3.59 \(\int \frac{(a+b x) (A+B x)}{x^6} \, dx\)

Optimal. Leaf size=33 \[ -\frac{a B+A b}{4 x^4}-\frac{a A}{5 x^5}-\frac{b B}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0419066, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{a B+A b}{4 x^4}-\frac{a A}{5 x^5}-\frac{b B}{3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.58898, size = 31, normalized size = 0.94 \[ - \frac{A a}{5 x^{5}} - \frac{B b}{3 x^{3}} - \frac{\frac{A b}{4} + \frac{B a}{4}}{x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(B*x+A)/x**6,x)

[Out]

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

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Mathematica [A]  time = 0.0155947, size = 31, normalized size = 0.94 \[ -\frac{3 a (4 A+5 B x)+5 b x (3 A+4 B x)}{60 x^5} \]

Antiderivative was successfully verified.

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

[Out]

-(5*b*x*(3*A + 4*B*x) + 3*a*(4*A + 5*B*x))/(60*x^5)

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Maple [A]  time = 0.007, size = 28, normalized size = 0.9 \[ -{\frac{Aa}{5\,{x}^{5}}}-{\frac{Bb}{3\,{x}^{3}}}-{\frac{Ab+Ba}{4\,{x}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)/x^6,x)

[Out]

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

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Maxima [A]  time = 1.35656, size = 36, normalized size = 1.09 \[ -\frac{20 \, B b x^{2} + 12 \, A a + 15 \,{\left (B a + A b\right )} x}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(20*B*b*x^2 + 12*A*a + 15*(B*a + A*b)*x)/x^5

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Fricas [A]  time = 0.194647, size = 36, normalized size = 1.09 \[ -\frac{20 \, B b x^{2} + 12 \, A a + 15 \,{\left (B a + A b\right )} x}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(20*B*b*x^2 + 12*A*a + 15*(B*a + A*b)*x)/x^5

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Sympy [A]  time = 2.32656, size = 31, normalized size = 0.94 \[ - \frac{12 A a + 20 B b x^{2} + x \left (15 A b + 15 B a\right )}{60 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(B*x+A)/x**6,x)

[Out]

-(12*A*a + 20*B*b*x**2 + x*(15*A*b + 15*B*a))/(60*x**5)

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GIAC/XCAS [A]  time = 0.27023, size = 36, normalized size = 1.09 \[ -\frac{20 \, B b x^{2} + 15 \, B a x + 15 \, A b x + 12 \, A a}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(20*B*b*x^2 + 15*B*a*x + 15*A*b*x + 12*A*a)/x^5