\(\int (a+\frac {b}{x^2})^2 x^6 \, dx\) [1817]

Optimal result

Integrand size = 13, antiderivative size = 30 \[ \int \left (a+\frac {b}{x^2}\right )^2 x^6 \, dx=\frac {b^2 x^3}{3}+\frac {2}{5} a b x^5+\frac {a^2 x^7}{7} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, 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} \[ \int \left (a+\frac {b}{x^2}\right )^2 x^6 \, dx=\frac {a^2 x^7}{7}+\frac {2}{5} a b x^5+\frac {b^2 x^3}{3} \]

[In]

Int[(a + b/x^2)^2*x^6,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x^2}\right )^2 x^6 \, dx=\frac {b^2 x^3}{3}+\frac {2}{5} a b x^5+\frac {a^2 x^7}{7} \]

[In]

Integrate[(a + b/x^2)^2*x^6,x]

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {1}{3} b^{2} x^{3}+\frac {2}{5} a b \,x^{5}+\frac {1}{7} a^{2} x^{7}\)	\(25\)
risch	\(\frac {1}{3} b^{2} x^{3}+\frac {2}{5} a b \,x^{5}+\frac {1}{7} a^{2} x^{7}\)	\(25\)
parallelrisch	\(\frac {1}{3} b^{2} x^{3}+\frac {2}{5} a b \,x^{5}+\frac {1}{7} a^{2} x^{7}\)	\(25\)
gosper	\(\frac {x^{3} \left (15 a^{2} x^{4}+42 a b \,x^{2}+35 b^{2}\right )}{105}\)	\(27\)
norman	\(\frac {\frac {1}{7} a^{2} x^{10}+\frac {1}{3} b^{2} x^{6}+\frac {2}{5} a b \,x^{8}}{x^{3}}\)	\(29\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \left (a+\frac {b}{x^2}\right )^2 x^6 \, dx=\frac {1}{7} \, a^{2} x^{7} + \frac {2}{5} \, a b x^{5} + \frac {1}{3} \, b^{2} x^{3} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \left (a+\frac {b}{x^2}\right )^2 x^6 \, dx=\frac {a^{2} x^{7}}{7} + \frac {2 a b x^{5}}{5} + \frac {b^{2} x^{3}}{3} \]

[In]

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

[Out]

a**2*x**7/7 + 2*a*b*x**5/5 + b**2*x**3/3

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \left (a+\frac {b}{x^2}\right )^2 x^6 \, dx=\frac {1}{7} \, a^{2} x^{7} + \frac {2}{5} \, a b x^{5} + \frac {1}{3} \, b^{2} x^{3} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \left (a+\frac {b}{x^2}\right )^2 x^6 \, dx=\frac {1}{7} \, a^{2} x^{7} + \frac {2}{5} \, a b x^{5} + \frac {1}{3} \, b^{2} x^{3} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.93 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \left (a+\frac {b}{x^2}\right )^2 x^6 \, dx=\frac {a^2\,x^7}{7}+\frac {2\,a\,b\,x^5}{5}+\frac {b^2\,x^3}{3} \]

[In]

int(x^6*(a + b/x^2)^2,x)

[Out]

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