3.1312 \(\int \frac{(c+d x)^{10}}{a+b x} \, dx\)

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

[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

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Rubi [A]  time = 0.225503, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{(b c-a d)^{10} \log (a+b x)}{b^{11}}+\frac{d x (b c-a d)^9}{b^{10}}+\frac{(c+d x)^2 (b c-a d)^8}{2 b^9}+\frac{(c+d x)^3 (b c-a d)^7}{3 b^8}+\frac{(c+d x)^4 (b c-a d)^6}{4 b^7}+\frac{(c+d x)^5 (b c-a d)^5}{5 b^6}+\frac{(c+d x)^6 (b c-a d)^4}{6 b^5}+\frac{(c+d x)^7 (b c-a d)^3}{7 b^4}+\frac{(c+d x)^8 (b c-a d)^2}{8 b^3}+\frac{(c+d x)^9 (b c-a d)}{9 b^2}+\frac{(c+d x)^{10}}{10 b} \]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\left (c + d x\right )^{10}}{10 b} - \frac{\left (c + d x\right )^{9} \left (a d - b c\right )}{9 b^{2}} + \frac{\left (c + d x\right )^{8} \left (a d - b c\right )^{2}}{8 b^{3}} - \frac{\left (c + d x\right )^{7} \left (a d - b c\right )^{3}}{7 b^{4}} + \frac{\left (c + d x\right )^{6} \left (a d - b c\right )^{4}}{6 b^{5}} - \frac{\left (c + d x\right )^{5} \left (a d - b c\right )^{5}}{5 b^{6}} + \frac{\left (c + d x\right )^{4} \left (a d - b c\right )^{6}}{4 b^{7}} - \frac{\left (c + d x\right )^{3} \left (a d - b c\right )^{7}}{3 b^{8}} + \frac{\left (c + d x\right )^{2} \left (a d - b c\right )^{8}}{2 b^{9}} - \frac{\left (a d - b c\right )^{9} \int d\, dx}{b^{10}} + \frac{\left (a d - b c\right )^{10} \log{\left (a + b x \right )}}{b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**10/(b*x+a),x)

[Out]

(c + d*x)**10/(10*b) - (c + d*x)**9*(a*d - b*c)/(9*b**2) + (c + d*x)**8*(a*d - b
*c)**2/(8*b**3) - (c + d*x)**7*(a*d - b*c)**3/(7*b**4) + (c + d*x)**6*(a*d - b*c
)**4/(6*b**5) - (c + d*x)**5*(a*d - b*c)**5/(5*b**6) + (c + d*x)**4*(a*d - b*c)*
*6/(4*b**7) - (c + d*x)**3*(a*d - b*c)**7/(3*b**8) + (c + d*x)**2*(a*d - b*c)**8
/(2*b**9) - (a*d - b*c)**9*Integral(d, x)/b**10 + (a*d - b*c)**10*log(a + b*x)/b
**11

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Mathematica [B]  time = 0.864701, size = 591, normalized size = 2.45 \[ \frac{d x \left (-2520 a^9 d^9+1260 a^8 b d^8 (20 c+d x)-840 a^7 b^2 d^7 \left (135 c^2+15 c d x+d^2 x^2\right )+210 a^6 b^3 d^6 \left (1440 c^3+270 c^2 d x+40 c d^2 x^2+3 d^3 x^3\right )-252 a^5 b^4 d^5 \left (2100 c^4+600 c^3 d x+150 c^2 d^2 x^2+25 c d^3 x^3+2 d^4 x^4\right )+210 a^4 b^5 d^4 \left (3024 c^5+1260 c^4 d x+480 c^3 d^2 x^2+135 c^2 d^3 x^3+24 c d^4 x^4+2 d^5 x^5\right )-120 a^3 b^6 d^3 \left (4410 c^6+2646 c^5 d x+1470 c^4 d^2 x^2+630 c^3 d^3 x^3+189 c^2 d^4 x^4+35 c d^5 x^5+3 d^6 x^6\right )+45 a^2 b^7 d^2 \left (6720 c^7+5880 c^6 d x+4704 c^5 d^2 x^2+2940 c^4 d^3 x^3+1344 c^3 d^4 x^4+420 c^2 d^5 x^5+80 c d^6 x^6+7 d^7 x^7\right )-10 a b^8 d \left (11340 c^8+15120 c^7 d x+17640 c^6 d^2 x^2+15876 c^5 d^3 x^3+10584 c^4 d^4 x^4+5040 c^3 d^5 x^5+1620 c^2 d^6 x^6+315 c d^7 x^7+28 d^8 x^8\right )+b^9 \left (25200 c^9+56700 c^8 d x+100800 c^7 d^2 x^2+132300 c^6 d^3 x^3+127008 c^5 d^4 x^4+88200 c^4 d^5 x^5+43200 c^3 d^6 x^6+14175 c^2 d^7 x^7+2800 c d^8 x^8+252 d^9 x^9\right )\right )}{2520 b^{10}}+\frac{(b c-a d)^{10} \log (a+b x)}{b^{11}} \]

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 + 1
5*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 + 105
84*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^1
1

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Maple [B]  time = 0.012, size = 1022, normalized size = 4.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^10/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.38896, size = 1169, normalized size = 4.85 \[ \text{result too large to display} \]

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*(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)*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 - 12
0*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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Fricas [A]  time = 0.209371, size = 1172, normalized size = 4.86 \[ \text{result too large to display} \]

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*(2
52*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 - 10*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 + 1
0*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 1260*(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)*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 + 2
10*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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Sympy [A]  time = 5.41552, size = 772, normalized size = 3.2 \[ \frac{d^{10} x^{10}}{10 b} - \frac{x^{9} \left (a d^{10} - 10 b c d^{9}\right )}{9 b^{2}} + \frac{x^{8} \left (a^{2} d^{10} - 10 a b c d^{9} + 45 b^{2} c^{2} d^{8}\right )}{8 b^{3}} - \frac{x^{7} \left (a^{3} d^{10} - 10 a^{2} b c d^{9} + 45 a b^{2} c^{2} d^{8} - 120 b^{3} c^{3} d^{7}\right )}{7 b^{4}} + \frac{x^{6} \left (a^{4} d^{10} - 10 a^{3} b c d^{9} + 45 a^{2} b^{2} c^{2} d^{8} - 120 a b^{3} c^{3} d^{7} + 210 b^{4} c^{4} d^{6}\right )}{6 b^{5}} - \frac{x^{5} \left (a^{5} d^{10} - 10 a^{4} b c d^{9} + 45 a^{3} b^{2} c^{2} d^{8} - 120 a^{2} b^{3} c^{3} d^{7} + 210 a b^{4} c^{4} d^{6} - 252 b^{5} c^{5} d^{5}\right )}{5 b^{6}} + \frac{x^{4} \left (a^{6} d^{10} - 10 a^{5} b c d^{9} + 45 a^{4} b^{2} c^{2} d^{8} - 120 a^{3} b^{3} c^{3} d^{7} + 210 a^{2} b^{4} c^{4} d^{6} - 252 a b^{5} c^{5} d^{5} + 210 b^{6} c^{6} d^{4}\right )}{4 b^{7}} - \frac{x^{3} \left (a^{7} d^{10} - 10 a^{6} b c d^{9} + 45 a^{5} b^{2} c^{2} d^{8} - 120 a^{4} b^{3} c^{3} d^{7} + 210 a^{3} b^{4} c^{4} d^{6} - 252 a^{2} b^{5} c^{5} d^{5} + 210 a b^{6} c^{6} d^{4} - 120 b^{7} c^{7} d^{3}\right )}{3 b^{8}} + \frac{x^{2} \left (a^{8} d^{10} - 10 a^{7} b c d^{9} + 45 a^{6} b^{2} c^{2} d^{8} - 120 a^{5} b^{3} c^{3} d^{7} + 210 a^{4} b^{4} c^{4} d^{6} - 252 a^{3} b^{5} c^{5} d^{5} + 210 a^{2} b^{6} c^{6} d^{4} - 120 a b^{7} c^{7} d^{3} + 45 b^{8} c^{8} d^{2}\right )}{2 b^{9}} - \frac{x \left (a^{9} d^{10} - 10 a^{8} b c d^{9} + 45 a^{7} b^{2} c^{2} d^{8} - 120 a^{6} b^{3} c^{3} d^{7} + 210 a^{5} b^{4} c^{4} d^{6} - 252 a^{4} b^{5} c^{5} d^{5} + 210 a^{3} b^{6} c^{6} d^{4} - 120 a^{2} b^{7} c^{7} d^{3} + 45 a b^{8} c^{8} d^{2} - 10 b^{9} c^{9} d\right )}{b^{10}} + \frac{\left (a d - b c\right )^{10} \log{\left (a + b x \right )}}{b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**10*x**10/(10*b) - x**9*(a*d**10 - 10*b*c*d**9)/(9*b**2) + x**8*(a**2*d**10 -
10*a*b*c*d**9 + 45*b**2*c**2*d**8)/(8*b**3) - x**7*(a**3*d**10 - 10*a**2*b*c*d**
9 + 45*a*b**2*c**2*d**8 - 120*b**3*c**3*d**7)/(7*b**4) + x**6*(a**4*d**10 - 10*a
**3*b*c*d**9 + 45*a**2*b**2*c**2*d**8 - 120*a*b**3*c**3*d**7 + 210*b**4*c**4*d**
6)/(6*b**5) - x**5*(a**5*d**10 - 10*a**4*b*c*d**9 + 45*a**3*b**2*c**2*d**8 - 120
*a**2*b**3*c**3*d**7 + 210*a*b**4*c**4*d**6 - 252*b**5*c**5*d**5)/(5*b**6) + x**
4*(a**6*d**10 - 10*a**5*b*c*d**9 + 45*a**4*b**2*c**2*d**8 - 120*a**3*b**3*c**3*d
**7 + 210*a**2*b**4*c**4*d**6 - 252*a*b**5*c**5*d**5 + 210*b**6*c**6*d**4)/(4*b*
*7) - x**3*(a**7*d**10 - 10*a**6*b*c*d**9 + 45*a**5*b**2*c**2*d**8 - 120*a**4*b*
*3*c**3*d**7 + 210*a**3*b**4*c**4*d**6 - 252*a**2*b**5*c**5*d**5 + 210*a*b**6*c*
*6*d**4 - 120*b**7*c**7*d**3)/(3*b**8) + x**2*(a**8*d**10 - 10*a**7*b*c*d**9 + 4
5*a**6*b**2*c**2*d**8 - 120*a**5*b**3*c**3*d**7 + 210*a**4*b**4*c**4*d**6 - 252*
a**3*b**5*c**5*d**5 + 210*a**2*b**6*c**6*d**4 - 120*a*b**7*c**7*d**3 + 45*b**8*c
**8*d**2)/(2*b**9) - x*(a**9*d**10 - 10*a**8*b*c*d**9 + 45*a**7*b**2*c**2*d**8 -
120*a**6*b**3*c**3*d**7 + 210*a**5*b**4*c**4*d**6 - 252*a**4*b**5*c**5*d**5 + 2
10*a**3*b**6*c**6*d**4 - 120*a**2*b**7*c**7*d**3 + 45*a*b**8*c**8*d**2 - 10*b**9
*c**9*d)/b**10 + (a*d - b*c)**10*log(a + b*x)/b**11

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GIAC/XCAS [A]  time = 0.21692, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done