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

Optimal. Leaf size=57 \[ \frac{x^4 (4 a B+A b)}{20 a^2 b (a+b x)^4}+\frac{x^4 (A b-a B)}{5 a b (a+b x)^5} \]

[Out]

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

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Rubi [A]  time = 0.0609395, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x^4 (4 a B+A b)}{20 a^2 b (a+b x)^4}+\frac{x^4 (A b-a B)}{5 a b (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.2757, size = 46, normalized size = 0.81 \[ \frac{x^{4} \left (A b - B a\right )}{5 a b \left (a + b x\right )^{5}} + \frac{x^{4} \left (A b + 4 B a\right )}{20 a^{2} b \left (a + b x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0583051, size = 76, normalized size = 1.33 \[ -\frac{4 a^4 B+a^3 b (A+20 B x)+5 a^2 b^2 x (A+8 B x)+10 a b^3 x^2 (A+4 B x)+10 b^4 x^3 (A+2 B x)}{20 b^5 (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 102, normalized size = 1.8 \[{\frac{a \left ( Ab-2\,Ba \right ) }{{b}^{5} \left ( bx+a \right ) ^{3}}}+{\frac{{a}^{3} \left ( Ab-Ba \right ) }{5\,{b}^{5} \left ( bx+a \right ) ^{5}}}-{\frac{{a}^{2} \left ( 3\,Ab-4\,Ba \right ) }{4\,{b}^{5} \left ( bx+a \right ) ^{4}}}-{\frac{Ab-4\,Ba}{2\,{b}^{5} \left ( bx+a \right ) ^{2}}}-{\frac{B}{ \left ( bx+a \right ){b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.704041, size = 188, normalized size = 3.3 \[ -\frac{20 \, B b^{4} x^{4} + 4 \, B a^{4} + A a^{3} b + 10 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{3} + 10 \,{\left (4 \, B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 5 \,{\left (4 \, B a^{3} b + A a^{2} b^{2}\right )} x}{20 \,{\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(20*B*b^4*x^4 + 4*B*a^4 + A*a^3*b + 10*(4*B*a*b^3 + A*b^4)*x^3 + 10*(4*B*a
^2*b^2 + A*a*b^3)*x^2 + 5*(4*B*a^3*b + A*a^2*b^2)*x)/(b^10*x^5 + 5*a*b^9*x^4 + 1
0*a^2*b^8*x^3 + 10*a^3*b^7*x^2 + 5*a^4*b^6*x + a^5*b^5)

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Fricas [A]  time = 0.267275, size = 188, normalized size = 3.3 \[ -\frac{20 \, B b^{4} x^{4} + 4 \, B a^{4} + A a^{3} b + 10 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{3} + 10 \,{\left (4 \, B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 5 \,{\left (4 \, B a^{3} b + A a^{2} b^{2}\right )} x}{20 \,{\left (b^{10} x^{5} + 5 \, a b^{9} x^{4} + 10 \, a^{2} b^{8} x^{3} + 10 \, a^{3} b^{7} x^{2} + 5 \, a^{4} b^{6} x + a^{5} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(20*B*b^4*x^4 + 4*B*a^4 + A*a^3*b + 10*(4*B*a*b^3 + A*b^4)*x^3 + 10*(4*B*a
^2*b^2 + A*a*b^3)*x^2 + 5*(4*B*a^3*b + A*a^2*b^2)*x)/(b^10*x^5 + 5*a*b^9*x^4 + 1
0*a^2*b^8*x^3 + 10*a^3*b^7*x^2 + 5*a^4*b^6*x + a^5*b^5)

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Sympy [A]  time = 6.55361, size = 146, normalized size = 2.56 \[ - \frac{A a^{3} b + 4 B a^{4} + 20 B b^{4} x^{4} + x^{3} \left (10 A b^{4} + 40 B a b^{3}\right ) + x^{2} \left (10 A a b^{3} + 40 B a^{2} b^{2}\right ) + x \left (5 A a^{2} b^{2} + 20 B a^{3} b\right )}{20 a^{5} b^{5} + 100 a^{4} b^{6} x + 200 a^{3} b^{7} x^{2} + 200 a^{2} b^{8} x^{3} + 100 a b^{9} x^{4} + 20 b^{10} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(A*a**3*b + 4*B*a**4 + 20*B*b**4*x**4 + x**3*(10*A*b**4 + 40*B*a*b**3) + x**2*(
10*A*a*b**3 + 40*B*a**2*b**2) + x*(5*A*a**2*b**2 + 20*B*a**3*b))/(20*a**5*b**5 +
100*a**4*b**6*x + 200*a**3*b**7*x**2 + 200*a**2*b**8*x**3 + 100*a*b**9*x**4 + 2
0*b**10*x**5)

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GIAC/XCAS [A]  time = 0.268174, size = 126, normalized size = 2.21 \[ -\frac{20 \, B b^{4} x^{4} + 40 \, B a b^{3} x^{3} + 10 \, A b^{4} x^{3} + 40 \, B a^{2} b^{2} x^{2} + 10 \, A a b^{3} x^{2} + 20 \, B a^{3} b x + 5 \, A a^{2} b^{2} x + 4 \, B a^{4} + A a^{3} b}{20 \,{\left (b x + a\right )}^{5} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*(20*B*b^4*x^4 + 40*B*a*b^3*x^3 + 10*A*b^4*x^3 + 40*B*a^2*b^2*x^2 + 10*A*a*
b^3*x^2 + 20*B*a^3*b*x + 5*A*a^2*b^2*x + 4*B*a^4 + A*a^3*b)/((b*x + a)^5*b^5)