3.634 \(\int \frac{x^4 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=121 \[ -\frac{a^4 (A b-a B)}{3 b^6 (a+b x)^3}+\frac{a^3 (4 A b-5 a B)}{2 b^6 (a+b x)^2}-\frac{2 a^2 (3 A b-5 a B)}{b^6 (a+b x)}-\frac{2 a (2 A b-5 a B) \log (a+b x)}{b^6}+\frac{x (A b-4 a B)}{b^5}+\frac{B x^2}{2 b^4} \]

[Out]

((A*b - 4*a*B)*x)/b^5 + (B*x^2)/(2*b^4) - (a^4*(A*b - a*B))/(3*b^6*(a + b*x)^3)
+ (a^3*(4*A*b - 5*a*B))/(2*b^6*(a + b*x)^2) - (2*a^2*(3*A*b - 5*a*B))/(b^6*(a +
b*x)) - (2*a*(2*A*b - 5*a*B)*Log[a + b*x])/b^6

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Rubi [A]  time = 0.258103, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{a^4 (A b-a B)}{3 b^6 (a+b x)^3}+\frac{a^3 (4 A b-5 a B)}{2 b^6 (a+b x)^2}-\frac{2 a^2 (3 A b-5 a B)}{b^6 (a+b x)}-\frac{2 a (2 A b-5 a B) \log (a+b x)}{b^6}+\frac{x (A b-4 a B)}{b^5}+\frac{B x^2}{2 b^4} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - 4*a*B)*x)/b^5 + (B*x^2)/(2*b^4) - (a^4*(A*b - a*B))/(3*b^6*(a + b*x)^3)
+ (a^3*(4*A*b - 5*a*B))/(2*b^6*(a + b*x)^2) - (2*a^2*(3*A*b - 5*a*B))/(b^6*(a +
b*x)) - (2*a*(2*A*b - 5*a*B)*Log[a + b*x])/b^6

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B \int x\, dx}{b^{4}} - \frac{a^{4} \left (A b - B a\right )}{3 b^{6} \left (a + b x\right )^{3}} + \frac{a^{3} \left (4 A b - 5 B a\right )}{2 b^{6} \left (a + b x\right )^{2}} - \frac{2 a^{2} \left (3 A b - 5 B a\right )}{b^{6} \left (a + b x\right )} - \frac{2 a \left (2 A b - 5 B a\right ) \log{\left (a + b x \right )}}{b^{6}} + \frac{16 \left (A b - 4 B a\right ) \int \frac{1}{16}\, dx}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

B*Integral(x, x)/b**4 - a**4*(A*b - B*a)/(3*b**6*(a + b*x)**3) + a**3*(4*A*b - 5
*B*a)/(2*b**6*(a + b*x)**2) - 2*a**2*(3*A*b - 5*B*a)/(b**6*(a + b*x)) - 2*a*(2*A
*b - 5*B*a)*log(a + b*x)/b**6 + 16*(A*b - 4*B*a)*Integral(1/16, x)/b**5

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Mathematica [A]  time = 0.0827681, size = 129, normalized size = 1.07 \[ \frac{2 \left (5 a^2 B-2 a A b\right ) \log (a+b x)}{b^6}+\frac{a^5 B-a^4 A b}{3 b^6 (a+b x)^3}+\frac{4 a^3 A b-5 a^4 B}{2 b^6 (a+b x)^2}+\frac{2 \left (5 a^3 B-3 a^2 A b\right )}{b^6 (a+b x)}+\frac{x (A b-4 a B)}{b^5}+\frac{B x^2}{2 b^4} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - 4*a*B)*x)/b^5 + (B*x^2)/(2*b^4) + (-(a^4*A*b) + a^5*B)/(3*b^6*(a + b*x)^
3) + (4*a^3*A*b - 5*a^4*B)/(2*b^6*(a + b*x)^2) + (2*(-3*a^2*A*b + 5*a^3*B))/(b^6
*(a + b*x)) + (2*(-2*a*A*b + 5*a^2*B)*Log[a + b*x])/b^6

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Maple [A]  time = 0.013, size = 149, normalized size = 1.2 \[{\frac{B{x}^{2}}{2\,{b}^{4}}}+{\frac{Ax}{{b}^{4}}}-4\,{\frac{aBx}{{b}^{5}}}-{\frac{{a}^{4}A}{3\,{b}^{5} \left ( bx+a \right ) ^{3}}}+{\frac{B{a}^{5}}{3\,{b}^{6} \left ( bx+a \right ) ^{3}}}-4\,{\frac{a\ln \left ( bx+a \right ) A}{{b}^{5}}}+10\,{\frac{{a}^{2}\ln \left ( bx+a \right ) B}{{b}^{6}}}+2\,{\frac{A{a}^{3}}{{b}^{5} \left ( bx+a \right ) ^{2}}}-{\frac{5\,B{a}^{4}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}-6\,{\frac{A{a}^{2}}{{b}^{5} \left ( bx+a \right ) }}+10\,{\frac{B{a}^{3}}{{b}^{6} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.694385, size = 193, normalized size = 1.6 \[ \frac{47 \, B a^{5} - 26 \, A a^{4} b + 12 \,{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 15 \,{\left (7 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x}{6 \,{\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} + \frac{B b x^{2} - 2 \,{\left (4 \, B a - A b\right )} x}{2 \, b^{5}} + \frac{2 \,{\left (5 \, B a^{2} - 2 \, A a b\right )} \log \left (b x + a\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(47*B*a^5 - 26*A*a^4*b + 12*(5*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 + 15*(7*B*a^4*b
- 4*A*a^3*b^2)*x)/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6) + 1/2*(B*b*x^2
- 2*(4*B*a - A*b)*x)/b^5 + 2*(5*B*a^2 - 2*A*a*b)*log(b*x + a)/b^6

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Fricas [A]  time = 0.283949, size = 312, normalized size = 2.58 \[ \frac{3 \, B b^{5} x^{5} + 47 \, B a^{5} - 26 \, A a^{4} b - 3 \,{\left (5 \, B a b^{4} - 2 \, A b^{5}\right )} x^{4} - 9 \,{\left (7 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{3} - 9 \,{\left (B a^{3} b^{2} + 2 \, A a^{2} b^{3}\right )} x^{2} + 27 \,{\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x + 12 \,{\left (5 \, B a^{5} - 2 \, A a^{4} b +{\left (5 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{3} + 3 \,{\left (5 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{2} + 3 \,{\left (5 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(3*B*b^5*x^5 + 47*B*a^5 - 26*A*a^4*b - 3*(5*B*a*b^4 - 2*A*b^5)*x^4 - 9*(7*B*
a^2*b^3 - 2*A*a*b^4)*x^3 - 9*(B*a^3*b^2 + 2*A*a^2*b^3)*x^2 + 27*(3*B*a^4*b - 2*A
*a^3*b^2)*x + 12*(5*B*a^5 - 2*A*a^4*b + (5*B*a^2*b^3 - 2*A*a*b^4)*x^3 + 3*(5*B*a
^3*b^2 - 2*A*a^2*b^3)*x^2 + 3*(5*B*a^4*b - 2*A*a^3*b^2)*x)*log(b*x + a))/(b^9*x^
3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6)

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Sympy [A]  time = 5.49893, size = 143, normalized size = 1.18 \[ \frac{B x^{2}}{2 b^{4}} + \frac{2 a \left (- 2 A b + 5 B a\right ) \log{\left (a + b x \right )}}{b^{6}} + \frac{- 26 A a^{4} b + 47 B a^{5} + x^{2} \left (- 36 A a^{2} b^{3} + 60 B a^{3} b^{2}\right ) + x \left (- 60 A a^{3} b^{2} + 105 B a^{4} b\right )}{6 a^{3} b^{6} + 18 a^{2} b^{7} x + 18 a b^{8} x^{2} + 6 b^{9} x^{3}} - \frac{x \left (- A b + 4 B a\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

B*x**2/(2*b**4) + 2*a*(-2*A*b + 5*B*a)*log(a + b*x)/b**6 + (-26*A*a**4*b + 47*B*
a**5 + x**2*(-36*A*a**2*b**3 + 60*B*a**3*b**2) + x*(-60*A*a**3*b**2 + 105*B*a**4
*b))/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) - x*(-A*b + 4
*B*a)/b**5

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GIAC/XCAS [A]  time = 0.274834, size = 167, normalized size = 1.38 \[ \frac{2 \,{\left (5 \, B a^{2} - 2 \, A a b\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{B b^{4} x^{2} - 8 \, B a b^{3} x + 2 \, A b^{4} x}{2 \, b^{8}} + \frac{47 \, B a^{5} - 26 \, A a^{4} b + 12 \,{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 15 \,{\left (7 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x}{6 \,{\left (b x + a\right )}^{3} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(5*B*a^2 - 2*A*a*b)*ln(abs(b*x + a))/b^6 + 1/2*(B*b^4*x^2 - 8*B*a*b^3*x + 2*A*
b^4*x)/b^8 + 1/6*(47*B*a^5 - 26*A*a^4*b + 12*(5*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 + 1
5*(7*B*a^4*b - 4*A*a^3*b^2)*x)/((b*x + a)^3*b^6)