\(\int \frac {(a+b x) (A+B x)}{x^4} \, dx\) [88]

Optimal result

Integrand size = 14, antiderivative size = 31 \[ \int \frac {(a+b x) (A+B x)}{x^4} \, dx=-\frac {a A}{3 x^3}-\frac {A b+a B}{2 x^2}-\frac {b B}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {77} \[ \int \frac {(a+b x) (A+B x)}{x^4} \, dx=-\frac {a B+A b}{2 x^2}-\frac {a A}{3 x^3}-\frac {b B}{x} \]

[In]

Int[((a + b*x)*(A + B*x))/x^4,x]

[Out]

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

Rule 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a A}{x^4}+\frac {A b+a B}{x^3}+\frac {b B}{x^2}\right ) \, dx \\ & = -\frac {a A}{3 x^3}-\frac {A b+a B}{2 x^2}-\frac {b B}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x) (A+B x)}{x^4} \, dx=-\frac {3 b x (A+2 B x)+a (2 A+3 B x)}{6 x^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90

method	result	size
gosper	\(-\frac {6 b B \,x^{2}+3 A b x +3 B a x +2 A a}{6 x^{3}}\)	\(28\)
default	\(-\frac {a A}{3 x^{3}}-\frac {b B}{x}-\frac {A b +B a}{2 x^{2}}\)	\(28\)
norman	\(\frac {-b B \,x^{2}+\left (-\frac {A b}{2}-\frac {B a}{2}\right ) x -\frac {A a}{3}}{x^{3}}\)	\(28\)
risch	\(\frac {-b B \,x^{2}+\left (-\frac {A b}{2}-\frac {B a}{2}\right ) x -\frac {A a}{3}}{x^{3}}\)	\(28\)
parallelrisch	\(-\frac {6 b B \,x^{2}+3 A b x +3 B a x +2 A a}{6 x^{3}}\)	\(28\)

[In]

int((b*x+a)*(B*x+A)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x) (A+B x)}{x^4} \, dx=-\frac {6 \, B b x^{2} + 2 \, A a + 3 \, {\left (B a + A b\right )} x}{6 \, x^{3}} \]

[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] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (A+B x)}{x^4} \, dx=\frac {- 2 A a - 6 B b x^{2} + x \left (- 3 A b - 3 B a\right )}{6 x^{3}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x) (A+B x)}{x^4} \, dx=-\frac {6 \, B b x^{2} + 2 \, A a + 3 \, {\left (B a + A b\right )} x}{6 \, x^{3}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x) (A+B x)}{x^4} \, dx=-\frac {6 \, B b x^{2} + 3 \, B a x + 3 \, A b x + 2 \, A a}{6 \, x^{3}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x) (A+B x)}{x^4} \, dx=-\frac {B\,b\,x^2+\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )\,x+\frac {A\,a}{3}}{x^3} \]

[In]

int(((A + B*x)*(a + b*x))/x^4,x)

[Out]

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