3.548 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^4} \, dx\)

Optimal. Leaf size=134 \[ -\frac{a^6 A}{3 x^3}-\frac{a^5 (a B+6 A b)}{2 x^2}-\frac{3 a^4 b (2 a B+5 A b)}{x}+5 a^3 b^2 \log (x) (3 a B+4 A b)+5 a^2 b^3 x (4 a B+3 A b)+\frac{1}{3} b^5 x^3 (6 a B+A b)+\frac{3}{2} a b^4 x^2 (5 a B+2 A b)+\frac{1}{4} b^6 B x^4 \]

[Out]

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

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Rubi [A]  time = 0.220495, antiderivative size = 134, 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 \[ -\frac{a^6 A}{3 x^3}-\frac{a^5 (a B+6 A b)}{2 x^2}-\frac{3 a^4 b (2 a B+5 A b)}{x}+5 a^3 b^2 \log (x) (3 a B+4 A b)+5 a^2 b^3 x (4 a B+3 A b)+\frac{1}{3} b^5 x^3 (6 a B+A b)+\frac{3}{2} a b^4 x^2 (5 a B+2 A b)+\frac{1}{4} b^6 B x^4 \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^4,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{6}}{3 x^{3}} + \frac{B b^{6} x^{4}}{4} - \frac{a^{5} \left (6 A b + B a\right )}{2 x^{2}} - \frac{3 a^{4} b \left (5 A b + 2 B a\right )}{x} + 5 a^{3} b^{2} \left (4 A b + 3 B a\right ) \log{\left (x \right )} + 15 a^{2} b^{3} x \left (A b + \frac{4 B a}{3}\right ) + 3 a b^{4} \left (2 A b + 5 B a\right ) \int x\, dx + \frac{b^{5} x^{3} \left (A b + 6 B a\right )}{3} \]

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**4,x)

[Out]

-A*a**6/(3*x**3) + B*b**6*x**4/4 - a**5*(6*A*b + B*a)/(2*x**2) - 3*a**4*b*(5*A*b
+ 2*B*a)/x + 5*a**3*b**2*(4*A*b + 3*B*a)*log(x) + 15*a**2*b**3*x*(A*b + 4*B*a/3
) + 3*a*b**4*(2*A*b + 5*B*a)*Integral(x, x) + b**5*x**3*(A*b + 6*B*a)/3

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Mathematica [A]  time = 0.0910316, size = 127, normalized size = 0.95 \[ -\frac{a^6 (2 A+3 B x)}{6 x^3}-\frac{3 a^5 b (A+2 B x)}{x^2}-\frac{15 a^4 A b^2}{x}+5 a^3 b^2 \log (x) (3 a B+4 A b)+20 a^3 b^3 B x+\frac{15}{2} a^2 b^4 x (2 A+B x)+a b^5 x^2 (3 A+2 B x)+\frac{1}{12} b^6 x^3 (4 A+3 B x) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^4,x]

[Out]

(-15*a^4*A*b^2)/x + 20*a^3*b^3*B*x + (15*a^2*b^4*x*(2*A + B*x))/2 - (3*a^5*b*(A
+ 2*B*x))/x^2 + a*b^5*x^2*(3*A + 2*B*x) - (a^6*(2*A + 3*B*x))/(6*x^3) + (b^6*x^3
*(4*A + 3*B*x))/12 + 5*a^3*b^2*(4*A*b + 3*a*B)*Log[x]

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Maple [A]  time = 0.011, size = 144, normalized size = 1.1 \[{\frac{{b}^{6}B{x}^{4}}{4}}+{\frac{A{x}^{3}{b}^{6}}{3}}+2\,B{x}^{3}a{b}^{5}+3\,A{x}^{2}a{b}^{5}+{\frac{15\,B{x}^{2}{a}^{2}{b}^{4}}{2}}+15\,Ax{a}^{2}{b}^{4}+20\,Bx{a}^{3}{b}^{3}+20\,A\ln \left ( x \right ){a}^{3}{b}^{3}+15\,B\ln \left ( x \right ){a}^{4}{b}^{2}-{\frac{A{a}^{6}}{3\,{x}^{3}}}-3\,{\frac{A{a}^{5}b}{{x}^{2}}}-{\frac{B{a}^{6}}{2\,{x}^{2}}}-15\,{\frac{A{b}^{2}{a}^{4}}{x}}-6\,{\frac{B{a}^{5}b}{x}} \]

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^4,x)

[Out]

1/4*b^6*B*x^4+1/3*A*x^3*b^6+2*B*x^3*a*b^5+3*A*x^2*a*b^5+15/2*B*x^2*a^2*b^4+15*A*
x*a^2*b^4+20*B*x*a^3*b^3+20*A*ln(x)*a^3*b^3+15*B*ln(x)*a^4*b^2-1/3*a^6*A/x^3-3*a
^5/x^2*A*b-1/2*a^6/x^2*B-15*a^4*b^2/x*A-6*a^5*b/x*B

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Maxima [A]  time = 0.6819, size = 196, normalized size = 1.46 \[ \frac{1}{4} \, B b^{6} x^{4} + \frac{1}{3} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{3} + \frac{3}{2} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{2} + 5 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x + 5 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} \log \left (x\right ) - \frac{2 \, A a^{6} + 18 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 3 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)/x^4,x, algorithm="maxima")

[Out]

1/4*B*b^6*x^4 + 1/3*(6*B*a*b^5 + A*b^6)*x^3 + 3/2*(5*B*a^2*b^4 + 2*A*a*b^5)*x^2
+ 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*log(x) - 1/6*(
2*A*a^6 + 18*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 3*(B*a^6 + 6*A*a^5*b)*x)/x^3

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Fricas [A]  time = 0.279916, size = 201, normalized size = 1.5 \[ \frac{3 \, B b^{6} x^{7} - 4 \, A a^{6} + 4 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 18 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 60 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 60 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} \log \left (x\right ) - 36 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 6 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)/x^4,x, algorithm="fricas")

[Out]

1/12*(3*B*b^6*x^7 - 4*A*a^6 + 4*(6*B*a*b^5 + A*b^6)*x^6 + 18*(5*B*a^2*b^4 + 2*A*
a*b^5)*x^5 + 60*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 60*(3*B*a^4*b^2 + 4*A*a^3*b^3)
*x^3*log(x) - 36*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 6*(B*a^6 + 6*A*a^5*b)*x)/x^3

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Sympy [A]  time = 3.76915, size = 146, normalized size = 1.09 \[ \frac{B b^{6} x^{4}}{4} + 5 a^{3} b^{2} \left (4 A b + 3 B a\right ) \log{\left (x \right )} + x^{3} \left (\frac{A b^{6}}{3} + 2 B a b^{5}\right ) + x^{2} \left (3 A a b^{5} + \frac{15 B a^{2} b^{4}}{2}\right ) + x \left (15 A a^{2} b^{4} + 20 B a^{3} b^{3}\right ) - \frac{2 A a^{6} + x^{2} \left (90 A a^{4} b^{2} + 36 B a^{5} b\right ) + x \left (18 A a^{5} b + 3 B a^{6}\right )}{6 x^{3}} \]

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**4,x)

[Out]

B*b**6*x**4/4 + 5*a**3*b**2*(4*A*b + 3*B*a)*log(x) + x**3*(A*b**6/3 + 2*B*a*b**5
) + x**2*(3*A*a*b**5 + 15*B*a**2*b**4/2) + x*(15*A*a**2*b**4 + 20*B*a**3*b**3) -
(2*A*a**6 + x**2*(90*A*a**4*b**2 + 36*B*a**5*b) + x*(18*A*a**5*b + 3*B*a**6))/(
6*x**3)

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GIAC/XCAS [A]  time = 0.269019, size = 196, normalized size = 1.46 \[ \frac{1}{4} \, B b^{6} x^{4} + 2 \, B a b^{5} x^{3} + \frac{1}{3} \, A b^{6} x^{3} + \frac{15}{2} \, B a^{2} b^{4} x^{2} + 3 \, A a b^{5} x^{2} + 20 \, B a^{3} b^{3} x + 15 \, A a^{2} b^{4} x + 5 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, A a^{6} + 18 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 3 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)/x^4,x, algorithm="giac")

[Out]

1/4*B*b^6*x^4 + 2*B*a*b^5*x^3 + 1/3*A*b^6*x^3 + 15/2*B*a^2*b^4*x^2 + 3*A*a*b^5*x
^2 + 20*B*a^3*b^3*x + 15*A*a^2*b^4*x + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*ln(abs(x))
- 1/6*(2*A*a^6 + 18*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 3*(B*a^6 + 6*A*a^5*b)*x)/x^3