3.25.54 \(\int \frac {12800 x-25600 x^3+14400 x^5-3200 x^7+250 x^9}{6400 x^2-6400 x^4+2400 x^6-400 x^8+25 x^{10}+\log ^5(4)} \, dx\)

Optimal. Leaf size=22 \[ \log \left (\frac {25 x^2 \left (4-x^2\right )^4}{\log ^4(4)}+\log (4)\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.41, number of steps used = 1, number of rules used = 1, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {1587} \begin {gather*} \log \left (25 x^{10}-400 x^8+2400 x^6-6400 x^4+6400 x^2+\log ^5(4)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12800*x - 25600*x^3 + 14400*x^5 - 3200*x^7 + 250*x^9)/(6400*x^2 - 6400*x^4 + 2400*x^6 - 400*x^8 + 25*x^10
+ Log[4]^5),x]

[Out]

Log[6400*x^2 - 6400*x^4 + 2400*x^6 - 400*x^8 + 25*x^10 + Log[4]^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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (6400 x^2-6400 x^4+2400 x^6-400 x^8+25 x^{10}+\log ^5(4)\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 24, normalized size = 1.09 \begin {gather*} \log \left (100 \left (-4+x^2\right )^4+25 \left (-4+x^2\right )^5+\log ^5(4)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12800*x - 25600*x^3 + 14400*x^5 - 3200*x^7 + 250*x^9)/(6400*x^2 - 6400*x^4 + 2400*x^6 - 400*x^8 + 2
5*x^10 + Log[4]^5),x]

[Out]

Log[100*(-4 + x^2)^4 + 25*(-4 + x^2)^5 + Log[4]^5]

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 33, normalized size = 1.50 \begin {gather*} \log \left (25 \, x^{10} - 400 \, x^{8} + 2400 \, x^{6} + 32 \, \log \relax (2)^{5} - 6400 \, x^{4} + 6400 \, x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((250*x^9-3200*x^7+14400*x^5-25600*x^3+12800*x)/(32*log(2)^5+25*x^10-400*x^8+2400*x^6-6400*x^4+6400*x
^2),x, algorithm="fricas")

[Out]

log(25*x^10 - 400*x^8 + 2400*x^6 + 32*log(2)^5 - 6400*x^4 + 6400*x^2)

________________________________________________________________________________________

giac [A]  time = 0.30, size = 33, normalized size = 1.50 \begin {gather*} \log \left (25 \, x^{10} - 400 \, x^{8} + 2400 \, x^{6} + 32 \, \log \relax (2)^{5} - 6400 \, x^{4} + 6400 \, x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((250*x^9-3200*x^7+14400*x^5-25600*x^3+12800*x)/(32*log(2)^5+25*x^10-400*x^8+2400*x^6-6400*x^4+6400*x
^2),x, algorithm="giac")

[Out]

log(25*x^10 - 400*x^8 + 2400*x^6 + 32*log(2)^5 - 6400*x^4 + 6400*x^2)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 34, normalized size = 1.55

method	result	size
derivativedivides	\(\ln \left (32 \ln \relax (2)^{5}+25 x^{10}-400 x^{8}+2400 x^{6}-6400 x^{4}+6400 x^{2}\right )\)	\(34\)
default	\(\ln \left (32 \ln \relax (2)^{5}+25 x^{10}-400 x^{8}+2400 x^{6}-6400 x^{4}+6400 x^{2}\right )\)	\(34\)
norman	\(\ln \left (32 \ln \relax (2)^{5}+25 x^{10}-400 x^{8}+2400 x^{6}-6400 x^{4}+6400 x^{2}\right )\)	\(34\)
risch	\(\ln \left (32 \ln \relax (2)^{5}+25 x^{10}-400 x^{8}+2400 x^{6}-6400 x^{4}+6400 x^{2}\right )\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((250*x^9-3200*x^7+14400*x^5-25600*x^3+12800*x)/(32*ln(2)^5+25*x^10-400*x^8+2400*x^6-6400*x^4+6400*x^2),x,m
ethod=_RETURNVERBOSE)

[Out]

ln(32*ln(2)^5+25*x^10-400*x^8+2400*x^6-6400*x^4+6400*x^2)

________________________________________________________________________________________

maxima [A]  time = 0.97, size = 33, normalized size = 1.50 \begin {gather*} \log \left (25 \, x^{10} - 400 \, x^{8} + 2400 \, x^{6} + 32 \, \log \relax (2)^{5} - 6400 \, x^{4} + 6400 \, x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((250*x^9-3200*x^7+14400*x^5-25600*x^3+12800*x)/(32*log(2)^5+25*x^10-400*x^8+2400*x^6-6400*x^4+6400*x
^2),x, algorithm="maxima")

[Out]

log(25*x^10 - 400*x^8 + 2400*x^6 + 32*log(2)^5 - 6400*x^4 + 6400*x^2)

________________________________________________________________________________________

mupad [B]  time = 0.12, size = 31, normalized size = 1.41 \begin {gather*} \ln \left (x^{10}-16\,x^8+96\,x^6-256\,x^4+256\,x^2+\frac {32\,{\ln \relax (2)}^5}{25}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12800*x - 25600*x^3 + 14400*x^5 - 3200*x^7 + 250*x^9)/(32*log(2)^5 + 6400*x^2 - 6400*x^4 + 2400*x^6 - 400
*x^8 + 25*x^10),x)

[Out]

log((32*log(2)^5)/25 + 256*x^2 - 256*x^4 + 96*x^6 - 16*x^8 + x^10)

________________________________________________________________________________________

sympy [A]  time = 0.40, size = 32, normalized size = 1.45 \begin {gather*} \log {\left (25 x^{10} - 400 x^{8} + 2400 x^{6} - 6400 x^{4} + 6400 x^{2} + 32 \log {\relax (2 )}^{5} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((250*x**9-3200*x**7+14400*x**5-25600*x**3+12800*x)/(32*ln(2)**5+25*x**10-400*x**8+2400*x**6-6400*x**
4+6400*x**2),x)

[Out]

log(25*x**10 - 400*x**8 + 2400*x**6 - 6400*x**4 + 6400*x**2 + 32*log(2)**5)

________________________________________________________________________________________