3.208 \(\int \frac{x^{11}}{(b x^2+c x^4)^3} \, dx\)

Optimal. Leaf size=49 \[ -\frac{b^2}{4 c^3 \left (b+c x^2\right )^2}+\frac{b}{c^3 \left (b+c x^2\right )}+\frac{\log \left (b+c x^2\right )}{2 c^3} \]

[Out]

-b^2/(4*c^3*(b + c*x^2)^2) + b/(c^3*(b + c*x^2)) + Log[b + c*x^2]/(2*c^3)

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Rubi [A]  time = 0.0447711, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1584, 266, 43} \[ -\frac{b^2}{4 c^3 \left (b+c x^2\right )^2}+\frac{b}{c^3 \left (b+c x^2\right )}+\frac{\log \left (b+c x^2\right )}{2 c^3} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(b*x^2 + c*x^4)^3,x]

[Out]

-b^2/(4*c^3*(b + c*x^2)^2) + b/(c^3*(b + c*x^2)) + Log[b + c*x^2]/(2*c^3)

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^{11}}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{x^5}{\left (b+c x^2\right )^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(b+c x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^2}{c^2 (b+c x)^3}-\frac{2 b}{c^2 (b+c x)^2}+\frac{1}{c^2 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{b^2}{4 c^3 \left (b+c x^2\right )^2}+\frac{b}{c^3 \left (b+c x^2\right )}+\frac{\log \left (b+c x^2\right )}{2 c^3}\\ \end{align*}

Mathematica [A]  time = 0.015503, size = 39, normalized size = 0.8 \[ \frac{\frac{b \left (3 b+4 c x^2\right )}{\left (b+c x^2\right )^2}+2 \log \left (b+c x^2\right )}{4 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(b*x^2 + c*x^4)^3,x]

[Out]

((b*(3*b + 4*c*x^2))/(b + c*x^2)^2 + 2*Log[b + c*x^2])/(4*c^3)

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Maple [A]  time = 0.051, size = 46, normalized size = 0.9 \begin{align*} -{\frac{{b}^{2}}{4\,{c}^{3} \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{b}{{c}^{3} \left ( c{x}^{2}+b \right ) }}+{\frac{\ln \left ( c{x}^{2}+b \right ) }{2\,{c}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(c*x^4+b*x^2)^3,x)

[Out]

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

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Maxima [A]  time = 1.02124, size = 74, normalized size = 1.51 \begin{align*} \frac{4 \, b c x^{2} + 3 \, b^{2}}{4 \,{\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} + \frac{\log \left (c x^{2} + b\right )}{2 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(c*x^4+b*x^2)^3,x, algorithm="maxima")

[Out]

1/4*(4*b*c*x^2 + 3*b^2)/(c^5*x^4 + 2*b*c^4*x^2 + b^2*c^3) + 1/2*log(c*x^2 + b)/c^3

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Fricas [A]  time = 1.45995, size = 143, normalized size = 2.92 \begin{align*} \frac{4 \, b c x^{2} + 3 \, b^{2} + 2 \,{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \log \left (c x^{2} + b\right )}{4 \,{\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(c*x^4+b*x^2)^3,x, algorithm="fricas")

[Out]

1/4*(4*b*c*x^2 + 3*b^2 + 2*(c^2*x^4 + 2*b*c*x^2 + b^2)*log(c*x^2 + b))/(c^5*x^4 + 2*b*c^4*x^2 + b^2*c^3)

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Sympy [A]  time = 0.50805, size = 53, normalized size = 1.08 \begin{align*} \frac{3 b^{2} + 4 b c x^{2}}{4 b^{2} c^{3} + 8 b c^{4} x^{2} + 4 c^{5} x^{4}} + \frac{\log{\left (b + c x^{2} \right )}}{2 c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(c*x**4+b*x**2)**3,x)

[Out]

(3*b**2 + 4*b*c*x**2)/(4*b**2*c**3 + 8*b*c**4*x**2 + 4*c**5*x**4) + log(b + c*x**2)/(2*c**3)

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Giac [A]  time = 1.2059, size = 57, normalized size = 1.16 \begin{align*} \frac{\log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{3}} - \frac{3 \, c x^{4} + 2 \, b x^{2}}{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^11/(c*x^4+b*x^2)^3,x, algorithm="giac")

[Out]

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