\(\int \frac {(a+\frac {b}{x^2})^2}{x^5} \, dx\) [1828]

Optimal result

Integrand size = 13, antiderivative size = 30 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^5} \, dx=-\frac {b^2}{8 x^8}-\frac {a b}{3 x^6}-\frac {a^2}{4 x^4} \]

[Out]

-1/8*b^2/x^8-1/3*a*b/x^6-1/4*a^2/x^4

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {269, 272, 45} \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^5} \, dx=-\frac {a^2}{4 x^4}-\frac {a b}{3 x^6}-\frac {b^2}{8 x^8} \]

[In]

Int[(a + b/x^2)^2/x^5,x]

[Out]

-1/8*b^2/x^8 - (a*b)/(3*x^6) - a^2/(4*x^4)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (b+a x^2\right )^2}{x^9} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {(b+a x)^2}{x^5} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {b^2}{x^5}+\frac {2 a b}{x^4}+\frac {a^2}{x^3}\right ) \, dx,x,x^2\right ) \\ & = -\frac {b^2}{8 x^8}-\frac {a b}{3 x^6}-\frac {a^2}{4 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^5} \, dx=-\frac {b^2}{8 x^8}-\frac {a b}{3 x^6}-\frac {a^2}{4 x^4} \]

[In]

Integrate[(a + b/x^2)^2/x^5,x]

[Out]

-1/8*b^2/x^8 - (a*b)/(3*x^6) - a^2/(4*x^4)

Maple [A] (verified)

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

method	result	size
default	\(-\frac {b^{2}}{8 x^{8}}-\frac {a b}{3 x^{6}}-\frac {a^{2}}{4 x^{4}}\)	\(25\)
norman	\(\frac {-\frac {1}{4} a^{2} x^{4}-\frac {1}{3} a b \,x^{2}-\frac {1}{8} b^{2}}{x^{8}}\)	\(26\)
risch	\(\frac {-\frac {1}{4} a^{2} x^{4}-\frac {1}{3} a b \,x^{2}-\frac {1}{8} b^{2}}{x^{8}}\)	\(26\)
gosper	\(-\frac {6 a^{2} x^{4}+8 a b \,x^{2}+3 b^{2}}{24 x^{8}}\)	\(27\)
parallelrisch	\(\frac {-6 a^{2} x^{4}-8 a b \,x^{2}-3 b^{2}}{24 x^{8}}\)	\(27\)

[In]

int((a+b/x^2)^2/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/8*b^2/x^8-1/3*a*b/x^6-1/4*a^2/x^4

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^5} \, dx=-\frac {6 \, a^{2} x^{4} + 8 \, a b x^{2} + 3 \, b^{2}}{24 \, x^{8}} \]

[In]

integrate((a+b/x^2)^2/x^5,x, algorithm="fricas")

[Out]

-1/24*(6*a^2*x^4 + 8*a*b*x^2 + 3*b^2)/x^8

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^5} \, dx=\frac {- 6 a^{2} x^{4} - 8 a b x^{2} - 3 b^{2}}{24 x^{8}} \]

[In]

integrate((a+b/x**2)**2/x**5,x)

[Out]

(-6*a**2*x**4 - 8*a*b*x**2 - 3*b**2)/(24*x**8)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^5} \, dx=-\frac {6 \, a^{2} x^{4} + 8 \, a b x^{2} + 3 \, b^{2}}{24 \, x^{8}} \]

[In]

integrate((a+b/x^2)^2/x^5,x, algorithm="maxima")

[Out]

-1/24*(6*a^2*x^4 + 8*a*b*x^2 + 3*b^2)/x^8

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^5} \, dx=-\frac {6 \, a^{2} x^{4} + 8 \, a b x^{2} + 3 \, b^{2}}{24 \, x^{8}} \]

[In]

integrate((a+b/x^2)^2/x^5,x, algorithm="giac")

[Out]

-1/24*(6*a^2*x^4 + 8*a*b*x^2 + 3*b^2)/x^8

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^2}{x^5} \, dx=-\frac {\frac {a^2\,x^4}{4}+\frac {a\,b\,x^2}{3}+\frac {b^2}{8}}{x^8} \]

[In]

int((a + b/x^2)^2/x^5,x)

[Out]

-(b^2/8 + (a^2*x^4)/4 + (a*b*x^2)/3)/x^8