3.1755 \(\int \frac{a c+(b c+a d) x+b d x^2}{(a+b x)^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0350269, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{(c+d x)^2}{2 (a+b x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.0948, size = 20, normalized size = 0.71 \[ \frac{\left (c + d x\right )^{2}}{2 \left (a + b x\right )^{2} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.016731, size = 26, normalized size = 0.93 \[ -\frac{a d+b (c+2 d x)}{2 b^2 (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 35, normalized size = 1.3 \[ -{\frac{-ad+bc}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{d}{{b}^{2} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.727923, size = 51, normalized size = 1.82 \[ -\frac{2 \, b d x + b c + a d}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.199302, size = 51, normalized size = 1.82 \[ -\frac{2 \, b d x + b c + a d}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.72876, size = 39, normalized size = 1.39 \[ - \frac{a d + b c + 2 b d x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209275, size = 32, normalized size = 1.14 \[ -\frac{2 \, b d x + b c + a d}{2 \,{\left (b x + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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