\(\int (a+\frac {b}{x^2})^3 x^4 \, dx\) [1832]

Optimal result

Integrand size = 13, antiderivative size = 34 \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=-\frac {b^3}{x}+3 a b^2 x+a^2 b x^3+\frac {a^3 x^5}{5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, 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} \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=\frac {a^3 x^5}{5}+a^2 b x^3+3 a b^2 x-\frac {b^3}{x} \]

[In]

Int[(a + b/x^2)^3*x^4,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=-\frac {b^3}{x}+3 a b^2 x+a^2 b x^3+\frac {a^3 x^5}{5} \]

[In]

Integrate[(a + b/x^2)^3*x^4,x]

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {b^{3}}{x}+3 a \,b^{2} x +a^{2} b \,x^{3}+\frac {a^{3} x^{5}}{5}\)	\(33\)
risch	\(-\frac {b^{3}}{x}+3 a \,b^{2} x +a^{2} b \,x^{3}+\frac {a^{3} x^{5}}{5}\)	\(33\)
gosper	\(\frac {x^{6} a^{3}+5 a^{2} b \,x^{4}+15 a \,b^{2} x^{2}-5 b^{3}}{5 x}\)	\(37\)
parallelrisch	\(\frac {x^{6} a^{3}+5 a^{2} b \,x^{4}+15 a \,b^{2} x^{2}-5 b^{3}}{5 x}\)	\(37\)
norman	\(\frac {a^{2} b \,x^{8}-b^{3} x^{4}+\frac {1}{5} x^{10} a^{3}+3 a \,b^{2} x^{6}}{x^{5}}\)	\(39\)

[In]

int((a+b/x^2)^3*x^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=\frac {a^{3} x^{6} + 5 \, a^{2} b x^{4} + 15 \, a b^{2} x^{2} - 5 \, b^{3}}{5 \, x} \]

[In]

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

[Out]

1/5*(a^3*x^6 + 5*a^2*b*x^4 + 15*a*b^2*x^2 - 5*b^3)/x

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=\frac {a^{3} x^{5}}{5} + a^{2} b x^{3} + 3 a b^{2} x - \frac {b^{3}}{x} \]

[In]

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

[Out]

a**3*x**5/5 + a**2*b*x**3 + 3*a*b**2*x - b**3/x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=\frac {1}{5} \, a^{3} x^{5} + a^{2} b x^{3} + 3 \, a b^{2} x - \frac {b^{3}}{x} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=\frac {1}{5} \, a^{3} x^{5} + a^{2} b x^{3} + 3 \, a b^{2} x - \frac {b^{3}}{x} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \left (a+\frac {b}{x^2}\right )^3 x^4 \, dx=\frac {a^3\,x^5}{5}-\frac {b^3}{x}+a^2\,b\,x^3+3\,a\,b^2\,x \]

[In]

int(x^4*(a + b/x^2)^3,x)

[Out]

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