3.1125 \(\int \frac{\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{11}} \, dx\)

Optimal. Leaf size=73 \[ -\frac{\left (b^2-4 a c\right )^2}{320 c^3 d^{11} (b+2 c x)^{10}}+\frac{b^2-4 a c}{128 c^3 d^{11} (b+2 c x)^8}-\frac{1}{192 c^3 d^{11} (b+2 c x)^6} \]

[Out]

-(b^2 - 4*a*c)^2/(320*c^3*d^11*(b + 2*c*x)^10) + (b^2 - 4*a*c)/(128*c^3*d^11*(b
+ 2*c*x)^8) - 1/(192*c^3*d^11*(b + 2*c*x)^6)

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Rubi [A]  time = 0.139063, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{\left (b^2-4 a c\right )^2}{320 c^3 d^{11} (b+2 c x)^{10}}+\frac{b^2-4 a c}{128 c^3 d^{11} (b+2 c x)^8}-\frac{1}{192 c^3 d^{11} (b+2 c x)^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^11,x]

[Out]

-(b^2 - 4*a*c)^2/(320*c^3*d^11*(b + 2*c*x)^10) + (b^2 - 4*a*c)/(128*c^3*d^11*(b
+ 2*c*x)^8) - 1/(192*c^3*d^11*(b + 2*c*x)^6)

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Rubi in Sympy [A]  time = 34.2515, size = 70, normalized size = 0.96 \[ - \frac{1}{192 c^{3} d^{11} \left (b + 2 c x\right )^{6}} + \frac{- 4 a c + b^{2}}{128 c^{3} d^{11} \left (b + 2 c x\right )^{8}} - \frac{\left (- 4 a c + b^{2}\right )^{2}}{320 c^{3} d^{11} \left (b + 2 c x\right )^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**11,x)

[Out]

-1/(192*c**3*d**11*(b + 2*c*x)**6) + (-4*a*c + b**2)/(128*c**3*d**11*(b + 2*c*x)
**8) - (-4*a*c + b**2)**2/(320*c**3*d**11*(b + 2*c*x)**10)

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Mathematica [A]  time = 0.0644372, size = 59, normalized size = 0.81 \[ \frac{15 \left (b^2-4 a c\right ) (b+2 c x)^2-6 \left (b^2-4 a c\right )^2-10 (b+2 c x)^4}{1920 c^3 d^{11} (b+2 c x)^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^11,x]

[Out]

(-6*(b^2 - 4*a*c)^2 + 15*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 10*(b + 2*c*x)^4)/(1920*c
^3*d^11*(b + 2*c*x)^10)

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Maple [A]  time = 0.009, size = 74, normalized size = 1. \[{\frac{1}{{d}^{11}} \left ( -{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{320\,{c}^{3} \left ( 2\,cx+b \right ) ^{10}}}-{\frac{4\,ac-{b}^{2}}{128\,{c}^{3} \left ( 2\,cx+b \right ) ^{8}}}-{\frac{1}{192\,{c}^{3} \left ( 2\,cx+b \right ) ^{6}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^11,x)

[Out]

1/d^11*(-1/320*(16*a^2*c^2-8*a*b^2*c+b^4)/c^3/(2*c*x+b)^10-1/128*(4*a*c-b^2)/c^3
/(2*c*x+b)^8-1/192/c^3/(2*c*x+b)^6)

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Maxima [A]  time = 0.731737, size = 297, normalized size = 4.07 \[ -\frac{160 \, c^{4} x^{4} + 320 \, b c^{3} x^{3} + b^{4} + 12 \, a b^{2} c + 96 \, a^{2} c^{2} + 60 \,{\left (3 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 20 \,{\left (b^{3} c + 12 \, a b c^{2}\right )} x}{1920 \,{\left (1024 \, c^{13} d^{11} x^{10} + 5120 \, b c^{12} d^{11} x^{9} + 11520 \, b^{2} c^{11} d^{11} x^{8} + 15360 \, b^{3} c^{10} d^{11} x^{7} + 13440 \, b^{4} c^{9} d^{11} x^{6} + 8064 \, b^{5} c^{8} d^{11} x^{5} + 3360 \, b^{6} c^{7} d^{11} x^{4} + 960 \, b^{7} c^{6} d^{11} x^{3} + 180 \, b^{8} c^{5} d^{11} x^{2} + 20 \, b^{9} c^{4} d^{11} x + b^{10} c^{3} d^{11}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2/(2*c*d*x + b*d)^11,x, algorithm="maxima")

[Out]

-1/1920*(160*c^4*x^4 + 320*b*c^3*x^3 + b^4 + 12*a*b^2*c + 96*a^2*c^2 + 60*(3*b^2
*c^2 + 4*a*c^3)*x^2 + 20*(b^3*c + 12*a*b*c^2)*x)/(1024*c^13*d^11*x^10 + 5120*b*c
^12*d^11*x^9 + 11520*b^2*c^11*d^11*x^8 + 15360*b^3*c^10*d^11*x^7 + 13440*b^4*c^9
*d^11*x^6 + 8064*b^5*c^8*d^11*x^5 + 3360*b^6*c^7*d^11*x^4 + 960*b^7*c^6*d^11*x^3
+ 180*b^8*c^5*d^11*x^2 + 20*b^9*c^4*d^11*x + b^10*c^3*d^11)

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Fricas [A]  time = 0.206807, size = 297, normalized size = 4.07 \[ -\frac{160 \, c^{4} x^{4} + 320 \, b c^{3} x^{3} + b^{4} + 12 \, a b^{2} c + 96 \, a^{2} c^{2} + 60 \,{\left (3 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 20 \,{\left (b^{3} c + 12 \, a b c^{2}\right )} x}{1920 \,{\left (1024 \, c^{13} d^{11} x^{10} + 5120 \, b c^{12} d^{11} x^{9} + 11520 \, b^{2} c^{11} d^{11} x^{8} + 15360 \, b^{3} c^{10} d^{11} x^{7} + 13440 \, b^{4} c^{9} d^{11} x^{6} + 8064 \, b^{5} c^{8} d^{11} x^{5} + 3360 \, b^{6} c^{7} d^{11} x^{4} + 960 \, b^{7} c^{6} d^{11} x^{3} + 180 \, b^{8} c^{5} d^{11} x^{2} + 20 \, b^{9} c^{4} d^{11} x + b^{10} c^{3} d^{11}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2/(2*c*d*x + b*d)^11,x, algorithm="fricas")

[Out]

-1/1920*(160*c^4*x^4 + 320*b*c^3*x^3 + b^4 + 12*a*b^2*c + 96*a^2*c^2 + 60*(3*b^2
*c^2 + 4*a*c^3)*x^2 + 20*(b^3*c + 12*a*b*c^2)*x)/(1024*c^13*d^11*x^10 + 5120*b*c
^12*d^11*x^9 + 11520*b^2*c^11*d^11*x^8 + 15360*b^3*c^10*d^11*x^7 + 13440*b^4*c^9
*d^11*x^6 + 8064*b^5*c^8*d^11*x^5 + 3360*b^6*c^7*d^11*x^4 + 960*b^7*c^6*d^11*x^3
+ 180*b^8*c^5*d^11*x^2 + 20*b^9*c^4*d^11*x + b^10*c^3*d^11)

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Sympy [A]  time = 26.2099, size = 233, normalized size = 3.19 \[ - \frac{96 a^{2} c^{2} + 12 a b^{2} c + b^{4} + 320 b c^{3} x^{3} + 160 c^{4} x^{4} + x^{2} \left (240 a c^{3} + 180 b^{2} c^{2}\right ) + x \left (240 a b c^{2} + 20 b^{3} c\right )}{1920 b^{10} c^{3} d^{11} + 38400 b^{9} c^{4} d^{11} x + 345600 b^{8} c^{5} d^{11} x^{2} + 1843200 b^{7} c^{6} d^{11} x^{3} + 6451200 b^{6} c^{7} d^{11} x^{4} + 15482880 b^{5} c^{8} d^{11} x^{5} + 25804800 b^{4} c^{9} d^{11} x^{6} + 29491200 b^{3} c^{10} d^{11} x^{7} + 22118400 b^{2} c^{11} d^{11} x^{8} + 9830400 b c^{12} d^{11} x^{9} + 1966080 c^{13} d^{11} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**11,x)

[Out]

-(96*a**2*c**2 + 12*a*b**2*c + b**4 + 320*b*c**3*x**3 + 160*c**4*x**4 + x**2*(24
0*a*c**3 + 180*b**2*c**2) + x*(240*a*b*c**2 + 20*b**3*c))/(1920*b**10*c**3*d**11
+ 38400*b**9*c**4*d**11*x + 345600*b**8*c**5*d**11*x**2 + 1843200*b**7*c**6*d**
11*x**3 + 6451200*b**6*c**7*d**11*x**4 + 15482880*b**5*c**8*d**11*x**5 + 2580480
0*b**4*c**9*d**11*x**6 + 29491200*b**3*c**10*d**11*x**7 + 22118400*b**2*c**11*d*
*11*x**8 + 9830400*b*c**12*d**11*x**9 + 1966080*c**13*d**11*x**10)

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GIAC/XCAS [A]  time = 0.214062, size = 117, normalized size = 1.6 \[ -\frac{160 \, c^{4} x^{4} + 320 \, b c^{3} x^{3} + 180 \, b^{2} c^{2} x^{2} + 240 \, a c^{3} x^{2} + 20 \, b^{3} c x + 240 \, a b c^{2} x + b^{4} + 12 \, a b^{2} c + 96 \, a^{2} c^{2}}{1920 \,{\left (2 \, c x + b\right )}^{10} c^{3} d^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2/(2*c*d*x + b*d)^11,x, algorithm="giac")

[Out]

-1/1920*(160*c^4*x^4 + 320*b*c^3*x^3 + 180*b^2*c^2*x^2 + 240*a*c^3*x^2 + 20*b^3*
c*x + 240*a*b*c^2*x + b^4 + 12*a*b^2*c + 96*a^2*c^2)/((2*c*x + b)^10*c^3*d^11)