3.63.10 \(\int \frac {-8192-25 x^3+25 x^5}{2048 x+25 x^4+25 x^6} \, dx\)

Optimal. Leaf size=19 \[ 4+\log \left (\frac {4096}{25 x^4}+\frac {2}{x}+2 x\right ) \]

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Rubi [A]  time = 0.15, antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1594, 6742, 1587} \begin {gather*} \log \left (25 x^5+25 x^3+2048\right )-4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8192 - 25*x^3 + 25*x^5)/(2048*x + 25*x^4 + 25*x^6),x]

[Out]

-4*Log[x] + Log[2048 + 25*x^3 + 25*x^5]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8192-25 x^3+25 x^5}{x \left (2048+25 x^3+25 x^5\right )} \, dx\\ &=\int \left (-\frac {4}{x}+\frac {25 x^2 \left (3+5 x^2\right )}{2048+25 x^3+25 x^5}\right ) \, dx\\ &=-4 \log (x)+25 \int \frac {x^2 \left (3+5 x^2\right )}{2048+25 x^3+25 x^5} \, dx\\ &=-4 \log (x)+\log \left (2048+25 x^3+25 x^5\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.95 \begin {gather*} -4 \log (x)+\log \left (2048+25 x^3+25 x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8192 - 25*x^3 + 25*x^5)/(2048*x + 25*x^4 + 25*x^6),x]

[Out]

-4*Log[x] + Log[2048 + 25*x^3 + 25*x^5]

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fricas [A]  time = 0.66, size = 18, normalized size = 0.95 \begin {gather*} \log \left (25 \, x^{5} + 25 \, x^{3} + 2048\right ) - 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x^5-25*x^3-8192)/(25*x^6+25*x^4+2048*x),x, algorithm="fricas")

[Out]

log(25*x^5 + 25*x^3 + 2048) - 4*log(x)

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giac [A]  time = 0.15, size = 20, normalized size = 1.05 \begin {gather*} \log \left ({\left | 25 \, x^{5} + 25 \, x^{3} + 2048 \right |}\right ) - 4 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x^5-25*x^3-8192)/(25*x^6+25*x^4+2048*x),x, algorithm="giac")

[Out]

log(abs(25*x^5 + 25*x^3 + 2048)) - 4*log(abs(x))

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maple [A]  time = 0.03, size = 19, normalized size = 1.00

method	result	size
default	\(\ln \left (25 x^{5}+25 x^{3}+2048\right )-4 \ln \relax (x )\)	\(19\)
norman	\(\ln \left (25 x^{5}+25 x^{3}+2048\right )-4 \ln \relax (x )\)	\(19\)
risch	\(\ln \left (25 x^{5}+25 x^{3}+2048\right )-4 \ln \relax (x )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((25*x^5-25*x^3-8192)/(25*x^6+25*x^4+2048*x),x,method=_RETURNVERBOSE)

[Out]

ln(25*x^5+25*x^3+2048)-4*ln(x)

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maxima [A]  time = 0.39, size = 18, normalized size = 0.95 \begin {gather*} \log \left (25 \, x^{5} + 25 \, x^{3} + 2048\right ) - 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x^5-25*x^3-8192)/(25*x^6+25*x^4+2048*x),x, algorithm="maxima")

[Out]

log(25*x^5 + 25*x^3 + 2048) - 4*log(x)

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mupad [B]  time = 0.08, size = 14, normalized size = 0.74 \begin {gather*} \ln \left (x^5+x^3+\frac {2048}{25}\right )-4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(25*x^3 - 25*x^5 + 8192)/(2048*x + 25*x^4 + 25*x^6),x)

[Out]

log(x^3 + x^5 + 2048/25) - 4*log(x)

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sympy [A]  time = 0.11, size = 17, normalized size = 0.89 \begin {gather*} - 4 \log {\relax (x )} + \log {\left (25 x^{5} + 25 x^{3} + 2048 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x**5-25*x**3-8192)/(25*x**6+25*x**4+2048*x),x)

[Out]

-4*log(x) + log(25*x**5 + 25*x**3 + 2048)

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