\(\int \frac {(a+b x) (A+B x)}{x^6} \, dx\) [90]

Optimal result

Integrand size = 14, antiderivative size = 33 \[ \int \frac {(a+b x) (A+B x)}{x^6} \, dx=-\frac {a A}{5 x^5}-\frac {A b+a B}{4 x^4}-\frac {b B}{3 x^3} \]

[Out]

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

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} \[ \int \frac {(a+b x) (A+B x)}{x^6} \, dx=-\frac {a B+A b}{4 x^4}-\frac {a A}{5 x^5}-\frac {b B}{3 x^3} \]

[In]

Int[((a + b*x)*(A + B*x))/x^6,x]

[Out]

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

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^6}+\frac {A b+a B}{x^5}+\frac {b B}{x^4}\right ) \, dx \\ & = -\frac {a A}{5 x^5}-\frac {A b+a B}{4 x^4}-\frac {b B}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (A+B x)}{x^6} \, dx=-\frac {5 b x (3 A+4 B x)+3 a (4 A+5 B x)}{60 x^5} \]

[In]

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

[Out]

-1/60*(5*b*x*(3*A + 4*B*x) + 3*a*(4*A + 5*B*x))/x^5

Maple [A] (verified)

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

method	result	size
gosper	\(-\frac {20 b B \,x^{2}+15 A b x +15 B a x +12 A a}{60 x^{5}}\)	\(28\)
default	\(-\frac {b B}{3 x^{3}}-\frac {A b +B a}{4 x^{4}}-\frac {a A}{5 x^{5}}\)	\(28\)
norman	\(\frac {-\frac {b B \,x^{2}}{3}+\left (-\frac {A b}{4}-\frac {B a}{4}\right ) x -\frac {A a}{5}}{x^{5}}\)	\(28\)
risch	\(\frac {-\frac {b B \,x^{2}}{3}+\left (-\frac {A b}{4}-\frac {B a}{4}\right ) x -\frac {A a}{5}}{x^{5}}\)	\(28\)
parallelrisch	\(-\frac {20 b B \,x^{2}+15 A b x +15 B a x +12 A a}{60 x^{5}}\)	\(28\)

[In]

int((b*x+a)*(B*x+A)/x^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x) (A+B x)}{x^6} \, dx=-\frac {20 \, B b x^{2} + 12 \, A a + 15 \, {\left (B a + A b\right )} x}{60 \, x^{5}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (A+B x)}{x^6} \, dx=\frac {- 12 A a - 20 B b x^{2} + x \left (- 15 A b - 15 B a\right )}{60 x^{5}} \]

[In]

integrate((b*x+a)*(B*x+A)/x**6,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x) (A+B x)}{x^6} \, dx=-\frac {20 \, B b x^{2} + 12 \, A a + 15 \, {\left (B a + A b\right )} x}{60 \, x^{5}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x) (A+B x)}{x^6} \, dx=-\frac {20 \, B b x^{2} + 15 \, B a x + 15 \, A b x + 12 \, A a}{60 \, x^{5}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x) (A+B x)}{x^6} \, dx=-\frac {\frac {B\,b\,x^2}{3}+\left (\frac {A\,b}{4}+\frac {B\,a}{4}\right )\,x+\frac {A\,a}{5}}{x^5} \]

[In]

int(((A + B*x)*(a + b*x))/x^6,x)

[Out]

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