3.649 \(\int \frac{A+B x}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=38 \[ -\frac{A b-a B}{5 b^2 (a+b x)^5}-\frac{B}{4 b^2 (a+b x)^4} \]

[Out]

-(A*b - a*B)/(5*b^2*(a + b*x)^5) - B/(4*b^2*(a + b*x)^4)

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Rubi [A]  time = 0.0553206, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{A b-a B}{5 b^2 (a+b x)^5}-\frac{B}{4 b^2 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(A*b - a*B)/(5*b^2*(a + b*x)^5) - B/(4*b^2*(a + b*x)^4)

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Rubi in Sympy [A]  time = 18.6347, size = 32, normalized size = 0.84 \[ - \frac{B}{4 b^{2} \left (a + b x\right )^{4}} - \frac{A b - B a}{5 b^{2} \left (a + b x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

-B/(4*b**2*(a + b*x)**4) - (A*b - B*a)/(5*b**2*(a + b*x)**5)

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Mathematica [A]  time = 0.0186675, size = 27, normalized size = 0.71 \[ -\frac{B (a+5 b x)+4 A b}{20 b^2 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(4*A*b + B*(a + 5*b*x))/(20*b^2*(a + b*x)^5)

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Maple [A]  time = 0.01, size = 35, normalized size = 0.9 \[ -{\frac{Ab-Ba}{5\,{b}^{2} \left ( bx+a \right ) ^{5}}}-{\frac{B}{4\,{b}^{2} \left ( bx+a \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-1/5*(A*b-B*a)/b^2/(b*x+a)^5-1/4*B/b^2/(b*x+a)^4

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Maxima [A]  time = 0.679854, size = 97, normalized size = 2.55 \[ -\frac{5 \, B b x + B a + 4 \, A b}{20 \,{\left (b^{7} x^{5} + 5 \, a b^{6} x^{4} + 10 \, a^{2} b^{5} x^{3} + 10 \, a^{3} b^{4} x^{2} + 5 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(5*B*b*x + B*a + 4*A*b)/(b^7*x^5 + 5*a*b^6*x^4 + 10*a^2*b^5*x^3 + 10*a^3*b
^4*x^2 + 5*a^4*b^3*x + a^5*b^2)

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Fricas [A]  time = 0.257385, size = 97, normalized size = 2.55 \[ -\frac{5 \, B b x + B a + 4 \, A b}{20 \,{\left (b^{7} x^{5} + 5 \, a b^{6} x^{4} + 10 \, a^{2} b^{5} x^{3} + 10 \, a^{3} b^{4} x^{2} + 5 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(5*B*b*x + B*a + 4*A*b)/(b^7*x^5 + 5*a*b^6*x^4 + 10*a^2*b^5*x^3 + 10*a^3*b
^4*x^2 + 5*a^4*b^3*x + a^5*b^2)

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Sympy [A]  time = 2.73942, size = 76, normalized size = 2. \[ - \frac{4 A b + B a + 5 B b x}{20 a^{5} b^{2} + 100 a^{4} b^{3} x + 200 a^{3} b^{4} x^{2} + 200 a^{2} b^{5} x^{3} + 100 a b^{6} x^{4} + 20 b^{7} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

-(4*A*b + B*a + 5*B*b*x)/(20*a**5*b**2 + 100*a**4*b**3*x + 200*a**3*b**4*x**2 +
200*a**2*b**5*x**3 + 100*a*b**6*x**4 + 20*b**7*x**5)

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GIAC/XCAS [A]  time = 0.270793, size = 34, normalized size = 0.89 \[ -\frac{5 \, B b x + B a + 4 \, A b}{20 \,{\left (b x + a\right )}^{5} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(5*B*b*x + B*a + 4*A*b)/((b*x + a)^5*b^2)