∫ −50x4+(−450+300x−50x2−150x4) log(x)+(450−150x−150x4) log2(x)+(450−150x−50x4) log3(x)__ −x3+25x5+(−3x3+75x5) log(x)+(225x−150x2+22x3+75x5) log2(x)+(225x−150x2+24x3+25x5) log3(x) dx

3.30.18 \(\int \frac {-50 x^4+(-450+300 x-50 x^2-150 x^4) \log (x)+(450-150 x-150 x^4) \log ^2(x)+(450-150 x-50 x^4) \log ^3(x)}{-x^3+25 x^5+(-3 x^3+75 x^5) \log (x)+(225 x-150 x^2+22 x^3+75 x^5) \log ^2(x)+(225 x-150 x^2+24 x^3+25 x^5) \log ^3(x)} \, dx\)

Optimal. Leaf size=35 \[ \log \left (\frac {5 \log (5)}{-\frac {1}{5}+5 \left (x^2+\frac {(3-x)^2}{\left (x+\frac {x}{\log (x)}\right )^2}\right )}\right ) \]

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Rubi [F] time = 129.54, 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 {-50 x^4+\left (-450+300 x-50 x^2-150 x^4\right ) \log (x)+\left (450-150 x-150 x^4\right ) \log ^2(x)+\left (450-150 x-50 x^4\right ) \log ^3(x)}{-x^3+25 x^5+\left (-3 x^3+75 x^5\right ) \log (x)+\left (225 x-150 x^2+22 x^3+75 x^5\right ) \log ^2(x)+\left (225 x-150 x^2+24 x^3+25 x^5\right ) \log ^3(x)} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(-50*x^4 + (-450 + 300*x - 50*x^2 - 150*x^4)*Log[x] + (450 - 150*x - 150*x^4)*Log[x]^2 + (450 - 150*x - 50

*x^4)*Log[x]^3)/(-x^3 + 25*x^5 + (-3*x^3 + 75*x^5)*Log[x] + (225*x - 150*x^2 + 22*x^3 + 75*x^5)*Log[x]^2 + (22

5*x - 150*x^2 + 24*x^3 + 25*x^5)*Log[x]^3),x]

[Out]

2*Log[x] - Log[225 - 150*x + 24*x^2 + 25*x^4] + 2*Log[1 + Log[x]] + 450*Defer[Int][(-x^2 + 25*x^4 - 2*x^2*Log[

x] + 50*x^4*Log[x] + 225*Log[x]^2 - 150*x*Log[x]^2 + 24*x^2*Log[x]^2 + 25*x^4*Log[x]^2)^(-1), x] - 48*Defer[In

t][x/(-x^2 + 25*x^4 - 2*x^2*Log[x] + 50*x^4*Log[x] + 225*Log[x]^2 - 150*x*Log[x]^2 + 24*x^2*Log[x]^2 + 25*x^4*

Log[x]^2), x] - 50*Defer[Int][x^3/(-x^2 + 25*x^4 - 2*x^2*Log[x] + 50*x^4*Log[x] + 225*Log[x]^2 - 150*x*Log[x]^

2 + 24*x^2*Log[x]^2 + 25*x^4*Log[x]^2), x] - 101250*Defer[Int][1/((225 - 150*x + 24*x^2 + 25*x^4)*(-x^2 + 25*x

^4 - 2*x^2*Log[x] + 50*x^4*Log[x] + 225*Log[x]^2 - 150*x*Log[x]^2 + 24*x^2*Log[x]^2 + 25*x^4*Log[x]^2)), x] +

79200*Defer[Int][x/((225 - 150*x + 24*x^2 + 25*x^4)*(-x^2 + 25*x^4 - 2*x^2*Log[x] + 50*x^4*Log[x] + 225*Log[x]

^2 - 150*x*Log[x]^2 + 24*x^2*Log[x]^2 + 25*x^4*Log[x]^2)), x] - 18450*Defer[Int][x^2/((225 - 150*x + 24*x^2 +

25*x^4)*(-x^2 + 25*x^4 - 2*x^2*Log[x] + 50*x^4*Log[x] + 225*Log[x]^2 - 150*x*Log[x]^2 + 24*x^2*Log[x]^2 + 25*x

^4*Log[x]^2)), x] - 21300*Defer[Int][x^3/((225 - 150*x + 24*x^2 + 25*x^4)*(-x^2 + 25*x^4 - 2*x^2*Log[x] + 50*x

^4*Log[x] + 225*Log[x]^2 - 150*x*Log[x]^2 + 24*x^2*Log[x]^2 + 25*x^4*Log[x]^2)), x] + 1200*Defer[Int][Log[x]/(

-x^2 + 25*x^4 - 2*x^2*Log[x] + 50*x^4*Log[x] + 225*Log[x]^2 - 150*x*Log[x]^2 + 24*x^2*Log[x]^2 + 25*x^4*Log[x]

^2), x] - 450*Defer[Int][Log[x]/(x*(-x^2 + 25*x^4 - 2*x^2*Log[x] + 50*x^4*Log[x] + 225*Log[x]^2 - 150*x*Log[x]

^2 + 24*x^2*Log[x]^2 + 25*x^4*Log[x]^2)), x] - 148*Defer[Int][(x*Log[x])/(-x^2 + 25*x^4 - 2*x^2*Log[x] + 50*x^

4*Log[x] + 225*Log[x]^2 - 150*x*Log[x]^2 + 24*x^2*Log[x]^2 + 25*x^4*Log[x]^2), x] - 50*Defer[Int][(x^3*Log[x])

/(-x^2 + 25*x^4 - 2*x^2*Log[x] + 50*x^4*Log[x] + 225*Log[x]^2 - 150*x*Log[x]^2 + 24*x^2*Log[x]^2 + 25*x^4*Log[

x]^2), x] - 202500*Defer[Int][Log[x]/((225 - 150*x + 24*x^2 + 25*x^4)*(-x^2 + 25*x^4 - 2*x^2*Log[x] + 50*x^4*L

og[x] + 225*Log[x]^2 - 150*x*Log[x]^2 + 24*x^2*Log[x]^2 + 25*x^4*Log[x]^2)), x] + 158400*Defer[Int][(x*Log[x])

/((225 - 150*x + 24*x^2 + 25*x^4)*(-x^2 + 25*x^4 - 2*x^2*Log[x] + 50*x^4*Log[x] + 225*Log[x]^2 - 150*x*Log[x]^

2 + 24*x^2*Log[x]^2 + 25*x^4*Log[x]^2)), x] - 36900*Defer[Int][(x^2*Log[x])/((225 - 150*x + 24*x^2 + 25*x^4)*(

-x^2 + 25*x^4 - 2*x^2*Log[x] + 50*x^4*Log[x] + 225*Log[x]^2 - 150*x*Log[x]^2 + 24*x^2*Log[x]^2 + 25*x^4*Log[x]

^2)), x] - 42600*Defer[Int][(x^3*Log[x])/((225 - 150*x + 24*x^2 + 25*x^4)*(-x^2 + 25*x^4 - 2*x^2*Log[x] + 50*x

^4*Log[x] + 225*Log[x]^2 - 150*x*Log[x]^2 + 24*x^2*Log[x]^2 + 25*x^4*Log[x]^2)), x]

Rubi steps

Aborted

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Mathematica [B] time = 0.14, size = 79, normalized size = 2.26 \begin {gather*} -50 \left (-\frac {\log (x)}{25}-\frac {1}{25} \log (1+\log (x))+\frac {1}{50} \log \left (-x^2+25 x^4-2 x^2 \log (x)+50 x^4 \log (x)+225 \log ^2(x)-150 x \log ^2(x)+24 x^2 \log ^2(x)+25 x^4 \log ^2(x)\right )\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-50*x^4 + (-450 + 300*x - 50*x^2 - 150*x^4)*Log[x] + (450 - 150*x - 150*x^4)*Log[x]^2 + (450 - 150*

x - 50*x^4)*Log[x]^3)/(-x^3 + 25*x^5 + (-3*x^3 + 75*x^5)*Log[x] + (225*x - 150*x^2 + 22*x^3 + 75*x^5)*Log[x]^2

+ (225*x - 150*x^2 + 24*x^3 + 25*x^5)*Log[x]^3),x]

[Out]

-50*(-1/25*Log[x] - Log[1 + Log[x]]/25 + Log[-x^2 + 25*x^4 - 2*x^2*Log[x] + 50*x^4*Log[x] + 225*Log[x]^2 - 150

*x*Log[x]^2 + 24*x^2*Log[x]^2 + 25*x^4*Log[x]^2]/50)

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fricas [B] time = 0.76, size = 97, normalized size = 2.77 \begin {gather*} -\log \left (25 \, x^{4} + 24 \, x^{2} - 150 \, x + 225\right ) + 2 \, \log \relax (x) - \log \left (\frac {25 \, x^{4} + {\left (25 \, x^{4} + 24 \, x^{2} - 150 \, x + 225\right )} \log \relax (x)^{2} - x^{2} + 2 \, {\left (25 \, x^{4} - x^{2}\right )} \log \relax (x)}{25 \, x^{4} + 24 \, x^{2} - 150 \, x + 225}\right ) + 2 \, \log \left (\log \relax (x) + 1\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*x^4-150*x+450)*log(x)^3+(-150*x^4-150*x+450)*log(x)^2+(-150*x^4-50*x^2+300*x-450)*log(x)-50*x^

4)/((25*x^5+24*x^3-150*x^2+225*x)*log(x)^3+(75*x^5+22*x^3-150*x^2+225*x)*log(x)^2+(75*x^5-3*x^3)*log(x)+25*x^5

-x^3),x, algorithm="fricas")

[Out]

-log(25*x^4 + 24*x^2 - 150*x + 225) + 2*log(x) - log((25*x^4 + (25*x^4 + 24*x^2 - 150*x + 225)*log(x)^2 - x^2

+ 2*(25*x^4 - x^2)*log(x))/(25*x^4 + 24*x^2 - 150*x + 225)) + 2*log(log(x) + 1)

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giac [B] time = 0.71, size = 71, normalized size = 2.03 \begin {gather*} -\log \left (25 \, x^{4} \log \relax (x)^{2} + 50 \, x^{4} \log \relax (x) + 25 \, x^{4} + 24 \, x^{2} \log \relax (x)^{2} - 2 \, x^{2} \log \relax (x) - 150 \, x \log \relax (x)^{2} - x^{2} + 225 \, \log \relax (x)^{2}\right ) + 2 \, \log \relax (x) + 2 \, \log \left (\log \relax (x) + 1\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*x^4-150*x+450)*log(x)^3+(-150*x^4-150*x+450)*log(x)^2+(-150*x^4-50*x^2+300*x-450)*log(x)-50*x^

4)/((25*x^5+24*x^3-150*x^2+225*x)*log(x)^3+(75*x^5+22*x^3-150*x^2+225*x)*log(x)^2+(75*x^5-3*x^3)*log(x)+25*x^5

-x^3),x, algorithm="giac")

[Out]

-log(25*x^4*log(x)^2 + 50*x^4*log(x) + 25*x^4 + 24*x^2*log(x)^2 - 2*x^2*log(x) - 150*x*log(x)^2 - x^2 + 225*lo

g(x)^2) + 2*log(x) + 2*log(log(x) + 1)

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maple [B] time = 0.09, size = 72, normalized size = 2.06


method	result	size

norman	\(2 \ln \relax (x )+2 \ln \left (\ln \relax (x )+1\right )-\ln \left (25 x^{4} \ln \relax (x )^{2}+50 x^{4} \ln \relax (x )+25 x^{4}+24 x^{2} \ln \relax (x )^{2}-2 x^{2} \ln \relax (x )-150 x \ln \relax (x )^{2}-x^{2}+225 \ln \relax (x )^{2}\right )\)	\(72\)
risch	\(2 \ln \relax (x )-\ln \left (25 x^{4}+24 x^{2}-150 x +225\right )+2 \ln \left (\ln \relax (x )+1\right )-\ln \left (\ln \relax (x )^{2}+\frac {2 x^{2} \left (25 x^{2}-1\right ) \ln \relax (x )}{25 x^{4}+24 x^{2}-150 x +225}+\frac {x^{2} \left (25 x^{2}-1\right )}{25 x^{4}+24 x^{2}-150 x +225}\right )\)	\(98\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((-50*x^4-150*x+450)*ln(x)^3+(-150*x^4-150*x+450)*ln(x)^2+(-150*x^4-50*x^2+300*x-450)*ln(x)-50*x^4)/((25*x

^5+24*x^3-150*x^2+225*x)*ln(x)^3+(75*x^5+22*x^3-150*x^2+225*x)*ln(x)^2+(75*x^5-3*x^3)*ln(x)+25*x^5-x^3),x,meth

od=_RETURNVERBOSE)

[Out]

2*ln(x)+2*ln(ln(x)+1)-ln(25*x^4*ln(x)^2+50*x^4*ln(x)+25*x^4+24*x^2*ln(x)^2-2*x^2*ln(x)-150*x*ln(x)^2-x^2+225*l

n(x)^2)

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maxima [B] time = 0.57, size = 97, normalized size = 2.77 \begin {gather*} -\log \left (25 \, x^{4} + 24 \, x^{2} - 150 \, x + 225\right ) + 2 \, \log \relax (x) - \log \left (\frac {25 \, x^{4} + {\left (25 \, x^{4} + 24 \, x^{2} - 150 \, x + 225\right )} \log \relax (x)^{2} - x^{2} + 2 \, {\left (25 \, x^{4} - x^{2}\right )} \log \relax (x)}{25 \, x^{4} + 24 \, x^{2} - 150 \, x + 225}\right ) + 2 \, \log \left (\log \relax (x) + 1\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*x^4-150*x+450)*log(x)^3+(-150*x^4-150*x+450)*log(x)^2+(-150*x^4-50*x^2+300*x-450)*log(x)-50*x^

4)/((25*x^5+24*x^3-150*x^2+225*x)*log(x)^3+(75*x^5+22*x^3-150*x^2+225*x)*log(x)^2+(75*x^5-3*x^3)*log(x)+25*x^5

-x^3),x, algorithm="maxima")

[Out]

-log(25*x^4 + 24*x^2 - 150*x + 225) + 2*log(x) - log((25*x^4 + (25*x^4 + 24*x^2 - 150*x + 225)*log(x)^2 - x^2

+ 2*(25*x^4 - x^2)*log(x))/(25*x^4 + 24*x^2 - 150*x + 225)) + 2*log(log(x) + 1)

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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\ln \relax (x)}^3\,\left (50\,x^4+150\,x-450\right )+{\ln \relax (x)}^2\,\left (150\,x^4+150\,x-450\right )+50\,x^4+\ln \relax (x)\,\left (150\,x^4+50\,x^2-300\,x+450\right )}{{\ln \relax (x)}^3\,\left (25\,x^5+24\,x^3-150\,x^2+225\,x\right )-\ln \relax (x)\,\left (3\,x^3-75\,x^5\right )+{\ln \relax (x)}^2\,\left (75\,x^5+22\,x^3-150\,x^2+225\,x\right )-x^3+25\,x^5} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)^3*(150*x + 50*x^4 - 450) + log(x)^2*(150*x + 150*x^4 - 450) + 50*x^4 + log(x)*(50*x^2 - 300*x + 1

50*x^4 + 450))/(log(x)^3*(225*x - 150*x^2 + 24*x^3 + 25*x^5) - log(x)*(3*x^3 - 75*x^5) + log(x)^2*(225*x - 150

*x^2 + 22*x^3 + 75*x^5) - x^3 + 25*x^5),x)

[Out]

int(-(log(x)^3*(150*x + 50*x^4 - 450) + log(x)^2*(150*x + 150*x^4 - 450) + 50*x^4 + log(x)*(50*x^2 - 300*x + 1

50*x^4 + 450))/(log(x)^3*(225*x - 150*x^2 + 24*x^3 + 25*x^5) - log(x)*(3*x^3 - 75*x^5) + log(x)^2*(225*x - 150

*x^2 + 22*x^3 + 75*x^5) - x^3 + 25*x^5), x)

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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**4-150*x+450)*ln(x)**3+(-150*x**4-150*x+450)*ln(x)**2+(-150*x**4-50*x**2+300*x-450)*ln(x)-50

*x**4)/((25*x**5+24*x**3-150*x**2+225*x)*ln(x)**3+(75*x**5+22*x**3-150*x**2+225*x)*ln(x)**2+(75*x**5-3*x**3)*l

n(x)+25*x**5-x**3),x)

[Out]

Exception raised: PolynomialError

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