3.7.74 \(\int \frac {x^7}{(a+c x^4)^3} \, dx\) [674]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {270} \begin {gather*} \frac {x^8}{8 a \left (a+c x^4\right )^2} \end {gather*}

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 270

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 = 0.01, size = 24, normalized size = 1.26 \begin {gather*} -\frac {a+2 c x^4}{8 c^2 \left (a+c x^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + c*x^4)^3,x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.15, size = 31, normalized size = 1.63

method	result	size
gosper	\(-\frac {2 x^{4} c +a}{8 c^{2} \left (x^{4} c +a \right )^{2}}\)	\(23\)
norman	\(\frac {-\frac {x^{4}}{4 c}-\frac {a}{8 c^{2}}}{\left (x^{4} c +a \right )^{2}}\)	\(26\)
risch	\(\frac {-\frac {x^{4}}{4 c}-\frac {a}{8 c^{2}}}{\left (x^{4} c +a \right )^{2}}\)	\(26\)
default	\(\frac {a}{8 c^{2} \left (x^{4} c +a \right )^{2}}-\frac {1}{4 c^{2} \left (x^{4} c +a \right )}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
time = 0.31, size = 36, normalized size = 1.89 \begin {gather*} -\frac {2 \, c x^{4} + a}{8 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} \end {gather*}

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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
time = 0.36, size = 36, normalized size = 1.89 \begin {gather*} -\frac {2 \, c x^{4} + a}{8 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} \end {gather*}

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)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (14) = 28\).
time = 0.18, size = 36, normalized size = 1.89 \begin {gather*} \frac {- a - 2 c x^{4}}{8 a^{2} c^{2} + 16 a c^{3} x^{4} + 8 c^{4} x^{8}} \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 0.67, size = 22, normalized size = 1.16 \begin {gather*} -\frac {2 \, c x^{4} + a}{8 \, {\left (c x^{4} + a\right )}^{2} c^{2}} \end {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 37, normalized size = 1.95 \begin {gather*} -\frac {\frac {a}{8\,c^2}+\frac {x^4}{4\,c}}{a^2+2\,a\,c\,x^4+c^2\,x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a + c*x^4)^3,x)

[Out]

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

________________________________________________________________________________________