∖int(c+dx)10 _____________

3.1333 \(\int \frac{(c+d x)^{10}}{(a+b x)^{22}} \, dx\)

Optimal. Leaf size=279 \[ -\frac{5 d^9 (b c-a d)}{6 b^{11} (a+b x)^{12}}-\frac{45 d^8 (b c-a d)^2}{13 b^{11} (a+b x)^{13}}-\frac{60 d^7 (b c-a d)^3}{7 b^{11} (a+b x)^{14}}-\frac{14 d^6 (b c-a d)^4}{b^{11} (a+b x)^{15}}-\frac{63 d^5 (b c-a d)^5}{4 b^{11} (a+b x)^{16}}-\frac{210 d^4 (b c-a d)^6}{17 b^{11} (a+b x)^{17}}-\frac{20 d^3 (b c-a d)^7}{3 b^{11} (a+b x)^{18}}-\frac{45 d^2 (b c-a d)^8}{19 b^{11} (a+b x)^{19}}-\frac{d (b c-a d)^9}{2 b^{11} (a+b x)^{20}}-\frac{(b c-a d)^{10}}{21 b^{11} (a+b x)^{21}}-\frac{d^{10}}{11 b^{11} (a+b x)^{11}} \]

[Out]

-(b*c - a*d)^10/(21*b^11*(a + b*x)^21) - (d*(b*c - a*d)^9)/(2*b^11*(a + b*x)^20)

- (45*d^2*(b*c - a*d)^8)/(19*b^11*(a + b*x)^19) - (20*d^3*(b*c - a*d)^7)/(3*b^1

1*(a + b*x)^18) - (210*d^4*(b*c - a*d)^6)/(17*b^11*(a + b*x)^17) - (63*d^5*(b*c

- a*d)^5)/(4*b^11*(a + b*x)^16) - (14*d^6*(b*c - a*d)^4)/(b^11*(a + b*x)^15) - (

60*d^7*(b*c - a*d)^3)/(7*b^11*(a + b*x)^14) - (45*d^8*(b*c - a*d)^2)/(13*b^11*(a

+ b*x)^13) - (5*d^9*(b*c - a*d))/(6*b^11*(a + b*x)^12) - d^10/(11*b^11*(a + b*x

)^11)

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Rubi [A] time = 0.814453, 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 \[ -\frac{5 d^9 (b c-a d)}{6 b^{11} (a+b x)^{12}}-\frac{45 d^8 (b c-a d)^2}{13 b^{11} (a+b x)^{13}}-\frac{60 d^7 (b c-a d)^3}{7 b^{11} (a+b x)^{14}}-\frac{14 d^6 (b c-a d)^4}{b^{11} (a+b x)^{15}}-\frac{63 d^5 (b c-a d)^5}{4 b^{11} (a+b x)^{16}}-\frac{210 d^4 (b c-a d)^6}{17 b^{11} (a+b x)^{17}}-\frac{20 d^3 (b c-a d)^7}{3 b^{11} (a+b x)^{18}}-\frac{45 d^2 (b c-a d)^8}{19 b^{11} (a+b x)^{19}}-\frac{d (b c-a d)^9}{2 b^{11} (a+b x)^{20}}-\frac{(b c-a d)^{10}}{21 b^{11} (a+b x)^{21}}-\frac{d^{10}}{11 b^{11} (a+b x)^{11}} \]

Antiderivative was successfully veriﬁed.

[In] Int[(c + d*x)^10/(a + b*x)^22,x]

[Out]

-(b*c - a*d)^10/(21*b^11*(a + b*x)^21) - (d*(b*c - a*d)^9)/(2*b^11*(a + b*x)^20)

- (45*d^2*(b*c - a*d)^8)/(19*b^11*(a + b*x)^19) - (20*d^3*(b*c - a*d)^7)/(3*b^1

1*(a + b*x)^18) - (210*d^4*(b*c - a*d)^6)/(17*b^11*(a + b*x)^17) - (63*d^5*(b*c

- a*d)^5)/(4*b^11*(a + b*x)^16) - (14*d^6*(b*c - a*d)^4)/(b^11*(a + b*x)^15) - (

60*d^7*(b*c - a*d)^3)/(7*b^11*(a + b*x)^14) - (45*d^8*(b*c - a*d)^2)/(13*b^11*(a

+ b*x)^13) - (5*d^9*(b*c - a*d))/(6*b^11*(a + b*x)^12) - d^10/(11*b^11*(a + b*x

)^11)

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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)**22,x)

[Out]

Timed out

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Mathematica [B] time = 0.792893, size = 692, normalized size = 2.48 \[ -\frac{a^{10} d^{10}+a^9 b d^9 (11 c+21 d x)+3 a^8 b^2 d^8 \left (22 c^2+77 c d x+70 d^2 x^2\right )+2 a^7 b^3 d^7 \left (143 c^3+693 c^2 d x+1155 c d^2 x^2+665 d^3 x^3\right )+7 a^6 b^4 d^6 \left (143 c^4+858 c^3 d x+1980 c^2 d^2 x^2+2090 c d^3 x^3+855 d^4 x^4\right )+21 a^5 b^5 d^5 \left (143 c^5+1001 c^4 d x+2860 c^3 d^2 x^2+4180 c^2 d^3 x^3+3135 c d^4 x^4+969 d^5 x^5\right )+7 a^4 b^6 d^4 \left (1144 c^6+9009 c^5 d x+30030 c^4 d^2 x^2+54340 c^3 d^3 x^3+56430 c^2 d^4 x^4+31977 c d^5 x^5+7752 d^6 x^6\right )+2 a^3 b^7 d^3 \left (9724 c^7+84084 c^6 d x+315315 c^5 d^2 x^2+665665 c^4 d^3 x^3+855855 c^3 d^4 x^4+671517 c^2 d^5 x^5+298452 c d^6 x^6+58140 d^7 x^7\right )+3 a^2 b^8 d^2 \left (14586 c^8+136136 c^7 d x+560560 c^6 d^2 x^2+1331330 c^5 d^3 x^3+1996995 c^4 d^4 x^4+1939938 c^3 d^5 x^5+1193808 c^2 d^6 x^6+426360 c d^7 x^7+67830 d^8 x^8\right )+a b^9 d \left (92378 c^9+918918 c^8 d x+4084080 c^7 d^2 x^2+10650640 c^6 d^3 x^3+17972955 c^5 d^4 x^4+20369349 c^4 d^5 x^5+15519504 c^3 d^6 x^6+7674480 c^2 d^7 x^7+2238390 c d^8 x^8+293930 d^9 x^9\right )+b^{10} \left (184756 c^{10}+1939938 c^9 d x+9189180 c^8 d^2 x^2+25865840 c^7 d^3 x^3+47927880 c^6 d^4 x^4+61108047 c^5 d^5 x^5+54318264 c^4 d^6 x^6+33256080 c^3 d^7 x^7+13430340 c^2 d^8 x^8+3233230 c d^9 x^9+352716 d^{10} x^{10}\right )}{3879876 b^{11} (a+b x)^{21}} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[(c + d*x)^10/(a + b*x)^22,x]

[Out]

-(a^10*d^10 + a^9*b*d^9*(11*c + 21*d*x) + 3*a^8*b^2*d^8*(22*c^2 + 77*c*d*x + 70*

d^2*x^2) + 2*a^7*b^3*d^7*(143*c^3 + 693*c^2*d*x + 1155*c*d^2*x^2 + 665*d^3*x^3)

+ 7*a^6*b^4*d^6*(143*c^4 + 858*c^3*d*x + 1980*c^2*d^2*x^2 + 2090*c*d^3*x^3 + 855

*d^4*x^4) + 21*a^5*b^5*d^5*(143*c^5 + 1001*c^4*d*x + 2860*c^3*d^2*x^2 + 4180*c^2

*d^3*x^3 + 3135*c*d^4*x^4 + 969*d^5*x^5) + 7*a^4*b^6*d^4*(1144*c^6 + 9009*c^5*d*

x + 30030*c^4*d^2*x^2 + 54340*c^3*d^3*x^3 + 56430*c^2*d^4*x^4 + 31977*c*d^5*x^5

+ 7752*d^6*x^6) + 2*a^3*b^7*d^3*(9724*c^7 + 84084*c^6*d*x + 315315*c^5*d^2*x^2 +

665665*c^4*d^3*x^3 + 855855*c^3*d^4*x^4 + 671517*c^2*d^5*x^5 + 298452*c*d^6*x^6

+ 58140*d^7*x^7) + 3*a^2*b^8*d^2*(14586*c^8 + 136136*c^7*d*x + 560560*c^6*d^2*x

^2 + 1331330*c^5*d^3*x^3 + 1996995*c^4*d^4*x^4 + 1939938*c^3*d^5*x^5 + 1193808*c

^2*d^6*x^6 + 426360*c*d^7*x^7 + 67830*d^8*x^8) + a*b^9*d*(92378*c^9 + 918918*c^8

*d*x + 4084080*c^7*d^2*x^2 + 10650640*c^6*d^3*x^3 + 17972955*c^5*d^4*x^4 + 20369

349*c^4*d^5*x^5 + 15519504*c^3*d^6*x^6 + 7674480*c^2*d^7*x^7 + 2238390*c*d^8*x^8

+ 293930*d^9*x^9) + b^10*(184756*c^10 + 1939938*c^9*d*x + 9189180*c^8*d^2*x^2 +

25865840*c^7*d^3*x^3 + 47927880*c^6*d^4*x^4 + 61108047*c^5*d^5*x^5 + 54318264*c

^4*d^6*x^6 + 33256080*c^3*d^7*x^7 + 13430340*c^2*d^8*x^8 + 3233230*c*d^9*x^9 + 3

52716*d^10*x^10))/(3879876*b^11*(a + b*x)^21)

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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)^22,x)

[Out]

-210/17*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)^17-45/19*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)^19+60/7*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)^14+1/2*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)/b^11/(b*x+a)^20+20/3*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)^18+5/6*d^9*(a*d-b*c)/b^

11/(b*x+a)^12+63/4*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)^16-1/21*(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)^21-14*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)^15-1/11*d^10/b^11/(b*x+a)^11-45/13*d^8*(a^2*d^2-2*a*b*c*d+b^2*

c^2)/b^11/(b*x+a)^13

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Maxima [A] time = 1.53213, size = 1465, normalized size = 5.25 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((d*x + c)^10/(b*x + a)^22,x, algorithm="maxima")

[Out]

-1/3879876*(352716*b^10*d^10*x^10 + 184756*b^10*c^10 + 92378*a*b^9*c^9*d + 43758

*a^2*b^8*c^8*d^2 + 19448*a^3*b^7*c^7*d^3 + 8008*a^4*b^6*c^6*d^4 + 3003*a^5*b^5*c

^5*d^5 + 1001*a^6*b^4*c^4*d^6 + 286*a^7*b^3*c^3*d^7 + 66*a^8*b^2*c^2*d^8 + 11*a^

9*b*c*d^9 + a^10*d^10 + 293930*(11*b^10*c*d^9 + a*b^9*d^10)*x^9 + 203490*(66*b^1

0*c^2*d^8 + 11*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 116280*(286*b^10*c^3*d^7 + 66*a

*b^9*c^2*d^8 + 11*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 54264*(1001*b^10*c^4*d^6 +

286*a*b^9*c^3*d^7 + 66*a^2*b^8*c^2*d^8 + 11*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 +

20349*(3003*b^10*c^5*d^5 + 1001*a*b^9*c^4*d^6 + 286*a^2*b^8*c^3*d^7 + 66*a^3*b^

7*c^2*d^8 + 11*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 5985*(8008*b^10*c^6*d^4 + 300

3*a*b^9*c^5*d^5 + 1001*a^2*b^8*c^4*d^6 + 286*a^3*b^7*c^3*d^7 + 66*a^4*b^6*c^2*d^

8 + 11*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 1330*(19448*b^10*c^7*d^3 + 8008*a*b^9

*c^6*d^4 + 3003*a^2*b^8*c^5*d^5 + 1001*a^3*b^7*c^4*d^6 + 286*a^4*b^6*c^3*d^7 + 6

6*a^5*b^5*c^2*d^8 + 11*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 210*(43758*b^10*c^8*d

^2 + 19448*a*b^9*c^7*d^3 + 8008*a^2*b^8*c^6*d^4 + 3003*a^3*b^7*c^5*d^5 + 1001*a^

4*b^6*c^4*d^6 + 286*a^5*b^5*c^3*d^7 + 66*a^6*b^4*c^2*d^8 + 11*a^7*b^3*c*d^9 + a^

8*b^2*d^10)*x^2 + 21*(92378*b^10*c^9*d + 43758*a*b^9*c^8*d^2 + 19448*a^2*b^8*c^7

*d^3 + 8008*a^3*b^7*c^6*d^4 + 3003*a^4*b^6*c^5*d^5 + 1001*a^5*b^5*c^4*d^6 + 286*

a^6*b^4*c^3*d^7 + 66*a^7*b^3*c^2*d^8 + 11*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^32*x

^21 + 21*a*b^31*x^20 + 210*a^2*b^30*x^19 + 1330*a^3*b^29*x^18 + 5985*a^4*b^28*x^

17 + 20349*a^5*b^27*x^16 + 54264*a^6*b^26*x^15 + 116280*a^7*b^25*x^14 + 203490*a

^8*b^24*x^13 + 293930*a^9*b^23*x^12 + 352716*a^10*b^22*x^11 + 352716*a^11*b^21*x

^10 + 293930*a^12*b^20*x^9 + 203490*a^13*b^19*x^8 + 116280*a^14*b^18*x^7 + 54264

*a^15*b^17*x^6 + 20349*a^16*b^16*x^5 + 5985*a^17*b^15*x^4 + 1330*a^18*b^14*x^3 +

210*a^19*b^13*x^2 + 21*a^20*b^12*x + a^21*b^11)

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Fricas [A] time = 0.203545, size = 1465, normalized size = 5.25 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((d*x + c)^10/(b*x + a)^22,x, algorithm="fricas")