3.1123 \(\int \frac{\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^9} \, dx\)

Optimal. Leaf size=73 \[ -\frac{\left (b^2-4 a c\right )^2}{256 c^3 d^9 (b+2 c x)^8}+\frac{b^2-4 a c}{96 c^3 d^9 (b+2 c x)^6}-\frac{1}{128 c^3 d^9 (b+2 c x)^4} \]

[Out]

-(b^2 - 4*a*c)^2/(256*c^3*d^9*(b + 2*c*x)^8) + (b^2 - 4*a*c)/(96*c^3*d^9*(b + 2*
c*x)^6) - 1/(128*c^3*d^9*(b + 2*c*x)^4)

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Rubi [A]  time = 0.139339, 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 \[ -\frac{\left (b^2-4 a c\right )^2}{256 c^3 d^9 (b+2 c x)^8}+\frac{b^2-4 a c}{96 c^3 d^9 (b+2 c x)^6}-\frac{1}{128 c^3 d^9 (b+2 c x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^9,x]

[Out]

-(b^2 - 4*a*c)^2/(256*c^3*d^9*(b + 2*c*x)^8) + (b^2 - 4*a*c)/(96*c^3*d^9*(b + 2*
c*x)^6) - 1/(128*c^3*d^9*(b + 2*c*x)^4)

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Rubi in Sympy [A]  time = 32.7456, size = 70, normalized size = 0.96 \[ - \frac{1}{128 c^{3} d^{9} \left (b + 2 c x\right )^{4}} + \frac{- 4 a c + b^{2}}{96 c^{3} d^{9} \left (b + 2 c x\right )^{6}} - \frac{\left (- 4 a c + b^{2}\right )^{2}}{256 c^{3} d^{9} \left (b + 2 c x\right )^{8}} \]

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)**9,x)

[Out]

-1/(128*c**3*d**9*(b + 2*c*x)**4) + (-4*a*c + b**2)/(96*c**3*d**9*(b + 2*c*x)**6
) - (-4*a*c + b**2)**2/(256*c**3*d**9*(b + 2*c*x)**8)

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Mathematica [A]  time = 0.0554329, size = 59, normalized size = 0.81 \[ \frac{8 \left (b^2-4 a c\right ) (b+2 c x)^2-3 \left (b^2-4 a c\right )^2-6 (b+2 c x)^4}{768 c^3 d^9 (b+2 c x)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^9,x]

[Out]

(-3*(b^2 - 4*a*c)^2 + 8*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 6*(b + 2*c*x)^4)/(768*c^3*
d^9*(b + 2*c*x)^8)

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Maple [A]  time = 0.007, size = 74, normalized size = 1. \[{\frac{1}{{d}^{9}} \left ( -{\frac{1}{128\,{c}^{3} \left ( 2\,cx+b \right ) ^{4}}}-{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{256\,{c}^{3} \left ( 2\,cx+b \right ) ^{8}}}-{\frac{4\,ac-{b}^{2}}{96\,{c}^{3} \left ( 2\,cx+b \right ) ^{6}}} \right ) } \]

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)^9,x)

[Out]

1/d^9*(-1/128/c^3/(2*c*x+b)^4-1/256*(16*a^2*c^2-8*a*b^2*c+b^4)/c^3/(2*c*x+b)^8-1
/96*(4*a*c-b^2)/c^3/(2*c*x+b)^6)

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Maxima [A]  time = 0.710437, size = 259, normalized size = 3.55 \[ -\frac{96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + b^{4} + 8 \, a b^{2} c + 48 \, a^{2} c^{2} + 16 \,{\left (7 \, b^{2} c^{2} + 8 \, a c^{3}\right )} x^{2} + 16 \,{\left (b^{3} c + 8 \, a b c^{2}\right )} x}{768 \,{\left (256 \, c^{11} d^{9} x^{8} + 1024 \, b c^{10} d^{9} x^{7} + 1792 \, b^{2} c^{9} d^{9} x^{6} + 1792 \, b^{3} c^{8} d^{9} x^{5} + 1120 \, b^{4} c^{7} d^{9} x^{4} + 448 \, b^{5} c^{6} d^{9} x^{3} + 112 \, b^{6} c^{5} d^{9} x^{2} + 16 \, b^{7} c^{4} d^{9} x + b^{8} c^{3} d^{9}\right )}} \]

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)^9,x, algorithm="maxima")

[Out]

-1/768*(96*c^4*x^4 + 192*b*c^3*x^3 + b^4 + 8*a*b^2*c + 48*a^2*c^2 + 16*(7*b^2*c^
2 + 8*a*c^3)*x^2 + 16*(b^3*c + 8*a*b*c^2)*x)/(256*c^11*d^9*x^8 + 1024*b*c^10*d^9
*x^7 + 1792*b^2*c^9*d^9*x^6 + 1792*b^3*c^8*d^9*x^5 + 1120*b^4*c^7*d^9*x^4 + 448*
b^5*c^6*d^9*x^3 + 112*b^6*c^5*d^9*x^2 + 16*b^7*c^4*d^9*x + b^8*c^3*d^9)

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Fricas [A]  time = 0.220274, size = 259, normalized size = 3.55 \[ -\frac{96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + b^{4} + 8 \, a b^{2} c + 48 \, a^{2} c^{2} + 16 \,{\left (7 \, b^{2} c^{2} + 8 \, a c^{3}\right )} x^{2} + 16 \,{\left (b^{3} c + 8 \, a b c^{2}\right )} x}{768 \,{\left (256 \, c^{11} d^{9} x^{8} + 1024 \, b c^{10} d^{9} x^{7} + 1792 \, b^{2} c^{9} d^{9} x^{6} + 1792 \, b^{3} c^{8} d^{9} x^{5} + 1120 \, b^{4} c^{7} d^{9} x^{4} + 448 \, b^{5} c^{6} d^{9} x^{3} + 112 \, b^{6} c^{5} d^{9} x^{2} + 16 \, b^{7} c^{4} d^{9} x + b^{8} c^{3} d^{9}\right )}} \]

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)^9,x, algorithm="fricas")

[Out]

-1/768*(96*c^4*x^4 + 192*b*c^3*x^3 + b^4 + 8*a*b^2*c + 48*a^2*c^2 + 16*(7*b^2*c^
2 + 8*a*c^3)*x^2 + 16*(b^3*c + 8*a*b*c^2)*x)/(256*c^11*d^9*x^8 + 1024*b*c^10*d^9
*x^7 + 1792*b^2*c^9*d^9*x^6 + 1792*b^3*c^8*d^9*x^5 + 1120*b^4*c^7*d^9*x^4 + 448*
b^5*c^6*d^9*x^3 + 112*b^6*c^5*d^9*x^2 + 16*b^7*c^4*d^9*x + b^8*c^3*d^9)

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Sympy [A]  time = 15.4375, size = 202, normalized size = 2.77 \[ - \frac{48 a^{2} c^{2} + 8 a b^{2} c + b^{4} + 192 b c^{3} x^{3} + 96 c^{4} x^{4} + x^{2} \left (128 a c^{3} + 112 b^{2} c^{2}\right ) + x \left (128 a b c^{2} + 16 b^{3} c\right )}{768 b^{8} c^{3} d^{9} + 12288 b^{7} c^{4} d^{9} x + 86016 b^{6} c^{5} d^{9} x^{2} + 344064 b^{5} c^{6} d^{9} x^{3} + 860160 b^{4} c^{7} d^{9} x^{4} + 1376256 b^{3} c^{8} d^{9} x^{5} + 1376256 b^{2} c^{9} d^{9} x^{6} + 786432 b c^{10} d^{9} x^{7} + 196608 c^{11} d^{9} x^{8}} \]

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)**9,x)

[Out]

-(48*a**2*c**2 + 8*a*b**2*c + b**4 + 192*b*c**3*x**3 + 96*c**4*x**4 + x**2*(128*
a*c**3 + 112*b**2*c**2) + x*(128*a*b*c**2 + 16*b**3*c))/(768*b**8*c**3*d**9 + 12
288*b**7*c**4*d**9*x + 86016*b**6*c**5*d**9*x**2 + 344064*b**5*c**6*d**9*x**3 +
860160*b**4*c**7*d**9*x**4 + 1376256*b**3*c**8*d**9*x**5 + 1376256*b**2*c**9*d**
9*x**6 + 786432*b*c**10*d**9*x**7 + 196608*c**11*d**9*x**8)

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GIAC/XCAS [A]  time = 0.213459, size = 117, normalized size = 1.6 \[ -\frac{96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + 112 \, b^{2} c^{2} x^{2} + 128 \, a c^{3} x^{2} + 16 \, b^{3} c x + 128 \, a b c^{2} x + b^{4} + 8 \, a b^{2} c + 48 \, a^{2} c^{2}}{768 \,{\left (2 \, c x + b\right )}^{8} c^{3} d^{9}} \]

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)^9,x, algorithm="giac")

[Out]

-1/768*(96*c^4*x^4 + 192*b*c^3*x^3 + 112*b^2*c^2*x^2 + 128*a*c^3*x^2 + 16*b^3*c*
x + 128*a*b*c^2*x + b^4 + 8*a*b^2*c + 48*a^2*c^2)/((2*c*x + b)^8*c^3*d^9)