3.1106 \(\int \frac{a+b x+c x^2}{(b d+2 c d x)^5} \, dx\)

Optimal. Leaf size=37 \[ \frac{\left (a+b x+c x^2\right )^2}{2 d^5 \left (b^2-4 a c\right ) (b+2 c x)^4} \]

[Out]

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

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Rubi [A]  time = 0.039003, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{\left (a+b x+c x^2\right )^2}{2 d^5 \left (b^2-4 a c\right ) (b+2 c x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.35374, size = 32, normalized size = 0.86 \[ \frac{\left (a + b x + c x^{2}\right )^{2}}{2 d^{5} \left (b + 2 c x\right )^{4} \left (- 4 a c + b^{2}\right )} \]

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)**5,x)

[Out]

(a + b*x + c*x**2)**2/(2*d**5*(b + 2*c*x)**4*(-4*a*c + b**2))

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Mathematica [A]  time = 0.0265915, size = 38, normalized size = 1.03 \[ -\frac{4 c \left (a+2 c x^2\right )+b^2+8 b c x}{32 c^2 d^5 (b+2 c x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^5,x]

[Out]

-(b^2 + 8*b*c*x + 4*c*(a + 2*c*x^2))/(32*c^2*d^5*(b + 2*c*x)^4)

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Maple [A]  time = 0.008, size = 42, normalized size = 1.1 \[{\frac{1}{{d}^{5}} \left ( -{\frac{1}{16\,{c}^{2} \left ( 2\,cx+b \right ) ^{2}}}-{\frac{4\,ac-{b}^{2}}{32\,{c}^{2} \left ( 2\,cx+b \right ) ^{4}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)/(2*c*d*x+b*d)^5,x)

[Out]

1/d^5*(-1/16/c^2/(2*c*x+b)^2-1/32*(4*a*c-b^2)/c^2/(2*c*x+b)^4)

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Maxima [A]  time = 0.689551, size = 115, normalized size = 3.11 \[ -\frac{8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c}{32 \,{\left (16 \, c^{6} d^{5} x^{4} + 32 \, b c^{5} d^{5} x^{3} + 24 \, b^{2} c^{4} d^{5} x^{2} + 8 \, b^{3} c^{3} d^{5} x + b^{4} c^{2} d^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.207861, size = 115, normalized size = 3.11 \[ -\frac{8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c}{32 \,{\left (16 \, c^{6} d^{5} x^{4} + 32 \, b c^{5} d^{5} x^{3} + 24 \, b^{2} c^{4} d^{5} x^{2} + 8 \, b^{3} c^{3} d^{5} x + b^{4} c^{2} d^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.96659, size = 90, normalized size = 2.43 \[ - \frac{4 a c + b^{2} + 8 b c x + 8 c^{2} x^{2}}{32 b^{4} c^{2} d^{5} + 256 b^{3} c^{3} d^{5} x + 768 b^{2} c^{4} d^{5} x^{2} + 1024 b c^{5} d^{5} x^{3} + 512 c^{6} d^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**5,x)

[Out]

-(4*a*c + b**2 + 8*b*c*x + 8*c**2*x**2)/(32*b**4*c**2*d**5 + 256*b**3*c**3*d**5*
x + 768*b**2*c**4*d**5*x**2 + 1024*b*c**5*d**5*x**3 + 512*c**6*d**5*x**4)

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GIAC/XCAS [A]  time = 0.215177, size = 84, normalized size = 2.27 \[ \frac{\frac{b^{2} d}{{\left (2 \, c d x + b d\right )}^{4} c^{2}} - \frac{4 \, a d}{{\left (2 \, c d x + b d\right )}^{4} c} - \frac{2}{{\left (2 \, c d x + b d\right )}^{2} c^{2} d}}{32 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^5,x, algorithm="giac")

[Out]

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