3.206 \(\int \frac{x^{13}}{(b x^2+c x^4)^3} \, dx\)

Optimal. Leaf size=65 \[ \frac{b^3}{4 c^4 \left (b+c x^2\right )^2}-\frac{3 b^2}{2 c^4 \left (b+c x^2\right )}-\frac{3 b \log \left (b+c x^2\right )}{2 c^4}+\frac{x^2}{2 c^3} \]

[Out]

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

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Rubi [A]  time = 0.0541817, antiderivative size = 65, 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} \[ \frac{b^3}{4 c^4 \left (b+c x^2\right )^2}-\frac{3 b^2}{2 c^4 \left (b+c x^2\right )}-\frac{3 b \log \left (b+c x^2\right )}{2 c^4}+\frac{x^2}{2 c^3} \]

Antiderivative was successfully verified.

[In]

Int[x^13/(b*x^2 + c*x^4)^3,x]

[Out]

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

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^{13}}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac{x^7}{\left (b+c x^2\right )^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{(b+c x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{c^3}-\frac{b^3}{c^3 (b+c x)^3}+\frac{3 b^2}{c^3 (b+c x)^2}-\frac{3 b}{c^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac{x^2}{2 c^3}+\frac{b^3}{4 c^4 \left (b+c x^2\right )^2}-\frac{3 b^2}{2 c^4 \left (b+c x^2\right )}-\frac{3 b \log \left (b+c x^2\right )}{2 c^4}\\ \end{align*}

Mathematica [A]  time = 0.0612385, size = 48, normalized size = 0.74 \[ -\frac{\frac{b^2 \left (5 b+6 c x^2\right )}{\left (b+c x^2\right )^2}+6 b \log \left (b+c x^2\right )-2 c x^2}{4 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^13/(b*x^2 + c*x^4)^3,x]

[Out]

-(-2*c*x^2 + (b^2*(5*b + 6*c*x^2))/(b + c*x^2)^2 + 6*b*Log[b + c*x^2])/(4*c^4)

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Maple [A]  time = 0.052, size = 58, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2\,{c}^{3}}}+{\frac{{b}^{3}}{4\,{c}^{4} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{3\,{b}^{2}}{2\,{c}^{4} \left ( c{x}^{2}+b \right ) }}-{\frac{3\,b\ln \left ( c{x}^{2}+b \right ) }{2\,{c}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^13/(c*x^4+b*x^2)^3,x)

[Out]

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

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Maxima [A]  time = 0.984314, size = 89, normalized size = 1.37 \begin{align*} -\frac{6 \, b^{2} c x^{2} + 5 \, b^{3}}{4 \,{\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}} + \frac{x^{2}}{2 \, c^{3}} - \frac{3 \, b \log \left (c x^{2} + b\right )}{2 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(c*x^4+b*x^2)^3,x, algorithm="maxima")

[Out]

-1/4*(6*b^2*c*x^2 + 5*b^3)/(c^6*x^4 + 2*b*c^5*x^2 + b^2*c^4) + 1/2*x^2/c^3 - 3/2*b*log(c*x^2 + b)/c^4

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Fricas [A]  time = 1.50194, size = 186, normalized size = 2.86 \begin{align*} \frac{2 \, c^{3} x^{6} + 4 \, b c^{2} x^{4} - 4 \, b^{2} c x^{2} - 5 \, b^{3} - 6 \,{\left (b c^{2} x^{4} + 2 \, b^{2} c x^{2} + b^{3}\right )} \log \left (c x^{2} + b\right )}{4 \,{\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(c*x^4+b*x^2)^3,x, algorithm="fricas")

[Out]

1/4*(2*c^3*x^6 + 4*b*c^2*x^4 - 4*b^2*c*x^2 - 5*b^3 - 6*(b*c^2*x^4 + 2*b^2*c*x^2 + b^3)*log(c*x^2 + b))/(c^6*x^
4 + 2*b*c^5*x^2 + b^2*c^4)

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Sympy [A]  time = 0.619005, size = 66, normalized size = 1.02 \begin{align*} - \frac{3 b \log{\left (b + c x^{2} \right )}}{2 c^{4}} - \frac{5 b^{3} + 6 b^{2} c x^{2}}{4 b^{2} c^{4} + 8 b c^{5} x^{2} + 4 c^{6} x^{4}} + \frac{x^{2}}{2 c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**13/(c*x**4+b*x**2)**3,x)

[Out]

-3*b*log(b + c*x**2)/(2*c**4) - (5*b**3 + 6*b**2*c*x**2)/(4*b**2*c**4 + 8*b*c**5*x**2 + 4*c**6*x**4) + x**2/(2
*c**3)

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Giac [A]  time = 1.19791, size = 84, normalized size = 1.29 \begin{align*} \frac{x^{2}}{2 \, c^{3}} - \frac{3 \, b \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{4}} + \frac{9 \, b c^{2} x^{4} + 12 \, b^{2} c x^{2} + 4 \, b^{3}}{4 \,{\left (c x^{2} + b\right )}^{2} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(c*x^4+b*x^2)^3,x, algorithm="giac")

[Out]

1/2*x^2/c^3 - 3/2*b*log(abs(c*x^2 + b))/c^4 + 1/4*(9*b*c^2*x^4 + 12*b^2*c*x^2 + 4*b^3)/((c*x^2 + b)^2*c^4)