3.674 \(\int \frac{x^7}{(a+c x^4)^3} \, dx\)

Optimal. Leaf size=19 \[ \frac{x^8}{8 a \left (a+c x^4\right )^2} \]

[Out]

x^8/(8*a*(a + c*x^4)^2)

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Rubi [A]  time = 0.0035922, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {264} \[ \frac{x^8}{8 a \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a + c*x^4)^3,x]

[Out]

x^8/(8*a*(a + c*x^4)^2)

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^7}{\left (a+c x^4\right )^3} \, dx &=\frac{x^8}{8 a \left (a+c x^4\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0087713, size = 24, normalized size = 1.26 \[ -\frac{a+2 c x^4}{8 c^2 \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + c*x^4)^3,x]

[Out]

-(a + 2*c*x^4)/(8*c^2*(a + c*x^4)^2)

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Maple [A]  time = 0.01, size = 31, normalized size = 1.6 \begin{align*} -{\frac{1}{4\,{c}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{a}{8\,{c}^{2} \left ( c{x}^{4}+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(c*x^4+a)^3,x)

[Out]

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

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Maxima [B]  time = 0.985189, size = 49, normalized size = 2.58 \begin{align*} -\frac{2 \, c x^{4} + a}{8 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^4+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.64476, size = 73, normalized size = 3.84 \begin{align*} -\frac{2 \, c x^{4} + a}{8 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^4+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.40093, size = 36, normalized size = 1.89 \begin{align*} - \frac{a + 2 c x^{4}}{8 a^{2} c^{2} + 16 a c^{3} x^{4} + 8 c^{4} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(c*x**4+a)**3,x)

[Out]

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

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Giac [A]  time = 1.11741, size = 30, normalized size = 1.58 \begin{align*} -\frac{2 \, c x^{4} + a}{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^7/(c*x^4+a)^3,x, algorithm="giac")

[Out]

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