3.1328 \(\int \frac{(c+d x)^{10}}{(a+b x)^{17}} \, dx\)

Optimal. Leaf size=182 \[ \frac{d^5 (c+d x)^{11}}{48048 (a+b x)^{11} (b c-a d)^6}-\frac{d^4 (c+d x)^{11}}{4368 (a+b x)^{12} (b c-a d)^5}+\frac{d^3 (c+d x)^{11}}{728 (a+b x)^{13} (b c-a d)^4}-\frac{d^2 (c+d x)^{11}}{168 (a+b x)^{14} (b c-a d)^3}+\frac{d (c+d x)^{11}}{48 (a+b x)^{15} (b c-a d)^2}-\frac{(c+d x)^{11}}{16 (a+b x)^{16} (b c-a d)} \]

[Out]

-(c + d*x)^11/(16*(b*c - a*d)*(a + b*x)^16) + (d*(c + d*x)^11)/(48*(b*c - a*d)^2
*(a + b*x)^15) - (d^2*(c + d*x)^11)/(168*(b*c - a*d)^3*(a + b*x)^14) + (d^3*(c +
d*x)^11)/(728*(b*c - a*d)^4*(a + b*x)^13) - (d^4*(c + d*x)^11)/(4368*(b*c - a*d
)^5*(a + b*x)^12) + (d^5*(c + d*x)^11)/(48048*(b*c - a*d)^6*(a + b*x)^11)

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Rubi [A]  time = 0.166074, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{d^5 (c+d x)^{11}}{48048 (a+b x)^{11} (b c-a d)^6}-\frac{d^4 (c+d x)^{11}}{4368 (a+b x)^{12} (b c-a d)^5}+\frac{d^3 (c+d x)^{11}}{728 (a+b x)^{13} (b c-a d)^4}-\frac{d^2 (c+d x)^{11}}{168 (a+b x)^{14} (b c-a d)^3}+\frac{d (c+d x)^{11}}{48 (a+b x)^{15} (b c-a d)^2}-\frac{(c+d x)^{11}}{16 (a+b x)^{16} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^10/(a + b*x)^17,x]

[Out]

-(c + d*x)^11/(16*(b*c - a*d)*(a + b*x)^16) + (d*(c + d*x)^11)/(48*(b*c - a*d)^2
*(a + b*x)^15) - (d^2*(c + d*x)^11)/(168*(b*c - a*d)^3*(a + b*x)^14) + (d^3*(c +
d*x)^11)/(728*(b*c - a*d)^4*(a + b*x)^13) - (d^4*(c + d*x)^11)/(4368*(b*c - a*d
)^5*(a + b*x)^12) + (d^5*(c + d*x)^11)/(48048*(b*c - a*d)^6*(a + b*x)^11)

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Rubi in Sympy [A]  time = 44.793, size = 155, normalized size = 0.85 \[ \frac{d^{5} \left (c + d x\right )^{11}}{48048 \left (a + b x\right )^{11} \left (a d - b c\right )^{6}} + \frac{d^{4} \left (c + d x\right )^{11}}{4368 \left (a + b x\right )^{12} \left (a d - b c\right )^{5}} + \frac{d^{3} \left (c + d x\right )^{11}}{728 \left (a + b x\right )^{13} \left (a d - b c\right )^{4}} + \frac{d^{2} \left (c + d x\right )^{11}}{168 \left (a + b x\right )^{14} \left (a d - b c\right )^{3}} + \frac{d \left (c + d x\right )^{11}}{48 \left (a + b x\right )^{15} \left (a d - b c\right )^{2}} + \frac{\left (c + d x\right )^{11}}{16 \left (a + b x\right )^{16} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**10/(b*x+a)**17,x)

[Out]

d**5*(c + d*x)**11/(48048*(a + b*x)**11*(a*d - b*c)**6) + d**4*(c + d*x)**11/(43
68*(a + b*x)**12*(a*d - b*c)**5) + d**3*(c + d*x)**11/(728*(a + b*x)**13*(a*d -
b*c)**4) + d**2*(c + d*x)**11/(168*(a + b*x)**14*(a*d - b*c)**3) + d*(c + d*x)**
11/(48*(a + b*x)**15*(a*d - b*c)**2) + (c + d*x)**11/(16*(a + b*x)**16*(a*d - b*
c))

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Mathematica [B]  time = 0.778752, size = 694, normalized size = 3.81 \[ -\frac{a^{10} d^{10}+2 a^9 b d^9 (3 c+8 d x)+3 a^8 b^2 d^8 \left (7 c^2+32 c d x+40 d^2 x^2\right )+8 a^7 b^3 d^7 \left (7 c^3+42 c^2 d x+90 c d^2 x^2+70 d^3 x^3\right )+14 a^6 b^4 d^6 \left (9 c^4+64 c^3 d x+180 c^2 d^2 x^2+240 c d^3 x^3+130 d^4 x^4\right )+84 a^5 b^5 d^5 \left (3 c^5+24 c^4 d x+80 c^3 d^2 x^2+140 c^2 d^3 x^3+130 c d^4 x^4+52 d^5 x^5\right )+14 a^4 b^6 d^4 \left (33 c^6+288 c^5 d x+1080 c^4 d^2 x^2+2240 c^3 d^3 x^3+2730 c^2 d^4 x^4+1872 c d^5 x^5+572 d^6 x^6\right )+8 a^3 b^7 d^3 \left (99 c^7+924 c^6 d x+3780 c^5 d^2 x^2+8820 c^4 d^3 x^3+12740 c^3 d^4 x^4+11466 c^2 d^5 x^5+6006 c d^6 x^6+1430 d^7 x^7\right )+3 a^2 b^8 d^2 \left (429 c^8+4224 c^7 d x+18480 c^6 d^2 x^2+47040 c^5 d^3 x^3+76440 c^4 d^4 x^4+81536 c^3 d^5 x^5+56056 c^2 d^6 x^6+22880 c d^7 x^7+4290 d^8 x^8\right )+2 a b^9 d \left (1001 c^9+10296 c^8 d x+47520 c^7 d^2 x^2+129360 c^6 d^3 x^3+229320 c^5 d^4 x^4+275184 c^4 d^5 x^5+224224 c^3 d^6 x^6+120120 c^2 d^7 x^7+38610 c d^8 x^8+5720 d^9 x^9\right )+b^{10} \left (3003 c^{10}+32032 c^9 d x+154440 c^8 d^2 x^2+443520 c^7 d^3 x^3+840840 c^6 d^4 x^4+1100736 c^5 d^5 x^5+1009008 c^4 d^6 x^6+640640 c^3 d^7 x^7+270270 c^2 d^8 x^8+68640 c d^9 x^9+8008 d^{10} x^{10}\right )}{48048 b^{11} (a+b x)^{16}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^10/(a + b*x)^17,x]

[Out]

-(a^10*d^10 + 2*a^9*b*d^9*(3*c + 8*d*x) + 3*a^8*b^2*d^8*(7*c^2 + 32*c*d*x + 40*d
^2*x^2) + 8*a^7*b^3*d^7*(7*c^3 + 42*c^2*d*x + 90*c*d^2*x^2 + 70*d^3*x^3) + 14*a^
6*b^4*d^6*(9*c^4 + 64*c^3*d*x + 180*c^2*d^2*x^2 + 240*c*d^3*x^3 + 130*d^4*x^4) +
84*a^5*b^5*d^5*(3*c^5 + 24*c^4*d*x + 80*c^3*d^2*x^2 + 140*c^2*d^3*x^3 + 130*c*d
^4*x^4 + 52*d^5*x^5) + 14*a^4*b^6*d^4*(33*c^6 + 288*c^5*d*x + 1080*c^4*d^2*x^2 +
2240*c^3*d^3*x^3 + 2730*c^2*d^4*x^4 + 1872*c*d^5*x^5 + 572*d^6*x^6) + 8*a^3*b^7
*d^3*(99*c^7 + 924*c^6*d*x + 3780*c^5*d^2*x^2 + 8820*c^4*d^3*x^3 + 12740*c^3*d^4
*x^4 + 11466*c^2*d^5*x^5 + 6006*c*d^6*x^6 + 1430*d^7*x^7) + 3*a^2*b^8*d^2*(429*c
^8 + 4224*c^7*d*x + 18480*c^6*d^2*x^2 + 47040*c^5*d^3*x^3 + 76440*c^4*d^4*x^4 +
81536*c^3*d^5*x^5 + 56056*c^2*d^6*x^6 + 22880*c*d^7*x^7 + 4290*d^8*x^8) + 2*a*b^
9*d*(1001*c^9 + 10296*c^8*d*x + 47520*c^7*d^2*x^2 + 129360*c^6*d^3*x^3 + 229320*
c^5*d^4*x^4 + 275184*c^4*d^5*x^5 + 224224*c^3*d^6*x^6 + 120120*c^2*d^7*x^7 + 386
10*c*d^8*x^8 + 5720*d^9*x^9) + b^10*(3003*c^10 + 32032*c^9*d*x + 154440*c^8*d^2*
x^2 + 443520*c^7*d^3*x^3 + 840840*c^6*d^4*x^4 + 1100736*c^5*d^5*x^5 + 1009008*c^
4*d^6*x^6 + 640640*c^3*d^7*x^7 + 270270*c^2*d^8*x^8 + 68640*c*d^9*x^9 + 8008*d^1
0*x^10))/(48048*b^11*(a + b*x)^16)

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Maple [B]  time = 0.015, size = 867, normalized size = 4.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^10/(b*x+a)^17,x)

[Out]

-45/14*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)^14-1/6*d^10/b^11/(b*x+a)^6-35/2*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)^12+
10/7*d^9*(a*d-b*c)/b^11/(b*x+a)^7-1/16*(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
)^16+2/3*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)^15-45/8*d^8*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^11/(b*x+a)^8
-21*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)^10+252/11*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)^11+120/13*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)^13+40/3*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)^9

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Maxima [A]  time = 1.4652, size = 1391, normalized size = 7.64 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^17,x, algorithm="maxima")

[Out]

-1/48048*(8008*b^10*d^10*x^10 + 3003*b^10*c^10 + 2002*a*b^9*c^9*d + 1287*a^2*b^8
*c^8*d^2 + 792*a^3*b^7*c^7*d^3 + 462*a^4*b^6*c^6*d^4 + 252*a^5*b^5*c^5*d^5 + 126
*a^6*b^4*c^4*d^6 + 56*a^7*b^3*c^3*d^7 + 21*a^8*b^2*c^2*d^8 + 6*a^9*b*c*d^9 + a^1
0*d^10 + 11440*(6*b^10*c*d^9 + a*b^9*d^10)*x^9 + 12870*(21*b^10*c^2*d^8 + 6*a*b^
9*c*d^9 + a^2*b^8*d^10)*x^8 + 11440*(56*b^10*c^3*d^7 + 21*a*b^9*c^2*d^8 + 6*a^2*
b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 8008*(126*b^10*c^4*d^6 + 56*a*b^9*c^3*d^7 + 21*a
^2*b^8*c^2*d^8 + 6*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 4368*(252*b^10*c^5*d^5 +
126*a*b^9*c^4*d^6 + 56*a^2*b^8*c^3*d^7 + 21*a^3*b^7*c^2*d^8 + 6*a^4*b^6*c*d^9 +
a^5*b^5*d^10)*x^5 + 1820*(462*b^10*c^6*d^4 + 252*a*b^9*c^5*d^5 + 126*a^2*b^8*c^4
*d^6 + 56*a^3*b^7*c^3*d^7 + 21*a^4*b^6*c^2*d^8 + 6*a^5*b^5*c*d^9 + a^6*b^4*d^10)
*x^4 + 560*(792*b^10*c^7*d^3 + 462*a*b^9*c^6*d^4 + 252*a^2*b^8*c^5*d^5 + 126*a^3
*b^7*c^4*d^6 + 56*a^4*b^6*c^3*d^7 + 21*a^5*b^5*c^2*d^8 + 6*a^6*b^4*c*d^9 + a^7*b
^3*d^10)*x^3 + 120*(1287*b^10*c^8*d^2 + 792*a*b^9*c^7*d^3 + 462*a^2*b^8*c^6*d^4
+ 252*a^3*b^7*c^5*d^5 + 126*a^4*b^6*c^4*d^6 + 56*a^5*b^5*c^3*d^7 + 21*a^6*b^4*c^
2*d^8 + 6*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 16*(2002*b^10*c^9*d + 1287*a*b^9*c
^8*d^2 + 792*a^2*b^8*c^7*d^3 + 462*a^3*b^7*c^6*d^4 + 252*a^4*b^6*c^5*d^5 + 126*a
^5*b^5*c^4*d^6 + 56*a^6*b^4*c^3*d^7 + 21*a^7*b^3*c^2*d^8 + 6*a^8*b^2*c*d^9 + a^9
*b*d^10)*x)/(b^27*x^16 + 16*a*b^26*x^15 + 120*a^2*b^25*x^14 + 560*a^3*b^24*x^13
+ 1820*a^4*b^23*x^12 + 4368*a^5*b^22*x^11 + 8008*a^6*b^21*x^10 + 11440*a^7*b^20*
x^9 + 12870*a^8*b^19*x^8 + 11440*a^9*b^18*x^7 + 8008*a^10*b^17*x^6 + 4368*a^11*b
^16*x^5 + 1820*a^12*b^15*x^4 + 560*a^13*b^14*x^3 + 120*a^14*b^13*x^2 + 16*a^15*b
^12*x + a^16*b^11)

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Fricas [A]  time = 0.214917, size = 1391, normalized size = 7.64 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^17,x, algorithm="fricas")

[Out]

-1/48048*(8008*b^10*d^10*x^10 + 3003*b^10*c^10 + 2002*a*b^9*c^9*d + 1287*a^2*b^8
*c^8*d^2 + 792*a^3*b^7*c^7*d^3 + 462*a^4*b^6*c^6*d^4 + 252*a^5*b^5*c^5*d^5 + 126
*a^6*b^4*c^4*d^6 + 56*a^7*b^3*c^3*d^7 + 21*a^8*b^2*c^2*d^8 + 6*a^9*b*c*d^9 + a^1
0*d^10 + 11440*(6*b^10*c*d^9 + a*b^9*d^10)*x^9 + 12870*(21*b^10*c^2*d^8 + 6*a*b^
9*c*d^9 + a^2*b^8*d^10)*x^8 + 11440*(56*b^10*c^3*d^7 + 21*a*b^9*c^2*d^8 + 6*a^2*
b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 8008*(126*b^10*c^4*d^6 + 56*a*b^9*c^3*d^7 + 21*a
^2*b^8*c^2*d^8 + 6*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 4368*(252*b^10*c^5*d^5 +
126*a*b^9*c^4*d^6 + 56*a^2*b^8*c^3*d^7 + 21*a^3*b^7*c^2*d^8 + 6*a^4*b^6*c*d^9 +
a^5*b^5*d^10)*x^5 + 1820*(462*b^10*c^6*d^4 + 252*a*b^9*c^5*d^5 + 126*a^2*b^8*c^4
*d^6 + 56*a^3*b^7*c^3*d^7 + 21*a^4*b^6*c^2*d^8 + 6*a^5*b^5*c*d^9 + a^6*b^4*d^10)
*x^4 + 560*(792*b^10*c^7*d^3 + 462*a*b^9*c^6*d^4 + 252*a^2*b^8*c^5*d^5 + 126*a^3
*b^7*c^4*d^6 + 56*a^4*b^6*c^3*d^7 + 21*a^5*b^5*c^2*d^8 + 6*a^6*b^4*c*d^9 + a^7*b
^3*d^10)*x^3 + 120*(1287*b^10*c^8*d^2 + 792*a*b^9*c^7*d^3 + 462*a^2*b^8*c^6*d^4
+ 252*a^3*b^7*c^5*d^5 + 126*a^4*b^6*c^4*d^6 + 56*a^5*b^5*c^3*d^7 + 21*a^6*b^4*c^
2*d^8 + 6*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 16*(2002*b^10*c^9*d + 1287*a*b^9*c
^8*d^2 + 792*a^2*b^8*c^7*d^3 + 462*a^3*b^7*c^6*d^4 + 252*a^4*b^6*c^5*d^5 + 126*a
^5*b^5*c^4*d^6 + 56*a^6*b^4*c^3*d^7 + 21*a^7*b^3*c^2*d^8 + 6*a^8*b^2*c*d^9 + a^9
*b*d^10)*x)/(b^27*x^16 + 16*a*b^26*x^15 + 120*a^2*b^25*x^14 + 560*a^3*b^24*x^13
+ 1820*a^4*b^23*x^12 + 4368*a^5*b^22*x^11 + 8008*a^6*b^21*x^10 + 11440*a^7*b^20*
x^9 + 12870*a^8*b^19*x^8 + 11440*a^9*b^18*x^7 + 8008*a^10*b^17*x^6 + 4368*a^11*b
^16*x^5 + 1820*a^12*b^15*x^4 + 560*a^13*b^14*x^3 + 120*a^14*b^13*x^2 + 16*a^15*b
^12*x + a^16*b^11)

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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)**17,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224133, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^17,x, algorithm="giac")

[Out]

Done