3.1327 \(\int \frac{(c+d x)^{10}}{(a+b x)^{16}} \, dx\)

Optimal. Leaf size=151 \[ -\frac{d^4 (c+d x)^{11}}{15015 (a+b x)^{11} (b c-a d)^5}+\frac{d^3 (c+d x)^{11}}{1365 (a+b x)^{12} (b c-a d)^4}-\frac{2 d^2 (c+d x)^{11}}{455 (a+b x)^{13} (b c-a d)^3}+\frac{2 d (c+d x)^{11}}{105 (a+b x)^{14} (b c-a d)^2}-\frac{(c+d x)^{11}}{15 (a+b x)^{15} (b c-a d)} \]

[Out]

-(c + d*x)^11/(15*(b*c - a*d)*(a + b*x)^15) + (2*d*(c + d*x)^11)/(105*(b*c - a*d
)^2*(a + b*x)^14) - (2*d^2*(c + d*x)^11)/(455*(b*c - a*d)^3*(a + b*x)^13) + (d^3
*(c + d*x)^11)/(1365*(b*c - a*d)^4*(a + b*x)^12) - (d^4*(c + d*x)^11)/(15015*(b*
c - a*d)^5*(a + b*x)^11)

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Rubi [A]  time = 0.124241, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{d^4 (c+d x)^{11}}{15015 (a+b x)^{11} (b c-a d)^5}+\frac{d^3 (c+d x)^{11}}{1365 (a+b x)^{12} (b c-a d)^4}-\frac{2 d^2 (c+d x)^{11}}{455 (a+b x)^{13} (b c-a d)^3}+\frac{2 d (c+d x)^{11}}{105 (a+b x)^{14} (b c-a d)^2}-\frac{(c+d x)^{11}}{15 (a+b x)^{15} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^10/(a + b*x)^16,x]

[Out]

-(c + d*x)^11/(15*(b*c - a*d)*(a + b*x)^15) + (2*d*(c + d*x)^11)/(105*(b*c - a*d
)^2*(a + b*x)^14) - (2*d^2*(c + d*x)^11)/(455*(b*c - a*d)^3*(a + b*x)^13) + (d^3
*(c + d*x)^11)/(1365*(b*c - a*d)^4*(a + b*x)^12) - (d^4*(c + d*x)^11)/(15015*(b*
c - a*d)^5*(a + b*x)^11)

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Rubi in Sympy [A]  time = 32.6443, size = 131, normalized size = 0.87 \[ \frac{d^{4} \left (c + d x\right )^{11}}{15015 \left (a + b x\right )^{11} \left (a d - b c\right )^{5}} + \frac{d^{3} \left (c + d x\right )^{11}}{1365 \left (a + b x\right )^{12} \left (a d - b c\right )^{4}} + \frac{2 d^{2} \left (c + d x\right )^{11}}{455 \left (a + b x\right )^{13} \left (a d - b c\right )^{3}} + \frac{2 d \left (c + d x\right )^{11}}{105 \left (a + b x\right )^{14} \left (a d - b c\right )^{2}} + \frac{\left (c + d x\right )^{11}}{15 \left (a + b x\right )^{15} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**10/(b*x+a)**16,x)

[Out]

d**4*(c + d*x)**11/(15015*(a + b*x)**11*(a*d - b*c)**5) + d**3*(c + d*x)**11/(13
65*(a + b*x)**12*(a*d - b*c)**4) + 2*d**2*(c + d*x)**11/(455*(a + b*x)**13*(a*d
- b*c)**3) + 2*d*(c + d*x)**11/(105*(a + b*x)**14*(a*d - b*c)**2) + (c + d*x)**1
1/(15*(a + b*x)**15*(a*d - b*c))

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Mathematica [B]  time = 0.613432, size = 690, normalized size = 4.57 \[ -\frac{a^{10} d^{10}+5 a^9 b d^9 (c+3 d x)+15 a^8 b^2 d^8 \left (c^2+5 c d x+7 d^2 x^2\right )+5 a^7 b^3 d^7 \left (7 c^3+45 c^2 d x+105 c d^2 x^2+91 d^3 x^3\right )+35 a^6 b^4 d^6 \left (2 c^4+15 c^3 d x+45 c^2 d^2 x^2+65 c d^3 x^3+39 d^4 x^4\right )+21 a^5 b^5 d^5 \left (6 c^5+50 c^4 d x+175 c^3 d^2 x^2+325 c^2 d^3 x^3+325 c d^4 x^4+143 d^5 x^5\right )+35 a^4 b^6 d^4 \left (6 c^6+54 c^5 d x+210 c^4 d^2 x^2+455 c^3 d^3 x^3+585 c^2 d^4 x^4+429 c d^5 x^5+143 d^6 x^6\right )+5 a^3 b^7 d^3 \left (66 c^7+630 c^6 d x+2646 c^5 d^2 x^2+6370 c^4 d^3 x^3+9555 c^3 d^4 x^4+9009 c^2 d^5 x^5+5005 c d^6 x^6+1287 d^7 x^7\right )+15 a^2 b^8 d^2 \left (33 c^8+330 c^7 d x+1470 c^6 d^2 x^2+3822 c^5 d^3 x^3+6370 c^4 d^4 x^4+7007 c^3 d^5 x^5+5005 c^2 d^6 x^6+2145 c d^7 x^7+429 d^8 x^8\right )+5 a b^9 d \left (143 c^9+1485 c^8 d x+6930 c^7 d^2 x^2+19110 c^6 d^3 x^3+34398 c^5 d^4 x^4+42042 c^4 d^5 x^5+35035 c^3 d^6 x^6+19305 c^2 d^7 x^7+6435 c d^8 x^8+1001 d^9 x^9\right )+b^{10} \left (1001 c^{10}+10725 c^9 d x+51975 c^8 d^2 x^2+150150 c^7 d^3 x^3+286650 c^6 d^4 x^4+378378 c^5 d^5 x^5+350350 c^4 d^6 x^6+225225 c^3 d^7 x^7+96525 c^2 d^8 x^8+25025 c d^9 x^9+3003 d^{10} x^{10}\right )}{15015 b^{11} (a+b x)^{15}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^10/(a + b*x)^16,x]

[Out]

-(a^10*d^10 + 5*a^9*b*d^9*(c + 3*d*x) + 15*a^8*b^2*d^8*(c^2 + 5*c*d*x + 7*d^2*x^
2) + 5*a^7*b^3*d^7*(7*c^3 + 45*c^2*d*x + 105*c*d^2*x^2 + 91*d^3*x^3) + 35*a^6*b^
4*d^6*(2*c^4 + 15*c^3*d*x + 45*c^2*d^2*x^2 + 65*c*d^3*x^3 + 39*d^4*x^4) + 21*a^5
*b^5*d^5*(6*c^5 + 50*c^4*d*x + 175*c^3*d^2*x^2 + 325*c^2*d^3*x^3 + 325*c*d^4*x^4
+ 143*d^5*x^5) + 35*a^4*b^6*d^4*(6*c^6 + 54*c^5*d*x + 210*c^4*d^2*x^2 + 455*c^3
*d^3*x^3 + 585*c^2*d^4*x^4 + 429*c*d^5*x^5 + 143*d^6*x^6) + 5*a^3*b^7*d^3*(66*c^
7 + 630*c^6*d*x + 2646*c^5*d^2*x^2 + 6370*c^4*d^3*x^3 + 9555*c^3*d^4*x^4 + 9009*
c^2*d^5*x^5 + 5005*c*d^6*x^6 + 1287*d^7*x^7) + 15*a^2*b^8*d^2*(33*c^8 + 330*c^7*
d*x + 1470*c^6*d^2*x^2 + 3822*c^5*d^3*x^3 + 6370*c^4*d^4*x^4 + 7007*c^3*d^5*x^5
+ 5005*c^2*d^6*x^6 + 2145*c*d^7*x^7 + 429*d^8*x^8) + 5*a*b^9*d*(143*c^9 + 1485*c
^8*d*x + 6930*c^7*d^2*x^2 + 19110*c^6*d^3*x^3 + 34398*c^5*d^4*x^4 + 42042*c^4*d^
5*x^5 + 35035*c^3*d^6*x^6 + 19305*c^2*d^7*x^7 + 6435*c*d^8*x^8 + 1001*d^9*x^9) +
b^10*(1001*c^10 + 10725*c^9*d*x + 51975*c^8*d^2*x^2 + 150150*c^7*d^3*x^3 + 2866
50*c^6*d^4*x^4 + 378378*c^5*d^5*x^5 + 350350*c^4*d^6*x^6 + 225225*c^3*d^7*x^7 +
96525*c^2*d^8*x^8 + 25025*c*d^9*x^9 + 3003*d^10*x^10))/(15015*b^11*(a + b*x)^15)

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Maple [B]  time = 0.014, size = 867, normalized size = 5.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^10/(b*x+a)^16,x)

[Out]

-1/5*d^10/b^11/(b*x+a)^5+5/7*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)^14+5/3*d^9*(a*d-b*c)/b^11/(b*x+a)^
6+10*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)^12-45/7*d^8*(a^2
*d^2-2*a*b*c*d+b^2*c^2)/b^11/(b*x+a)^7-1/15*(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)^15+15*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)^8+12
6/5*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)^10-210/11*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)^11-45
/13*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
)^13-70/3*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)^9

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Maxima [A]  time = 1.59941, size = 1376, normalized size = 9.11 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^16,x, algorithm="maxima")

[Out]

-1/15015*(3003*b^10*d^10*x^10 + 1001*b^10*c^10 + 715*a*b^9*c^9*d + 495*a^2*b^8*c
^8*d^2 + 330*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 70*a^
6*b^4*c^4*d^6 + 35*a^7*b^3*c^3*d^7 + 15*a^8*b^2*c^2*d^8 + 5*a^9*b*c*d^9 + a^10*d
^10 + 5005*(5*b^10*c*d^9 + a*b^9*d^10)*x^9 + 6435*(15*b^10*c^2*d^8 + 5*a*b^9*c*d
^9 + a^2*b^8*d^10)*x^8 + 6435*(35*b^10*c^3*d^7 + 15*a*b^9*c^2*d^8 + 5*a^2*b^8*c*
d^9 + a^3*b^7*d^10)*x^7 + 5005*(70*b^10*c^4*d^6 + 35*a*b^9*c^3*d^7 + 15*a^2*b^8*
c^2*d^8 + 5*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 3003*(126*b^10*c^5*d^5 + 70*a*b^
9*c^4*d^6 + 35*a^2*b^8*c^3*d^7 + 15*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 + a^5*b^5*
d^10)*x^5 + 1365*(210*b^10*c^6*d^4 + 126*a*b^9*c^5*d^5 + 70*a^2*b^8*c^4*d^6 + 35
*a^3*b^7*c^3*d^7 + 15*a^4*b^6*c^2*d^8 + 5*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 45
5*(330*b^10*c^7*d^3 + 210*a*b^9*c^6*d^4 + 126*a^2*b^8*c^5*d^5 + 70*a^3*b^7*c^4*d
^6 + 35*a^4*b^6*c^3*d^7 + 15*a^5*b^5*c^2*d^8 + 5*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x
^3 + 105*(495*b^10*c^8*d^2 + 330*a*b^9*c^7*d^3 + 210*a^2*b^8*c^6*d^4 + 126*a^3*b
^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 35*a^5*b^5*c^3*d^7 + 15*a^6*b^4*c^2*d^8 + 5*a^
7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 15*(715*b^10*c^9*d + 495*a*b^9*c^8*d^2 + 330*a
^2*b^8*c^7*d^3 + 210*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 70*a^5*b^5*c^4*d^6
+ 35*a^6*b^4*c^3*d^7 + 15*a^7*b^3*c^2*d^8 + 5*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^
26*x^15 + 15*a*b^25*x^14 + 105*a^2*b^24*x^13 + 455*a^3*b^23*x^12 + 1365*a^4*b^22
*x^11 + 3003*a^5*b^21*x^10 + 5005*a^6*b^20*x^9 + 6435*a^7*b^19*x^8 + 6435*a^8*b^
18*x^7 + 5005*a^9*b^17*x^6 + 3003*a^10*b^16*x^5 + 1365*a^11*b^15*x^4 + 455*a^12*
b^14*x^3 + 105*a^13*b^13*x^2 + 15*a^14*b^12*x + a^15*b^11)

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Fricas [A]  time = 0.208941, size = 1376, normalized size = 9.11 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^16,x, algorithm="fricas")

[Out]

-1/15015*(3003*b^10*d^10*x^10 + 1001*b^10*c^10 + 715*a*b^9*c^9*d + 495*a^2*b^8*c
^8*d^2 + 330*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 70*a^
6*b^4*c^4*d^6 + 35*a^7*b^3*c^3*d^7 + 15*a^8*b^2*c^2*d^8 + 5*a^9*b*c*d^9 + a^10*d
^10 + 5005*(5*b^10*c*d^9 + a*b^9*d^10)*x^9 + 6435*(15*b^10*c^2*d^8 + 5*a*b^9*c*d
^9 + a^2*b^8*d^10)*x^8 + 6435*(35*b^10*c^3*d^7 + 15*a*b^9*c^2*d^8 + 5*a^2*b^8*c*
d^9 + a^3*b^7*d^10)*x^7 + 5005*(70*b^10*c^4*d^6 + 35*a*b^9*c^3*d^7 + 15*a^2*b^8*
c^2*d^8 + 5*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 3003*(126*b^10*c^5*d^5 + 70*a*b^
9*c^4*d^6 + 35*a^2*b^8*c^3*d^7 + 15*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 + a^5*b^5*
d^10)*x^5 + 1365*(210*b^10*c^6*d^4 + 126*a*b^9*c^5*d^5 + 70*a^2*b^8*c^4*d^6 + 35
*a^3*b^7*c^3*d^7 + 15*a^4*b^6*c^2*d^8 + 5*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 45
5*(330*b^10*c^7*d^3 + 210*a*b^9*c^6*d^4 + 126*a^2*b^8*c^5*d^5 + 70*a^3*b^7*c^4*d
^6 + 35*a^4*b^6*c^3*d^7 + 15*a^5*b^5*c^2*d^8 + 5*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x
^3 + 105*(495*b^10*c^8*d^2 + 330*a*b^9*c^7*d^3 + 210*a^2*b^8*c^6*d^4 + 126*a^3*b
^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 35*a^5*b^5*c^3*d^7 + 15*a^6*b^4*c^2*d^8 + 5*a^
7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 15*(715*b^10*c^9*d + 495*a*b^9*c^8*d^2 + 330*a
^2*b^8*c^7*d^3 + 210*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 70*a^5*b^5*c^4*d^6
+ 35*a^6*b^4*c^3*d^7 + 15*a^7*b^3*c^2*d^8 + 5*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^
26*x^15 + 15*a*b^25*x^14 + 105*a^2*b^24*x^13 + 455*a^3*b^23*x^12 + 1365*a^4*b^22
*x^11 + 3003*a^5*b^21*x^10 + 5005*a^6*b^20*x^9 + 6435*a^7*b^19*x^8 + 6435*a^8*b^
18*x^7 + 5005*a^9*b^17*x^6 + 3003*a^10*b^16*x^5 + 1365*a^11*b^15*x^4 + 455*a^12*
b^14*x^3 + 105*a^13*b^13*x^2 + 15*a^14*b^12*x + a^15*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)**16,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215968, 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)^16,x, algorithm="giac")

[Out]

Done