3.1840 \(\int \frac {(a+\frac {b}{x^2})^3}{x^4} \, dx\)

Optimal. Leaf size=43 \[ -\frac {a^3}{3 x^3}-\frac {3 a^2 b}{5 x^5}-\frac {3 a b^2}{7 x^7}-\frac {b^3}{9 x^9} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {263, 270} \[ -\frac {3 a^2 b}{5 x^5}-\frac {a^3}{3 x^3}-\frac {3 a b^2}{7 x^7}-\frac {b^3}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^3/x^4,x]

[Out]

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

Rule 263

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 270

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*} \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^4} \, dx &=\int \frac {\left (b+a x^2\right )^3}{x^{10}} \, dx\\ &=\int \left (\frac {b^3}{x^{10}}+\frac {3 a b^2}{x^8}+\frac {3 a^2 b}{x^6}+\frac {a^3}{x^4}\right ) \, dx\\ &=-\frac {b^3}{9 x^9}-\frac {3 a b^2}{7 x^7}-\frac {3 a^2 b}{5 x^5}-\frac {a^3}{3 x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 43, normalized size = 1.00 \[ -\frac {a^3}{3 x^3}-\frac {3 a^2 b}{5 x^5}-\frac {3 a b^2}{7 x^7}-\frac {b^3}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^3/x^4,x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.83, size = 37, normalized size = 0.86 \[ -\frac {105 \, a^{3} x^{6} + 189 \, a^{2} b x^{4} + 135 \, a b^{2} x^{2} + 35 \, b^{3}}{315 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^3/x^4,x, algorithm="fricas")

[Out]

-1/315*(105*a^3*x^6 + 189*a^2*b*x^4 + 135*a*b^2*x^2 + 35*b^3)/x^9

________________________________________________________________________________________

giac [A]  time = 0.15, size = 37, normalized size = 0.86 \[ -\frac {105 \, a^{3} x^{6} + 189 \, a^{2} b x^{4} + 135 \, a b^{2} x^{2} + 35 \, b^{3}}{315 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^3/x^4,x, algorithm="giac")

[Out]

-1/315*(105*a^3*x^6 + 189*a^2*b*x^4 + 135*a*b^2*x^2 + 35*b^3)/x^9

________________________________________________________________________________________

maple [A]  time = 0.00, size = 36, normalized size = 0.84 \[ -\frac {a^{3}}{3 x^{3}}-\frac {3 a^{2} b}{5 x^{5}}-\frac {3 a \,b^{2}}{7 x^{7}}-\frac {b^{3}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^2)^3/x^4,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.87, size = 37, normalized size = 0.86 \[ -\frac {105 \, a^{3} x^{6} + 189 \, a^{2} b x^{4} + 135 \, a b^{2} x^{2} + 35 \, b^{3}}{315 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^3/x^4,x, algorithm="maxima")

[Out]

-1/315*(105*a^3*x^6 + 189*a^2*b*x^4 + 135*a*b^2*x^2 + 35*b^3)/x^9

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 37, normalized size = 0.86 \[ -\frac {\frac {a^3\,x^6}{3}+\frac {3\,a^2\,b\,x^4}{5}+\frac {3\,a\,b^2\,x^2}{7}+\frac {b^3}{9}}{x^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^2)^3/x^4,x)

[Out]

-(b^3/9 + (a^3*x^6)/3 + (3*a*b^2*x^2)/7 + (3*a^2*b*x^4)/5)/x^9

________________________________________________________________________________________

sympy [A]  time = 0.29, size = 39, normalized size = 0.91 \[ \frac {- 105 a^{3} x^{6} - 189 a^{2} b x^{4} - 135 a b^{2} x^{2} - 35 b^{3}}{315 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)**3/x**4,x)

[Out]

(-105*a**3*x**6 - 189*a**2*b*x**4 - 135*a*b**2*x**2 - 35*b**3)/(315*x**9)

________________________________________________________________________________________