3.1326 \(\int \frac{(c+d x)^{10}}{(a+b x)^{15}} \, dx\)

Optimal. Leaf size=120 \[ \frac{d^3 (c+d x)^{11}}{4004 (a+b x)^{11} (b c-a d)^4}-\frac{d^2 (c+d x)^{11}}{364 (a+b x)^{12} (b c-a d)^3}+\frac{3 d (c+d x)^{11}}{182 (a+b x)^{13} (b c-a d)^2}-\frac{(c+d x)^{11}}{14 (a+b x)^{14} (b c-a d)} \]

[Out]

-(c + d*x)^11/(14*(b*c - a*d)*(a + b*x)^14) + (3*d*(c + d*x)^11)/(182*(b*c - a*d
)^2*(a + b*x)^13) - (d^2*(c + d*x)^11)/(364*(b*c - a*d)^3*(a + b*x)^12) + (d^3*(
c + d*x)^11)/(4004*(b*c - a*d)^4*(a + b*x)^11)

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Rubi [A]  time = 0.0921148, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{d^3 (c+d x)^{11}}{4004 (a+b x)^{11} (b c-a d)^4}-\frac{d^2 (c+d x)^{11}}{364 (a+b x)^{12} (b c-a d)^3}+\frac{3 d (c+d x)^{11}}{182 (a+b x)^{13} (b c-a d)^2}-\frac{(c+d x)^{11}}{14 (a+b x)^{14} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^10/(a + b*x)^15,x]

[Out]

-(c + d*x)^11/(14*(b*c - a*d)*(a + b*x)^14) + (3*d*(c + d*x)^11)/(182*(b*c - a*d
)^2*(a + b*x)^13) - (d^2*(c + d*x)^11)/(364*(b*c - a*d)^3*(a + b*x)^12) + (d^3*(
c + d*x)^11)/(4004*(b*c - a*d)^4*(a + b*x)^11)

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Rubi in Sympy [A]  time = 22.6875, size = 102, normalized size = 0.85 \[ \frac{d^{3} \left (c + d x\right )^{11}}{4004 \left (a + b x\right )^{11} \left (a d - b c\right )^{4}} + \frac{d^{2} \left (c + d x\right )^{11}}{364 \left (a + b x\right )^{12} \left (a d - b c\right )^{3}} + \frac{3 d \left (c + d x\right )^{11}}{182 \left (a + b x\right )^{13} \left (a d - b c\right )^{2}} + \frac{\left (c + d x\right )^{11}}{14 \left (a + b x\right )^{14} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**10/(b*x+a)**15,x)

[Out]

d**3*(c + d*x)**11/(4004*(a + b*x)**11*(a*d - b*c)**4) + d**2*(c + d*x)**11/(364
*(a + b*x)**12*(a*d - b*c)**3) + 3*d*(c + d*x)**11/(182*(a + b*x)**13*(a*d - b*c
)**2) + (c + d*x)**11/(14*(a + b*x)**14*(a*d - b*c))

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Mathematica [B]  time = 0.751314, size = 692, normalized size = 5.77 \[ -\frac{a^{10} d^{10}+2 a^9 b d^9 (2 c+7 d x)+a^8 b^2 d^8 \left (10 c^2+56 c d x+91 d^2 x^2\right )+4 a^7 b^3 d^7 \left (5 c^3+35 c^2 d x+91 c d^2 x^2+91 d^3 x^3\right )+7 a^6 b^4 d^6 \left (5 c^4+40 c^3 d x+130 c^2 d^2 x^2+208 c d^3 x^3+143 d^4 x^4\right )+14 a^5 b^5 d^5 \left (4 c^5+35 c^4 d x+130 c^3 d^2 x^2+260 c^2 d^3 x^3+286 c d^4 x^4+143 d^5 x^5\right )+7 a^4 b^6 d^4 \left (12 c^6+112 c^5 d x+455 c^4 d^2 x^2+1040 c^3 d^3 x^3+1430 c^2 d^4 x^4+1144 c d^5 x^5+429 d^6 x^6\right )+4 a^3 b^7 d^3 \left (30 c^7+294 c^6 d x+1274 c^5 d^2 x^2+3185 c^4 d^3 x^3+5005 c^3 d^4 x^4+5005 c^2 d^5 x^5+3003 c d^6 x^6+858 d^7 x^7\right )+a^2 b^8 d^2 \left (165 c^8+1680 c^7 d x+7644 c^6 d^2 x^2+20384 c^5 d^3 x^3+35035 c^4 d^4 x^4+40040 c^3 d^5 x^5+30030 c^2 d^6 x^6+13728 c d^7 x^7+3003 d^8 x^8\right )+2 a b^9 d \left (110 c^9+1155 c^8 d x+5460 c^7 d^2 x^2+15288 c^6 d^3 x^3+28028 c^5 d^4 x^4+35035 c^4 d^5 x^5+30030 c^3 d^6 x^6+17160 c^2 d^7 x^7+6006 c d^8 x^8+1001 d^9 x^9\right )+b^{10} \left (286 c^{10}+3080 c^9 d x+15015 c^8 d^2 x^2+43680 c^7 d^3 x^3+84084 c^6 d^4 x^4+112112 c^5 d^5 x^5+105105 c^4 d^6 x^6+68640 c^3 d^7 x^7+30030 c^2 d^8 x^8+8008 c d^9 x^9+1001 d^{10} x^{10}\right )}{4004 b^{11} (a+b x)^{14}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^10/(a + b*x)^15,x]

[Out]

-(a^10*d^10 + 2*a^9*b*d^9*(2*c + 7*d*x) + a^8*b^2*d^8*(10*c^2 + 56*c*d*x + 91*d^
2*x^2) + 4*a^7*b^3*d^7*(5*c^3 + 35*c^2*d*x + 91*c*d^2*x^2 + 91*d^3*x^3) + 7*a^6*
b^4*d^6*(5*c^4 + 40*c^3*d*x + 130*c^2*d^2*x^2 + 208*c*d^3*x^3 + 143*d^4*x^4) + 1
4*a^5*b^5*d^5*(4*c^5 + 35*c^4*d*x + 130*c^3*d^2*x^2 + 260*c^2*d^3*x^3 + 286*c*d^
4*x^4 + 143*d^5*x^5) + 7*a^4*b^6*d^4*(12*c^6 + 112*c^5*d*x + 455*c^4*d^2*x^2 + 1
040*c^3*d^3*x^3 + 1430*c^2*d^4*x^4 + 1144*c*d^5*x^5 + 429*d^6*x^6) + 4*a^3*b^7*d
^3*(30*c^7 + 294*c^6*d*x + 1274*c^5*d^2*x^2 + 3185*c^4*d^3*x^3 + 5005*c^3*d^4*x^
4 + 5005*c^2*d^5*x^5 + 3003*c*d^6*x^6 + 858*d^7*x^7) + a^2*b^8*d^2*(165*c^8 + 16
80*c^7*d*x + 7644*c^6*d^2*x^2 + 20384*c^5*d^3*x^3 + 35035*c^4*d^4*x^4 + 40040*c^
3*d^5*x^5 + 30030*c^2*d^6*x^6 + 13728*c*d^7*x^7 + 3003*d^8*x^8) + 2*a*b^9*d*(110
*c^9 + 1155*c^8*d*x + 5460*c^7*d^2*x^2 + 15288*c^6*d^3*x^3 + 28028*c^5*d^4*x^4 +
35035*c^4*d^5*x^5 + 30030*c^3*d^6*x^6 + 17160*c^2*d^7*x^7 + 6006*c*d^8*x^8 + 10
01*d^9*x^9) + b^10*(286*c^10 + 3080*c^9*d*x + 15015*c^8*d^2*x^2 + 43680*c^7*d^3*
x^3 + 84084*c^6*d^4*x^4 + 112112*c^5*d^5*x^5 + 105105*c^4*d^6*x^6 + 68640*c^3*d^
7*x^7 + 30030*c^2*d^8*x^8 + 8008*c*d^9*x^9 + 1001*d^10*x^10))/(4004*b^11*(a + b*
x)^14)

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Maple [B]  time = 0.013, size = 867, normalized size = 7.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^10/(b*x+a)^15,x)

[Out]

2*d^9*(a*d-b*c)/b^11/(b*x+a)^5-1/14*(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)^1
4-15/2*d^8*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^11/(b*x+a)^6-15/4*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)^12+120/7*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)^7-1/4*d^10/b^11/(b*x+a)^4-105/4
*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
)^8-21*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)^10+120/11*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)^11+10/13*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)^13+28*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)^9

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Maxima [A]  time = 1.45072, size = 1361, normalized size = 11.34 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^15,x, algorithm="maxima")

[Out]

-1/4004*(1001*b^10*d^10*x^10 + 286*b^10*c^10 + 220*a*b^9*c^9*d + 165*a^2*b^8*c^8
*d^2 + 120*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 56*a^5*b^5*c^5*d^5 + 35*a^6*b^
4*c^4*d^6 + 20*a^7*b^3*c^3*d^7 + 10*a^8*b^2*c^2*d^8 + 4*a^9*b*c*d^9 + a^10*d^10
+ 2002*(4*b^10*c*d^9 + a*b^9*d^10)*x^9 + 3003*(10*b^10*c^2*d^8 + 4*a*b^9*c*d^9 +
a^2*b^8*d^10)*x^8 + 3432*(20*b^10*c^3*d^7 + 10*a*b^9*c^2*d^8 + 4*a^2*b^8*c*d^9
+ a^3*b^7*d^10)*x^7 + 3003*(35*b^10*c^4*d^6 + 20*a*b^9*c^3*d^7 + 10*a^2*b^8*c^2*
d^8 + 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 2002*(56*b^10*c^5*d^5 + 35*a*b^9*c^4
*d^6 + 20*a^2*b^8*c^3*d^7 + 10*a^3*b^7*c^2*d^8 + 4*a^4*b^6*c*d^9 + a^5*b^5*d^10)
*x^5 + 1001*(84*b^10*c^6*d^4 + 56*a*b^9*c^5*d^5 + 35*a^2*b^8*c^4*d^6 + 20*a^3*b^
7*c^3*d^7 + 10*a^4*b^6*c^2*d^8 + 4*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 364*(120*
b^10*c^7*d^3 + 84*a*b^9*c^6*d^4 + 56*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6 + 20*a
^4*b^6*c^3*d^7 + 10*a^5*b^5*c^2*d^8 + 4*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 91*(
165*b^10*c^8*d^2 + 120*a*b^9*c^7*d^3 + 84*a^2*b^8*c^6*d^4 + 56*a^3*b^7*c^5*d^5 +
35*a^4*b^6*c^4*d^6 + 20*a^5*b^5*c^3*d^7 + 10*a^6*b^4*c^2*d^8 + 4*a^7*b^3*c*d^9
+ a^8*b^2*d^10)*x^2 + 14*(220*b^10*c^9*d + 165*a*b^9*c^8*d^2 + 120*a^2*b^8*c^7*d
^3 + 84*a^3*b^7*c^6*d^4 + 56*a^4*b^6*c^5*d^5 + 35*a^5*b^5*c^4*d^6 + 20*a^6*b^4*c
^3*d^7 + 10*a^7*b^3*c^2*d^8 + 4*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^25*x^14 + 14*a
*b^24*x^13 + 91*a^2*b^23*x^12 + 364*a^3*b^22*x^11 + 1001*a^4*b^21*x^10 + 2002*a^
5*b^20*x^9 + 3003*a^6*b^19*x^8 + 3432*a^7*b^18*x^7 + 3003*a^8*b^17*x^6 + 2002*a^
9*b^16*x^5 + 1001*a^10*b^15*x^4 + 364*a^11*b^14*x^3 + 91*a^12*b^13*x^2 + 14*a^13
*b^12*x + a^14*b^11)

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Fricas [A]  time = 0.220981, size = 1361, normalized size = 11.34 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^15,x, algorithm="fricas")

[Out]

-1/4004*(1001*b^10*d^10*x^10 + 286*b^10*c^10 + 220*a*b^9*c^9*d + 165*a^2*b^8*c^8
*d^2 + 120*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 56*a^5*b^5*c^5*d^5 + 35*a^6*b^
4*c^4*d^6 + 20*a^7*b^3*c^3*d^7 + 10*a^8*b^2*c^2*d^8 + 4*a^9*b*c*d^9 + a^10*d^10
+ 2002*(4*b^10*c*d^9 + a*b^9*d^10)*x^9 + 3003*(10*b^10*c^2*d^8 + 4*a*b^9*c*d^9 +
a^2*b^8*d^10)*x^8 + 3432*(20*b^10*c^3*d^7 + 10*a*b^9*c^2*d^8 + 4*a^2*b^8*c*d^9
+ a^3*b^7*d^10)*x^7 + 3003*(35*b^10*c^4*d^6 + 20*a*b^9*c^3*d^7 + 10*a^2*b^8*c^2*
d^8 + 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 2002*(56*b^10*c^5*d^5 + 35*a*b^9*c^4
*d^6 + 20*a^2*b^8*c^3*d^7 + 10*a^3*b^7*c^2*d^8 + 4*a^4*b^6*c*d^9 + a^5*b^5*d^10)
*x^5 + 1001*(84*b^10*c^6*d^4 + 56*a*b^9*c^5*d^5 + 35*a^2*b^8*c^4*d^6 + 20*a^3*b^
7*c^3*d^7 + 10*a^4*b^6*c^2*d^8 + 4*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 364*(120*
b^10*c^7*d^3 + 84*a*b^9*c^6*d^4 + 56*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6 + 20*a
^4*b^6*c^3*d^7 + 10*a^5*b^5*c^2*d^8 + 4*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 91*(
165*b^10*c^8*d^2 + 120*a*b^9*c^7*d^3 + 84*a^2*b^8*c^6*d^4 + 56*a^3*b^7*c^5*d^5 +
35*a^4*b^6*c^4*d^6 + 20*a^5*b^5*c^3*d^7 + 10*a^6*b^4*c^2*d^8 + 4*a^7*b^3*c*d^9
+ a^8*b^2*d^10)*x^2 + 14*(220*b^10*c^9*d + 165*a*b^9*c^8*d^2 + 120*a^2*b^8*c^7*d
^3 + 84*a^3*b^7*c^6*d^4 + 56*a^4*b^6*c^5*d^5 + 35*a^5*b^5*c^4*d^6 + 20*a^6*b^4*c
^3*d^7 + 10*a^7*b^3*c^2*d^8 + 4*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^25*x^14 + 14*a
*b^24*x^13 + 91*a^2*b^23*x^12 + 364*a^3*b^22*x^11 + 1001*a^4*b^21*x^10 + 2002*a^
5*b^20*x^9 + 3003*a^6*b^19*x^8 + 3432*a^7*b^18*x^7 + 3003*a^8*b^17*x^6 + 2002*a^
9*b^16*x^5 + 1001*a^10*b^15*x^4 + 364*a^11*b^14*x^3 + 91*a^12*b^13*x^2 + 14*a^13
*b^12*x + a^14*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)**15,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219462, 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)^15,x, algorithm="giac")

[Out]

Done