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

Optimal. Leaf size=61 \[ -\frac{A b-2 a B}{4 b^3 (a+b x)^4}+\frac{a (A b-a B)}{5 b^3 (a+b x)^5}-\frac{B}{3 b^3 (a+b x)^3} \]

[Out]

(a*(A*b - a*B))/(5*b^3*(a + b*x)^5) - (A*b - 2*a*B)/(4*b^3*(a + b*x)^4) - B/(3*b
^3*(a + b*x)^3)

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Rubi [A]  time = 0.100236, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{A b-2 a B}{4 b^3 (a+b x)^4}+\frac{a (A b-a B)}{5 b^3 (a+b x)^5}-\frac{B}{3 b^3 (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(a*(A*b - a*B))/(5*b^3*(a + b*x)^5) - (A*b - 2*a*B)/(4*b^3*(a + b*x)^4) - B/(3*b
^3*(a + b*x)^3)

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Rubi in Sympy [A]  time = 25.6645, size = 53, normalized size = 0.87 \[ - \frac{B}{3 b^{3} \left (a + b x\right )^{3}} + \frac{a \left (A b - B a\right )}{5 b^{3} \left (a + b x\right )^{5}} - \frac{A b - 2 B a}{4 b^{3} \left (a + b x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-B/(3*b**3*(a + b*x)**3) + a*(A*b - B*a)/(5*b**3*(a + b*x)**5) - (A*b - 2*B*a)/(
4*b**3*(a + b*x)**4)

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Mathematica [A]  time = 0.0292669, size = 46, normalized size = 0.75 \[ -\frac{2 a^2 B+a b (3 A+10 B x)+5 b^2 x (3 A+4 B x)}{60 b^3 (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

-(2*a^2*B + 5*b^2*x*(3*A + 4*B*x) + a*b*(3*A + 10*B*x))/(60*b^3*(a + b*x)^5)

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Maple [A]  time = 0.008, size = 56, normalized size = 0.9 \[ -{\frac{B}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}}-{\frac{Ab-2\,Ba}{4\,{b}^{3} \left ( bx+a \right ) ^{4}}}+{\frac{a \left ( Ab-Ba \right ) }{5\,{b}^{3} \left ( bx+a \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.69188, size = 128, normalized size = 2.1 \[ -\frac{20 \, B b^{2} x^{2} + 2 \, B a^{2} + 3 \, A a b + 5 \,{\left (2 \, B a b + 3 \, A b^{2}\right )} x}{60 \,{\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(20*B*b^2*x^2 + 2*B*a^2 + 3*A*a*b + 5*(2*B*a*b + 3*A*b^2)*x)/(b^8*x^5 + 5*
a*b^7*x^4 + 10*a^2*b^6*x^3 + 10*a^3*b^5*x^2 + 5*a^4*b^4*x + a^5*b^3)

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Fricas [A]  time = 0.266446, size = 128, normalized size = 2.1 \[ -\frac{20 \, B b^{2} x^{2} + 2 \, B a^{2} + 3 \, A a b + 5 \,{\left (2 \, B a b + 3 \, A b^{2}\right )} x}{60 \,{\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(20*B*b^2*x^2 + 2*B*a^2 + 3*A*a*b + 5*(2*B*a*b + 3*A*b^2)*x)/(b^8*x^5 + 5*
a*b^7*x^4 + 10*a^2*b^6*x^3 + 10*a^3*b^5*x^2 + 5*a^4*b^4*x + a^5*b^3)

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Sympy [A]  time = 3.3617, size = 100, normalized size = 1.64 \[ - \frac{3 A a b + 2 B a^{2} + 20 B b^{2} x^{2} + x \left (15 A b^{2} + 10 B a b\right )}{60 a^{5} b^{3} + 300 a^{4} b^{4} x + 600 a^{3} b^{5} x^{2} + 600 a^{2} b^{6} x^{3} + 300 a b^{7} x^{4} + 60 b^{8} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(3*A*a*b + 2*B*a**2 + 20*B*b**2*x**2 + x*(15*A*b**2 + 10*B*a*b))/(60*a**5*b**3
+ 300*a**4*b**4*x + 600*a**3*b**5*x**2 + 600*a**2*b**6*x**3 + 300*a*b**7*x**4 +
60*b**8*x**5)

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GIAC/XCAS [A]  time = 0.267565, size = 62, normalized size = 1.02 \[ -\frac{20 \, B b^{2} x^{2} + 10 \, B a b x + 15 \, A b^{2} x + 2 \, B a^{2} + 3 \, A a b}{60 \,{\left (b x + a\right )}^{5} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(20*B*b^2*x^2 + 10*B*a*b*x + 15*A*b^2*x + 2*B*a^2 + 3*A*a*b)/((b*x + a)^5*
b^3)