*b^9*d^10*x^2 + 11340*b^16*c^8*d^2*x - 60480*a*b^15*c^7*d^3*x + 158
760*a^2*b^14*c^6*d^4*x - 254016*a^3*b^13*c^5*d^5*x + 264600*a^4*b^12*c^4*d^6*x - 181440*a^5*b^11*c^3*d^7*x + 7
9380*a^6*b^10*c^2*d^8*x - 20160*a^7*b^9*c*d^9*x + 2268*a^8*b^8*d^10*x)/b^18

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Mupad [B]
time = 0.35, size = 3475, normalized size = 13.47

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^2,x)

[Out]

x^7*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/(7*b) - (a^2*d^10)/(7*b^4) + (45*c^2*d^8)/(7*b^2)) - x^5*((2*a*((
120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (
a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/(5*b) - (42*c^4*d^6)/b^2 + (a^2*((2*a*((2*a*d^10)/b^3 - (10*c*d^9
)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/(5*b^2)) - x^8*((a*d^10)/(4*b^3) - (5*c*d^9)/(4*b^2)) + x^3*((
70*c^6*d^4)/b^2 - (2*a*((2*a*((2*a*((120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2
*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/b - (210*c^4*d^6)/b^2 + (a^2
*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b^2))/b - (a^2*((120*c^3*d^7
)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*((2*a*d
^10)/b^3 - (10*c*d^9)/b^2))/b^2))/b^2 + (252*c^5*d^5)/b^2))/(3*b) + (a^2*((2*a*((120*c^3*d^7)/b^2 - (2*a*((2*a
*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*((2*a*d^10)/b^3 - (10*c*d
^9)/b^2))/b^2))/b - (210*c^4*d^6)/b^2 + (a^2*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45
*c^2*d^8)/b^2))/b^2))/(3*b^2)) - x^2*((a*((210*c^6*d^4)/b^2 - (2*a*((2*a*((2*a*((120*c^3*d^7)/b^2 - (2*a*((2*a
*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*((2*a*d^10)/b^3 - (10*c*d
^9)/b^2))/b^2))/b - (210*c^4*d^6)/b^2 + (a^2*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45
*c^2*d^8)/b^2))/b^2))/b - (a^2*((120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^1
0)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/b^2 + (252*c^5*d^5)/b^2))/b + (a
^2*((2*a*((120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/
b^2))/b + (a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/b - (210*c^4*d^6)/b^2 + (a^2*((2*a*((2*a*d^10)/b^3 - (
10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b^2))/b^2))/b - (60*c^7*d^3)/b^2 + (a^2*((2*a*((2*a*((
120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (
a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/b - (210*c^4*d^6)/b^2 + (a^2*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b
^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b^2))/b - (a^2*((120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 -
(10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/b^2
+ (252*c^5*d^5)/b^2))/(2*b^2)) + x^6*((20*c^3*d^7)/b^2 - (a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2
*d^10)/b^4 + (45*c^2*d^8)/b^2))/(3*b) + (a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/(6*b^2)) + x*((45*c^8*d^2)/b^2
- (a^2*((210*c^6*d^4)/b^2 - (2*a*((2*a*((2*a*((120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2
))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/b - (210*c^4*d^6)
/b^2 + (a^2*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b^2))/b - (a^2*((
120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (
a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/b^2 + (252*c^5*d^5)/b^2))/b + (a^2*((2*a*((120*c^3*d^7)/b^2 - (2*
a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*((2*a*d^10)/b^3 -
(10*c*d^9)/b^2))/b^2))/b - (210*c^4*d^6)/b^2 + (a^2*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^
4 + (45*c^2*d^8)/b^2))/b^2))/b^2))/b^2 + (2*a*((2*a*((210*c^6*d^4)/b^2 - (2*a*((2*a*((2*a*((120*c^3*d^7)/b^2 -
(2*a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*((2*a*d^10)/b^
3 - (10*c*d^9)/b^2))/b^2))/b - (210*c^4*d^6)/b^2 + (a^2*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10
)/b^4 + (45*c^2*d^8)/b^2))/b^2))/b - (a^2*((120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b
- (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/b^2 + (252*c^5*d^5)/b
^2))/b + (a^2*((2*a*((120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (4
5*c^2*d^8)/b^2))/b + (a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/b - (210*c^4*d^6)/b^2 + (a^2*((2*a*((2*a*d^
10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b^2))/b^2))/b - (120*c^7*d^3)/b^2 + (a^2*((
2*a*((2*a*((120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)
/b^2))/b + (a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/b - (210*c^4*d^6)/b^2 + (a^2*((2*a*((2*a*d^10)/b^3 -
(10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b^2))/b - (a^2*((120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*
d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2
))/b^2))/b^2 + (252*c^5*d^5)/b^2))/b^2))/b) + x^4*((a*((2*a*((120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 -
(10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b + (a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/b -
(210*c^4*d^6)/b^2 + (a^2*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^2*d^8)/b^2))/b^2
))/(2*b) - (a^2*((120*c^3*d^7)/b^2 - (2*a*((2*a*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b - (a^2*d^10)/b^4 + (45*c^
2*d^8)/b^2))/b + (a^2*((2*a*d^10)/b^3 - (10*c*d^9)/b^2))/b^2))/(4*b^2) + (63*c^5*d^5)/b^2) - (log(a + b*x)*(10
*a^9*d^10 - 10*b^9*c^9*d + 90*a*b^8*c^8*d^2 - 360*a^2*b^7*c^7*d^3 + 840*a^3*b^6*c^6*d^4 - 1260*a^4*b^5*c^5*d^5
+ 1260*a^5*b^4*c^4*d^6 - 840*a^6*b^3*c^3*d^7 + 360*a^7*b^2*c^2*d^8 - 90*a^8*b*c*d^9))/b^11 - (a^10*d^10 + b^1
0*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*(a*b^10 + b^11*x)) + (d
^10*x^9)/(9*b^2)

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