3.13.49 \(\int \frac {-14950+17450 x+32400 x^2+209952 x^3+(7475 x+88776 x^2-104976 x^3) \log (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2})}{14950 x+177552 x^2-209952 x^3+(-7475 x-88776 x^2+104976 x^3) \log (\frac {89401 x^2-193752 x^3+104976 x^4}{625+16200 x+104976 x^2})} \, dx\)

Optimal. Leaf size=23 \[ -x+\log \left (-2+\log \left (\left (-x+\frac {x}{\frac {25}{324}+x}\right )^2\right )\right ) \]

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Rubi [A]  time = 0.67, antiderivative size = 28, normalized size of antiderivative = 1.22, number of steps used = 4, number of rules used = 3, integrand size = 122, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {6688, 6728, 6684} \begin {gather*} \log \left (2-\log \left (\frac {(299-324 x)^2 x^2}{(324 x+25)^2}\right )\right )-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-14950 + 17450*x + 32400*x^2 + 209952*x^3 + (7475*x + 88776*x^2 - 104976*x^3)*Log[(89401*x^2 - 193752*x^3
+ 104976*x^4)/(625 + 16200*x + 104976*x^2)])/(14950*x + 177552*x^2 - 209952*x^3 + (-7475*x - 88776*x^2 + 1049
76*x^3)*Log[(89401*x^2 - 193752*x^3 + 104976*x^4)/(625 + 16200*x + 104976*x^2)]),x]

[Out]

-x + Log[2 - Log[((299 - 324*x)^2*x^2)/(25 + 324*x)^2]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6688

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

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-7475+8725 x+16200 x^2+104976 x^3\right )+x \left (7475+88776 x-104976 x^2\right ) \log \left (\frac {x^2 (-299+324 x)^2}{(25+324 x)^2}\right )}{x \left (7475+88776 x-104976 x^2\right ) \left (2-\log \left (\frac {x^2 (-299+324 x)^2}{(25+324 x)^2}\right )\right )} \, dx\\ &=\int \left (-1+\frac {2 \left (-7475+16200 x+104976 x^2\right )}{x (-299+324 x) (25+324 x) \left (-2+\log \left (\frac {x^2 (-299+324 x)^2}{(25+324 x)^2}\right )\right )}\right ) \, dx\\ &=-x+2 \int \frac {-7475+16200 x+104976 x^2}{x (-299+324 x) (25+324 x) \left (-2+\log \left (\frac {x^2 (-299+324 x)^2}{(25+324 x)^2}\right )\right )} \, dx\\ &=-x+\log \left (2-\log \left (\frac {(299-324 x)^2 x^2}{(25+324 x)^2}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 28, normalized size = 1.22 \begin {gather*} -x+\log \left (2-\log \left (\frac {x^2 (-299+324 x)^2}{(25+324 x)^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-14950 + 17450*x + 32400*x^2 + 209952*x^3 + (7475*x + 88776*x^2 - 104976*x^3)*Log[(89401*x^2 - 1937
52*x^3 + 104976*x^4)/(625 + 16200*x + 104976*x^2)])/(14950*x + 177552*x^2 - 209952*x^3 + (-7475*x - 88776*x^2
+ 104976*x^3)*Log[(89401*x^2 - 193752*x^3 + 104976*x^4)/(625 + 16200*x + 104976*x^2)]),x]

[Out]

-x + Log[2 - Log[(x^2*(-299 + 324*x)^2)/(25 + 324*x)^2]]

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fricas [A]  time = 0.67, size = 37, normalized size = 1.61 \begin {gather*} -x + \log \left (\log \left (\frac {104976 \, x^{4} - 193752 \, x^{3} + 89401 \, x^{2}}{104976 \, x^{2} + 16200 \, x + 625}\right ) - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-104976*x^3+88776*x^2+7475*x)*log((104976*x^4-193752*x^3+89401*x^2)/(104976*x^2+16200*x+625))+2099
52*x^3+32400*x^2+17450*x-14950)/((104976*x^3-88776*x^2-7475*x)*log((104976*x^4-193752*x^3+89401*x^2)/(104976*x
^2+16200*x+625))-209952*x^3+177552*x^2+14950*x),x, algorithm="fricas")

[Out]

-x + log(log((104976*x^4 - 193752*x^3 + 89401*x^2)/(104976*x^2 + 16200*x + 625)) - 2)

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giac [A]  time = 0.73, size = 37, normalized size = 1.61 \begin {gather*} -x + \log \left (\log \left (\frac {104976 \, x^{4} - 193752 \, x^{3} + 89401 \, x^{2}}{104976 \, x^{2} + 16200 \, x + 625}\right ) - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-104976*x^3+88776*x^2+7475*x)*log((104976*x^4-193752*x^3+89401*x^2)/(104976*x^2+16200*x+625))+2099
52*x^3+32400*x^2+17450*x-14950)/((104976*x^3-88776*x^2-7475*x)*log((104976*x^4-193752*x^3+89401*x^2)/(104976*x
^2+16200*x+625))-209952*x^3+177552*x^2+14950*x),x, algorithm="giac")

[Out]

-x + log(log((104976*x^4 - 193752*x^3 + 89401*x^2)/(104976*x^2 + 16200*x + 625)) - 2)

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maple [A]  time = 0.06, size = 38, normalized size = 1.65

method	result	size
norman	\(-x +\ln \left (\ln \left (\frac {104976 x^{4}-193752 x^{3}+89401 x^{2}}{104976 x^{2}+16200 x +625}\right )-2\right )\)	\(38\)
risch	\(-x +\ln \left (\ln \left (\frac {104976 x^{4}-193752 x^{3}+89401 x^{2}}{104976 x^{2}+16200 x +625}\right )-2\right )\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-104976*x^3+88776*x^2+7475*x)*ln((104976*x^4-193752*x^3+89401*x^2)/(104976*x^2+16200*x+625))+209952*x^3+
32400*x^2+17450*x-14950)/((104976*x^3-88776*x^2-7475*x)*ln((104976*x^4-193752*x^3+89401*x^2)/(104976*x^2+16200
*x+625))-209952*x^3+177552*x^2+14950*x),x,method=_RETURNVERBOSE)

[Out]

-x+ln(ln((104976*x^4-193752*x^3+89401*x^2)/(104976*x^2+16200*x+625))-2)

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maxima [A]  time = 0.44, size = 25, normalized size = 1.09 \begin {gather*} -x + \log \left (\log \left (324 \, x + 25\right ) - \log \left (324 \, x - 299\right ) - \log \relax (x) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-104976*x^3+88776*x^2+7475*x)*log((104976*x^4-193752*x^3+89401*x^2)/(104976*x^2+16200*x+625))+2099
52*x^3+32400*x^2+17450*x-14950)/((104976*x^3-88776*x^2-7475*x)*log((104976*x^4-193752*x^3+89401*x^2)/(104976*x
^2+16200*x+625))-209952*x^3+177552*x^2+14950*x),x, algorithm="maxima")

[Out]

-x + log(log(324*x + 25) - log(324*x - 299) - log(x) + 1)

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mupad [B]  time = 1.34, size = 37, normalized size = 1.61 \begin {gather*} \ln \left (\ln \left (\frac {104976\,x^4-193752\,x^3+89401\,x^2}{104976\,x^2+16200\,x+625}\right )-2\right )-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((17450*x + log((89401*x^2 - 193752*x^3 + 104976*x^4)/(16200*x + 104976*x^2 + 625))*(7475*x + 88776*x^2 - 1
04976*x^3) + 32400*x^2 + 209952*x^3 - 14950)/(14950*x - log((89401*x^2 - 193752*x^3 + 104976*x^4)/(16200*x + 1
04976*x^2 + 625))*(7475*x + 88776*x^2 - 104976*x^3) + 177552*x^2 - 209952*x^3),x)

[Out]

log(log((89401*x^2 - 193752*x^3 + 104976*x^4)/(16200*x + 104976*x^2 + 625)) - 2) - x

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sympy [B]  time = 0.33, size = 31, normalized size = 1.35 \begin {gather*} - x + \log {\left (\log {\left (\frac {104976 x^{4} - 193752 x^{3} + 89401 x^{2}}{104976 x^{2} + 16200 x + 625} \right )} - 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-104976*x**3+88776*x**2+7475*x)*ln((104976*x**4-193752*x**3+89401*x**2)/(104976*x**2+16200*x+625))
+209952*x**3+32400*x**2+17450*x-14950)/((104976*x**3-88776*x**2-7475*x)*ln((104976*x**4-193752*x**3+89401*x**2
)/(104976*x**2+16200*x+625))-209952*x**3+177552*x**2+14950*x),x)

[Out]

-x + log(log((104976*x**4 - 193752*x**3 + 89401*x**2)/(104976*x**2 + 16200*x + 625)) - 2)

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