\(\int \frac {x^9}{(b x^2+c x^4)^3} \, dx\) [210]

Optimal result

Integrand size = 17, antiderivative size = 19 \[ \int \frac {x^9}{\left (b x^2+c x^4\right )^3} \, dx=\frac {x^4}{4 b \left (b+c x^2\right )^2} \]

[Out]

1/4*x^4/b/(c*x^2+b)^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 270} \[ \int \frac {x^9}{\left (b x^2+c x^4\right )^3} \, dx=\frac {x^4}{4 b \left (b+c x^2\right )^2} \]

[In]

Int[x^9/(b*x^2 + c*x^4)^3,x]

[Out]

x^4/(4*b*(b + c*x^2)^2)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3}{\left (b+c x^2\right )^3} \, dx \\ & = \frac {x^4}{4 b \left (b+c x^2\right )^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {x^9}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {b+2 c x^2}{4 c^2 \left (b+c x^2\right )^2} \]

[In]

Integrate[x^9/(b*x^2 + c*x^4)^3,x]

[Out]

-1/4*(b + 2*c*x^2)/(c^2*(b + c*x^2)^2)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21

method	result	size
gosper	\(-\frac {2 c \,x^{2}+b}{4 \left (c \,x^{2}+b \right )^{2} c^{2}}\)	\(23\)
parallelrisch	\(\frac {-2 c \,x^{2}-b}{4 c^{2} \left (c \,x^{2}+b \right )^{2}}\)	\(25\)
risch	\(\frac {-\frac {x^{2}}{2 c}-\frac {b}{4 c^{2}}}{\left (c \,x^{2}+b \right )^{2}}\)	\(26\)
default	\(\frac {b}{4 c^{2} \left (c \,x^{2}+b \right )^{2}}-\frac {1}{2 c^{2} \left (c \,x^{2}+b \right )}\)	\(31\)
norman	\(\frac {-\frac {x^{7}}{2 c}-\frac {b \,x^{5}}{4 c^{2}}}{x^{5} \left (c \,x^{2}+b \right )^{2}}\)	\(32\)

[In]

int(x^9/(c*x^4+b*x^2)^3,x,method=_RETURNVERBOSE)

[Out]

-1/4*(2*c*x^2+b)/(c*x^2+b)^2/c^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {x^9}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {2 \, c x^{2} + b}{4 \, {\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )}} \]

[In]

integrate(x^9/(c*x^4+b*x^2)^3,x, algorithm="fricas")

[Out]

-1/4*(2*c*x^2 + b)/(c^4*x^4 + 2*b*c^3*x^2 + b^2*c^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (14) = 28\).

Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {x^9}{\left (b x^2+c x^4\right )^3} \, dx=\frac {- b - 2 c x^{2}}{4 b^{2} c^{2} + 8 b c^{3} x^{2} + 4 c^{4} x^{4}} \]

[In]

integrate(x**9/(c*x**4+b*x**2)**3,x)

[Out]

(-b - 2*c*x**2)/(4*b**2*c**2 + 8*b*c**3*x**2 + 4*c**4*x**4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).

Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {x^9}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {2 \, c x^{2} + b}{4 \, {\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )}} \]

[In]

integrate(x^9/(c*x^4+b*x^2)^3,x, algorithm="maxima")

[Out]

-1/4*(2*c*x^2 + b)/(c^4*x^4 + 2*b*c^3*x^2 + b^2*c^2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {x^9}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {2 \, c x^{2} + b}{4 \, {\left (c x^{2} + b\right )}^{2} c^{2}} \]

[In]

integrate(x^9/(c*x^4+b*x^2)^3,x, algorithm="giac")

[Out]

-1/4*(2*c*x^2 + b)/((c*x^2 + b)^2*c^2)

Mupad [B] (verification not implemented)

Time = 12.88 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.95 \[ \int \frac {x^9}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {\frac {b}{4\,c^2}+\frac {x^2}{2\,c}}{b^2+2\,b\,c\,x^2+c^2\,x^4} \]

[In]

int(x^9/(b*x^2 + c*x^4)^3,x)

[Out]

-(b/(4*c^2) + x^2/(2*c))/(b^2 + c^2*x^4 + 2*b*c*x^2)