3.69 \(\int \frac{(a+b x)^2 (A+B x)}{x^5} \, dx\)

Optimal. Leaf size=44 \[ \frac{(a+b x)^3 (A b-4 a B)}{12 a^2 x^3}-\frac{A (a+b x)^3}{4 a x^4} \]

[Out]

-(A*(a + b*x)^3)/(4*a*x^4) + ((A*b - 4*a*B)*(a + b*x)^3)/(12*a^2*x^3)

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Rubi [A]  time = 0.0526766, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{(a+b x)^3 (A b-4 a B)}{12 a^2 x^3}-\frac{A (a+b x)^3}{4 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(A*(a + b*x)^3)/(4*a*x^4) + ((A*b - 4*a*B)*(a + b*x)^3)/(12*a^2*x^3)

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Rubi in Sympy [A]  time = 15.5182, size = 48, normalized size = 1.09 \[ - \frac{A a^{2}}{4 x^{4}} - \frac{B b^{2}}{x} - \frac{a \left (2 A b + B a\right )}{3 x^{3}} - \frac{b \left (A b + 2 B a\right )}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0249638, size = 47, normalized size = 1.07 \[ -\frac{a^2 (3 A+4 B x)+4 a b x (2 A+3 B x)+6 b^2 x^2 (A+2 B x)}{12 x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(6*b^2*x^2*(A + 2*B*x) + 4*a*b*x*(2*A + 3*B*x) + a^2*(3*A + 4*B*x))/(12*x^4)

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Maple [A]  time = 0.007, size = 48, normalized size = 1.1 \[ -{\frac{b \left ( Ab+2\,Ba \right ) }{2\,{x}^{2}}}-{\frac{B{b}^{2}}{x}}-{\frac{a \left ( 2\,Ab+Ba \right ) }{3\,{x}^{3}}}-{\frac{A{a}^{2}}{4\,{x}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(B*x+A)/x^5,x)

[Out]

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

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Maxima [A]  time = 1.33655, size = 69, normalized size = 1.57 \[ -\frac{12 \, B b^{2} x^{3} + 3 \, A a^{2} + 6 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^2/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.195127, size = 69, normalized size = 1.57 \[ -\frac{12 \, B b^{2} x^{3} + 3 \, A a^{2} + 6 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^2/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.34528, size = 54, normalized size = 1.23 \[ - \frac{3 A a^{2} + 12 B b^{2} x^{3} + x^{2} \left (6 A b^{2} + 12 B a b\right ) + x \left (8 A a b + 4 B a^{2}\right )}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(3*A*a**2 + 12*B*b**2*x**3 + x**2*(6*A*b**2 + 12*B*a*b) + x*(8*A*a*b + 4*B*a**2
))/(12*x**4)

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GIAC/XCAS [A]  time = 0.305717, size = 69, normalized size = 1.57 \[ -\frac{12 \, B b^{2} x^{3} + 12 \, B a b x^{2} + 6 \, A b^{2} x^{2} + 4 \, B a^{2} x + 8 \, A a b x + 3 \, A a^{2}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^2/x^5,x, algorithm="giac")

[Out]

-1/12*(12*B*b^2*x^3 + 12*B*a*b*x^2 + 6*A*b^2*x^2 + 4*B*a^2*x + 8*A*a*b*x + 3*A*a
^2)/x^4