3.22.64 \(\int \frac {(320+1264 x+1740 x^2+925 x^3+125 x^4) \log (\frac {5+x}{5})+(250 x+50 x^2+(250 x+50 x^2) \log (2)) \log ^2(\frac {5+x}{5})+\log (x) (-64 x-240 x^2-300 x^3-125 x^4+(-250 x-50 x^2) \log (\frac {5+x}{5}))}{(320 x+1264 x^2+1740 x^3+925 x^4+125 x^5) \log (x) \log (\frac {5+x}{5})+(-320 x-1264 x^2-1740 x^3-925 x^4-125 x^5+(-320 x-1264 x^2-1740 x^3-925 x^4-125 x^5) \log (2)) \log ^2(\frac {5+x}{5})} \, dx\) [2164]

Optimal. Leaf size=30 \[ \frac {5}{(4+5 x)^2}+\log \left (-1-\log (2)+\frac {\log (x)}{\log \left (1+\frac {x}{5}\right )}\right ) \]

[Out]

5/(4+5*x)^2+ln(ln(x)/ln(1+1/5*x)-1-ln(2))

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Rubi [F]
time = 5.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (320+1264 x+1740 x^2+925 x^3+125 x^4\right ) \log \left (\frac {5+x}{5}\right )+\left (250 x+50 x^2+\left (250 x+50 x^2\right ) \log (2)\right ) \log ^2\left (\frac {5+x}{5}\right )+\log (x) \left (-64 x-240 x^2-300 x^3-125 x^4+\left (-250 x-50 x^2\right ) \log \left (\frac {5+x}{5}\right )\right )}{\left (320 x+1264 x^2+1740 x^3+925 x^4+125 x^5\right ) \log (x) \log \left (\frac {5+x}{5}\right )+\left (-320 x-1264 x^2-1740 x^3-925 x^4-125 x^5+\left (-320 x-1264 x^2-1740 x^3-925 x^4-125 x^5\right ) \log (2)\right ) \log ^2\left (\frac {5+x}{5}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((320 + 1264*x + 1740*x^2 + 925*x^3 + 125*x^4)*Log[(5 + x)/5] + (250*x + 50*x^2 + (250*x + 50*x^2)*Log[2])
*Log[(5 + x)/5]^2 + Log[x]*(-64*x - 240*x^2 - 300*x^3 - 125*x^4 + (-250*x - 50*x^2)*Log[(5 + x)/5]))/((320*x +
1264*x^2 + 1740*x^3 + 925*x^4 + 125*x^5)*Log[x]*Log[(5 + x)/5] + (-320*x - 1264*x^2 - 1740*x^3 - 925*x^4 - 12
5*x^5 + (-320*x - 1264*x^2 - 1740*x^3 - 925*x^4 - 125*x^5)*Log[2])*Log[(5 + x)/5]^2),x]

[Out]

5/(4 + 5*x)^2 - Defer[Int][1/(x*Log[1 + x/5]), x]/(1 + Log[2]) + Defer[Int][Log[x]/(x*Log[1 + x/5]*(Log[x] - (
1 + Log[2])*Log[(5 + x)/5])), x]/(1 + Log[2]) - Defer[Int][Log[x]/((5 + x)*Log[1 + x/5]*(Log[x] - (1 + Log[2])
*Log[(5 + x)/5])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x \log (x) \left ((4+5 x)^3+50 (5+x) \log \left (\frac {5+x}{5}\right )\right )+(5+x) \log \left (\frac {5+x}{5}\right ) \left ((4+5 x)^3+50 x (1+\log (2)) \log \left (\frac {5+x}{5}\right )\right )}{x (5+x) (4+5 x)^3 \log \left (1+\frac {x}{5}\right ) \left (\log (x)-(1+\log (2)) \log \left (\frac {5+x}{5}\right )\right )} \, dx\\ &=\int \left (-\frac {50}{(4+5 x)^3}+\frac {1}{x (-1-\log (2)) \log \left (1+\frac {x}{5}\right )}+\frac {(5-x \log (2)) \log (x)}{x (5+x) (1+\log (2)) \log \left (1+\frac {x}{5}\right ) \left (\log (x)-(1+\log (2)) \log \left (\frac {5+x}{5}\right )\right )}\right ) \, dx\\ &=\frac {5}{(4+5 x)^2}-\frac {\int \frac {1}{x \log \left (1+\frac {x}{5}\right )} \, dx}{1+\log (2)}+\frac {\int \frac {(5-x \log (2)) \log (x)}{x (5+x) \log \left (1+\frac {x}{5}\right ) \left (\log (x)-(1+\log (2)) \log \left (\frac {5+x}{5}\right )\right )} \, dx}{1+\log (2)}\\ &=\frac {5}{(4+5 x)^2}-\frac {\int \frac {1}{x \log \left (1+\frac {x}{5}\right )} \, dx}{1+\log (2)}+\frac {\int \left (\frac {\log (x)}{x \log \left (1+\frac {x}{5}\right ) \left (\log (x)-(1+\log (2)) \log \left (\frac {5+x}{5}\right )\right )}+\frac {(-1-\log (2)) \log (x)}{(5+x) \log \left (1+\frac {x}{5}\right ) \left (\log (x)-(1+\log (2)) \log \left (\frac {5+x}{5}\right )\right )}\right ) \, dx}{1+\log (2)}\\ &=\frac {5}{(4+5 x)^2}-\frac {\int \frac {1}{x \log \left (1+\frac {x}{5}\right )} \, dx}{1+\log (2)}+\frac {\int \frac {\log (x)}{x \log \left (1+\frac {x}{5}\right ) \left (\log (x)-(1+\log (2)) \log \left (\frac {5+x}{5}\right )\right )} \, dx}{1+\log (2)}-\int \frac {\log (x)}{(5+x) \log \left (1+\frac {x}{5}\right ) \left (\log (x)-(1+\log (2)) \log \left (\frac {5+x}{5}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.05, size = 40, normalized size = 1.33 \begin {gather*} \frac {5}{(4+5 x)^2}-\log \left (\log \left (\frac {5+x}{5}\right )\right )+\log \left (-\log (x)+(1+\log (2)) \log \left (\frac {5+x}{5}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((320 + 1264*x + 1740*x^2 + 925*x^3 + 125*x^4)*Log[(5 + x)/5] + (250*x + 50*x^2 + (250*x + 50*x^2)*L
og[2])*Log[(5 + x)/5]^2 + Log[x]*(-64*x - 240*x^2 - 300*x^3 - 125*x^4 + (-250*x - 50*x^2)*Log[(5 + x)/5]))/((3
20*x + 1264*x^2 + 1740*x^3 + 925*x^4 + 125*x^5)*Log[x]*Log[(5 + x)/5] + (-320*x - 1264*x^2 - 1740*x^3 - 925*x^
4 - 125*x^5 + (-320*x - 1264*x^2 - 1740*x^3 - 925*x^4 - 125*x^5)*Log[2])*Log[(5 + x)/5]^2),x]

[Out]

5/(4 + 5*x)^2 - Log[Log[(5 + x)/5]] + Log[-Log[x] + (1 + Log[2])*Log[(5 + x)/5]]

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Maple [A]
time = 0.49, size = 47, normalized size = 1.57

method	result	size
risch	\(\frac {5}{25 x^{2}+40 x +16}-\ln \left (\ln \left (1+\frac {x}{5}\right )\right )+\ln \left (\ln \left (1+\frac {x}{5}\right )-\frac {\ln \left (x \right )}{1+\ln \left (2\right )}\right )\)	\(43\)
default	\(\frac {5}{\left (4+5 x \right )^{2}}-\ln \left (-\ln \left (5\right )+\ln \left (5+x \right )\right )+\ln \left (\ln \left (5+x \right )-\frac {\ln \left (2\right ) \ln \left (5\right )+\ln \left (5\right )+\ln \left (x \right )}{1+\ln \left (2\right )}\right )\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-50*x^2-250*x)*ln(1+1/5*x)-125*x^4-300*x^3-240*x^2-64*x)*ln(x)+((50*x^2+250*x)*ln(2)+50*x^2+250*x)*ln(1
+1/5*x)^2+(125*x^4+925*x^3+1740*x^2+1264*x+320)*ln(1+1/5*x))/((125*x^5+925*x^4+1740*x^3+1264*x^2+320*x)*ln(1+1
/5*x)*ln(x)+((-125*x^5-925*x^4-1740*x^3-1264*x^2-320*x)*ln(2)-125*x^5-925*x^4-1740*x^3-1264*x^2-320*x)*ln(1+1/
5*x)^2),x,method=_RETURNVERBOSE)

[Out]

5/(4+5*x)^2-ln(-ln(5)+ln(5+x))+ln(ln(5+x)-(ln(2)*ln(5)+ln(5)+ln(x))/(1+ln(2)))

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Maxima [A]
time = 0.53, size = 56, normalized size = 1.87 \begin {gather*} \frac {5}{25 \, x^{2} + 40 \, x + 16} + \log \left (-\frac {\log \left (5\right ) \log \left (2\right ) - {\left (\log \left (2\right ) + 1\right )} \log \left (x + 5\right ) + \log \left (5\right ) + \log \left (x\right )}{\log \left (2\right ) + 1}\right ) - \log \left (-\log \left (5\right ) + \log \left (x + 5\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-50*x^2-250*x)*log(1+1/5*x)-125*x^4-300*x^3-240*x^2-64*x)*log(x)+((50*x^2+250*x)*log(2)+50*x^2+25
0*x)*log(1+1/5*x)^2+(125*x^4+925*x^3+1740*x^2+1264*x+320)*log(1+1/5*x))/((125*x^5+925*x^4+1740*x^3+1264*x^2+32
0*x)*log(1+1/5*x)*log(x)+((-125*x^5-925*x^4-1740*x^3-1264*x^2-320*x)*log(2)-125*x^5-925*x^4-1740*x^3-1264*x^2-
320*x)*log(1+1/5*x)^2),x, algorithm="maxima")

[Out]

5/(25*x^2 + 40*x + 16) + log(-(log(5)*log(2) - (log(2) + 1)*log(x + 5) + log(5) + log(x))/(log(2) + 1)) - log(
-log(5) + log(x + 5))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (28) = 56\).
time = 0.38, size = 61, normalized size = 2.03 \begin {gather*} \frac {{\left (25 \, x^{2} + 40 \, x + 16\right )} \log \left (-{\left (\log \left (2\right ) + 1\right )} \log \left (\frac {1}{5} \, x + 1\right ) + \log \left (x\right )\right ) - {\left (25 \, x^{2} + 40 \, x + 16\right )} \log \left (\log \left (\frac {1}{5} \, x + 1\right )\right ) + 5}{25 \, x^{2} + 40 \, x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-50*x^2-250*x)*log(1+1/5*x)-125*x^4-300*x^3-240*x^2-64*x)*log(x)+((50*x^2+250*x)*log(2)+50*x^2+25
0*x)*log(1+1/5*x)^2+(125*x^4+925*x^3+1740*x^2+1264*x+320)*log(1+1/5*x))/((125*x^5+925*x^4+1740*x^3+1264*x^2+32
0*x)*log(1+1/5*x)*log(x)+((-125*x^5-925*x^4-1740*x^3-1264*x^2-320*x)*log(2)-125*x^5-925*x^4-1740*x^3-1264*x^2-
320*x)*log(1+1/5*x)^2),x, algorithm="fricas")

[Out]

((25*x^2 + 40*x + 16)*log(-(log(2) + 1)*log(1/5*x + 1) + log(x)) - (25*x^2 + 40*x + 16)*log(log(1/5*x + 1)) +
5)/(25*x^2 + 40*x + 16)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-50*x**2-250*x)*ln(1+1/5*x)-125*x**4-300*x**3-240*x**2-64*x)*ln(x)+((50*x**2+250*x)*ln(2)+50*x**2
+250*x)*ln(1+1/5*x)**2+(125*x**4+925*x**3+1740*x**2+1264*x+320)*ln(1+1/5*x))/((125*x**5+925*x**4+1740*x**3+126
4*x**2+320*x)*ln(1+1/5*x)*ln(x)+((-125*x**5-925*x**4-1740*x**3-1264*x**2-320*x)*ln(2)-125*x**5-925*x**4-1740*x
**3-1264*x**2-320*x)*ln(1+1/5*x)**2),x)

[Out]

Exception raised: PolynomialError >> 1/(x**2*log(2) + x**2 + 5*x*log(2) + 5*x) contains an element of the set
of generators.

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Giac [A]
time = 0.45, size = 48, normalized size = 1.60 \begin {gather*} \frac {5}{25 \, {\left (x + 5\right )}^{2} - 210 \, x - 609} + \log \left (-\log \left (2\right ) \log \left (\frac {1}{5} \, x + 1\right ) + \log \left (x\right ) - \log \left (\frac {1}{5} \, x + 1\right )\right ) - \log \left (\log \left (\frac {1}{5} \, x + 1\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-50*x^2-250*x)*log(1+1/5*x)-125*x^4-300*x^3-240*x^2-64*x)*log(x)+((50*x^2+250*x)*log(2)+50*x^2+25
0*x)*log(1+1/5*x)^2+(125*x^4+925*x^3+1740*x^2+1264*x+320)*log(1+1/5*x))/((125*x^5+925*x^4+1740*x^3+1264*x^2+32
0*x)*log(1+1/5*x)*log(x)+((-125*x^5-925*x^4-1740*x^3-1264*x^2-320*x)*log(2)-125*x^5-925*x^4-1740*x^3-1264*x^2-
320*x)*log(1+1/5*x)^2),x, algorithm="giac")

[Out]

5/(25*(x + 5)^2 - 210*x - 609) + log(-log(2)*log(1/5*x + 1) + log(x) - log(1/5*x + 1)) - log(log(1/5*x + 1))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\ln \left (\frac {x}{5}+1\right )}^2\,\left (250\,x+\ln \left (2\right )\,\left (50\,x^2+250\,x\right )+50\,x^2\right )-\ln \left (x\right )\,\left (64\,x+\ln \left (\frac {x}{5}+1\right )\,\left (50\,x^2+250\,x\right )+240\,x^2+300\,x^3+125\,x^4\right )+\ln \left (\frac {x}{5}+1\right )\,\left (125\,x^4+925\,x^3+1740\,x^2+1264\,x+320\right )}{{\ln \left (\frac {x}{5}+1\right )}^2\,\left (320\,x+1264\,x^2+1740\,x^3+925\,x^4+125\,x^5+\ln \left (2\right )\,\left (125\,x^5+925\,x^4+1740\,x^3+1264\,x^2+320\,x\right )\right )-\ln \left (\frac {x}{5}+1\right )\,\ln \left (x\right )\,\left (125\,x^5+925\,x^4+1740\,x^3+1264\,x^2+320\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x/5 + 1)^2*(250*x + log(2)*(250*x + 50*x^2) + 50*x^2) - log(x)*(64*x + log(x/5 + 1)*(250*x + 50*x^2)
+ 240*x^2 + 300*x^3 + 125*x^4) + log(x/5 + 1)*(1264*x + 1740*x^2 + 925*x^3 + 125*x^4 + 320))/(log(x/5 + 1)^2*
(320*x + 1264*x^2 + 1740*x^3 + 925*x^4 + 125*x^5 + log(2)*(320*x + 1264*x^2 + 1740*x^3 + 925*x^4 + 125*x^5)) -
log(x/5 + 1)*log(x)*(320*x + 1264*x^2 + 1740*x^3 + 925*x^4 + 125*x^5)),x)

[Out]

int(-(log(x/5 + 1)^2*(250*x + log(2)*(250*x + 50*x^2) + 50*x^2) - log(x)*(64*x + log(x/5 + 1)*(250*x + 50*x^2)
+ 240*x^2 + 300*x^3 + 125*x^4) + log(x/5 + 1)*(1264*x + 1740*x^2 + 925*x^3 + 125*x^4 + 320))/(log(x/5 + 1)^2*
(320*x + 1264*x^2 + 1740*x^3 + 925*x^4 + 125*x^5 + log(2)*(320*x + 1264*x^2 + 1740*x^3 + 925*x^4 + 125*x^5)) -
log(x/5 + 1)*log(x)*(320*x + 1264*x^2 + 1740*x^3 + 925*x^4 + 125*x^5)), x)

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