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

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

[Out]

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

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Rubi [A]  time = 0.0744248, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{4 x^4}-\frac{b (2 a B+A b)}{3 x^3}-\frac{b^2 B}{2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.0546, size = 51, normalized size = 0.93 \[ - \frac{A a^{2}}{5 x^{5}} - \frac{B b^{2}}{2 x^{2}} - \frac{a \left (2 A b + B a\right )}{4 x^{4}} - \frac{b \left (A b + 2 B a\right )}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.023793, size = 50, normalized size = 0.91 \[ -\frac{3 a^2 (4 A+5 B x)+10 a b x (3 A+4 B x)+10 b^2 x^2 (2 A+3 B x)}{60 x^5} \]

Antiderivative was successfully verified.

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

[Out]

-(10*b^2*x^2*(2*A + 3*B*x) + 10*a*b*x*(3*A + 4*B*x) + 3*a^2*(4*A + 5*B*x))/(60*x
^5)

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Maple [A]  time = 0.009, size = 48, normalized size = 0.9 \[ -{\frac{A{a}^{2}}{5\,{x}^{5}}}-{\frac{a \left ( 2\,Ab+Ba \right ) }{4\,{x}^{4}}}-{\frac{b \left ( Ab+2\,Ba \right ) }{3\,{x}^{3}}}-{\frac{B{b}^{2}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(B*x+A)/x^6,x)

[Out]

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

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Maxima [A]  time = 1.32855, size = 69, normalized size = 1.25 \[ -\frac{30 \, B b^{2} x^{3} + 12 \, A a^{2} + 20 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(30*B*b^2*x^3 + 12*A*a^2 + 20*(2*B*a*b + A*b^2)*x^2 + 15*(B*a^2 + 2*A*a*b)
*x)/x^5

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Fricas [A]  time = 0.196755, size = 69, normalized size = 1.25 \[ -\frac{30 \, B b^{2} x^{3} + 12 \, A a^{2} + 20 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(30*B*b^2*x^3 + 12*A*a^2 + 20*(2*B*a*b + A*b^2)*x^2 + 15*(B*a^2 + 2*A*a*b)
*x)/x^5

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Sympy [A]  time = 4.00108, size = 54, normalized size = 0.98 \[ - \frac{12 A a^{2} + 30 B b^{2} x^{3} + x^{2} \left (20 A b^{2} + 40 B a b\right ) + x \left (30 A a b + 15 B a^{2}\right )}{60 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(12*A*a**2 + 30*B*b**2*x**3 + x**2*(20*A*b**2 + 40*B*a*b) + x*(30*A*a*b + 15*B*
a**2))/(60*x**5)

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GIAC/XCAS [A]  time = 0.264698, size = 69, normalized size = 1.25 \[ -\frac{30 \, B b^{2} x^{3} + 40 \, B a b x^{2} + 20 \, A b^{2} x^{2} + 15 \, B a^{2} x + 30 \, A a b x + 12 \, A a^{2}}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(30*B*b^2*x^3 + 40*B*a*b*x^2 + 20*A*b^2*x^2 + 15*B*a^2*x + 30*A*a*b*x + 12
*A*a^2)/x^5