3.1108 \(\int \frac{a+b x+c x^2}{(b d+2 c d x)^7} \, dx\)

Optimal. Leaf size=45 \[ \frac{b^2-4 a c}{48 c^2 d^7 (b+2 c x)^6}-\frac{1}{32 c^2 d^7 (b+2 c x)^4} \]

[Out]

(b^2 - 4*a*c)/(48*c^2*d^7*(b + 2*c*x)^6) - 1/(32*c^2*d^7*(b + 2*c*x)^4)

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Rubi [A]  time = 0.0843846, 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 \[ \frac{b^2-4 a c}{48 c^2 d^7 (b+2 c x)^6}-\frac{1}{32 c^2 d^7 (b+2 c x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^7,x]

[Out]

(b^2 - 4*a*c)/(48*c^2*d^7*(b + 2*c*x)^6) - 1/(32*c^2*d^7*(b + 2*c*x)^4)

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Rubi in Sympy [A]  time = 17.1447, size = 42, normalized size = 0.93 \[ - \frac{1}{32 c^{2} d^{7} \left (b + 2 c x\right )^{4}} + \frac{- a c + \frac{b^{2}}{4}}{12 c^{2} d^{7} \left (b + 2 c x\right )^{6}} \]

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)**7,x)

[Out]

-1/(32*c**2*d**7*(b + 2*c*x)**4) + (-a*c + b**2/4)/(12*c**2*d**7*(b + 2*c*x)**6)

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Mathematica [A]  time = 0.0249318, size = 43, normalized size = 0.96 \[ \frac{\frac{b^2-4 a c}{48 c^2 (b+2 c x)^6}-\frac{1}{32 c^2 (b+2 c x)^4}}{d^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^7,x]

[Out]

((b^2 - 4*a*c)/(48*c^2*(b + 2*c*x)^6) - 1/(32*c^2*(b + 2*c*x)^4))/d^7

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Maple [A]  time = 0.007, size = 42, normalized size = 0.9 \[{\frac{1}{{d}^{7}} \left ( -{\frac{1}{32\,{c}^{2} \left ( 2\,cx+b \right ) ^{4}}}-{\frac{4\,ac-{b}^{2}}{48\,{c}^{2} \left ( 2\,cx+b \right ) ^{6}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)/(2*c*d*x+b*d)^7,x)

[Out]

1/d^7*(-1/32/c^2/(2*c*x+b)^4-1/48*(4*a*c-b^2)/c^2/(2*c*x+b)^6)

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Maxima [A]  time = 0.68909, size = 153, normalized size = 3.4 \[ -\frac{12 \, c^{2} x^{2} + 12 \, b c x + b^{2} + 8 \, a c}{96 \,{\left (64 \, c^{8} d^{7} x^{6} + 192 \, b c^{7} d^{7} x^{5} + 240 \, b^{2} c^{6} d^{7} x^{4} + 160 \, b^{3} c^{5} d^{7} x^{3} + 60 \, b^{4} c^{4} d^{7} x^{2} + 12 \, b^{5} c^{3} d^{7} x + b^{6} c^{2} d^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^7,x, algorithm="maxima")

[Out]

-1/96*(12*c^2*x^2 + 12*b*c*x + b^2 + 8*a*c)/(64*c^8*d^7*x^6 + 192*b*c^7*d^7*x^5
+ 240*b^2*c^6*d^7*x^4 + 160*b^3*c^5*d^7*x^3 + 60*b^4*c^4*d^7*x^2 + 12*b^5*c^3*d^
7*x + b^6*c^2*d^7)

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Fricas [A]  time = 0.205423, size = 153, normalized size = 3.4 \[ -\frac{12 \, c^{2} x^{2} + 12 \, b c x + b^{2} + 8 \, a c}{96 \,{\left (64 \, c^{8} d^{7} x^{6} + 192 \, b c^{7} d^{7} x^{5} + 240 \, b^{2} c^{6} d^{7} x^{4} + 160 \, b^{3} c^{5} d^{7} x^{3} + 60 \, b^{4} c^{4} d^{7} x^{2} + 12 \, b^{5} c^{3} d^{7} x + b^{6} c^{2} d^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^7,x, algorithm="fricas")

[Out]

-1/96*(12*c^2*x^2 + 12*b*c*x + b^2 + 8*a*c)/(64*c^8*d^7*x^6 + 192*b*c^7*d^7*x^5
+ 240*b^2*c^6*d^7*x^4 + 160*b^3*c^5*d^7*x^3 + 60*b^4*c^4*d^7*x^2 + 12*b^5*c^3*d^
7*x + b^6*c^2*d^7)

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Sympy [A]  time = 4.06013, size = 121, normalized size = 2.69 \[ - \frac{8 a c + b^{2} + 12 b c x + 12 c^{2} x^{2}}{96 b^{6} c^{2} d^{7} + 1152 b^{5} c^{3} d^{7} x + 5760 b^{4} c^{4} d^{7} x^{2} + 15360 b^{3} c^{5} d^{7} x^{3} + 23040 b^{2} c^{6} d^{7} x^{4} + 18432 b c^{7} d^{7} x^{5} + 6144 c^{8} d^{7} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**7,x)

[Out]

-(8*a*c + b**2 + 12*b*c*x + 12*c**2*x**2)/(96*b**6*c**2*d**7 + 1152*b**5*c**3*d*
*7*x + 5760*b**4*c**4*d**7*x**2 + 15360*b**3*c**5*d**7*x**3 + 23040*b**2*c**6*d*
*7*x**4 + 18432*b*c**7*d**7*x**5 + 6144*c**8*d**7*x**6)

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GIAC/XCAS [A]  time = 0.212065, size = 50, normalized size = 1.11 \[ -\frac{12 \, c^{2} x^{2} + 12 \, b c x + b^{2} + 8 \, a c}{96 \,{\left (2 \, c x + b\right )}^{6} c^{2} d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^7,x, algorithm="giac")

[Out]

-1/96*(12*c^2*x^2 + 12*b*c*x + b^2 + 8*a*c)/((2*c*x + b)^6*c^2*d^7)