3.91.9 \(\int \frac {2 x+(25+x) \log (5)+(50+2 x) \log (16)+(-25-x) \log (625+50 x+x^2)}{25 x^2+x^3+(50 x+2 x^2) \log (5)+(25+x) \log ^2(5)+(100 x+4 x^2+(100+4 x) \log (5)) \log (16)+(100+4 x) \log ^2(16)+(-50 x-2 x^2+(-50-2 x) \log (5)+(-100-4 x) \log (16)) \log (625+50 x+x^2)+(25+x) \log ^2(625+50 x+x^2)} \, dx\)

Optimal. Leaf size=22 \[ 2+\frac {x}{x+\log (5)+2 \log (16)-\log \left ((25+x)^2\right )} \]

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Rubi [F]  time = 0.65, 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 {2 x+(25+x) \log (5)+(50+2 x) \log (16)+(-25-x) \log \left (625+50 x+x^2\right )}{25 x^2+x^3+\left (50 x+2 x^2\right ) \log (5)+(25+x) \log ^2(5)+\left (100 x+4 x^2+(100+4 x) \log (5)\right ) \log (16)+(100+4 x) \log ^2(16)+\left (-50 x-2 x^2+(-50-2 x) \log (5)+(-100-4 x) \log (16)\right ) \log \left (625+50 x+x^2\right )+(25+x) \log ^2\left (625+50 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x + (25 + x)*Log[5] + (50 + 2*x)*Log[16] + (-25 - x)*Log[625 + 50*x + x^2])/(25*x^2 + x^3 + (50*x + 2*x
^2)*Log[5] + (25 + x)*Log[5]^2 + (100*x + 4*x^2 + (100 + 4*x)*Log[5])*Log[16] + (100 + 4*x)*Log[16]^2 + (-50*x
- 2*x^2 + (-50 - 2*x)*Log[5] + (-100 - 4*x)*Log[16])*Log[625 + 50*x + x^2] + (25 + x)*Log[625 + 50*x + x^2]^2
),x]

[Out]

2*Defer[Int][(x + Log[1280] - Log[(25 + x)^2])^(-2), x] - Defer[Int][x/(x + Log[1280] - Log[(25 + x)^2])^2, x]
- 50*Defer[Int][1/((25 + x)*(x + Log[1280] - Log[(25 + x)^2])^2), x] + Defer[Int][(x + Log[1280] - Log[(25 +
x)^2])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 \log (1280)+x (2+\log (1280))-(25+x) \log \left ((25+x)^2\right )}{(25+x) \left (x+\log (1280)-\log \left ((25+x)^2\right )\right )^2} \, dx\\ &=\int \left (-\frac {x (23+x)}{(25+x) \left (x+\log (1280)-\log \left ((25+x)^2\right )\right )^2}+\frac {1}{x+\log (1280)-\log \left ((25+x)^2\right )}\right ) \, dx\\ &=-\int \frac {x (23+x)}{(25+x) \left (x+\log (1280)-\log \left ((25+x)^2\right )\right )^2} \, dx+\int \frac {1}{x+\log (1280)-\log \left ((25+x)^2\right )} \, dx\\ &=-\int \left (-\frac {2}{\left (x+\log (1280)-\log \left ((25+x)^2\right )\right )^2}+\frac {x}{\left (x+\log (1280)-\log \left ((25+x)^2\right )\right )^2}+\frac {50}{(25+x) \left (x+\log (1280)-\log \left ((25+x)^2\right )\right )^2}\right ) \, dx+\int \frac {1}{x+\log (1280)-\log \left ((25+x)^2\right )} \, dx\\ &=2 \int \frac {1}{\left (x+\log (1280)-\log \left ((25+x)^2\right )\right )^2} \, dx-50 \int \frac {1}{(25+x) \left (x+\log (1280)-\log \left ((25+x)^2\right )\right )^2} \, dx-\int \frac {x}{\left (x+\log (1280)-\log \left ((25+x)^2\right )\right )^2} \, dx+\int \frac {1}{x+\log (1280)-\log \left ((25+x)^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.58, size = 16, normalized size = 0.73 \begin {gather*} \frac {x}{x+\log (1280)-\log \left ((25+x)^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x + (25 + x)*Log[5] + (50 + 2*x)*Log[16] + (-25 - x)*Log[625 + 50*x + x^2])/(25*x^2 + x^3 + (50*x
+ 2*x^2)*Log[5] + (25 + x)*Log[5]^2 + (100*x + 4*x^2 + (100 + 4*x)*Log[5])*Log[16] + (100 + 4*x)*Log[16]^2 +
(-50*x - 2*x^2 + (-50 - 2*x)*Log[5] + (-100 - 4*x)*Log[16])*Log[625 + 50*x + x^2] + (25 + x)*Log[625 + 50*x +
x^2]^2),x]

[Out]

x/(x + Log[1280] - Log[(25 + x)^2])

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fricas [A]  time = 0.76, size = 23, normalized size = 1.05 \begin {gather*} \frac {x}{x + \log \relax (5) + 8 \, \log \relax (2) - \log \left (x^{2} + 50 \, x + 625\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-25)*log(x^2+50*x+625)+4*(2*x+50)*log(2)+(x+25)*log(5)+2*x)/((x+25)*log(x^2+50*x+625)^2+(4*(-4*x
-100)*log(2)+(-2*x-50)*log(5)-2*x^2-50*x)*log(x^2+50*x+625)+16*(4*x+100)*log(2)^2+4*((4*x+100)*log(5)+4*x^2+10
0*x)*log(2)+(x+25)*log(5)^2+(2*x^2+50*x)*log(5)+x^3+25*x^2),x, algorithm="fricas")

[Out]

x/(x + log(5) + 8*log(2) - log(x^2 + 50*x + 625))

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giac [A]  time = 0.25, size = 23, normalized size = 1.05 \begin {gather*} \frac {x}{x + \log \relax (5) + 8 \, \log \relax (2) - \log \left (x^{2} + 50 \, x + 625\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-25)*log(x^2+50*x+625)+4*(2*x+50)*log(2)+(x+25)*log(5)+2*x)/((x+25)*log(x^2+50*x+625)^2+(4*(-4*x
-100)*log(2)+(-2*x-50)*log(5)-2*x^2-50*x)*log(x^2+50*x+625)+16*(4*x+100)*log(2)^2+4*((4*x+100)*log(5)+4*x^2+10
0*x)*log(2)+(x+25)*log(5)^2+(2*x^2+50*x)*log(5)+x^3+25*x^2),x, algorithm="giac")

[Out]

x/(x + log(5) + 8*log(2) - log(x^2 + 50*x + 625))

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maple [A]  time = 0.32, size = 24, normalized size = 1.09

method	result	size
risch	\(\frac {x}{\ln \relax (5)+8 \ln \relax (2)-\ln \left (x^{2}+50 x +625\right )+x}\)	\(24\)
norman	\(\frac {\ln \left (x^{2}+50 x +625\right )-\ln \relax (5)-8 \ln \relax (2)}{\ln \relax (5)+8 \ln \relax (2)-\ln \left (x^{2}+50 x +625\right )+x}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x-25)*ln(x^2+50*x+625)+4*(2*x+50)*ln(2)+(x+25)*ln(5)+2*x)/((x+25)*ln(x^2+50*x+625)^2+(4*(-4*x-100)*ln(2
)+(-2*x-50)*ln(5)-2*x^2-50*x)*ln(x^2+50*x+625)+16*(4*x+100)*ln(2)^2+4*((4*x+100)*ln(5)+4*x^2+100*x)*ln(2)+(x+2
5)*ln(5)^2+(2*x^2+50*x)*ln(5)+x^3+25*x^2),x,method=_RETURNVERBOSE)

[Out]

x/(ln(5)+8*ln(2)-ln(x^2+50*x+625)+x)

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maxima [A]  time = 0.47, size = 18, normalized size = 0.82 \begin {gather*} \frac {x}{x + \log \relax (5) + 8 \, \log \relax (2) - 2 \, \log \left (x + 25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-25)*log(x^2+50*x+625)+4*(2*x+50)*log(2)+(x+25)*log(5)+2*x)/((x+25)*log(x^2+50*x+625)^2+(4*(-4*x
-100)*log(2)+(-2*x-50)*log(5)-2*x^2-50*x)*log(x^2+50*x+625)+16*(4*x+100)*log(2)^2+4*((4*x+100)*log(5)+4*x^2+10
0*x)*log(2)+(x+25)*log(5)^2+(2*x^2+50*x)*log(5)+x^3+25*x^2),x, algorithm="maxima")

[Out]

x/(x + log(5) + 8*log(2) - 2*log(x + 25))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {2\,x+4\,\ln \relax (2)\,\left (2\,x+50\right )+\ln \relax (5)\,\left (x+25\right )-\ln \left (x^2+50\,x+625\right )\,\left (x+25\right )}{{\ln \relax (5)}^2\,\left (x+25\right )+{\ln \left (x^2+50\,x+625\right )}^2\,\left (x+25\right )+\ln \relax (5)\,\left (2\,x^2+50\,x\right )+4\,\ln \relax (2)\,\left (100\,x+\ln \relax (5)\,\left (4\,x+100\right )+4\,x^2\right )-\ln \left (x^2+50\,x+625\right )\,\left (50\,x+\ln \relax (5)\,\left (2\,x+50\right )+4\,\ln \relax (2)\,\left (4\,x+100\right )+2\,x^2\right )+16\,{\ln \relax (2)}^2\,\left (4\,x+100\right )+25\,x^2+x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 4*log(2)*(2*x + 50) + log(5)*(x + 25) - log(50*x + x^2 + 625)*(x + 25))/(log(5)^2*(x + 25) + log(50
*x + x^2 + 625)^2*(x + 25) + log(5)*(50*x + 2*x^2) + 4*log(2)*(100*x + log(5)*(4*x + 100) + 4*x^2) - log(50*x
+ x^2 + 625)*(50*x + log(5)*(2*x + 50) + 4*log(2)*(4*x + 100) + 2*x^2) + 16*log(2)^2*(4*x + 100) + 25*x^2 + x^
3),x)

[Out]

int((2*x + 4*log(2)*(2*x + 50) + log(5)*(x + 25) - log(50*x + x^2 + 625)*(x + 25))/(log(5)^2*(x + 25) + log(50
*x + x^2 + 625)^2*(x + 25) + log(5)*(50*x + 2*x^2) + 4*log(2)*(100*x + log(5)*(4*x + 100) + 4*x^2) - log(50*x
+ x^2 + 625)*(50*x + log(5)*(2*x + 50) + 4*log(2)*(4*x + 100) + 2*x^2) + 16*log(2)^2*(4*x + 100) + 25*x^2 + x^
3), x)

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sympy [A]  time = 0.20, size = 22, normalized size = 1.00 \begin {gather*} - \frac {x}{- x + \log {\left (x^{2} + 50 x + 625 \right )} - 8 \log {\relax (2 )} - \log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-25)*ln(x**2+50*x+625)+4*(2*x+50)*ln(2)+(x+25)*ln(5)+2*x)/((x+25)*ln(x**2+50*x+625)**2+(4*(-4*x-
100)*ln(2)+(-2*x-50)*ln(5)-2*x**2-50*x)*ln(x**2+50*x+625)+16*(4*x+100)*ln(2)**2+4*((4*x+100)*ln(5)+4*x**2+100*
x)*ln(2)+(x+25)*ln(5)**2+(2*x**2+50*x)*ln(5)+x**3+25*x**2),x)

[Out]

-x/(-x + log(x**2 + 50*x + 625) - 8*log(2) - log(5))

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