3.7.62 \(\int \frac {x^3}{(a+c x^4)^2} \, dx\) [662]

Optimal. Leaf size=16 \[ -\frac {1}{4 c \left (a+c x^4\right )} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, 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 = {267} \begin {gather*} -\frac {1}{4 c \left (a+c x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + c*x^4)^2,x]

[Out]

-1/4*1/(c*(a + c*x^4))

Rule 267

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} -\frac {1}{4 c \left (a+c x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + c*x^4)^2,x]

[Out]

-1/4*1/(c*(a + c*x^4))

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Maple [A]
time = 0.14, size = 15, normalized size = 0.94

method	result	size
gosper	\(-\frac {1}{4 c \left (x^{4} c +a \right )}\)	\(15\)
derivativedivides	\(-\frac {1}{4 c \left (x^{4} c +a \right )}\)	\(15\)
default	\(-\frac {1}{4 c \left (x^{4} c +a \right )}\)	\(15\)
norman	\(-\frac {1}{4 c \left (x^{4} c +a \right )}\)	\(15\)
risch	\(-\frac {1}{4 c \left (x^{4} c +a \right )}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(c*x^4+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{4 \, {\left (c x^{4} + a\right )} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/((c*x^4 + a)*c)

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Fricas [A]
time = 0.37, size = 15, normalized size = 0.94 \begin {gather*} -\frac {1}{4 \, {\left (c^{2} x^{4} + a c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/(c^2*x^4 + a*c)

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Sympy [A]
time = 0.09, size = 15, normalized size = 0.94 \begin {gather*} - \frac {1}{4 a c + 4 c^{2} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(4*a*c + 4*c**2*x**4)

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Giac [A]
time = 2.15, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{4 \, {\left (c x^{4} + a\right )} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/((c*x^4 + a)*c)

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Mupad [B]
time = 0.95, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{4\,c\,\left (c\,x^4+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + c*x^4)^2,x)

[Out]

-1/(4*c*(a + c*x^4))

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