∖int(c+dx)10 _____________

3.1332 \(\int \frac{(c+d x)^{10}}{(a+b x)^{21}} \, dx\)

Optimal. Leaf size=279 \[ -\frac{10 d^9 (b c-a d)}{11 b^{11} (a+b x)^{11}}-\frac{15 d^8 (b c-a d)^2}{4 b^{11} (a+b x)^{12}}-\frac{120 d^7 (b c-a d)^3}{13 b^{11} (a+b x)^{13}}-\frac{15 d^6 (b c-a d)^4}{b^{11} (a+b x)^{14}}-\frac{84 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^{15}}-\frac{105 d^4 (b c-a d)^6}{8 b^{11} (a+b x)^{16}}-\frac{120 d^3 (b c-a d)^7}{17 b^{11} (a+b x)^{17}}-\frac{5 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^{18}}-\frac{10 d (b c-a d)^9}{19 b^{11} (a+b x)^{19}}-\frac{(b c-a d)^{10}}{20 b^{11} (a+b x)^{20}}-\frac{d^{10}}{10 b^{11} (a+b x)^{10}} \]

[Out]

-(b*c - a*d)^10/(20*b^11*(a + b*x)^20) - (10*d*(b*c - a*d)^9)/(19*b^11*(a + b*x)

^19) - (5*d^2*(b*c - a*d)^8)/(2*b^11*(a + b*x)^18) - (120*d^3*(b*c - a*d)^7)/(17

*b^11*(a + b*x)^17) - (105*d^4*(b*c - a*d)^6)/(8*b^11*(a + b*x)^16) - (84*d^5*(b

*c - a*d)^5)/(5*b^11*(a + b*x)^15) - (15*d^6*(b*c - a*d)^4)/(b^11*(a + b*x)^14)

- (120*d^7*(b*c - a*d)^3)/(13*b^11*(a + b*x)^13) - (15*d^8*(b*c - a*d)^2)/(4*b^1

1*(a + b*x)^12) - (10*d^9*(b*c - a*d))/(11*b^11*(a + b*x)^11) - d^10/(10*b^11*(a

+ b*x)^10)

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Rubi [A] time = 0.815827, antiderivative size = 279, 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 \[ -\frac{10 d^9 (b c-a d)}{11 b^{11} (a+b x)^{11}}-\frac{15 d^8 (b c-a d)^2}{4 b^{11} (a+b x)^{12}}-\frac{120 d^7 (b c-a d)^3}{13 b^{11} (a+b x)^{13}}-\frac{15 d^6 (b c-a d)^4}{b^{11} (a+b x)^{14}}-\frac{84 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^{15}}-\frac{105 d^4 (b c-a d)^6}{8 b^{11} (a+b x)^{16}}-\frac{120 d^3 (b c-a d)^7}{17 b^{11} (a+b x)^{17}}-\frac{5 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^{18}}-\frac{10 d (b c-a d)^9}{19 b^{11} (a+b x)^{19}}-\frac{(b c-a d)^{10}}{20 b^{11} (a+b x)^{20}}-\frac{d^{10}}{10 b^{11} (a+b x)^{10}} \]

Antiderivative was successfully veriﬁed.

[In] Int[(c + d*x)^10/(a + b*x)^21,x]

[Out]

-(b*c - a*d)^10/(20*b^11*(a + b*x)^20) - (10*d*(b*c - a*d)^9)/(19*b^11*(a + b*x)

^19) - (5*d^2*(b*c - a*d)^8)/(2*b^11*(a + b*x)^18) - (120*d^3*(b*c - a*d)^7)/(17

*b^11*(a + b*x)^17) - (105*d^4*(b*c - a*d)^6)/(8*b^11*(a + b*x)^16) - (84*d^5*(b

*c - a*d)^5)/(5*b^11*(a + b*x)^15) - (15*d^6*(b*c - a*d)^4)/(b^11*(a + b*x)^14)

- (120*d^7*(b*c - a*d)^3)/(13*b^11*(a + b*x)^13) - (15*d^8*(b*c - a*d)^2)/(4*b^1

1*(a + b*x)^12) - (10*d^9*(b*c - a*d))/(11*b^11*(a + b*x)^11) - d^10/(10*b^11*(a

+ b*x)^10)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate((d*x+c)**10/(b*x+a)**21,x)

[Out]

Timed out

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Mathematica [B] time = 0.932526, size = 692, normalized size = 2.48 \[ -\frac{a^{10} d^{10}+10 a^9 b d^9 (c+2 d x)+5 a^8 b^2 d^8 \left (11 c^2+40 c d x+38 d^2 x^2\right )+20 a^7 b^3 d^7 \left (11 c^3+55 c^2 d x+95 c d^2 x^2+57 d^3 x^3\right )+5 a^6 b^4 d^6 \left (143 c^4+880 c^3 d x+2090 c^2 d^2 x^2+2280 c d^3 x^3+969 d^4 x^4\right )+2 a^5 b^5 d^5 \left (1001 c^5+7150 c^4 d x+20900 c^3 d^2 x^2+31350 c^2 d^3 x^3+24225 c d^4 x^4+7752 d^5 x^5\right )+5 a^4 b^6 d^4 \left (1001 c^6+8008 c^5 d x+27170 c^4 d^2 x^2+50160 c^3 d^3 x^3+53295 c^2 d^4 x^4+31008 c d^5 x^5+7752 d^6 x^6\right )+20 a^3 b^7 d^3 \left (572 c^7+5005 c^6 d x+19019 c^5 d^2 x^2+40755 c^4 d^3 x^3+53295 c^3 d^4 x^4+42636 c^2 d^5 x^5+19380 c d^6 x^6+3876 d^7 x^7\right )+5 a^2 b^8 d^2 \left (4862 c^8+45760 c^7 d x+190190 c^6 d^2 x^2+456456 c^5 d^3 x^3+692835 c^4 d^4 x^4+682176 c^3 d^5 x^5+426360 c^2 d^6 x^6+155040 c d^7 x^7+25194 d^8 x^8\right )+10 a b^9 d \left (4862 c^9+48620 c^8 d x+217360 c^7 d^2 x^2+570570 c^6 d^3 x^3+969969 c^5 d^4 x^4+1108536 c^4 d^5 x^5+852720 c^3 d^6 x^6+426360 c^2 d^7 x^7+125970 c d^8 x^8+16796 d^9 x^9\right )+b^{10} \left (92378 c^{10}+972400 c^9 d x+4618900 c^8 d^2 x^2+13041600 c^7 d^3 x^3+24249225 c^6 d^4 x^4+31039008 c^5 d^5 x^5+27713400 c^4 d^6 x^6+17054400 c^3 d^7 x^7+6928350 c^2 d^8 x^8+1679600 c d^9 x^9+184756 d^{10} x^{10}\right )}{1847560 b^{11} (a+b x)^{20}} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[(c + d*x)^10/(a + b*x)^21,x]

[Out]

-(a^10*d^10 + 10*a^9*b*d^9*(c + 2*d*x) + 5*a^8*b^2*d^8*(11*c^2 + 40*c*d*x + 38*d

^2*x^2) + 20*a^7*b^3*d^7*(11*c^3 + 55*c^2*d*x + 95*c*d^2*x^2 + 57*d^3*x^3) + 5*a

^6*b^4*d^6*(143*c^4 + 880*c^3*d*x + 2090*c^2*d^2*x^2 + 2280*c*d^3*x^3 + 969*d^4*

x^4) + 2*a^5*b^5*d^5*(1001*c^5 + 7150*c^4*d*x + 20900*c^3*d^2*x^2 + 31350*c^2*d^

3*x^3 + 24225*c*d^4*x^4 + 7752*d^5*x^5) + 5*a^4*b^6*d^4*(1001*c^6 + 8008*c^5*d*x

+ 27170*c^4*d^2*x^2 + 50160*c^3*d^3*x^3 + 53295*c^2*d^4*x^4 + 31008*c*d^5*x^5 +

7752*d^6*x^6) + 20*a^3*b^7*d^3*(572*c^7 + 5005*c^6*d*x + 19019*c^5*d^2*x^2 + 40

755*c^4*d^3*x^3 + 53295*c^3*d^4*x^4 + 42636*c^2*d^5*x^5 + 19380*c*d^6*x^6 + 3876

*d^7*x^7) + 5*a^2*b^8*d^2*(4862*c^8 + 45760*c^7*d*x + 190190*c^6*d^2*x^2 + 45645

6*c^5*d^3*x^3 + 692835*c^4*d^4*x^4 + 682176*c^3*d^5*x^5 + 426360*c^2*d^6*x^6 + 1

55040*c*d^7*x^7 + 25194*d^8*x^8) + 10*a*b^9*d*(4862*c^9 + 48620*c^8*d*x + 217360

*c^7*d^2*x^2 + 570570*c^6*d^3*x^3 + 969969*c^5*d^4*x^4 + 1108536*c^4*d^5*x^5 + 8

52720*c^3*d^6*x^6 + 426360*c^2*d^7*x^7 + 125970*c*d^8*x^8 + 16796*d^9*x^9) + b^1

0*(92378*c^10 + 972400*c^9*d*x + 4618900*c^8*d^2*x^2 + 13041600*c^7*d^3*x^3 + 24

249225*c^6*d^4*x^4 + 31039008*c^5*d^5*x^5 + 27713400*c^4*d^6*x^6 + 17054400*c^3*

d^7*x^7 + 6928350*c^2*d^8*x^8 + 1679600*c*d^9*x^9 + 184756*d^10*x^10))/(1847560*

b^11*(a + b*x)^20)

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Maple [B] time = 0.014, size = 867, normalized size = 3.1 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int((d*x+c)^10/(b*x+a)^21,x)

[Out]

120/17*d^3*(a^7*d^7-7*a^6*b*c*d^6+21*a^5*b^2*c^2*d^5-35*a^4*b^3*c^3*d^4+35*a^3*b

^4*c^4*d^3-21*a^2*b^5*c^5*d^2+7*a*b^6*c^6*d-b^7*c^7)/b^11/(b*x+a)^17+10/19*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^5-12

6*a^4*b^5*c^5*d^4+84*a^3*b^6*c^6*d^3-36*a^2*b^7*c^7*d^2+9*a*b^8*c^8*d-b^9*c^9)/b

^11/(b*x+a)^19-15*d^6*(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4

*c^4)/b^11/(b*x+a)^14-1/20*(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)^20-5/2*d^2

*(a^8*d^8-8*a^7*b*c*d^7+28*a^6*b^2*c^2*d^6-56*a^5*b^3*c^3*d^5+70*a^4*b^4*c^4*d^4

-56*a^3*b^5*c^5*d^3+28*a^2*b^6*c^6*d^2-8*a*b^7*c^7*d+b^8*c^8)/b^11/(b*x+a)^18-15

/4*d^8*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^11/(b*x+a)^12-105/8*d^4*(a^6*d^6-6*a^5*b*c*

d^5+15*a^4*b^2*c^2*d^4-20*a^3*b^3*c^3*d^3+15*a^2*b^4*c^4*d^2-6*a*b^5*c^5*d+b^6*c

^6)/b^11/(b*x+a)^16+84/5*d^5*(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c^2*d^3-10*a^2*b^

3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)/b^11/(b*x+a)^15-1/10*d^10/b^11/(b*x+a)^10+10/11

*d^9*(a*d-b*c)/b^11/(b*x+a)^11+120/13*d^7*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b

^3*c^3)/b^11/(b*x+a)^13

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Maxima [A] time = 1.49415, size = 1450, normalized size = 5.2 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((d*x + c)^10/(b*x + a)^21,x, algorithm="maxima")

[Out]

-1/1847560*(184756*b^10*d^10*x^10 + 92378*b^10*c^10 + 48620*a*b^9*c^9*d + 24310*

a^2*b^8*c^8*d^2 + 11440*a^3*b^7*c^7*d^3 + 5005*a^4*b^6*c^6*d^4 + 2002*a^5*b^5*c^

5*d^5 + 715*a^6*b^4*c^4*d^6 + 220*a^7*b^3*c^3*d^7 + 55*a^8*b^2*c^2*d^8 + 10*a^9*

b*c*d^9 + a^10*d^10 + 167960*(10*b^10*c*d^9 + a*b^9*d^10)*x^9 + 125970*(55*b^10*

c^2*d^8 + 10*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 77520*(220*b^10*c^3*d^7 + 55*a*b^

9*c^2*d^8 + 10*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 38760*(715*b^10*c^4*d^6 + 220

*a*b^9*c^3*d^7 + 55*a^2*b^8*c^2*d^8 + 10*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 155

04*(2002*b^10*c^5*d^5 + 715*a*b^9*c^4*d^6 + 220*a^2*b^8*c^3*d^7 + 55*a^3*b^7*c^2

*d^8 + 10*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 4845*(5005*b^10*c^6*d^4 + 2002*a*b

^9*c^5*d^5 + 715*a^2*b^8*c^4*d^6 + 220*a^3*b^7*c^3*d^7 + 55*a^4*b^6*c^2*d^8 + 10

*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 1140*(11440*b^10*c^7*d^3 + 5005*a*b^9*c^6*d

^4 + 2002*a^2*b^8*c^5*d^5 + 715*a^3*b^7*c^4*d^6 + 220*a^4*b^6*c^3*d^7 + 55*a^5*b

^5*c^2*d^8 + 10*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 190*(24310*b^10*c^8*d^2 + 11

440*a*b^9*c^7*d^3 + 5005*a^2*b^8*c^6*d^4 + 2002*a^3*b^7*c^5*d^5 + 715*a^4*b^6*c^

4*d^6 + 220*a^5*b^5*c^3*d^7 + 55*a^6*b^4*c^2*d^8 + 10*a^7*b^3*c*d^9 + a^8*b^2*d^

10)*x^2 + 20*(48620*b^10*c^9*d + 24310*a*b^9*c^8*d^2 + 11440*a^2*b^8*c^7*d^3 + 5

005*a^3*b^7*c^6*d^4 + 2002*a^4*b^6*c^5*d^5 + 715*a^5*b^5*c^4*d^6 + 220*a^6*b^4*c

^3*d^7 + 55*a^7*b^3*c^2*d^8 + 10*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^31*x^20 + 20*

a*b^30*x^19 + 190*a^2*b^29*x^18 + 1140*a^3*b^28*x^17 + 4845*a^4*b^27*x^16 + 1550

4*a^5*b^26*x^15 + 38760*a^6*b^25*x^14 + 77520*a^7*b^24*x^13 + 125970*a^8*b^23*x^

12 + 167960*a^9*b^22*x^11 + 184756*a^10*b^21*x^10 + 167960*a^11*b^20*x^9 + 12597

0*a^12*b^19*x^8 + 77520*a^13*b^18*x^7 + 38760*a^14*b^17*x^6 + 15504*a^15*b^16*x^

5 + 4845*a^16*b^15*x^4 + 1140*a^17*b^14*x^3 + 190*a^18*b^13*x^2 + 20*a^19*b^12*x

+ a^20*b^11)

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Fricas [A] time = 0.211397, size = 1450, normalized size = 5.2 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((d*x + c)^10/(b*x + a)^21,x, algorithm="fricas")