3.549 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^5} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.218825, 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 \[ -\frac{a^6 A}{4 x^4}-\frac{a^5 (a B+6 A b)}{3 x^3}-\frac{3 a^4 b (2 a B+5 A b)}{2 x^2}-\frac{5 a^3 b^2 (3 a B+4 A b)}{x}+5 a^2 b^3 \log (x) (4 a B+3 A b)+\frac{1}{2} b^5 x^2 (6 a B+A b)+3 a b^4 x (5 a B+2 A b)+\frac{1}{3} b^6 B x^3 \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^5,x]

[Out]

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

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

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**5,x)

[Out]

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

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Mathematica [A]  time = 0.103384, size = 128, normalized size = 0.96 \[ -\frac{a^6 (3 A+4 B x)}{12 x^4}-\frac{a^5 b (2 A+3 B x)}{x^3}-\frac{15 a^4 b^2 (A+2 B x)}{2 x^2}-\frac{20 a^3 A b^3}{x}+5 a^2 b^3 \log (x) (4 a B+3 A b)+15 a^2 b^4 B x+3 a b^5 x (2 A+B x)+\frac{1}{6} b^6 x^2 (3 A+2 B x) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^5,x]

[Out]

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

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

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^5,x)

[Out]

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

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Maxima [A]  time = 0.676473, size = 196, normalized size = 1.46 \[ \frac{1}{3} \, B b^{6} x^{3} + \frac{1}{2} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{2} + 3 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x + 5 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} \log \left (x\right ) - \frac{3 \, A a^{6} + 60 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 18 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 4 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{4}} \]

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^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.281772, size = 201, normalized size = 1.5 \[ \frac{4 \, B b^{6} x^{7} - 3 \, A a^{6} + 6 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 36 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 60 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} \log \left (x\right ) - 60 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 18 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 4 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{4}} \]

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^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.79811, size = 144, normalized size = 1.07 \[ \frac{B b^{6} x^{3}}{3} + 5 a^{2} b^{3} \left (3 A b + 4 B a\right ) \log{\left (x \right )} + x^{2} \left (\frac{A b^{6}}{2} + 3 B a b^{5}\right ) + x \left (6 A a b^{5} + 15 B a^{2} b^{4}\right ) - \frac{3 A a^{6} + x^{3} \left (240 A a^{3} b^{3} + 180 B a^{4} b^{2}\right ) + x^{2} \left (90 A a^{4} b^{2} + 36 B a^{5} b\right ) + x \left (24 A a^{5} b + 4 B a^{6}\right )}{12 x^{4}} \]

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**5,x)

[Out]

B*b**6*x**3/3 + 5*a**2*b**3*(3*A*b + 4*B*a)*log(x) + x**2*(A*b**6/2 + 3*B*a*b**5
) + x*(6*A*a*b**5 + 15*B*a**2*b**4) - (3*A*a**6 + x**3*(240*A*a**3*b**3 + 180*B*
a**4*b**2) + x**2*(90*A*a**4*b**2 + 36*B*a**5*b) + x*(24*A*a**5*b + 4*B*a**6))/(
12*x**4)

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GIAC/XCAS [A]  time = 0.269371, size = 196, normalized size = 1.46 \[ \frac{1}{3} \, B b^{6} x^{3} + 3 \, B a b^{5} x^{2} + \frac{1}{2} \, A b^{6} x^{2} + 15 \, B a^{2} b^{4} x + 6 \, A a b^{5} x + 5 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{3 \, A a^{6} + 60 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 18 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 4 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{12 \, x^{4}} \]

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^5,x, algorithm="giac")

[Out]

1/3*B*b^6*x^3 + 3*B*a*b^5*x^2 + 1/2*A*b^6*x^2 + 15*B*a^2*b^4*x + 6*A*a*b^5*x + 5
*(4*B*a^3*b^3 + 3*A*a^2*b^4)*ln(abs(x)) - 1/12*(3*A*a^6 + 60*(3*B*a^4*b^2 + 4*A*
a^3*b^3)*x^3 + 18*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 4*(B*a^6 + 6*A*a^5*b)*x)/x^4