\(\int \frac {(a+\frac {b}{x^2})^3}{x^6} \, dx\) [1842]

Optimal result

Integrand size = 13, antiderivative size = 43 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^6} \, dx=-\frac {b^3}{11 x^{11}}-\frac {a b^2}{3 x^9}-\frac {3 a^2 b}{7 x^7}-\frac {a^3}{5 x^5} \]

[Out]

-1/11*b^3/x^11-1/3*a*b^2/x^9-3/7*a^2*b/x^7-1/5*a^3/x^5

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, 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} \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^6} \, dx=-\frac {a^3}{5 x^5}-\frac {3 a^2 b}{7 x^7}-\frac {a b^2}{3 x^9}-\frac {b^3}{11 x^{11}} \]

[In]

Int[(a + b/x^2)^3/x^6,x]

[Out]

-1/11*b^3/x^11 - (a*b^2)/(3*x^9) - (3*a^2*b)/(7*x^7) - a^3/(5*x^5)

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^{12}} \, dx \\ & = \int \left (\frac {b^3}{x^{12}}+\frac {3 a b^2}{x^{10}}+\frac {3 a^2 b}{x^8}+\frac {a^3}{x^6}\right ) \, dx \\ & = -\frac {b^3}{11 x^{11}}-\frac {a b^2}{3 x^9}-\frac {3 a^2 b}{7 x^7}-\frac {a^3}{5 x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^6} \, dx=-\frac {b^3}{11 x^{11}}-\frac {a b^2}{3 x^9}-\frac {3 a^2 b}{7 x^7}-\frac {a^3}{5 x^5} \]

[In]

Integrate[(a + b/x^2)^3/x^6,x]

[Out]

-1/11*b^3/x^11 - (a*b^2)/(3*x^9) - (3*a^2*b)/(7*x^7) - a^3/(5*x^5)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84

method	result	size
default	\(-\frac {b^{3}}{11 x^{11}}-\frac {a \,b^{2}}{3 x^{9}}-\frac {3 a^{2} b}{7 x^{7}}-\frac {a^{3}}{5 x^{5}}\)	\(36\)
norman	\(\frac {-\frac {1}{5} x^{6} a^{3}-\frac {3}{7} a^{2} b \,x^{4}-\frac {1}{3} a \,b^{2} x^{2}-\frac {1}{11} b^{3}}{x^{11}}\)	\(37\)
risch	\(\frac {-\frac {1}{5} x^{6} a^{3}-\frac {3}{7} a^{2} b \,x^{4}-\frac {1}{3} a \,b^{2} x^{2}-\frac {1}{11} b^{3}}{x^{11}}\)	\(37\)
gosper	\(-\frac {231 x^{6} a^{3}+495 a^{2} b \,x^{4}+385 a \,b^{2} x^{2}+105 b^{3}}{1155 x^{11}}\)	\(38\)
parallelrisch	\(\frac {-231 x^{6} a^{3}-495 a^{2} b \,x^{4}-385 a \,b^{2} x^{2}-105 b^{3}}{1155 x^{11}}\)	\(38\)

[In]

int((a+b/x^2)^3/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/11*b^3/x^11-1/3*a*b^2/x^9-3/7*a^2*b/x^7-1/5*a^3/x^5

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^6} \, dx=-\frac {231 \, a^{3} x^{6} + 495 \, a^{2} b x^{4} + 385 \, a b^{2} x^{2} + 105 \, b^{3}}{1155 \, x^{11}} \]

[In]

integrate((a+b/x^2)^3/x^6,x, algorithm="fricas")

[Out]

-1/1155*(231*a^3*x^6 + 495*a^2*b*x^4 + 385*a*b^2*x^2 + 105*b^3)/x^11

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^6} \, dx=\frac {- 231 a^{3} x^{6} - 495 a^{2} b x^{4} - 385 a b^{2} x^{2} - 105 b^{3}}{1155 x^{11}} \]

[In]

integrate((a+b/x**2)**3/x**6,x)

[Out]

(-231*a**3*x**6 - 495*a**2*b*x**4 - 385*a*b**2*x**2 - 105*b**3)/(1155*x**11)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^6} \, dx=-\frac {231 \, a^{3} x^{6} + 495 \, a^{2} b x^{4} + 385 \, a b^{2} x^{2} + 105 \, b^{3}}{1155 \, x^{11}} \]

[In]

integrate((a+b/x^2)^3/x^6,x, algorithm="maxima")

[Out]

-1/1155*(231*a^3*x^6 + 495*a^2*b*x^4 + 385*a*b^2*x^2 + 105*b^3)/x^11

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^6} \, dx=-\frac {231 \, a^{3} x^{6} + 495 \, a^{2} b x^{4} + 385 \, a b^{2} x^{2} + 105 \, b^{3}}{1155 \, x^{11}} \]

[In]

integrate((a+b/x^2)^3/x^6,x, algorithm="giac")

[Out]

-1/1155*(231*a^3*x^6 + 495*a^2*b*x^4 + 385*a*b^2*x^2 + 105*b^3)/x^11

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^6} \, dx=-\frac {\frac {a^3\,x^6}{5}+\frac {3\,a^2\,b\,x^4}{7}+\frac {a\,b^2\,x^2}{3}+\frac {b^3}{11}}{x^{11}} \]

[In]

int((a + b/x^2)^3/x^6,x)

[Out]

-(b^3/11 + (a^3*x^6)/5 + (a*b^2*x^2)/3 + (3*a^2*b*x^4)/7)/x^11