3.1811 \(\int \frac{(a+b x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx\)

Optimal. Leaf size=28 \[ \frac{(a+b x)^2}{2 (c+d x)^2 (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0313347, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{(a+b x)^2}{2 (c+d x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.47573, size = 22, normalized size = 0.79 \[ - \frac{\left (a + b x\right )^{2}}{2 \left (c + d x\right )^{2} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**4/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0172292, size = 26, normalized size = 0.93 \[ -\frac{a d+b (c+2 d x)}{2 d^2 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.007, size = 35, normalized size = 1.3 \[ -{\frac{b}{{d}^{2} \left ( dx+c \right ) }}-{\frac{ad-bc}{2\,{d}^{2} \left ( dx+c \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^4/(a*c+(a*d+b*c)*x+x^2*b*d)^3,x)

[Out]

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

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Maxima [A]  time = 0.763088, size = 51, normalized size = 1.82 \[ -\frac{2 \, b d x + b c + a d}{2 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.200592, size = 51, normalized size = 1.82 \[ -\frac{2 \, b d x + b c + a d}{2 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.87418, size = 39, normalized size = 1.39 \[ - \frac{a d + b c + 2 b d x}{2 c^{2} d^{2} + 4 c d^{3} x + 2 d^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**4/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)

[Out]

-(a*d + b*c + 2*b*d*x)/(2*c**2*d**2 + 4*c*d**3*x + 2*d**4*x**2)

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GIAC/XCAS [A]  time = 0.214342, size = 32, normalized size = 1.14 \[ -\frac{2 \, b d x + b c + a d}{2 \,{\left (d x + c\right )}^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4/(b*d*x^2 + a*c + (b*c + a*d)*x)^3,x, algorithm="giac")

[Out]

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