3.17.55 \(\int \frac {125-3000 x+1275 x^2-600 x^3+500 x^4+125 x^5+50 x^6+25 x^7+(-125 x-50 x^2-75 x^3) \log (x)}{1250 x+1000 x^3+300 x^5+40 x^7+2 x^9+(100 x+40 x^3+4 x^5) \log (x)+2 x \log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ \frac {25 (-1-x)}{2 \left (5+x^2+\frac {\log (x)}{5+x^2}\right )} \]

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Rubi [F]  time = 1.47, 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 {125-3000 x+1275 x^2-600 x^3+500 x^4+125 x^5+50 x^6+25 x^7+\left (-125 x-50 x^2-75 x^3\right ) \log (x)}{1250 x+1000 x^3+300 x^5+40 x^7+2 x^9+\left (100 x+40 x^3+4 x^5\right ) \log (x)+2 x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(125 - 3000*x + 1275*x^2 - 600*x^3 + 500*x^4 + 125*x^5 + 50*x^6 + 25*x^7 + (-125*x - 50*x^2 - 75*x^3)*Log[
x])/(1250*x + 1000*x^3 + 300*x^5 + 40*x^7 + 2*x^9 + (100*x + 40*x^3 + 4*x^5)*Log[x] + 2*x*Log[x]^2),x]

[Out]

(125*Defer[Int][(25 + 10*x^2 + x^4 + Log[x])^(-2), x])/2 + (125*Defer[Int][1/(x*(25 + 10*x^2 + x^4 + Log[x])^2
), x])/2 + (2525*Defer[Int][x/(25 + 10*x^2 + x^4 + Log[x])^2, x])/2 + (2525*Defer[Int][x^2/(25 + 10*x^2 + x^4
+ Log[x])^2, x])/2 + 500*Defer[Int][x^3/(25 + 10*x^2 + x^4 + Log[x])^2, x] + 500*Defer[Int][x^4/(25 + 10*x^2 +
x^4 + Log[x])^2, x] + 50*Defer[Int][x^5/(25 + 10*x^2 + x^4 + Log[x])^2, x] + 50*Defer[Int][x^6/(25 + 10*x^2 +
x^4 + Log[x])^2, x] - (125*Defer[Int][(25 + 10*x^2 + x^4 + Log[x])^(-1), x])/2 - 25*Defer[Int][x/(25 + 10*x^2
+ x^4 + Log[x]), x] - (75*Defer[Int][x^2/(25 + 10*x^2 + x^4 + Log[x]), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 \left (5-120 x+51 x^2-24 x^3+20 x^4+5 x^5+2 x^6+x^7-x \left (5+2 x+3 x^2\right ) \log (x)\right )}{2 x \left (\left (5+x^2\right )^2+\log (x)\right )^2} \, dx\\ &=\frac {25}{2} \int \frac {5-120 x+51 x^2-24 x^3+20 x^4+5 x^5+2 x^6+x^7-x \left (5+2 x+3 x^2\right ) \log (x)}{x \left (\left (5+x^2\right )^2+\log (x)\right )^2} \, dx\\ &=\frac {25}{2} \int \left (\frac {5+5 x+101 x^2+101 x^3+40 x^4+40 x^5+4 x^6+4 x^7}{x \left (25+10 x^2+x^4+\log (x)\right )^2}+\frac {-5-2 x-3 x^2}{25+10 x^2+x^4+\log (x)}\right ) \, dx\\ &=\frac {25}{2} \int \frac {5+5 x+101 x^2+101 x^3+40 x^4+40 x^5+4 x^6+4 x^7}{x \left (25+10 x^2+x^4+\log (x)\right )^2} \, dx+\frac {25}{2} \int \frac {-5-2 x-3 x^2}{25+10 x^2+x^4+\log (x)} \, dx\\ &=\frac {25}{2} \int \left (\frac {5}{\left (25+10 x^2+x^4+\log (x)\right )^2}+\frac {5}{x \left (25+10 x^2+x^4+\log (x)\right )^2}+\frac {101 x}{\left (25+10 x^2+x^4+\log (x)\right )^2}+\frac {101 x^2}{\left (25+10 x^2+x^4+\log (x)\right )^2}+\frac {40 x^3}{\left (25+10 x^2+x^4+\log (x)\right )^2}+\frac {40 x^4}{\left (25+10 x^2+x^4+\log (x)\right )^2}+\frac {4 x^5}{\left (25+10 x^2+x^4+\log (x)\right )^2}+\frac {4 x^6}{\left (25+10 x^2+x^4+\log (x)\right )^2}\right ) \, dx+\frac {25}{2} \int \left (-\frac {5}{25+10 x^2+x^4+\log (x)}-\frac {2 x}{25+10 x^2+x^4+\log (x)}-\frac {3 x^2}{25+10 x^2+x^4+\log (x)}\right ) \, dx\\ &=-\left (25 \int \frac {x}{25+10 x^2+x^4+\log (x)} \, dx\right )-\frac {75}{2} \int \frac {x^2}{25+10 x^2+x^4+\log (x)} \, dx+50 \int \frac {x^5}{\left (25+10 x^2+x^4+\log (x)\right )^2} \, dx+50 \int \frac {x^6}{\left (25+10 x^2+x^4+\log (x)\right )^2} \, dx+\frac {125}{2} \int \frac {1}{\left (25+10 x^2+x^4+\log (x)\right )^2} \, dx+\frac {125}{2} \int \frac {1}{x \left (25+10 x^2+x^4+\log (x)\right )^2} \, dx-\frac {125}{2} \int \frac {1}{25+10 x^2+x^4+\log (x)} \, dx+500 \int \frac {x^3}{\left (25+10 x^2+x^4+\log (x)\right )^2} \, dx+500 \int \frac {x^4}{\left (25+10 x^2+x^4+\log (x)\right )^2} \, dx+\frac {2525}{2} \int \frac {x}{\left (25+10 x^2+x^4+\log (x)\right )^2} \, dx+\frac {2525}{2} \int \frac {x^2}{\left (25+10 x^2+x^4+\log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.73, size = 27, normalized size = 1.04 \begin {gather*} -\frac {25 \left (5+5 x+x^2+x^3\right )}{2 \left (\left (5+x^2\right )^2+\log (x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(125 - 3000*x + 1275*x^2 - 600*x^3 + 500*x^4 + 125*x^5 + 50*x^6 + 25*x^7 + (-125*x - 50*x^2 - 75*x^3
)*Log[x])/(1250*x + 1000*x^3 + 300*x^5 + 40*x^7 + 2*x^9 + (100*x + 40*x^3 + 4*x^5)*Log[x] + 2*x*Log[x]^2),x]

[Out]

(-25*(5 + 5*x + x^2 + x^3))/(2*((5 + x^2)^2 + Log[x]))

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fricas [A]  time = 0.72, size = 27, normalized size = 1.04 \begin {gather*} -\frac {25 \, {\left (x^{3} + x^{2} + 5 \, x + 5\right )}}{2 \, {\left (x^{4} + 10 \, x^{2} + \log \relax (x) + 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-75*x^3-50*x^2-125*x)*log(x)+25*x^7+50*x^6+125*x^5+500*x^4-600*x^3+1275*x^2-3000*x+125)/(2*x*log(x
)^2+(4*x^5+40*x^3+100*x)*log(x)+2*x^9+40*x^7+300*x^5+1000*x^3+1250*x),x, algorithm="fricas")

[Out]

-25/2*(x^3 + x^2 + 5*x + 5)/(x^4 + 10*x^2 + log(x) + 25)

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giac [A]  time = 0.17, size = 27, normalized size = 1.04 \begin {gather*} -\frac {25 \, {\left (x^{3} + x^{2} + 5 \, x + 5\right )}}{2 \, {\left (x^{4} + 10 \, x^{2} + \log \relax (x) + 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-75*x^3-50*x^2-125*x)*log(x)+25*x^7+50*x^6+125*x^5+500*x^4-600*x^3+1275*x^2-3000*x+125)/(2*x*log(x
)^2+(4*x^5+40*x^3+100*x)*log(x)+2*x^9+40*x^7+300*x^5+1000*x^3+1250*x),x, algorithm="giac")

[Out]

-25/2*(x^3 + x^2 + 5*x + 5)/(x^4 + 10*x^2 + log(x) + 25)

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maple [A]  time = 0.02, size = 28, normalized size = 1.08

method	result	size
risch	\(-\frac {25 \left (x^{3}+x^{2}+5 x +5\right )}{2 \left (x^{4}+10 x^{2}+\ln \relax (x )+25\right )}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-75*x^3-50*x^2-125*x)*ln(x)+25*x^7+50*x^6+125*x^5+500*x^4-600*x^3+1275*x^2-3000*x+125)/(2*x*ln(x)^2+(4*x
^5+40*x^3+100*x)*ln(x)+2*x^9+40*x^7+300*x^5+1000*x^3+1250*x),x,method=_RETURNVERBOSE)

[Out]

-25/2*(x^3+x^2+5*x+5)/(x^4+10*x^2+ln(x)+25)

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maxima [A]  time = 0.63, size = 27, normalized size = 1.04 \begin {gather*} -\frac {25 \, {\left (x^{3} + x^{2} + 5 \, x + 5\right )}}{2 \, {\left (x^{4} + 10 \, x^{2} + \log \relax (x) + 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-75*x^3-50*x^2-125*x)*log(x)+25*x^7+50*x^6+125*x^5+500*x^4-600*x^3+1275*x^2-3000*x+125)/(2*x*log(x
)^2+(4*x^5+40*x^3+100*x)*log(x)+2*x^9+40*x^7+300*x^5+1000*x^3+1250*x),x, algorithm="maxima")

[Out]

-25/2*(x^3 + x^2 + 5*x + 5)/(x^4 + 10*x^2 + log(x) + 25)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1275\,x^2-3000\,x-600\,x^3+500\,x^4+125\,x^5+50\,x^6+25\,x^7-\ln \relax (x)\,\left (75\,x^3+50\,x^2+125\,x\right )+125}{1250\,x+2\,x\,{\ln \relax (x)}^2+1000\,x^3+300\,x^5+40\,x^7+2\,x^9+\ln \relax (x)\,\left (4\,x^5+40\,x^3+100\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1275*x^2 - 3000*x - 600*x^3 + 500*x^4 + 125*x^5 + 50*x^6 + 25*x^7 - log(x)*(125*x + 50*x^2 + 75*x^3) + 12
5)/(1250*x + 2*x*log(x)^2 + 1000*x^3 + 300*x^5 + 40*x^7 + 2*x^9 + log(x)*(100*x + 40*x^3 + 4*x^5)),x)

[Out]

int((1275*x^2 - 3000*x - 600*x^3 + 500*x^4 + 125*x^5 + 50*x^6 + 25*x^7 - log(x)*(125*x + 50*x^2 + 75*x^3) + 12
5)/(1250*x + 2*x*log(x)^2 + 1000*x^3 + 300*x^5 + 40*x^7 + 2*x^9 + log(x)*(100*x + 40*x^3 + 4*x^5)), x)

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sympy [A]  time = 0.14, size = 32, normalized size = 1.23 \begin {gather*} \frac {- 25 x^{3} - 25 x^{2} - 125 x - 125}{2 x^{4} + 20 x^{2} + 2 \log {\relax (x )} + 50} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-75*x**3-50*x**2-125*x)*ln(x)+25*x**7+50*x**6+125*x**5+500*x**4-600*x**3+1275*x**2-3000*x+125)/(2*
x*ln(x)**2+(4*x**5+40*x**3+100*x)*ln(x)+2*x**9+40*x**7+300*x**5+1000*x**3+1250*x),x)

[Out]

(-25*x**3 - 25*x**2 - 125*x - 125)/(2*x**4 + 20*x**2 + 2*log(x) + 50)

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