\(\int \frac {(a+\frac {b}{x^2})^3}{x^2} \, dx\) [1838]

Optimal result

Integrand size = 13, antiderivative size = 39 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^2} \, dx=-\frac {b^3}{7 x^7}-\frac {3 a b^2}{5 x^5}-\frac {a^2 b}{x^3}-\frac {a^3}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, 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} \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^2} \, dx=-\frac {a^3}{x}-\frac {a^2 b}{x^3}-\frac {3 a b^2}{5 x^5}-\frac {b^3}{7 x^7} \]

[In]

Int[(a + b/x^2)^3/x^2,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^2} \, dx=-\frac {b^3}{7 x^7}-\frac {3 a b^2}{5 x^5}-\frac {a^2 b}{x^3}-\frac {a^3}{x} \]

[In]

Integrate[(a + b/x^2)^3/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {b^{3}}{7 x^{7}}-\frac {3 a \,b^{2}}{5 x^{5}}-\frac {a^{2} b}{x^{3}}-\frac {a^{3}}{x}\)	\(36\)
norman	\(\frac {-x^{6} a^{3}-a^{2} b \,x^{4}-\frac {3}{5} a \,b^{2} x^{2}-\frac {1}{7} b^{3}}{x^{7}}\)	\(37\)
risch	\(\frac {-x^{6} a^{3}-a^{2} b \,x^{4}-\frac {3}{5} a \,b^{2} x^{2}-\frac {1}{7} b^{3}}{x^{7}}\)	\(37\)
gosper	\(-\frac {35 x^{6} a^{3}+35 a^{2} b \,x^{4}+21 a \,b^{2} x^{2}+5 b^{3}}{35 x^{7}}\)	\(38\)
parallelrisch	\(\frac {-35 x^{6} a^{3}-35 a^{2} b \,x^{4}-21 a \,b^{2} x^{2}-5 b^{3}}{35 x^{7}}\)	\(38\)

[In]

int((a+b/x^2)^3/x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^2} \, dx=-\frac {35 \, a^{3} x^{6} + 35 \, a^{2} b x^{4} + 21 \, a b^{2} x^{2} + 5 \, b^{3}}{35 \, x^{7}} \]

[In]

integrate((a+b/x^2)^3/x^2,x, algorithm="fricas")

[Out]

-1/35*(35*a^3*x^6 + 35*a^2*b*x^4 + 21*a*b^2*x^2 + 5*b^3)/x^7

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^2} \, dx=\frac {- 35 a^{3} x^{6} - 35 a^{2} b x^{4} - 21 a b^{2} x^{2} - 5 b^{3}}{35 x^{7}} \]

[In]

integrate((a+b/x**2)**3/x**2,x)

[Out]

(-35*a**3*x**6 - 35*a**2*b*x**4 - 21*a*b**2*x**2 - 5*b**3)/(35*x**7)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^2} \, dx=-\frac {35 \, a^{3} x^{6} + 35 \, a^{2} b x^{4} + 21 \, a b^{2} x^{2} + 5 \, b^{3}}{35 \, x^{7}} \]

[In]

integrate((a+b/x^2)^3/x^2,x, algorithm="maxima")

[Out]

-1/35*(35*a^3*x^6 + 35*a^2*b*x^4 + 21*a*b^2*x^2 + 5*b^3)/x^7

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^2} \, dx=-\frac {35 \, a^{3} x^{6} + 35 \, a^{2} b x^{4} + 21 \, a b^{2} x^{2} + 5 \, b^{3}}{35 \, x^{7}} \]

[In]

integrate((a+b/x^2)^3/x^2,x, algorithm="giac")

[Out]

-1/35*(35*a^3*x^6 + 35*a^2*b*x^4 + 21*a*b^2*x^2 + 5*b^3)/x^7

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^2} \, dx=-\frac {a^3\,x^6+a^2\,b\,x^4+\frac {3\,a\,b^2\,x^2}{5}+\frac {b^3}{7}}{x^7} \]

[In]

int((a + b/x^2)^3/x^2,x)

[Out]

-(b^3/7 + a^3*x^6 + (3*a*b^2*x^2)/5 + a^2*b*x^4)/x^7