3.57 \(\int \frac{(a+b x) (A+B x)}{x^4} \, dx\)

Optimal. Leaf size=31 \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{3 x^3}-\frac{b B}{x} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0418669, antiderivative size = 31, 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 \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{3 x^3}-\frac{b B}{x} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(A + B*x))/x^4,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.64497, size = 27, normalized size = 0.87 \[ - \frac{A a}{3 x^{3}} - \frac{B b}{x} - \frac{\frac{A b}{2} + \frac{B a}{2}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(B*x+A)/x**4,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0151285, size = 28, normalized size = 0.9 \[ -\frac{a (2 A+3 B x)+3 b x (A+2 B x)}{6 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(A + B*x))/x^4,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 28, normalized size = 0.9 \[ -{\frac{Ab+Ba}{2\,{x}^{2}}}-{\frac{Bb}{x}}-{\frac{Aa}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)/x^4,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34636, size = 36, normalized size = 1.16 \[ -\frac{6 \, B b x^{2} + 2 \, A a + 3 \,{\left (B a + A b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/x^4,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.194757, size = 36, normalized size = 1.16 \[ -\frac{6 \, B b x^{2} + 2 \, A a + 3 \,{\left (B a + A b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/x^4,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.78957, size = 31, normalized size = 1. \[ - \frac{2 A a + 6 B b x^{2} + x \left (3 A b + 3 B a\right )}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(B*x+A)/x**4,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.317351, size = 36, normalized size = 1.16 \[ -\frac{6 \, B b x^{2} + 3 \, B a x + 3 \, A b x + 2 \, A a}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/x^4,x, algorithm="giac")

[Out]

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