3.1321 \(\int \frac{(c+d x)^{10}}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=257 \[ \frac{10 d^9 (b c-a d) \log (a+b x)}{b^{11}}-\frac{45 d^8 (b c-a d)^2}{b^{11} (a+b x)}-\frac{60 d^7 (b c-a d)^3}{b^{11} (a+b x)^2}-\frac{70 d^6 (b c-a d)^4}{b^{11} (a+b x)^3}-\frac{63 d^5 (b c-a d)^5}{b^{11} (a+b x)^4}-\frac{42 d^4 (b c-a d)^6}{b^{11} (a+b x)^5}-\frac{20 d^3 (b c-a d)^7}{b^{11} (a+b x)^6}-\frac{45 d^2 (b c-a d)^8}{7 b^{11} (a+b x)^7}-\frac{5 d (b c-a d)^9}{4 b^{11} (a+b x)^8}-\frac{(b c-a d)^{10}}{9 b^{11} (a+b x)^9}+\frac{d^{10} x}{b^{10}} \]

[Out]

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

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Rubi [A]  time = 0.814336, antiderivative size = 257, 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 \[ \frac{10 d^9 (b c-a d) \log (a+b x)}{b^{11}}-\frac{45 d^8 (b c-a d)^2}{b^{11} (a+b x)}-\frac{60 d^7 (b c-a d)^3}{b^{11} (a+b x)^2}-\frac{70 d^6 (b c-a d)^4}{b^{11} (a+b x)^3}-\frac{63 d^5 (b c-a d)^5}{b^{11} (a+b x)^4}-\frac{42 d^4 (b c-a d)^6}{b^{11} (a+b x)^5}-\frac{20 d^3 (b c-a d)^7}{b^{11} (a+b x)^6}-\frac{45 d^2 (b c-a d)^8}{7 b^{11} (a+b x)^7}-\frac{5 d (b c-a d)^9}{4 b^{11} (a+b x)^8}-\frac{(b c-a d)^{10}}{9 b^{11} (a+b x)^9}+\frac{d^{10} x}{b^{10}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{10} \int \frac{1}{b^{10}}\, dx - \frac{10 d^{9} \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{11}} - \frac{45 d^{8} \left (a d - b c\right )^{2}}{b^{11} \left (a + b x\right )} + \frac{60 d^{7} \left (a d - b c\right )^{3}}{b^{11} \left (a + b x\right )^{2}} - \frac{70 d^{6} \left (a d - b c\right )^{4}}{b^{11} \left (a + b x\right )^{3}} + \frac{63 d^{5} \left (a d - b c\right )^{5}}{b^{11} \left (a + b x\right )^{4}} - \frac{42 d^{4} \left (a d - b c\right )^{6}}{b^{11} \left (a + b x\right )^{5}} + \frac{20 d^{3} \left (a d - b c\right )^{7}}{b^{11} \left (a + b x\right )^{6}} - \frac{45 d^{2} \left (a d - b c\right )^{8}}{7 b^{11} \left (a + b x\right )^{7}} + \frac{5 d \left (a d - b c\right )^{9}}{4 b^{11} \left (a + b x\right )^{8}} - \frac{\left (a d - b c\right )^{10}}{9 b^{11} \left (a + b x\right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 2.46671, size = 708, normalized size = 2.75 \[ -\frac{4861 a^{10} d^{10}+a^9 b d^9 (41229 d x-7129 c)+9 a^8 b^2 d^8 \left (140 c^2-6849 c d x+17064 d^2 x^2\right )+12 a^7 b^3 d^7 \left (35 c^3+945 c^2 d x-19602 c d^2 x^2+27342 d^3 x^3\right )+42 a^6 b^4 d^6 \left (5 c^4+90 c^3 d x+1080 c^2 d^2 x^2-12348 c d^3 x^3+10458 d^4 x^4\right )+126 a^5 b^5 d^5 \left (c^5+15 c^4 d x+120 c^3 d^2 x^2+840 c^2 d^3 x^3-5754 c d^4 x^4+2982 d^5 x^5\right )+42 a^4 b^6 d^4 \left (2 c^6+27 c^5 d x+180 c^4 d^2 x^2+840 c^3 d^3 x^3+3780 c^2 d^4 x^4-15750 c d^5 x^5+4704 d^6 x^6\right )+12 a^3 b^7 d^3 \left (5 c^7+63 c^6 d x+378 c^5 d^2 x^2+1470 c^4 d^3 x^3+4410 c^3 d^4 x^4+13230 c^2 d^5 x^5-32340 c d^6 x^6+4536 d^7 x^7\right )+9 a^2 b^8 d^2 \left (5 c^8+60 c^7 d x+336 c^6 d^2 x^2+1176 c^5 d^3 x^3+2940 c^4 d^4 x^4+5880 c^3 d^5 x^5+11760 c^2 d^6 x^6-15120 c d^7 x^7+252 d^8 x^8\right )+a b^9 d \left (35 c^9+405 c^8 d x+2160 c^7 d^2 x^2+7056 c^6 d^3 x^3+15876 c^5 d^4 x^4+26460 c^4 d^5 x^5+35280 c^3 d^6 x^6+45360 c^2 d^7 x^7-22680 c d^8 x^8-2268 d^9 x^9\right )+2520 d^9 (a+b x)^9 (a d-b c) \log (a+b x)+b^{10} \left (28 c^{10}+315 c^9 d x+1620 c^8 d^2 x^2+5040 c^7 d^3 x^3+10584 c^6 d^4 x^4+15876 c^5 d^5 x^5+17640 c^4 d^6 x^6+15120 c^3 d^7 x^7+11340 c^2 d^8 x^8-252 d^{10} x^{10}\right )}{252 b^{11} (a+b x)^9} \]

Antiderivative was successfully verified.

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

[Out]

-(4861*a^10*d^10 + a^9*b*d^9*(-7129*c + 41229*d*x) + 9*a^8*b^2*d^8*(140*c^2 - 68
49*c*d*x + 17064*d^2*x^2) + 12*a^7*b^3*d^7*(35*c^3 + 945*c^2*d*x - 19602*c*d^2*x
^2 + 27342*d^3*x^3) + 42*a^6*b^4*d^6*(5*c^4 + 90*c^3*d*x + 1080*c^2*d^2*x^2 - 12
348*c*d^3*x^3 + 10458*d^4*x^4) + 126*a^5*b^5*d^5*(c^5 + 15*c^4*d*x + 120*c^3*d^2
*x^2 + 840*c^2*d^3*x^3 - 5754*c*d^4*x^4 + 2982*d^5*x^5) + 42*a^4*b^6*d^4*(2*c^6
+ 27*c^5*d*x + 180*c^4*d^2*x^2 + 840*c^3*d^3*x^3 + 3780*c^2*d^4*x^4 - 15750*c*d^
5*x^5 + 4704*d^6*x^6) + 12*a^3*b^7*d^3*(5*c^7 + 63*c^6*d*x + 378*c^5*d^2*x^2 + 1
470*c^4*d^3*x^3 + 4410*c^3*d^4*x^4 + 13230*c^2*d^5*x^5 - 32340*c*d^6*x^6 + 4536*
d^7*x^7) + 9*a^2*b^8*d^2*(5*c^8 + 60*c^7*d*x + 336*c^6*d^2*x^2 + 1176*c^5*d^3*x^
3 + 2940*c^4*d^4*x^4 + 5880*c^3*d^5*x^5 + 11760*c^2*d^6*x^6 - 15120*c*d^7*x^7 +
252*d^8*x^8) + a*b^9*d*(35*c^9 + 405*c^8*d*x + 2160*c^7*d^2*x^2 + 7056*c^6*d^3*x
^3 + 15876*c^5*d^4*x^4 + 26460*c^4*d^5*x^5 + 35280*c^3*d^6*x^6 + 45360*c^2*d^7*x
^7 - 22680*c*d^8*x^8 - 2268*d^9*x^9) + b^10*(28*c^10 + 315*c^9*d*x + 1620*c^8*d^
2*x^2 + 5040*c^7*d^3*x^3 + 10584*c^6*d^4*x^4 + 15876*c^5*d^5*x^5 + 17640*c^4*d^6
*x^6 + 15120*c^3*d^7*x^7 + 11340*c^2*d^8*x^8 - 252*d^10*x^10) + 2520*d^9*(-(b*c)
+ a*d)*(a + b*x)^9*Log[a + b*x])/(252*b^11*(a + b*x)^9)

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Maple [B]  time = 0.025, size = 1266, normalized size = 4.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^10/(b*x+a)^10,x)

[Out]

-42/b^5*d^4/(b*x+a)^5*c^6-10/b^11*d^10*ln(b*x+a)*a+10/b^10*d^9*ln(b*x+a)*c-45/b^
11*d^10/(b*x+a)*a^2-45/b^9*d^8/(b*x+a)*c^2-60/b^8*d^7/(b*x+a)^2*c^3+5/4/b^11*d^1
0/(b*x+a)^8*a^9-5/4/b^2*d/(b*x+a)^8*c^9-1/9/b^11/(b*x+a)^9*a^10*d^10+20/b^11*d^1
0/(b*x+a)^6*a^7-20/b^4*d^3/(b*x+a)^6*c^7-45/7/b^11*d^10/(b*x+a)^7*a^8-45/7/b^3*d
^2/(b*x+a)^7*c^8+63/b^11*d^10/(b*x+a)^4*a^5-63/b^6*d^5/(b*x+a)^4*c^5-70/b^11*d^1
0/(b*x+a)^3*a^4-70/b^7*d^6/(b*x+a)^3*c^4+60/b^11*d^10/(b*x+a)^2*a^3-42/b^11*d^10
/(b*x+a)^5*a^6+d^10*x/b^10+252/b^6*d^5/(b*x+a)^5*a*c^5-140/b^10*d^9/(b*x+a)^6*a^
6*c+420/b^9*d^8/(b*x+a)^6*a^5*c^2-700/b^8*d^7/(b*x+a)^6*a^4*c^3+700/b^7*d^6/(b*x
+a)^6*a^3*c^4-420/b^6*d^5/(b*x+a)^6*a^2*c^5+140/b^5*d^4/(b*x+a)^6*a*c^6+360/7/b^
10*d^9/(b*x+a)^7*a^7*c-180/b^9*d^8/(b*x+a)^7*a^6*c^2+360/b^8*d^7/(b*x+a)^7*a^5*c
^3-450/b^7*d^6/(b*x+a)^7*a^4*c^4+360/b^6*d^5/(b*x+a)^7*a^3*c^5+252/b^10*d^9/(b*x
+a)^5*a^5*c-630/b^9*d^8/(b*x+a)^5*a^4*c^2+840/b^8*d^7/(b*x+a)^5*a^3*c^3+315/2/b^
7*d^6/(b*x+a)^8*a^5*c^4+180/b^9*d^8/(b*x+a)^2*a*c^2-45/4/b^10*d^9/(b*x+a)^8*a^8*
c+45/b^9*d^8/(b*x+a)^8*a^7*c^2-105/b^8*d^7/(b*x+a)^8*a^6*c^3-180/b^10*d^9/(b*x+a
)^2*a^2*c-630/b^8*d^7/(b*x+a)^4*a^2*c^3+315/b^7*d^6/(b*x+a)^4*a*c^4+280/b^10*d^9
/(b*x+a)^3*a^3*c-420/b^9*d^8/(b*x+a)^3*a^2*c^2+280/b^8*d^7/(b*x+a)^3*a*c^3-5/b^3
/(b*x+a)^9*a^2*c^8*d^2+10/9/b^2/(b*x+a)^9*a*c^9*d+90/b^10*d^9/(b*x+a)*a*c-180/b^
5*d^4/(b*x+a)^7*a^2*c^6+360/7/b^4*d^3/(b*x+a)^7*a*c^7-315/b^10*d^9/(b*x+a)^4*a^4
*c+630/b^9*d^8/(b*x+a)^4*a^3*c^2-70/3/b^5/(b*x+a)^9*a^4*c^6*d^4+40/3/b^4/(b*x+a)
^9*a^3*c^7*d^3+40/3/b^8/(b*x+a)^9*a^7*c^3*d^7-70/3/b^7/(b*x+a)^9*a^6*c^4*d^6+28/
b^6/(b*x+a)^9*a^5*c^5*d^5-5/b^9/(b*x+a)^9*a^8*c^2*d^8+10/9/b^10/(b*x+a)^9*a^9*c*
d^9-45/b^4*d^3/(b*x+a)^8*a^2*c^7+105/b^5*d^4/(b*x+a)^8*a^3*c^6-315/2/b^6*d^5/(b*
x+a)^8*a^4*c^5-1/9/b/(b*x+a)^9*c^10-630/b^7*d^6/(b*x+a)^5*a^2*c^4+45/4/b^3*d^2/(
b*x+a)^8*a*c^8

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Maxima [A]  time = 1.46947, size = 1292, normalized size = 5.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^10*x/b^10 - 1/252*(28*b^10*c^10 + 35*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 + 60*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 + 210*a^6*b^4*c^4*d^6 +
420*a^7*b^3*c^3*d^7 + 1260*a^8*b^2*c^2*d^8 - 7129*a^9*b*c*d^9 + 4861*a^10*d^10 +
11340*(b^10*c^2*d^8 - 2*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 15120*(b^10*c^3*d^7 +
3*a*b^9*c^2*d^8 - 9*a^2*b^8*c*d^9 + 5*a^3*b^7*d^10)*x^7 + 17640*(b^10*c^4*d^6 +
2*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 22*a^3*b^7*c*d^9 + 13*a^4*b^6*d^10)*x^6 +
5292*(3*b^10*c^5*d^5 + 5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 + 30*a^3*b^7*c^2*d^
8 - 125*a^4*b^6*c*d^9 + 77*a^5*b^5*d^10)*x^5 + 5292*(2*b^10*c^6*d^4 + 3*a*b^9*c^
5*d^5 + 5*a^2*b^8*c^4*d^6 + 10*a^3*b^7*c^3*d^7 + 30*a^4*b^6*c^2*d^8 - 137*a^5*b^
5*c*d^9 + 87*a^6*b^4*d^10)*x^4 + 504*(10*b^10*c^7*d^3 + 14*a*b^9*c^6*d^4 + 21*a^
2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6 + 70*a^4*b^6*c^3*d^7 + 210*a^5*b^5*c^2*d^8 -
1029*a^6*b^4*c*d^9 + 669*a^7*b^3*d^10)*x^3 + 108*(15*b^10*c^8*d^2 + 20*a*b^9*c^7
*d^3 + 28*a^2*b^8*c^6*d^4 + 42*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 140*a^5*b^
5*c^3*d^7 + 420*a^6*b^4*c^2*d^8 - 2178*a^7*b^3*c*d^9 + 1443*a^8*b^2*d^10)*x^2 +
9*(35*b^10*c^9*d + 45*a*b^9*c^8*d^2 + 60*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 + 210*a^5*b^5*c^4*d^6 + 420*a^6*b^4*c^3*d^7 + 1260*a^7*b^3*c
^2*d^8 - 6849*a^8*b^2*c*d^9 + 4609*a^9*b*d^10)*x)/(b^20*x^9 + 9*a*b^19*x^8 + 36*
a^2*b^18*x^7 + 84*a^3*b^17*x^6 + 126*a^4*b^16*x^5 + 126*a^5*b^15*x^4 + 84*a^6*b^
14*x^3 + 36*a^7*b^13*x^2 + 9*a^8*b^12*x + a^9*b^11) + 10*(b*c*d^9 - a*d^10)*log(
b*x + a)/b^11

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Fricas [A]  time = 0.232824, size = 1642, normalized size = 6.39 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/252*(252*b^10*d^10*x^10 + 2268*a*b^9*d^10*x^9 - 28*b^10*c^10 - 35*a*b^9*c^9*d
- 45*a^2*b^8*c^8*d^2 - 60*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 - 210*a^6*b^4*c^4*d^6 - 420*a^7*b^3*c^3*d^7 - 1260*a^8*b^2*c^2*d^8 + 7129*a
^9*b*c*d^9 - 4861*a^10*d^10 - 2268*(5*b^10*c^2*d^8 - 10*a*b^9*c*d^9 + a^2*b^8*d^
10)*x^8 - 3024*(5*b^10*c^3*d^7 + 15*a*b^9*c^2*d^8 - 45*a^2*b^8*c*d^9 + 18*a^3*b^
7*d^10)*x^7 - 3528*(5*b^10*c^4*d^6 + 10*a*b^9*c^3*d^7 + 30*a^2*b^8*c^2*d^8 - 110
*a^3*b^7*c*d^9 + 56*a^4*b^6*d^10)*x^6 - 5292*(3*b^10*c^5*d^5 + 5*a*b^9*c^4*d^6 +
10*a^2*b^8*c^3*d^7 + 30*a^3*b^7*c^2*d^8 - 125*a^4*b^6*c*d^9 + 71*a^5*b^5*d^10)*
x^5 - 5292*(2*b^10*c^6*d^4 + 3*a*b^9*c^5*d^5 + 5*a^2*b^8*c^4*d^6 + 10*a^3*b^7*c^
3*d^7 + 30*a^4*b^6*c^2*d^8 - 137*a^5*b^5*c*d^9 + 83*a^6*b^4*d^10)*x^4 - 504*(10*
b^10*c^7*d^3 + 14*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6 + 70*a
^4*b^6*c^3*d^7 + 210*a^5*b^5*c^2*d^8 - 1029*a^6*b^4*c*d^9 + 651*a^7*b^3*d^10)*x^
3 - 108*(15*b^10*c^8*d^2 + 20*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 42*a^3*b^7*c^
5*d^5 + 70*a^4*b^6*c^4*d^6 + 140*a^5*b^5*c^3*d^7 + 420*a^6*b^4*c^2*d^8 - 2178*a^
7*b^3*c*d^9 + 1422*a^8*b^2*d^10)*x^2 - 9*(35*b^10*c^9*d + 45*a*b^9*c^8*d^2 + 60*
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 + 210*a^5*b^5*c^4*d^6
+ 420*a^6*b^4*c^3*d^7 + 1260*a^7*b^3*c^2*d^8 - 6849*a^8*b^2*c*d^9 + 4581*a^9*b*
d^10)*x + 2520*(a^9*b*c*d^9 - a^10*d^10 + (b^10*c*d^9 - a*b^9*d^10)*x^9 + 9*(a*b
^9*c*d^9 - a^2*b^8*d^10)*x^8 + 36*(a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 84*(a^3*b
^7*c*d^9 - a^4*b^6*d^10)*x^6 + 126*(a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 + 126*(a^5
*b^5*c*d^9 - a^6*b^4*d^10)*x^4 + 84*(a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 36*(a^7
*b^3*c*d^9 - a^8*b^2*d^10)*x^2 + 9*(a^8*b^2*c*d^9 - a^9*b*d^10)*x)*log(b*x + a))
/(b^20*x^9 + 9*a*b^19*x^8 + 36*a^2*b^18*x^7 + 84*a^3*b^17*x^6 + 126*a^4*b^16*x^5
+ 126*a^5*b^15*x^4 + 84*a^6*b^14*x^3 + 36*a^7*b^13*x^2 + 9*a^8*b^12*x + a^9*b^1
1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218816, size = 1170, normalized size = 4.55 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^10*x/b^10 + 10*(b*c*d^9 - a*d^10)*ln(abs(b*x + a))/b^11 - 1/252*(28*b^10*c^10
+ 35*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 + 60*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 + 210*a^6*b^4*c^4*d^6 + 420*a^7*b^3*c^3*d^7 + 1260*a^8*b^2
*c^2*d^8 - 7129*a^9*b*c*d^9 + 4861*a^10*d^10 + 11340*(b^10*c^2*d^8 - 2*a*b^9*c*d
^9 + a^2*b^8*d^10)*x^8 + 15120*(b^10*c^3*d^7 + 3*a*b^9*c^2*d^8 - 9*a^2*b^8*c*d^9
+ 5*a^3*b^7*d^10)*x^7 + 17640*(b^10*c^4*d^6 + 2*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d
^8 - 22*a^3*b^7*c*d^9 + 13*a^4*b^6*d^10)*x^6 + 5292*(3*b^10*c^5*d^5 + 5*a*b^9*c^
4*d^6 + 10*a^2*b^8*c^3*d^7 + 30*a^3*b^7*c^2*d^8 - 125*a^4*b^6*c*d^9 + 77*a^5*b^5
*d^10)*x^5 + 5292*(2*b^10*c^6*d^4 + 3*a*b^9*c^5*d^5 + 5*a^2*b^8*c^4*d^6 + 10*a^3
*b^7*c^3*d^7 + 30*a^4*b^6*c^2*d^8 - 137*a^5*b^5*c*d^9 + 87*a^6*b^4*d^10)*x^4 + 5
04*(10*b^10*c^7*d^3 + 14*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6
+ 70*a^4*b^6*c^3*d^7 + 210*a^5*b^5*c^2*d^8 - 1029*a^6*b^4*c*d^9 + 669*a^7*b^3*d
^10)*x^3 + 108*(15*b^10*c^8*d^2 + 20*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 42*a^3
*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 140*a^5*b^5*c^3*d^7 + 420*a^6*b^4*c^2*d^8 -
2178*a^7*b^3*c*d^9 + 1443*a^8*b^2*d^10)*x^2 + 9*(35*b^10*c^9*d + 45*a*b^9*c^8*d^
2 + 60*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 + 210*a^5*b^5*
c^4*d^6 + 420*a^6*b^4*c^3*d^7 + 1260*a^7*b^3*c^2*d^8 - 6849*a^8*b^2*c*d^9 + 4609
*a^9*b*d^10)*x)/((b*x + a)^9*b^11)