3.672 \(\int \frac{x^{11}}{(a+c x^4)^3} \, dx\)

Optimal. Leaf size=52 \[ -\frac{a^2}{8 c^3 \left (a+c x^4\right )^2}+\frac{a}{2 c^3 \left (a+c x^4\right )}+\frac{\log \left (a+c x^4\right )}{4 c^3} \]

[Out]

-a^2/(8*c^3*(a + c*x^4)^2) + a/(2*c^3*(a + c*x^4)) + Log[a + c*x^4]/(4*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0362498, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac{a^2}{8 c^3 \left (a+c x^4\right )^2}+\frac{a}{2 c^3 \left (a+c x^4\right )}+\frac{\log \left (a+c x^4\right )}{4 c^3} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a + c*x^4)^3,x]

[Out]

-a^2/(8*c^3*(a + c*x^4)^2) + a/(2*c^3*(a + c*x^4)) + Log[a + c*x^4]/(4*c^3)

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 (a+c x^4\right )^3} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{(a+c x)^3} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^2}{c^2 (a+c x)^3}-\frac{2 a}{c^2 (a+c x)^2}+\frac{1}{c^2 (a+c x)}\right ) \, dx,x,x^4\right )\\ &=-\frac{a^2}{8 c^3 \left (a+c x^4\right )^2}+\frac{a}{2 c^3 \left (a+c x^4\right )}+\frac{\log \left (a+c x^4\right )}{4 c^3}\\ \end{align*}

Mathematica [A]  time = 0.017793, size = 39, normalized size = 0.75 \[ \frac{\frac{a \left (3 a+4 c x^4\right )}{\left (a+c x^4\right )^2}+2 \log \left (a+c x^4\right )}{8 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + c*x^4)^3,x]

[Out]

((a*(3*a + 4*c*x^4))/(a + c*x^4)^2 + 2*Log[a + c*x^4])/(8*c^3)

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 47, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{8\,{c}^{3} \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{a}{2\,{c}^{3} \left ( c{x}^{4}+a \right ) }}+{\frac{\ln \left ( c{x}^{4}+a \right ) }{4\,{c}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*a^2/c^3/(c*x^4+a)^2+1/2*a/c^3/(c*x^4+a)+1/4*ln(c*x^4+a)/c^3

________________________________________________________________________________________

Maxima [A]  time = 1.0194, size = 74, normalized size = 1.42 \begin{align*} \frac{4 \, a c x^{4} + 3 \, a^{2}}{8 \,{\left (c^{5} x^{8} + 2 \, a c^{4} x^{4} + a^{2} c^{3}\right )}} + \frac{\log \left (c x^{4} + a\right )}{4 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*a*c*x^4 + 3*a^2)/(c^5*x^8 + 2*a*c^4*x^4 + a^2*c^3) + 1/4*log(c*x^4 + a)/c^3

________________________________________________________________________________________

Fricas [A]  time = 1.67201, size = 143, normalized size = 2.75 \begin{align*} \frac{4 \, a c x^{4} + 3 \, a^{2} + 2 \,{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \log \left (c x^{4} + a\right )}{8 \,{\left (c^{5} x^{8} + 2 \, a c^{4} x^{4} + a^{2} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*a*c*x^4 + 3*a^2 + 2*(c^2*x^8 + 2*a*c*x^4 + a^2)*log(c*x^4 + a))/(c^5*x^8 + 2*a*c^4*x^4 + a^2*c^3)

________________________________________________________________________________________

Sympy [A]  time = 1.65684, size = 53, normalized size = 1.02 \begin{align*} \frac{3 a^{2} + 4 a c x^{4}}{8 a^{2} c^{3} + 16 a c^{4} x^{4} + 8 c^{5} x^{8}} + \frac{\log{\left (a + c x^{4} \right )}}{4 c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*a**2 + 4*a*c*x**4)/(8*a**2*c**3 + 16*a*c**4*x**4 + 8*c**5*x**8) + log(a + c*x**4)/(4*c**3)

________________________________________________________________________________________

Giac [A]  time = 1.11675, size = 57, normalized size = 1.1 \begin{align*} \frac{\log \left ({\left | c x^{4} + a \right |}\right )}{4 \, c^{3}} - \frac{3 \, c x^{8} + 2 \, a x^{4}}{8 \,{\left (c x^{4} + a\right )}^{2} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(c*x^4+a)^3,x, algorithm="giac")

[Out]

1/4*log(abs(c*x^4 + a))/c^3 - 1/8*(3*c*x^8 + 2*a*x^4)/((c*x^4 + a)^2*c^2)