\(\int \frac {(a+\frac {b}{x^2})^2}{x^4} \, dx\) [1827]

Optimal result

Integrand size = 13, antiderivative size = 30 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^4} \, dx=-\frac {b^2}{7 x^7}-\frac {2 a b}{5 x^5}-\frac {a^2}{3 x^3} \]

[Out]

-1/7*b^2/x^7-2/5*a*b/x^5-1/3*a^2/x^3

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 276} \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^4} \, dx=-\frac {a^2}{3 x^3}-\frac {2 a b}{5 x^5}-\frac {b^2}{7 x^7} \]

[In]

Int[(a + b/x^2)^2/x^4,x]

[Out]

-1/7*b^2/x^7 - (2*a*b)/(5*x^5) - a^2/(3*x^3)

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (b+a x^2\right )^2}{x^8} \, dx \\ & = \int \left (\frac {b^2}{x^8}+\frac {2 a b}{x^6}+\frac {a^2}{x^4}\right ) \, dx \\ & = -\frac {b^2}{7 x^7}-\frac {2 a b}{5 x^5}-\frac {a^2}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^4} \, dx=-\frac {b^2}{7 x^7}-\frac {2 a b}{5 x^5}-\frac {a^2}{3 x^3} \]

[In]

Integrate[(a + b/x^2)^2/x^4,x]

[Out]

-1/7*b^2/x^7 - (2*a*b)/(5*x^5) - a^2/(3*x^3)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83

method	result	size
default	\(-\frac {b^{2}}{7 x^{7}}-\frac {2 a b}{5 x^{5}}-\frac {a^{2}}{3 x^{3}}\)	\(25\)
norman	\(\frac {-\frac {1}{3} a^{2} x^{4}-\frac {2}{5} a b \,x^{2}-\frac {1}{7} b^{2}}{x^{7}}\)	\(26\)
risch	\(\frac {-\frac {1}{3} a^{2} x^{4}-\frac {2}{5} a b \,x^{2}-\frac {1}{7} b^{2}}{x^{7}}\)	\(26\)
gosper	\(-\frac {35 a^{2} x^{4}+42 a b \,x^{2}+15 b^{2}}{105 x^{7}}\)	\(27\)
parallelrisch	\(\frac {-35 a^{2} x^{4}-42 a b \,x^{2}-15 b^{2}}{105 x^{7}}\)	\(27\)

[In]

int((a+b/x^2)^2/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/7*b^2/x^7-2/5*a*b/x^5-1/3*a^2/x^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^4} \, dx=-\frac {35 \, a^{2} x^{4} + 42 \, a b x^{2} + 15 \, b^{2}}{105 \, x^{7}} \]

[In]

integrate((a+b/x^2)^2/x^4,x, algorithm="fricas")

[Out]

-1/105*(35*a^2*x^4 + 42*a*b*x^2 + 15*b^2)/x^7

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^4} \, dx=\frac {- 35 a^{2} x^{4} - 42 a b x^{2} - 15 b^{2}}{105 x^{7}} \]

[In]

integrate((a+b/x**2)**2/x**4,x)

[Out]

(-35*a**2*x**4 - 42*a*b*x**2 - 15*b**2)/(105*x**7)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^4} \, dx=-\frac {35 \, a^{2} x^{4} + 42 \, a b x^{2} + 15 \, b^{2}}{105 \, x^{7}} \]

[In]

integrate((a+b/x^2)^2/x^4,x, algorithm="maxima")

[Out]

-1/105*(35*a^2*x^4 + 42*a*b*x^2 + 15*b^2)/x^7

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^4} \, dx=-\frac {35 \, a^{2} x^{4} + 42 \, a b x^{2} + 15 \, b^{2}}{105 \, x^{7}} \]

[In]

integrate((a+b/x^2)^2/x^4,x, algorithm="giac")

[Out]

-1/105*(35*a^2*x^4 + 42*a*b*x^2 + 15*b^2)/x^7

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^4} \, dx=-\frac {\frac {a^2\,x^4}{3}+\frac {2\,a\,b\,x^2}{5}+\frac {b^2}{7}}{x^7} \]

[In]

int((a + b/x^2)^2/x^4,x)

[Out]

-(b^2/7 + (a^2*x^4)/3 + (2*a*b*x^2)/5)/x^7