3.533 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^6} \, dx\)

Optimal. Leaf size=71 \[ -\frac{a^4 B}{4 x^4}-\frac{4 a^3 b B}{3 x^3}-\frac{3 a^2 b^2 B}{x^2}-\frac{A (a+b x)^5}{5 a x^5}-\frac{4 a b^3 B}{x}+b^4 B \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0772855, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{a^4 B}{4 x^4}-\frac{4 a^3 b B}{3 x^3}-\frac{3 a^2 b^2 B}{x^2}-\frac{A (a+b x)^5}{5 a x^5}-\frac{4 a b^3 B}{x}+b^4 B \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 26.1669, size = 70, normalized size = 0.99 \[ - \frac{A \left (a + b x\right )^{5}}{5 a x^{5}} - \frac{B a^{4}}{4 x^{4}} - \frac{4 B a^{3} b}{3 x^{3}} - \frac{3 B a^{2} b^{2}}{x^{2}} - \frac{4 B a b^{3}}{x} + B b^{4} \log{\left (x \right )} \]

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)**2/x**6,x)

[Out]

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

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Mathematica [A]  time = 0.102021, size = 87, normalized size = 1.23 \[ b^4 B \log (x)-\frac{3 a^4 (4 A+5 B x)+20 a^3 b x (3 A+4 B x)+60 a^2 b^2 x^2 (2 A+3 B x)+120 a b^3 x^3 (A+2 B x)+60 A b^4 x^4}{60 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^6,x]

[Out]

-(60*A*b^4*x^4 + 120*a*b^3*x^3*(A + 2*B*x) + 60*a^2*b^2*x^2*(2*A + 3*B*x) + 20*a
^3*b*x*(3*A + 4*B*x) + 3*a^4*(4*A + 5*B*x))/(60*x^5) + b^4*B*Log[x]

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Maple [A]  time = 0.01, size = 100, normalized size = 1.4 \[{b}^{4}B\ln \left ( x \right ) -{\frac{A{a}^{3}b}{{x}^{4}}}-{\frac{B{a}^{4}}{4\,{x}^{4}}}-2\,{\frac{A{a}^{2}{b}^{2}}{{x}^{3}}}-{\frac{4\,B{a}^{3}b}{3\,{x}^{3}}}-2\,{\frac{Aa{b}^{3}}{{x}^{2}}}-3\,{\frac{B{a}^{2}{b}^{2}}{{x}^{2}}}-{\frac{A{a}^{4}}{5\,{x}^{5}}}-{\frac{A{b}^{4}}{x}}-4\,{\frac{Ba{b}^{3}}{x}} \]

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)^2/x^6,x)

[Out]

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

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Maxima [A]  time = 0.675605, size = 132, normalized size = 1.86 \[ B b^{4} \log \left (x\right ) - \frac{12 \, A a^{4} + 60 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 60 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 40 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 15 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/x^6,x, algorithm="maxima")

[Out]

B*b^4*log(x) - 1/60*(12*A*a^4 + 60*(4*B*a*b^3 + A*b^4)*x^4 + 60*(3*B*a^2*b^2 + 2
*A*a*b^3)*x^3 + 40*(2*B*a^3*b + 3*A*a^2*b^2)*x^2 + 15*(B*a^4 + 4*A*a^3*b)*x)/x^5

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Fricas [A]  time = 0.280469, size = 136, normalized size = 1.92 \[ \frac{60 \, B b^{4} x^{5} \log \left (x\right ) - 12 \, A a^{4} - 60 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} - 60 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} - 40 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} - 15 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/x^6,x, algorithm="fricas")

[Out]

1/60*(60*B*b^4*x^5*log(x) - 12*A*a^4 - 60*(4*B*a*b^3 + A*b^4)*x^4 - 60*(3*B*a^2*
b^2 + 2*A*a*b^3)*x^3 - 40*(2*B*a^3*b + 3*A*a^2*b^2)*x^2 - 15*(B*a^4 + 4*A*a^3*b)
*x)/x^5

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Sympy [A]  time = 7.28149, size = 99, normalized size = 1.39 \[ B b^{4} \log{\left (x \right )} - \frac{12 A a^{4} + x^{4} \left (60 A b^{4} + 240 B a b^{3}\right ) + x^{3} \left (120 A a b^{3} + 180 B a^{2} b^{2}\right ) + x^{2} \left (120 A a^{2} b^{2} + 80 B a^{3} b\right ) + x \left (60 A a^{3} b + 15 B a^{4}\right )}{60 x^{5}} \]

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)**2/x**6,x)

[Out]

B*b**4*log(x) - (12*A*a**4 + x**4*(60*A*b**4 + 240*B*a*b**3) + x**3*(120*A*a*b**
3 + 180*B*a**2*b**2) + x**2*(120*A*a**2*b**2 + 80*B*a**3*b) + x*(60*A*a**3*b + 1
5*B*a**4))/(60*x**5)

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GIAC/XCAS [A]  time = 0.270862, size = 134, normalized size = 1.89 \[ B b^{4}{\rm ln}\left ({\left | x \right |}\right ) - \frac{12 \, A a^{4} + 60 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 60 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 40 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 15 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/x^6,x, algorithm="giac")

[Out]

B*b^4*ln(abs(x)) - 1/60*(12*A*a^4 + 60*(4*B*a*b^3 + A*b^4)*x^4 + 60*(3*B*a^2*b^2
+ 2*A*a*b^3)*x^3 + 40*(2*B*a^3*b + 3*A*a^2*b^2)*x^2 + 15*(B*a^4 + 4*A*a^3*b)*x)
/x^5