3.1122 \(\int \frac{\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx\)

Optimal. Leaf size=73 \[ -\frac{\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac{b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac{1}{96 c^3 d^8 (b+2 c x)^3} \]

[Out]

-(b^2 - 4*a*c)^2/(224*c^3*d^8*(b + 2*c*x)^7) + (b^2 - 4*a*c)/(80*c^3*d^8*(b + 2*
c*x)^5) - 1/(96*c^3*d^8*(b + 2*c*x)^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.141904, 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 \[ -\frac{\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac{b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac{1}{96 c^3 d^8 (b+2 c x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^8,x]

[Out]

-(b^2 - 4*a*c)^2/(224*c^3*d^8*(b + 2*c*x)^7) + (b^2 - 4*a*c)/(80*c^3*d^8*(b + 2*
c*x)^5) - 1/(96*c^3*d^8*(b + 2*c*x)^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.5542, size = 70, normalized size = 0.96 \[ - \frac{1}{96 c^{3} d^{8} \left (b + 2 c x\right )^{3}} + \frac{- 4 a c + b^{2}}{80 c^{3} d^{8} \left (b + 2 c x\right )^{5}} - \frac{\left (- 4 a c + b^{2}\right )^{2}}{224 c^{3} d^{8} \left (b + 2 c x\right )^{7}} \]

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)**8,x)

[Out]

-1/(96*c**3*d**8*(b + 2*c*x)**3) + (-4*a*c + b**2)/(80*c**3*d**8*(b + 2*c*x)**5)
- (-4*a*c + b**2)**2/(224*c**3*d**8*(b + 2*c*x)**7)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0624556, size = 59, normalized size = 0.81 \[ \frac{42 \left (b^2-4 a c\right ) (b+2 c x)^2-15 \left (b^2-4 a c\right )^2-35 (b+2 c x)^4}{3360 c^3 d^8 (b+2 c x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^8,x]

[Out]

(-15*(b^2 - 4*a*c)^2 + 42*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 35*(b + 2*c*x)^4)/(3360*
c^3*d^8*(b + 2*c*x)^7)

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 74, normalized size = 1. \[{\frac{1}{{d}^{8}} \left ( -{\frac{4\,ac-{b}^{2}}{80\,{c}^{3} \left ( 2\,cx+b \right ) ^{5}}}-{\frac{1}{96\,{c}^{3} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{224\,{c}^{3} \left ( 2\,cx+b \right ) ^{7}}} \right ) } \]

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)^8,x)

[Out]

1/d^8*(-1/80*(4*a*c-b^2)/c^3/(2*c*x+b)^5-1/96/c^3/(2*c*x+b)^3-1/224*(16*a^2*c^2-
8*a*b^2*c+b^4)/c^3/(2*c*x+b)^7)

_______________________________________________________________________________________

Maxima [A]  time = 0.711782, size = 238, normalized size = 3.26 \[ -\frac{70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2} + 84 \,{\left (b^{2} c^{2} + a c^{3}\right )} x^{2} + 14 \,{\left (b^{3} c + 6 \, a b c^{2}\right )} x}{420 \,{\left (128 \, c^{10} d^{8} x^{7} + 448 \, b c^{9} d^{8} x^{6} + 672 \, b^{2} c^{8} d^{8} x^{5} + 560 \, b^{3} c^{7} d^{8} x^{4} + 280 \, b^{4} c^{6} d^{8} x^{3} + 84 \, b^{5} c^{5} d^{8} x^{2} + 14 \, b^{6} c^{4} d^{8} x + b^{7} c^{3} d^{8}\right )}} \]

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)^8,x, algorithm="maxima")

[Out]

-1/420*(70*c^4*x^4 + 140*b*c^3*x^3 + b^4 + 6*a*b^2*c + 30*a^2*c^2 + 84*(b^2*c^2
+ a*c^3)*x^2 + 14*(b^3*c + 6*a*b*c^2)*x)/(128*c^10*d^8*x^7 + 448*b*c^9*d^8*x^6 +
672*b^2*c^8*d^8*x^5 + 560*b^3*c^7*d^8*x^4 + 280*b^4*c^6*d^8*x^3 + 84*b^5*c^5*d^
8*x^2 + 14*b^6*c^4*d^8*x + b^7*c^3*d^8)

_______________________________________________________________________________________

Fricas [A]  time = 0.22354, size = 238, normalized size = 3.26 \[ -\frac{70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2} + 84 \,{\left (b^{2} c^{2} + a c^{3}\right )} x^{2} + 14 \,{\left (b^{3} c + 6 \, a b c^{2}\right )} x}{420 \,{\left (128 \, c^{10} d^{8} x^{7} + 448 \, b c^{9} d^{8} x^{6} + 672 \, b^{2} c^{8} d^{8} x^{5} + 560 \, b^{3} c^{7} d^{8} x^{4} + 280 \, b^{4} c^{6} d^{8} x^{3} + 84 \, b^{5} c^{5} d^{8} x^{2} + 14 \, b^{6} c^{4} d^{8} x + b^{7} c^{3} d^{8}\right )}} \]

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)^8,x, algorithm="fricas")

[Out]

-1/420*(70*c^4*x^4 + 140*b*c^3*x^3 + b^4 + 6*a*b^2*c + 30*a^2*c^2 + 84*(b^2*c^2
+ a*c^3)*x^2 + 14*(b^3*c + 6*a*b*c^2)*x)/(128*c^10*d^8*x^7 + 448*b*c^9*d^8*x^6 +
672*b^2*c^8*d^8*x^5 + 560*b^3*c^7*d^8*x^4 + 280*b^4*c^6*d^8*x^3 + 84*b^5*c^5*d^
8*x^2 + 14*b^6*c^4*d^8*x + b^7*c^3*d^8)

_______________________________________________________________________________________

Sympy [A]  time = 8.77141, size = 187, normalized size = 2.56 \[ - \frac{30 a^{2} c^{2} + 6 a b^{2} c + b^{4} + 140 b c^{3} x^{3} + 70 c^{4} x^{4} + x^{2} \left (84 a c^{3} + 84 b^{2} c^{2}\right ) + x \left (84 a b c^{2} + 14 b^{3} c\right )}{420 b^{7} c^{3} d^{8} + 5880 b^{6} c^{4} d^{8} x + 35280 b^{5} c^{5} d^{8} x^{2} + 117600 b^{4} c^{6} d^{8} x^{3} + 235200 b^{3} c^{7} d^{8} x^{4} + 282240 b^{2} c^{8} d^{8} x^{5} + 188160 b c^{9} d^{8} x^{6} + 53760 c^{10} d^{8} x^{7}} \]

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)**8,x)

[Out]

-(30*a**2*c**2 + 6*a*b**2*c + b**4 + 140*b*c**3*x**3 + 70*c**4*x**4 + x**2*(84*a
*c**3 + 84*b**2*c**2) + x*(84*a*b*c**2 + 14*b**3*c))/(420*b**7*c**3*d**8 + 5880*
b**6*c**4*d**8*x + 35280*b**5*c**5*d**8*x**2 + 117600*b**4*c**6*d**8*x**3 + 2352
00*b**3*c**7*d**8*x**4 + 282240*b**2*c**8*d**8*x**5 + 188160*b*c**9*d**8*x**6 +
53760*c**10*d**8*x**7)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.213857, size = 117, normalized size = 1.6 \[ -\frac{70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + 84 \, b^{2} c^{2} x^{2} + 84 \, a c^{3} x^{2} + 14 \, b^{3} c x + 84 \, a b c^{2} x + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2}}{420 \,{\left (2 \, c x + b\right )}^{7} c^{3} d^{8}} \]

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)^8,x, algorithm="giac")

[Out]

-1/420*(70*c^4*x^4 + 140*b*c^3*x^3 + 84*b^2*c^2*x^2 + 84*a*c^3*x^2 + 14*b^3*c*x
+ 84*a*b*c^2*x + b^4 + 6*a*b^2*c + 30*a^2*c^2)/((2*c*x + b)^7*c^3*d^8)