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

Optimal. Leaf size=19 \[ \frac{x^4}{4 b \left (b+c x^2\right )^2} \]

[Out]

x^4/(4*b*(b + c*x^2)^2)

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Rubi [A]  time = 0.0110066, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1584, 264} \[ \frac{x^4}{4 b \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x^9/(b*x^2 + c*x^4)^3,x]

[Out]

x^4/(4*b*(b + c*x^2)^2)

Rule 1584

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]

Rule 264

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]

Rubi steps

\begin{align*} \int \frac{x^9}{\left (b x^2+c x^4\right )^3} \, dx &=\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]  time = 0.0067747, size = 24, normalized size = 1.26 \[ -\frac{b+2 c x^2}{4 c^2 \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(b*x^2 + c*x^4)^3,x]

[Out]

-(b + 2*c*x^2)/(4*c^2*(b + c*x^2)^2)

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Maple [A]  time = 0.049, size = 31, normalized size = 1.6 \begin{align*} -{\frac{1}{2\,{c}^{2} \left ( c{x}^{2}+b \right ) }}+{\frac{b}{4\,{c}^{2} \left ( c{x}^{2}+b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.00143, size = 49, normalized size = 2.58 \begin{align*} -\frac{2 \, c x^{2} + b}{4 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.47332, size = 73, normalized size = 3.84 \begin{align*} -\frac{2 \, c x^{2} + b}{4 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 0.535455, size = 36, normalized size = 1.89 \begin{align*} - \frac{b + 2 c x^{2}}{4 b^{2} c^{2} + 8 b c^{3} x^{2} + 4 c^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.28051, size = 30, normalized size = 1.58 \begin{align*} -\frac{2 \, c x^{2} + b}{4 \,{\left (c x^{2} + b\right )}^{2} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)