\(\int \frac {(a+b x) (A+B x)}{x^5} \, dx\) [89]

Optimal result

Integrand size = 14, antiderivative size = 33 \[ \int \frac {(a+b x) (A+B x)}{x^5} \, dx=-\frac {a A}{4 x^4}-\frac {A b+a B}{3 x^3}-\frac {b B}{2 x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, 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} \[ \int \frac {(a+b x) (A+B x)}{x^5} \, dx=-\frac {a B+A b}{3 x^3}-\frac {a A}{4 x^4}-\frac {b B}{2 x^2} \]

[In]

Int[((a + b*x)*(A + B*x))/x^5,x]

[Out]

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

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^5}+\frac {A b+a B}{x^4}+\frac {b B}{x^3}\right ) \, dx \\ & = -\frac {a A}{4 x^4}-\frac {A b+a B}{3 x^3}-\frac {b B}{2 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x) (A+B x)}{x^5} \, dx=-\frac {3 a A+4 A b x+4 a B x+6 b B x^2}{12 x^4} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85

method	result	size
gosper	\(-\frac {6 b B \,x^{2}+4 A b x +4 B a x +3 A a}{12 x^{4}}\)	\(28\)
default	\(-\frac {A b +B a}{3 x^{3}}-\frac {b B}{2 x^{2}}-\frac {a A}{4 x^{4}}\)	\(28\)
norman	\(\frac {-\frac {b B \,x^{2}}{2}+\left (-\frac {A b}{3}-\frac {B a}{3}\right ) x -\frac {A a}{4}}{x^{4}}\)	\(28\)
risch	\(\frac {-\frac {b B \,x^{2}}{2}+\left (-\frac {A b}{3}-\frac {B a}{3}\right ) x -\frac {A a}{4}}{x^{4}}\)	\(28\)
parallelrisch	\(-\frac {6 b B \,x^{2}+4 A b x +4 B a x +3 A a}{12 x^{4}}\)	\(28\)

[In]

int((b*x+a)*(B*x+A)/x^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x) (A+B x)}{x^5} \, dx=-\frac {6 \, B b x^{2} + 3 \, A a + 4 \, {\left (B a + A b\right )} x}{12 \, x^{4}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (A+B x)}{x^5} \, dx=\frac {- 3 A a - 6 B b x^{2} + x \left (- 4 A b - 4 B a\right )}{12 x^{4}} \]

[In]

integrate((b*x+a)*(B*x+A)/x**5,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x) (A+B x)}{x^5} \, dx=-\frac {6 \, B b x^{2} + 3 \, A a + 4 \, {\left (B a + A b\right )} x}{12 \, x^{4}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x) (A+B x)}{x^5} \, dx=-\frac {6 \, B b x^{2} + 4 \, B a x + 4 \, A b x + 3 \, A a}{12 \, x^{4}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x) (A+B x)}{x^5} \, dx=-\frac {\frac {B\,b\,x^2}{2}+\left (\frac {A\,b}{3}+\frac {B\,a}{3}\right )\,x+\frac {A\,a}{4}}{x^4} \]

[In]

int(((A + B*x)*(a + b*x))/x^5,x)

[Out]

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