3.5.43 \(\int \frac {x+2 x^3}{(x^2+x^4)^3} \, dx\) [443]

Optimal. Leaf size=13 \[ -\frac {1}{4 \left (x^2+x^4\right )^2} \]

[Out]

-1/4/(x^4+x^2)^2

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Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1602} \begin {gather*} -\frac {1}{4 \left (x^4+x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x + 2*x^3)/(x^2 + x^4)^3,x]

[Out]

-1/4*1/(x^2 + x^4)^2

Rule 1602

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*x^(p - q +
1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 14, normalized size = 1.08 \begin {gather*} -\frac {1}{4 x^4 \left (1+x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + 2*x^3)/(x^2 + x^4)^3,x]

[Out]

-1/4*1/(x^4*(1 + x^2)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(29\) vs. \(2(11)=22\).
time = 0.22, size = 30, normalized size = 2.31

method	result	size
gosper	\(-\frac {1}{4 x^{4} \left (x^{2}+1\right )^{2}}\)	\(13\)
norman	\(-\frac {1}{4 x^{4} \left (x^{2}+1\right )^{2}}\)	\(13\)
risch	\(-\frac {1}{4 x^{4} \left (x^{2}+1\right )^{2}}\)	\(13\)
default	\(-\frac {1}{2 \left (x^{2}+1\right )}-\frac {1}{4 \left (x^{2}+1\right )^{2}}-\frac {1}{4 x^{4}}+\frac {1}{2 x^{2}}\)	\(30\)
meijerg	\(\frac {x^{2} \left (5 x^{2}+6\right )}{2 \left (x^{2}+1\right )^{2}}-\frac {3}{4}+\frac {1}{2 x^{2}}-\frac {x^{2} \left (7 x^{2}+8\right )}{4 \left (x^{2}+1\right )^{2}}-\frac {1}{4 x^{4}}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(x^2+1)-1/4/(x^2+1)^2-1/4/x^4+1/2/x^2

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Maxima [A]
time = 0.28, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{4 \, {\left (x^{4} + x^{2}\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/(x^4 + x^2)^2

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Fricas [A]
time = 0.39, size = 16, normalized size = 1.23 \begin {gather*} -\frac {1}{4 \, {\left (x^{8} + 2 \, x^{6} + x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/(x^8 + 2*x^6 + x^4)

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Sympy [A]
time = 0.04, size = 17, normalized size = 1.31 \begin {gather*} - \frac {1}{4 x^{8} + 8 x^{6} + 4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(4*x**8 + 8*x**6 + 4*x**4)

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Giac [A]
time = 3.90, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{4 \, {\left (x^{4} + x^{2}\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/(x^4 + x^2)^2

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Mupad [B]
time = 2.24, size = 20, normalized size = 1.54 \begin {gather*} -\frac {1}{4\,x^8+8\,x^6+4\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 2*x^3)/(x^2 + x^4)^3,x)

[Out]

-1/(4*x^4 + 8*x^6 + 4*x^8)

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