3.53.35 \(\int \frac {-115+122 x+26 x^2+15 x^3+3 x^4+(-27 x-36 x^2-6 x^3) \log (x)+(15 x+3 x^2) \log ^2(x)+(-15+12 x+3 x^2) \log (5+x)}{-115 x^2+112 x^3+42 x^4+3 x^5+(115 x-217 x^2-78 x^3-6 x^4) \log (x)+(105 x+36 x^2+3 x^3) \log ^2(x)+(-15 x^2-3 x^3+(15 x+3 x^2) \log (x)) \log (5+x)} \, dx\)

Optimal. Leaf size=25 \[ \log \left (7+x-\frac {\frac {23}{3}-2 x+\log (5+x)}{x-\log (x)}\right ) \]

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Rubi [A]  time = 7.19, antiderivative size = 36, normalized size of antiderivative = 1.44, number of steps used = 5, number of rules used = 3, integrand size = 159, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6688, 6742, 6684} \begin {gather*} \log \left (-3 x^2-27 x+3 x \log (x)+21 \log (x)+3 \log (x+5)+23\right )-\log (x-\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-115 + 122*x + 26*x^2 + 15*x^3 + 3*x^4 + (-27*x - 36*x^2 - 6*x^3)*Log[x] + (15*x + 3*x^2)*Log[x]^2 + (-15
+ 12*x + 3*x^2)*Log[5 + x])/(-115*x^2 + 112*x^3 + 42*x^4 + 3*x^5 + (115*x - 217*x^2 - 78*x^3 - 6*x^4)*Log[x]
+ (105*x + 36*x^2 + 3*x^3)*Log[x]^2 + (-15*x^2 - 3*x^3 + (15*x + 3*x^2)*Log[x])*Log[5 + x]),x]

[Out]

-Log[x - Log[x]] + Log[23 - 27*x - 3*x^2 + 21*Log[x] + 3*x*Log[x] + 3*Log[5 + x]]

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 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 {115-122 x-26 x^2-15 x^3-3 x^4+3 x \left (9+12 x+2 x^2\right ) \log (x)-3 x (5+x) \log ^2(x)-3 \left (-5+4 x+x^2\right ) \log (5+x)}{x (5+x) (x-\log (x)) \left (23-27 x-3 x^2+3 (7+x) \log (x)+3 \log (5+x)\right )} \, dx\\ &=\int \left (\frac {1-x}{x (x-\log (x))}+\frac {3 \left (-35+32 x+18 x^2+2 x^3-5 x \log (x)-x^2 \log (x)\right )}{x (5+x) \left (-23+27 x+3 x^2-21 \log (x)-3 x \log (x)-3 \log (5+x)\right )}\right ) \, dx\\ &=3 \int \frac {-35+32 x+18 x^2+2 x^3-5 x \log (x)-x^2 \log (x)}{x (5+x) \left (-23+27 x+3 x^2-21 \log (x)-3 x \log (x)-3 \log (5+x)\right )} \, dx+\int \frac {1-x}{x (x-\log (x))} \, dx\\ &=-\log (x-\log (x))+\log \left (23-27 x-3 x^2+21 \log (x)+3 x \log (x)+3 \log (5+x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 36, normalized size = 1.44 \begin {gather*} -\log (x-\log (x))+\log \left (23-27 x-3 x^2+21 \log (x)+3 x \log (x)+3 \log (5+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-115 + 122*x + 26*x^2 + 15*x^3 + 3*x^4 + (-27*x - 36*x^2 - 6*x^3)*Log[x] + (15*x + 3*x^2)*Log[x]^2
+ (-15 + 12*x + 3*x^2)*Log[5 + x])/(-115*x^2 + 112*x^3 + 42*x^4 + 3*x^5 + (115*x - 217*x^2 - 78*x^3 - 6*x^4)*L
og[x] + (105*x + 36*x^2 + 3*x^3)*Log[x]^2 + (-15*x^2 - 3*x^3 + (15*x + 3*x^2)*Log[x])*Log[5 + x]),x]

[Out]

-Log[x - Log[x]] + Log[23 - 27*x - 3*x^2 + 21*Log[x] + 3*x*Log[x] + 3*Log[5 + x]]

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fricas [A]  time = 0.46, size = 34, normalized size = 1.36 \begin {gather*} \log \left (-3 \, x^{2} + 3 \, {\left (x + 7\right )} \log \relax (x) - 27 \, x + 3 \, \log \left (x + 5\right ) + 23\right ) - \log \left (-x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+12*x-15)*log(5+x)+(3*x^2+15*x)*log(x)^2+(-6*x^3-36*x^2-27*x)*log(x)+3*x^4+15*x^3+26*x^2+122*
x-115)/(((3*x^2+15*x)*log(x)-3*x^3-15*x^2)*log(5+x)+(3*x^3+36*x^2+105*x)*log(x)^2+(-6*x^4-78*x^3-217*x^2+115*x
)*log(x)+3*x^5+42*x^4+112*x^3-115*x^2),x, algorithm="fricas")

[Out]

log(-3*x^2 + 3*(x + 7)*log(x) - 27*x + 3*log(x + 5) + 23) - log(-x + log(x))

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giac [A]  time = 0.21, size = 36, normalized size = 1.44 \begin {gather*} \log \left (-3 \, x^{2} + 3 \, x \log \relax (x) - 27 \, x + 3 \, \log \left (x + 5\right ) + 21 \, \log \relax (x) + 23\right ) - \log \left (x - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+12*x-15)*log(5+x)+(3*x^2+15*x)*log(x)^2+(-6*x^3-36*x^2-27*x)*log(x)+3*x^4+15*x^3+26*x^2+122*
x-115)/(((3*x^2+15*x)*log(x)-3*x^3-15*x^2)*log(5+x)+(3*x^3+36*x^2+105*x)*log(x)^2+(-6*x^4-78*x^3-217*x^2+115*x
)*log(x)+3*x^5+42*x^4+112*x^3-115*x^2),x, algorithm="giac")

[Out]

log(-3*x^2 + 3*x*log(x) - 27*x + 3*log(x + 5) + 21*log(x) + 23) - log(x - log(x))

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maple [A]  time = 0.04, size = 34, normalized size = 1.36

method	result	size
risch	\(-\ln \left (\ln \relax (x )-x \right )+\ln \left (-x^{2}+x \ln \relax (x )-9 x +7 \ln \relax (x )+\ln \left (5+x \right )+\frac {23}{3}\right )\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^2+12*x-15)*ln(5+x)+(3*x^2+15*x)*ln(x)^2+(-6*x^3-36*x^2-27*x)*ln(x)+3*x^4+15*x^3+26*x^2+122*x-115)/((
(3*x^2+15*x)*ln(x)-3*x^3-15*x^2)*ln(5+x)+(3*x^3+36*x^2+105*x)*ln(x)^2+(-6*x^4-78*x^3-217*x^2+115*x)*ln(x)+3*x^
5+42*x^4+112*x^3-115*x^2),x,method=_RETURNVERBOSE)

[Out]

-ln(ln(x)-x)+ln(-x^2+x*ln(x)-9*x+7*ln(x)+ln(5+x)+23/3)

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maxima [A]  time = 0.40, size = 31, normalized size = 1.24 \begin {gather*} \log \left (-x^{2} + {\left (x + 7\right )} \log \relax (x) - 9 \, x + \log \left (x + 5\right ) + \frac {23}{3}\right ) - \log \left (-x + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+12*x-15)*log(5+x)+(3*x^2+15*x)*log(x)^2+(-6*x^3-36*x^2-27*x)*log(x)+3*x^4+15*x^3+26*x^2+122*
x-115)/(((3*x^2+15*x)*log(x)-3*x^3-15*x^2)*log(5+x)+(3*x^3+36*x^2+105*x)*log(x)^2+(-6*x^4-78*x^3-217*x^2+115*x
)*log(x)+3*x^5+42*x^4+112*x^3-115*x^2),x, algorithm="maxima")

[Out]

log(-x^2 + (x + 7)*log(x) - 9*x + log(x + 5) + 23/3) - log(-x + log(x))

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mupad [B]  time = 4.39, size = 33, normalized size = 1.32 \begin {gather*} \ln \left (\ln \left (x+5\right )-9\,x+7\,\ln \relax (x)+x\,\ln \relax (x)-x^2+\frac {23}{3}\right )-\ln \left (\ln \relax (x)-x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((122*x + log(x + 5)*(12*x + 3*x^2 - 15) + log(x)^2*(15*x + 3*x^2) + 26*x^2 + 15*x^3 + 3*x^4 - log(x)*(27*x
+ 36*x^2 + 6*x^3) - 115)/(log(x)^2*(105*x + 36*x^2 + 3*x^3) - log(x + 5)*(15*x^2 - log(x)*(15*x + 3*x^2) + 3*
x^3) - log(x)*(217*x^2 - 115*x + 78*x^3 + 6*x^4) - 115*x^2 + 112*x^3 + 42*x^4 + 3*x^5),x)

[Out]

log(log(x + 5) - 9*x + 7*log(x) + x*log(x) - x^2 + 23/3) - log(log(x) - x)

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sympy [A]  time = 0.59, size = 32, normalized size = 1.28 \begin {gather*} - \log {\left (- x + \log {\relax (x )} \right )} + \log {\left (- x^{2} + x \log {\relax (x )} - 9 x + 7 \log {\relax (x )} + \log {\left (x + 5 \right )} + \frac {23}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**2+12*x-15)*ln(5+x)+(3*x**2+15*x)*ln(x)**2+(-6*x**3-36*x**2-27*x)*ln(x)+3*x**4+15*x**3+26*x**2
+122*x-115)/(((3*x**2+15*x)*ln(x)-3*x**3-15*x**2)*ln(5+x)+(3*x**3+36*x**2+105*x)*ln(x)**2+(-6*x**4-78*x**3-217
*x**2+115*x)*ln(x)+3*x**5+42*x**4+112*x**3-115*x**2),x)

[Out]

-log(-x + log(x)) + log(-x**2 + x*log(x) - 9*x + 7*log(x) + log(x + 5) + 23/3)

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