3.1109 \(\int \frac{a+b x+c x^2}{(b d+2 c d x)^8} \, dx\)

Optimal. Leaf size=45 \[ \frac{b^2-4 a c}{56 c^2 d^8 (b+2 c x)^7}-\frac{1}{40 c^2 d^8 (b+2 c x)^5} \]

[Out]

(b^2 - 4*a*c)/(56*c^2*d^8*(b + 2*c*x)^7) - 1/(40*c^2*d^8*(b + 2*c*x)^5)

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Rubi [A]  time = 0.0815275, 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 \[ \frac{b^2-4 a c}{56 c^2 d^8 (b+2 c x)^7}-\frac{1}{40 c^2 d^8 (b+2 c x)^5} \]

Antiderivative was successfully verified.

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

[Out]

(b^2 - 4*a*c)/(56*c^2*d^8*(b + 2*c*x)^7) - 1/(40*c^2*d^8*(b + 2*c*x)^5)

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Rubi in Sympy [A]  time = 17.7967, size = 42, normalized size = 0.93 \[ - \frac{1}{40 c^{2} d^{8} \left (b + 2 c x\right )^{5}} + \frac{- a c + \frac{b^{2}}{4}}{14 c^{2} d^{8} \left (b + 2 c x\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(40*c**2*d**8*(b + 2*c*x)**5) + (-a*c + b**2/4)/(14*c**2*d**8*(b + 2*c*x)**7)

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Mathematica [A]  time = 0.0275477, size = 39, normalized size = 0.87 \[ \frac{5 \left (b^2-4 a c\right )-7 (b+2 c x)^2}{280 c^2 d^8 (b+2 c x)^7} \]

Antiderivative was successfully verified.

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

[Out]

(5*(b^2 - 4*a*c) - 7*(b + 2*c*x)^2)/(280*c^2*d^8*(b + 2*c*x)^7)

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Maple [A]  time = 0.007, size = 42, normalized size = 0.9 \[{\frac{1}{{d}^{8}} \left ( -{\frac{1}{40\,{c}^{2} \left ( 2\,cx+b \right ) ^{5}}}-{\frac{4\,ac-{b}^{2}}{56\,{c}^{2} \left ( 2\,cx+b \right ) ^{7}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/d^8*(-1/40/c^2/(2*c*x+b)^5-1/56*(4*a*c-b^2)/c^2/(2*c*x+b)^7)

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Maxima [A]  time = 0.689621, size = 171, normalized size = 3.8 \[ -\frac{14 \, c^{2} x^{2} + 14 \, b c x + b^{2} + 10 \, a c}{140 \,{\left (128 \, c^{9} d^{8} x^{7} + 448 \, b c^{8} d^{8} x^{6} + 672 \, b^{2} c^{7} d^{8} x^{5} + 560 \, b^{3} c^{6} d^{8} x^{4} + 280 \, b^{4} c^{5} d^{8} x^{3} + 84 \, b^{5} c^{4} d^{8} x^{2} + 14 \, b^{6} c^{3} d^{8} x + b^{7} c^{2} d^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^8,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.211773, size = 171, normalized size = 3.8 \[ -\frac{14 \, c^{2} x^{2} + 14 \, b c x + b^{2} + 10 \, a c}{140 \,{\left (128 \, c^{9} d^{8} x^{7} + 448 \, b c^{8} d^{8} x^{6} + 672 \, b^{2} c^{7} d^{8} x^{5} + 560 \, b^{3} c^{6} d^{8} x^{4} + 280 \, b^{4} c^{5} d^{8} x^{3} + 84 \, b^{5} c^{4} d^{8} x^{2} + 14 \, b^{6} c^{3} d^{8} x + b^{7} c^{2} d^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^8,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.13545, size = 136, normalized size = 3.02 \[ - \frac{10 a c + b^{2} + 14 b c x + 14 c^{2} x^{2}}{140 b^{7} c^{2} d^{8} + 1960 b^{6} c^{3} d^{8} x + 11760 b^{5} c^{4} d^{8} x^{2} + 39200 b^{4} c^{5} d^{8} x^{3} + 78400 b^{3} c^{6} d^{8} x^{4} + 94080 b^{2} c^{7} d^{8} x^{5} + 62720 b c^{8} d^{8} x^{6} + 17920 c^{9} d^{8} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(10*a*c + b**2 + 14*b*c*x + 14*c**2*x**2)/(140*b**7*c**2*d**8 + 1960*b**6*c**3*
d**8*x + 11760*b**5*c**4*d**8*x**2 + 39200*b**4*c**5*d**8*x**3 + 78400*b**3*c**6
*d**8*x**4 + 94080*b**2*c**7*d**8*x**5 + 62720*b*c**8*d**8*x**6 + 17920*c**9*d**
8*x**7)

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GIAC/XCAS [A]  time = 0.211122, size = 50, normalized size = 1.11 \[ -\frac{14 \, c^{2} x^{2} + 14 \, b c x + b^{2} + 10 \, a c}{140 \,{\left (2 \, c x + b\right )}^{7} c^{2} d^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^8,x, algorithm="giac")

[Out]

-1/140*(14*c^2*x^2 + 14*b*c*x + b^2 + 10*a*c)/((2*c*x + b)^7*c^2*d^8)