3.521 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^7} \, dx\)

Optimal. Leaf size=55 \[ -\frac{a^2 A}{6 x^6}-\frac{a (a B+2 A b)}{5 x^5}-\frac{b (2 a B+A b)}{4 x^4}-\frac{b^2 B}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0712196, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{a^2 A}{6 x^6}-\frac{a (a B+2 A b)}{5 x^5}-\frac{b (2 a B+A b)}{4 x^4}-\frac{b^2 B}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 20.7997, size = 51, normalized size = 0.93 \[ - \frac{A a^{2}}{6 x^{6}} - \frac{B b^{2}}{3 x^{3}} - \frac{a \left (2 A b + B a\right )}{5 x^{5}} - \frac{b \left (A b + 2 B a\right )}{4 x^{4}} \]

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)/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.02766, size = 50, normalized size = 0.91 \[ -\frac{2 a^2 (5 A+6 B x)+6 a b x (4 A+5 B x)+5 b^2 x^2 (3 A+4 B x)}{60 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/x^7,x]

[Out]

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

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Maple [A]  time = 0.007, size = 48, normalized size = 0.9 \[ -{\frac{A{a}^{2}}{6\,{x}^{6}}}-{\frac{a \left ( 2\,Ab+Ba \right ) }{5\,{x}^{5}}}-{\frac{b \left ( Ab+2\,Ba \right ) }{4\,{x}^{4}}}-{\frac{{b}^{2}B}{3\,{x}^{3}}} \]

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)/x^7,x)

[Out]

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

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Maxima [A]  time = 0.679177, size = 69, normalized size = 1.25 \[ -\frac{20 \, B b^{2} x^{3} + 10 \, A a^{2} + 15 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 12 \,{\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/x^7,x, algorithm="maxima")

[Out]

-1/60*(20*B*b^2*x^3 + 10*A*a^2 + 15*(2*B*a*b + A*b^2)*x^2 + 12*(B*a^2 + 2*A*a*b)
*x)/x^6

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Fricas [A]  time = 0.300565, size = 69, normalized size = 1.25 \[ -\frac{20 \, B b^{2} x^{3} + 10 \, A a^{2} + 15 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 12 \,{\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/x^7,x, algorithm="fricas")

[Out]

-1/60*(20*B*b^2*x^3 + 10*A*a^2 + 15*(2*B*a*b + A*b^2)*x^2 + 12*(B*a^2 + 2*A*a*b)
*x)/x^6

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Sympy [A]  time = 4.64784, size = 54, normalized size = 0.98 \[ - \frac{10 A a^{2} + 20 B b^{2} x^{3} + x^{2} \left (15 A b^{2} + 30 B a b\right ) + x \left (24 A a b + 12 B a^{2}\right )}{60 x^{6}} \]

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)/x**7,x)

[Out]

-(10*A*a**2 + 20*B*b**2*x**3 + x**2*(15*A*b**2 + 30*B*a*b) + x*(24*A*a*b + 12*B*
a**2))/(60*x**6)

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GIAC/XCAS [A]  time = 0.27034, size = 69, normalized size = 1.25 \[ -\frac{20 \, B b^{2} x^{3} + 30 \, B a b x^{2} + 15 \, A b^{2} x^{2} + 12 \, B a^{2} x + 24 \, A a b x + 10 \, A a^{2}}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/x^7,x, algorithm="giac")

[Out]

-1/60*(20*B*b^2*x^3 + 30*B*a*b*x^2 + 15*A*b^2*x^2 + 12*B*a^2*x + 24*A*a*b*x + 10
*A*a^2)/x^6