3.1120 \(\int \frac{\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx\)

Optimal. Leaf size=73 \[ -\frac{\left (b^2-4 a c\right )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac{b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac{1}{32 c^3 d^6 (b+2 c x)} \]

[Out]

-(b^2 - 4*a*c)^2/(160*c^3*d^6*(b + 2*c*x)^5) + (b^2 - 4*a*c)/(48*c^3*d^6*(b + 2*
c*x)^3) - 1/(32*c^3*d^6*(b + 2*c*x))

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Rubi [A]  time = 0.139244, 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 \[ -\frac{\left (b^2-4 a c\right )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac{b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac{1}{32 c^3 d^6 (b+2 c x)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^6,x]

[Out]

-(b^2 - 4*a*c)^2/(160*c^3*d^6*(b + 2*c*x)^5) + (b^2 - 4*a*c)/(48*c^3*d^6*(b + 2*
c*x)^3) - 1/(32*c^3*d^6*(b + 2*c*x))

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Rubi in Sympy [A]  time = 31.7652, size = 68, normalized size = 0.93 \[ - \frac{1}{32 c^{3} d^{6} \left (b + 2 c x\right )} + \frac{- 4 a c + b^{2}}{48 c^{3} d^{6} \left (b + 2 c x\right )^{3}} - \frac{\left (- 4 a c + b^{2}\right )^{2}}{160 c^{3} d^{6} \left (b + 2 c x\right )^{5}} \]

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)**6,x)

[Out]

-1/(32*c**3*d**6*(b + 2*c*x)) + (-4*a*c + b**2)/(48*c**3*d**6*(b + 2*c*x)**3) -
(-4*a*c + b**2)**2/(160*c**3*d**6*(b + 2*c*x)**5)

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Mathematica [A]  time = 0.067599, size = 59, normalized size = 0.81 \[ \frac{10 \left (b^2-4 a c\right ) (b+2 c x)^2-3 \left (b^2-4 a c\right )^2-15 (b+2 c x)^4}{480 c^3 d^6 (b+2 c x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^6,x]

[Out]

(-3*(b^2 - 4*a*c)^2 + 10*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 15*(b + 2*c*x)^4)/(480*c^
3*d^6*(b + 2*c*x)^5)

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Maple [A]  time = 0.009, size = 74, normalized size = 1. \[{\frac{1}{{d}^{6}} \left ( -{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{160\,{c}^{3} \left ( 2\,cx+b \right ) ^{5}}}-{\frac{4\,ac-{b}^{2}}{48\,{c}^{3} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{1}{32\,{c}^{3} \left ( 2\,cx+b \right ) }} \right ) } \]

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)^6,x)

[Out]

1/d^6*(-1/160*(16*a^2*c^2-8*a*b^2*c+b^4)/c^3/(2*c*x+b)^5-1/48*(4*a*c-b^2)/c^3/(2
*c*x+b)^3-1/32/c^3/(2*c*x+b))

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Maxima [A]  time = 0.699996, size = 201, normalized size = 2.75 \[ -\frac{30 \, c^{4} x^{4} + 60 \, b c^{3} x^{3} + b^{4} + 2 \, a b^{2} c + 6 \, a^{2} c^{2} + 20 \,{\left (2 \, b^{2} c^{2} + a c^{3}\right )} x^{2} + 10 \,{\left (b^{3} c + 2 \, a b c^{2}\right )} x}{60 \,{\left (32 \, c^{8} d^{6} x^{5} + 80 \, b c^{7} d^{6} x^{4} + 80 \, b^{2} c^{6} d^{6} x^{3} + 40 \, b^{3} c^{5} d^{6} x^{2} + 10 \, b^{4} c^{4} d^{6} x + b^{5} c^{3} d^{6}\right )}} \]

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)^6,x, algorithm="maxima")

[Out]

-1/60*(30*c^4*x^4 + 60*b*c^3*x^3 + b^4 + 2*a*b^2*c + 6*a^2*c^2 + 20*(2*b^2*c^2 +
a*c^3)*x^2 + 10*(b^3*c + 2*a*b*c^2)*x)/(32*c^8*d^6*x^5 + 80*b*c^7*d^6*x^4 + 80*
b^2*c^6*d^6*x^3 + 40*b^3*c^5*d^6*x^2 + 10*b^4*c^4*d^6*x + b^5*c^3*d^6)

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Fricas [A]  time = 0.20492, size = 201, normalized size = 2.75 \[ -\frac{30 \, c^{4} x^{4} + 60 \, b c^{3} x^{3} + b^{4} + 2 \, a b^{2} c + 6 \, a^{2} c^{2} + 20 \,{\left (2 \, b^{2} c^{2} + a c^{3}\right )} x^{2} + 10 \,{\left (b^{3} c + 2 \, a b c^{2}\right )} x}{60 \,{\left (32 \, c^{8} d^{6} x^{5} + 80 \, b c^{7} d^{6} x^{4} + 80 \, b^{2} c^{6} d^{6} x^{3} + 40 \, b^{3} c^{5} d^{6} x^{2} + 10 \, b^{4} c^{4} d^{6} x + b^{5} c^{3} d^{6}\right )}} \]

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)^6,x, algorithm="fricas")

[Out]

-1/60*(30*c^4*x^4 + 60*b*c^3*x^3 + b^4 + 2*a*b^2*c + 6*a^2*c^2 + 20*(2*b^2*c^2 +
a*c^3)*x^2 + 10*(b^3*c + 2*a*b*c^2)*x)/(32*c^8*d^6*x^5 + 80*b*c^7*d^6*x^4 + 80*
b^2*c^6*d^6*x^3 + 40*b^3*c^5*d^6*x^2 + 10*b^4*c^4*d^6*x + b^5*c^3*d^6)

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Sympy [A]  time = 6.31246, size = 156, normalized size = 2.14 \[ - \frac{6 a^{2} c^{2} + 2 a b^{2} c + b^{4} + 60 b c^{3} x^{3} + 30 c^{4} x^{4} + x^{2} \left (20 a c^{3} + 40 b^{2} c^{2}\right ) + x \left (20 a b c^{2} + 10 b^{3} c\right )}{60 b^{5} c^{3} d^{6} + 600 b^{4} c^{4} d^{6} x + 2400 b^{3} c^{5} d^{6} x^{2} + 4800 b^{2} c^{6} d^{6} x^{3} + 4800 b c^{7} d^{6} x^{4} + 1920 c^{8} d^{6} x^{5}} \]

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)**6,x)

[Out]

-(6*a**2*c**2 + 2*a*b**2*c + b**4 + 60*b*c**3*x**3 + 30*c**4*x**4 + x**2*(20*a*c
**3 + 40*b**2*c**2) + x*(20*a*b*c**2 + 10*b**3*c))/(60*b**5*c**3*d**6 + 600*b**4
*c**4*d**6*x + 2400*b**3*c**5*d**6*x**2 + 4800*b**2*c**6*d**6*x**3 + 4800*b*c**7
*d**6*x**4 + 1920*c**8*d**6*x**5)

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GIAC/XCAS [A]  time = 0.213128, size = 117, normalized size = 1.6 \[ -\frac{30 \, c^{4} x^{4} + 60 \, b c^{3} x^{3} + 40 \, b^{2} c^{2} x^{2} + 20 \, a c^{3} x^{2} + 10 \, b^{3} c x + 20 \, a b c^{2} x + b^{4} + 2 \, a b^{2} c + 6 \, a^{2} c^{2}}{60 \,{\left (2 \, c x + b\right )}^{5} c^{3} d^{6}} \]

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)^6,x, algorithm="giac")

[Out]

-1/60*(30*c^4*x^4 + 60*b*c^3*x^3 + 40*b^2*c^2*x^2 + 20*a*c^3*x^2 + 10*b^3*c*x +
20*a*b*c^2*x + b^4 + 2*a*b^2*c + 6*a^2*c^2)/((2*c*x + b)^5*c^3*d^6)