3.1107 \(\int \frac{a+b x+c x^2}{(b d+2 c d x)^6} \, dx\)

Optimal. Leaf size=45 \[ \frac{b^2-4 a c}{40 c^2 d^6 (b+2 c x)^5}-\frac{1}{24 c^2 d^6 (b+2 c x)^3} \]

[Out]

(b^2 - 4*a*c)/(40*c^2*d^6*(b + 2*c*x)^5) - 1/(24*c^2*d^6*(b + 2*c*x)^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.0846704, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{b^2-4 a c}{40 c^2 d^6 (b+2 c x)^5}-\frac{1}{24 c^2 d^6 (b+2 c x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(b^2 - 4*a*c)/(40*c^2*d^6*(b + 2*c*x)^5) - 1/(24*c^2*d^6*(b + 2*c*x)^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.8256, size = 42, normalized size = 0.93 \[ - \frac{1}{24 c^{2} d^{6} \left (b + 2 c x\right )^{3}} + \frac{- a c + \frac{b^{2}}{4}}{10 c^{2} d^{6} \left (b + 2 c x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(24*c**2*d**6*(b + 2*c*x)**3) + (-a*c + b**2/4)/(10*c**2*d**6*(b + 2*c*x)**5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0311865, size = 43, normalized size = 0.96 \[ \frac{\frac{b^2-4 a c}{40 c^2 (b+2 c x)^5}-\frac{1}{24 c^2 (b+2 c x)^3}}{d^6} \]

Antiderivative was successfully verified.

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

[Out]

((b^2 - 4*a*c)/(40*c^2*(b + 2*c*x)^5) - 1/(24*c^2*(b + 2*c*x)^3))/d^6

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 42, normalized size = 0.9 \[{\frac{1}{{d}^{6}} \left ( -{\frac{4\,ac-{b}^{2}}{40\,{c}^{2} \left ( 2\,cx+b \right ) ^{5}}}-{\frac{1}{24\,{c}^{2} \left ( 2\,cx+b \right ) ^{3}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/d^6*(-1/40*(4*a*c-b^2)/c^2/(2*c*x+b)^5-1/24/c^2/(2*c*x+b)^3)

_______________________________________________________________________________________

Maxima [A]  time = 0.687037, size = 134, normalized size = 2.98 \[ -\frac{10 \, c^{2} x^{2} + 10 \, b c x + b^{2} + 6 \, a c}{60 \,{\left (32 \, c^{7} d^{6} x^{5} + 80 \, b c^{6} d^{6} x^{4} + 80 \, b^{2} c^{5} d^{6} x^{3} + 40 \, b^{3} c^{4} d^{6} x^{2} + 10 \, b^{4} c^{3} d^{6} x + b^{5} c^{2} d^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^6,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.200036, size = 134, normalized size = 2.98 \[ -\frac{10 \, c^{2} x^{2} + 10 \, b c x + b^{2} + 6 \, a c}{60 \,{\left (32 \, c^{7} d^{6} x^{5} + 80 \, b c^{6} d^{6} x^{4} + 80 \, b^{2} c^{5} d^{6} x^{3} + 40 \, b^{3} c^{4} d^{6} x^{2} + 10 \, b^{4} c^{3} d^{6} x + b^{5} c^{2} d^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^6,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.11249, size = 105, normalized size = 2.33 \[ - \frac{6 a c + b^{2} + 10 b c x + 10 c^{2} x^{2}}{60 b^{5} c^{2} d^{6} + 600 b^{4} c^{3} d^{6} x + 2400 b^{3} c^{4} d^{6} x^{2} + 4800 b^{2} c^{5} d^{6} x^{3} + 4800 b c^{6} d^{6} x^{4} + 1920 c^{7} d^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212229, size = 50, normalized size = 1.11 \[ -\frac{10 \, c^{2} x^{2} + 10 \, b c x + b^{2} + 6 \, a c}{60 \,{\left (2 \, c x + b\right )}^{5} c^{2} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^6,x, algorithm="giac")

[Out]

-1/60*(10*c^2*x^2 + 10*b*c*x + b^2 + 6*a*c)/((2*c*x + b)^5*c^2*d^6)