3.1325 \(\int \frac{(c+d x)^{10}}{(a+b x)^{14}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{d^2 (c+d x)^{11}}{858 (a+b x)^{11} (b c-a d)^3}+\frac{d (c+d x)^{11}}{78 (a+b x)^{12} (b c-a d)^2}-\frac{(c+d x)^{11}}{13 (a+b x)^{13} (b c-a d)} \]

[Out]

-(c + d*x)^11/(13*(b*c - a*d)*(a + b*x)^13) + (d*(c + d*x)^11)/(78*(b*c - a*d)^2
*(a + b*x)^12) - (d^2*(c + d*x)^11)/(858*(b*c - a*d)^3*(a + b*x)^11)

_______________________________________________________________________________________

Rubi [A]  time = 0.0621759, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{d^2 (c+d x)^{11}}{858 (a+b x)^{11} (b c-a d)^3}+\frac{d (c+d x)^{11}}{78 (a+b x)^{12} (b c-a d)^2}-\frac{(c+d x)^{11}}{13 (a+b x)^{13} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^10/(a + b*x)^14,x]

[Out]

-(c + d*x)^11/(13*(b*c - a*d)*(a + b*x)^13) + (d*(c + d*x)^11)/(78*(b*c - a*d)^2
*(a + b*x)^12) - (d^2*(c + d*x)^11)/(858*(b*c - a*d)^3*(a + b*x)^11)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.4839, size = 73, normalized size = 0.82 \[ \frac{d^{2} \left (c + d x\right )^{11}}{858 \left (a + b x\right )^{11} \left (a d - b c\right )^{3}} + \frac{d \left (c + d x\right )^{11}}{78 \left (a + b x\right )^{12} \left (a d - b c\right )^{2}} + \frac{\left (c + d x\right )^{11}}{13 \left (a + b x\right )^{13} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**10/(b*x+a)**14,x)

[Out]

d**2*(c + d*x)**11/(858*(a + b*x)**11*(a*d - b*c)**3) + d*(c + d*x)**11/(78*(a +
b*x)**12*(a*d - b*c)**2) + (c + d*x)**11/(13*(a + b*x)**13*(a*d - b*c))

_______________________________________________________________________________________

Mathematica [B]  time = 0.774554, size = 690, normalized size = 7.75 \[ -\frac{a^{10} d^{10}+a^9 b d^9 (3 c+13 d x)+3 a^8 b^2 d^8 \left (2 c^2+13 c d x+26 d^2 x^2\right )+2 a^7 b^3 d^7 \left (5 c^3+39 c^2 d x+117 c d^2 x^2+143 d^3 x^3\right )+a^6 b^4 d^6 \left (15 c^4+130 c^3 d x+468 c^2 d^2 x^2+858 c d^3 x^3+715 d^4 x^4\right )+3 a^5 b^5 d^5 \left (7 c^5+65 c^4 d x+260 c^3 d^2 x^2+572 c^2 d^3 x^3+715 c d^4 x^4+429 d^5 x^5\right )+a^4 b^6 d^4 \left (28 c^6+273 c^5 d x+1170 c^4 d^2 x^2+2860 c^3 d^3 x^3+4290 c^2 d^4 x^4+3861 c d^5 x^5+1716 d^6 x^6\right )+2 a^3 b^7 d^3 \left (18 c^7+182 c^6 d x+819 c^5 d^2 x^2+2145 c^4 d^3 x^3+3575 c^3 d^4 x^4+3861 c^2 d^5 x^5+2574 c d^6 x^6+858 d^7 x^7\right )+3 a^2 b^8 d^2 \left (15 c^8+156 c^7 d x+728 c^6 d^2 x^2+2002 c^5 d^3 x^3+3575 c^4 d^4 x^4+4290 c^3 d^5 x^5+3432 c^2 d^6 x^6+1716 c d^7 x^7+429 d^8 x^8\right )+a b^9 d \left (55 c^9+585 c^8 d x+2808 c^7 d^2 x^2+8008 c^6 d^3 x^3+15015 c^5 d^4 x^4+19305 c^4 d^5 x^5+17160 c^3 d^6 x^6+10296 c^2 d^7 x^7+3861 c d^8 x^8+715 d^9 x^9\right )+b^{10} \left (66 c^{10}+715 c^9 d x+3510 c^8 d^2 x^2+10296 c^7 d^3 x^3+20020 c^6 d^4 x^4+27027 c^5 d^5 x^5+25740 c^4 d^6 x^6+17160 c^3 d^7 x^7+7722 c^2 d^8 x^8+2145 c d^9 x^9+286 d^{10} x^{10}\right )}{858 b^{11} (a+b x)^{13}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^10/(a + b*x)^14,x]

[Out]

-(a^10*d^10 + a^9*b*d^9*(3*c + 13*d*x) + 3*a^8*b^2*d^8*(2*c^2 + 13*c*d*x + 26*d^
2*x^2) + 2*a^7*b^3*d^7*(5*c^3 + 39*c^2*d*x + 117*c*d^2*x^2 + 143*d^3*x^3) + a^6*
b^4*d^6*(15*c^4 + 130*c^3*d*x + 468*c^2*d^2*x^2 + 858*c*d^3*x^3 + 715*d^4*x^4) +
3*a^5*b^5*d^5*(7*c^5 + 65*c^4*d*x + 260*c^3*d^2*x^2 + 572*c^2*d^3*x^3 + 715*c*d
^4*x^4 + 429*d^5*x^5) + a^4*b^6*d^4*(28*c^6 + 273*c^5*d*x + 1170*c^4*d^2*x^2 + 2
860*c^3*d^3*x^3 + 4290*c^2*d^4*x^4 + 3861*c*d^5*x^5 + 1716*d^6*x^6) + 2*a^3*b^7*
d^3*(18*c^7 + 182*c^6*d*x + 819*c^5*d^2*x^2 + 2145*c^4*d^3*x^3 + 3575*c^3*d^4*x^
4 + 3861*c^2*d^5*x^5 + 2574*c*d^6*x^6 + 858*d^7*x^7) + 3*a^2*b^8*d^2*(15*c^8 + 1
56*c^7*d*x + 728*c^6*d^2*x^2 + 2002*c^5*d^3*x^3 + 3575*c^4*d^4*x^4 + 4290*c^3*d^
5*x^5 + 3432*c^2*d^6*x^6 + 1716*c*d^7*x^7 + 429*d^8*x^8) + a*b^9*d*(55*c^9 + 585
*c^8*d*x + 2808*c^7*d^2*x^2 + 8008*c^6*d^3*x^3 + 15015*c^5*d^4*x^4 + 19305*c^4*d
^5*x^5 + 17160*c^3*d^6*x^6 + 10296*c^2*d^7*x^7 + 3861*c*d^8*x^8 + 715*d^9*x^9) +
b^10*(66*c^10 + 715*c^9*d*x + 3510*c^8*d^2*x^2 + 10296*c^7*d^3*x^3 + 20020*c^6*
d^4*x^4 + 27027*c^5*d^5*x^5 + 25740*c^4*d^6*x^6 + 17160*c^3*d^7*x^7 + 7722*c^2*d
^8*x^8 + 2145*c*d^9*x^9 + 286*d^10*x^10))/(858*b^11*(a + b*x)^13)

_______________________________________________________________________________________

Maple [B]  time = 0.013, size = 867, normalized size = 9.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^10/(b*x+a)^14,x)

[Out]

-9*d^8*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^11/(b*x+a)^5+20*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)^6+5/6*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)^12-30*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)^7+5/2*d^9*(a*d
-b*c)/b^11/(b*x+a)^4-1/3*d^10/b^11/(b*x+a)^3+63/2*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)^8+12*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)^10-45/11*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)^11-1/13*(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)^13-70/3*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)
^9

_______________________________________________________________________________________

Maxima [A]  time = 1.46696, size = 1346, normalized size = 15.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^14,x, algorithm="maxima")

[Out]

-1/858*(286*b^10*d^10*x^10 + 66*b^10*c^10 + 55*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2
+ 36*a^3*b^7*c^7*d^3 + 28*a^4*b^6*c^6*d^4 + 21*a^5*b^5*c^5*d^5 + 15*a^6*b^4*c^4*
d^6 + 10*a^7*b^3*c^3*d^7 + 6*a^8*b^2*c^2*d^8 + 3*a^9*b*c*d^9 + a^10*d^10 + 715*(
3*b^10*c*d^9 + a*b^9*d^10)*x^9 + 1287*(6*b^10*c^2*d^8 + 3*a*b^9*c*d^9 + a^2*b^8*
d^10)*x^8 + 1716*(10*b^10*c^3*d^7 + 6*a*b^9*c^2*d^8 + 3*a^2*b^8*c*d^9 + a^3*b^7*
d^10)*x^7 + 1716*(15*b^10*c^4*d^6 + 10*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 + 3*a^3
*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 1287*(21*b^10*c^5*d^5 + 15*a*b^9*c^4*d^6 + 10*a
^2*b^8*c^3*d^7 + 6*a^3*b^7*c^2*d^8 + 3*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 715*(
28*b^10*c^6*d^4 + 21*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 10*a^3*b^7*c^3*d^7 + 6
*a^4*b^6*c^2*d^8 + 3*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 286*(36*b^10*c^7*d^3 +
28*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 15*a^3*b^7*c^4*d^6 + 10*a^4*b^6*c^3*d^7
+ 6*a^5*b^5*c^2*d^8 + 3*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 78*(45*b^10*c^8*d^2
+ 36*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 21*a^3*b^7*c^5*d^5 + 15*a^4*b^6*c^4*d^
6 + 10*a^5*b^5*c^3*d^7 + 6*a^6*b^4*c^2*d^8 + 3*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2
+ 13*(55*b^10*c^9*d + 45*a*b^9*c^8*d^2 + 36*a^2*b^8*c^7*d^3 + 28*a^3*b^7*c^6*d^
4 + 21*a^4*b^6*c^5*d^5 + 15*a^5*b^5*c^4*d^6 + 10*a^6*b^4*c^3*d^7 + 6*a^7*b^3*c^2
*d^8 + 3*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^24*x^13 + 13*a*b^23*x^12 + 78*a^2*b^2
2*x^11 + 286*a^3*b^21*x^10 + 715*a^4*b^20*x^9 + 1287*a^5*b^19*x^8 + 1716*a^6*b^1
8*x^7 + 1716*a^7*b^17*x^6 + 1287*a^8*b^16*x^5 + 715*a^9*b^15*x^4 + 286*a^10*b^14
*x^3 + 78*a^11*b^13*x^2 + 13*a^12*b^12*x + a^13*b^11)

_______________________________________________________________________________________

Fricas [A]  time = 0.220577, size = 1346, normalized size = 15.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^14,x, algorithm="fricas")

[Out]

-1/858*(286*b^10*d^10*x^10 + 66*b^10*c^10 + 55*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2
+ 36*a^3*b^7*c^7*d^3 + 28*a^4*b^6*c^6*d^4 + 21*a^5*b^5*c^5*d^5 + 15*a^6*b^4*c^4*
d^6 + 10*a^7*b^3*c^3*d^7 + 6*a^8*b^2*c^2*d^8 + 3*a^9*b*c*d^9 + a^10*d^10 + 715*(
3*b^10*c*d^9 + a*b^9*d^10)*x^9 + 1287*(6*b^10*c^2*d^8 + 3*a*b^9*c*d^9 + a^2*b^8*
d^10)*x^8 + 1716*(10*b^10*c^3*d^7 + 6*a*b^9*c^2*d^8 + 3*a^2*b^8*c*d^9 + a^3*b^7*
d^10)*x^7 + 1716*(15*b^10*c^4*d^6 + 10*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 + 3*a^3
*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 1287*(21*b^10*c^5*d^5 + 15*a*b^9*c^4*d^6 + 10*a
^2*b^8*c^3*d^7 + 6*a^3*b^7*c^2*d^8 + 3*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 715*(
28*b^10*c^6*d^4 + 21*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 10*a^3*b^7*c^3*d^7 + 6
*a^4*b^6*c^2*d^8 + 3*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 286*(36*b^10*c^7*d^3 +
28*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 15*a^3*b^7*c^4*d^6 + 10*a^4*b^6*c^3*d^7
+ 6*a^5*b^5*c^2*d^8 + 3*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 78*(45*b^10*c^8*d^2
+ 36*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 21*a^3*b^7*c^5*d^5 + 15*a^4*b^6*c^4*d^
6 + 10*a^5*b^5*c^3*d^7 + 6*a^6*b^4*c^2*d^8 + 3*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2
+ 13*(55*b^10*c^9*d + 45*a*b^9*c^8*d^2 + 36*a^2*b^8*c^7*d^3 + 28*a^3*b^7*c^6*d^
4 + 21*a^4*b^6*c^5*d^5 + 15*a^5*b^5*c^4*d^6 + 10*a^6*b^4*c^3*d^7 + 6*a^7*b^3*c^2
*d^8 + 3*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^24*x^13 + 13*a*b^23*x^12 + 78*a^2*b^2
2*x^11 + 286*a^3*b^21*x^10 + 715*a^4*b^20*x^9 + 1287*a^5*b^19*x^8 + 1716*a^6*b^1
8*x^7 + 1716*a^7*b^17*x^6 + 1287*a^8*b^16*x^5 + 715*a^9*b^15*x^4 + 286*a^10*b^14
*x^3 + 78*a^11*b^13*x^2 + 13*a^12*b^12*x + a^13*b^11)

_______________________________________________________________________________________

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)**14,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216198, 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)^14,x, algorithm="giac")

[Out]

Done