3.80.61 \(\int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+(25 x^2+80 x^3+114 x^4+80 x^5+25 x^6) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+(25 x^2+80 x^3+114 x^4+80 x^5+25 x^6) \log (x)} \, dx\)

Optimal. Leaf size=29 \[ 2+x+\log \left (\frac {6}{x}+\frac {5}{-\frac {2 x}{5}+(1+x)^2}-\log (x)\right ) \]

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Rubi [F]  time = 3.20, 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 {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(150 + 355*x + 359*x^2 - 40*x^3 - 375*x^4 - 125*x^5 + (25*x^2 + 80*x^3 + 114*x^4 + 80*x^5 + 25*x^6)*Log[x]
)/(-150*x - 605*x^2 - 884*x^3 - 605*x^4 - 150*x^5 + (25*x^2 + 80*x^3 + 114*x^4 + 80*x^5 + 25*x^6)*Log[x]),x]

[Out]

x + 103*Defer[Int][(-30 - 73*x - 30*x^2 + 5*x*Log[x] + 8*x^2*Log[x] + 5*x^3*Log[x])^(-1), x] - ((1250*I)/3)*De
fer[Int][1/(((-8 + 6*I) - 10*x)*(-30 - 73*x - 30*x^2 + 5*x*Log[x] + 8*x^2*Log[x] + 5*x^3*Log[x])), x] + 30*Def
er[Int][1/(x*(-30 - 73*x - 30*x^2 + 5*x*Log[x] + 8*x^2*Log[x] + 5*x^3*Log[x])), x] + 38*Defer[Int][x/(-30 - 73
*x - 30*x^2 + 5*x*Log[x] + 8*x^2*Log[x] + 5*x^3*Log[x]), x] + 5*Defer[Int][x^2/(-30 - 73*x - 30*x^2 + 5*x*Log[
x] + 8*x^2*Log[x] + 5*x^3*Log[x]), x] - (200 + (800*I)/3)*Defer[Int][1/(((8 - 6*I) + 10*x)*(-30 - 73*x - 30*x^
2 + 5*x*Log[x] + 8*x^2*Log[x] + 5*x^3*Log[x])), x] - (200 + 150*I)*Defer[Int][1/(((8 + 6*I) + 10*x)*(-30 - 73*
x - 30*x^2 + 5*x*Log[x] + 8*x^2*Log[x] + 5*x^3*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-150-355 x-359 x^2+40 x^3+375 x^4+125 x^5-\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{x \left (5+8 x+5 x^2\right ) \left (30+73 x+30 x^2-5 x \log (x)-8 x^2 \log (x)-5 x^3 \log (x)\right )} \, dx\\ &=\int \left (1+\frac {150+505 x+964 x^2+844 x^3+230 x^4+25 x^5}{x \left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}\right ) \, dx\\ &=x+\int \frac {150+505 x+964 x^2+844 x^3+230 x^4+25 x^5}{x \left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx\\ &=x+\int \left (\frac {103}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)}+\frac {30}{x \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}+\frac {38 x}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)}+\frac {5 x^2}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)}-\frac {50 (5+4 x)}{\left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}\right ) \, dx\\ &=x+5 \int \frac {x^2}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+30 \int \frac {1}{x \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx+38 \int \frac {x}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx-50 \int \frac {5+4 x}{\left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx+103 \int \frac {1}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx\\ &=x+5 \int \frac {x^2}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+30 \int \frac {1}{x \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx+38 \int \frac {x}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx-50 \int \left (\frac {5}{\left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}+\frac {4 x}{\left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}\right ) \, dx+103 \int \frac {1}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx\\ &=x+5 \int \frac {x^2}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+30 \int \frac {1}{x \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx+38 \int \frac {x}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+103 \int \frac {1}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx-200 \int \frac {x}{\left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx-250 \int \frac {1}{\left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx\\ &=x+5 \int \frac {x^2}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+30 \int \frac {1}{x \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx+38 \int \frac {x}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+103 \int \frac {1}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx-200 \int \left (\frac {1+\frac {4 i}{3}}{((8-6 i)+10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}+\frac {1-\frac {4 i}{3}}{((8+6 i)+10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}\right ) \, dx-250 \int \left (\frac {5 i}{3 ((-8+6 i)-10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}+\frac {5 i}{3 ((8+6 i)+10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}\right ) \, dx\\ &=x-\frac {1250}{3} i \int \frac {1}{((-8+6 i)-10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx-\frac {1250}{3} i \int \frac {1}{((8+6 i)+10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx+5 \int \frac {x^2}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+30 \int \frac {1}{x \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx+38 \int \frac {x}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+103 \int \frac {1}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx-\left (200-\frac {800 i}{3}\right ) \int \frac {1}{((8+6 i)+10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx-\left (200+\frac {800 i}{3}\right ) \int \frac {1}{((8-6 i)+10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 47, normalized size = 1.62 \begin {gather*} x-\log \left (x \left (5+8 x+5 x^2\right )\right )+\log \left (30+73 x+30 x^2-5 x \log (x)-8 x^2 \log (x)-5 x^3 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(150 + 355*x + 359*x^2 - 40*x^3 - 375*x^4 - 125*x^5 + (25*x^2 + 80*x^3 + 114*x^4 + 80*x^5 + 25*x^6)*
Log[x])/(-150*x - 605*x^2 - 884*x^3 - 605*x^4 - 150*x^5 + (25*x^2 + 80*x^3 + 114*x^4 + 80*x^5 + 25*x^6)*Log[x]
),x]

[Out]

x - Log[x*(5 + 8*x + 5*x^2)] + Log[30 + 73*x + 30*x^2 - 5*x*Log[x] - 8*x^2*Log[x] - 5*x^3*Log[x]]

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fricas [A]  time = 0.99, size = 49, normalized size = 1.69 \begin {gather*} x + \log \left (-\frac {30 \, x^{2} - {\left (5 \, x^{3} + 8 \, x^{2} + 5 \, x\right )} \log \relax (x) + 73 \, x + 30}{5 \, x^{3} + 8 \, x^{2} + 5 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x^6+80*x^5+114*x^4+80*x^3+25*x^2)*log(x)-125*x^5-375*x^4-40*x^3+359*x^2+355*x+150)/((25*x^6+80*
x^5+114*x^4+80*x^3+25*x^2)*log(x)-150*x^5-605*x^4-884*x^3-605*x^2-150*x),x, algorithm="fricas")

[Out]

x + log(-(30*x^2 - (5*x^3 + 8*x^2 + 5*x)*log(x) + 73*x + 30)/(5*x^3 + 8*x^2 + 5*x))

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giac [A]  time = 0.20, size = 49, normalized size = 1.69 \begin {gather*} x + \log \left (5 \, x^{3} \log \relax (x) + 8 \, x^{2} \log \relax (x) - 30 \, x^{2} + 5 \, x \log \relax (x) - 73 \, x - 30\right ) - \log \left (5 \, x^{2} + 8 \, x + 5\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x^6+80*x^5+114*x^4+80*x^3+25*x^2)*log(x)-125*x^5-375*x^4-40*x^3+359*x^2+355*x+150)/((25*x^6+80*
x^5+114*x^4+80*x^3+25*x^2)*log(x)-150*x^5-605*x^4-884*x^3-605*x^2-150*x),x, algorithm="giac")

[Out]

x + log(5*x^3*log(x) + 8*x^2*log(x) - 30*x^2 + 5*x*log(x) - 73*x - 30) - log(5*x^2 + 8*x + 5) - log(x)

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maple [A]  time = 0.07, size = 34, normalized size = 1.17

method	result	size
risch	\(x +\ln \left (\ln \relax (x )-\frac {30 x^{2}+73 x +30}{x \left (5 x^{2}+8 x +5\right )}\right )\)	\(34\)
norman	\(x -\ln \relax (x )-\ln \left (5 x^{2}+8 x +5\right )+\ln \left (5 x^{3} \ln \relax (x )+8 x^{2} \ln \relax (x )-30 x^{2}+5 x \ln \relax (x )-73 x -30\right )\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((25*x^6+80*x^5+114*x^4+80*x^3+25*x^2)*ln(x)-125*x^5-375*x^4-40*x^3+359*x^2+355*x+150)/((25*x^6+80*x^5+114
*x^4+80*x^3+25*x^2)*ln(x)-150*x^5-605*x^4-884*x^3-605*x^2-150*x),x,method=_RETURNVERBOSE)

[Out]

x+ln(ln(x)-(30*x^2+73*x+30)/x/(5*x^2+8*x+5))

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maxima [A]  time = 0.41, size = 49, normalized size = 1.69 \begin {gather*} x + \log \left (-\frac {30 \, x^{2} - {\left (5 \, x^{3} + 8 \, x^{2} + 5 \, x\right )} \log \relax (x) + 73 \, x + 30}{5 \, x^{3} + 8 \, x^{2} + 5 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x^6+80*x^5+114*x^4+80*x^3+25*x^2)*log(x)-125*x^5-375*x^4-40*x^3+359*x^2+355*x+150)/((25*x^6+80*
x^5+114*x^4+80*x^3+25*x^2)*log(x)-150*x^5-605*x^4-884*x^3-605*x^2-150*x),x, algorithm="maxima")

[Out]

x + log(-(30*x^2 - (5*x^3 + 8*x^2 + 5*x)*log(x) + 73*x + 30)/(5*x^3 + 8*x^2 + 5*x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {355\,x+\ln \relax (x)\,\left (25\,x^6+80\,x^5+114\,x^4+80\,x^3+25\,x^2\right )+359\,x^2-40\,x^3-375\,x^4-125\,x^5+150}{150\,x-\ln \relax (x)\,\left (25\,x^6+80\,x^5+114\,x^4+80\,x^3+25\,x^2\right )+605\,x^2+884\,x^3+605\,x^4+150\,x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(355*x + log(x)*(25*x^2 + 80*x^3 + 114*x^4 + 80*x^5 + 25*x^6) + 359*x^2 - 40*x^3 - 375*x^4 - 125*x^5 + 15
0)/(150*x - log(x)*(25*x^2 + 80*x^3 + 114*x^4 + 80*x^5 + 25*x^6) + 605*x^2 + 884*x^3 + 605*x^4 + 150*x^5),x)

[Out]

int(-(355*x + log(x)*(25*x^2 + 80*x^3 + 114*x^4 + 80*x^5 + 25*x^6) + 359*x^2 - 40*x^3 - 375*x^4 - 125*x^5 + 15
0)/(150*x - log(x)*(25*x^2 + 80*x^3 + 114*x^4 + 80*x^5 + 25*x^6) + 605*x^2 + 884*x^3 + 605*x^4 + 150*x^5), x)

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sympy [A]  time = 0.49, size = 31, normalized size = 1.07 \begin {gather*} x + \log {\left (\frac {- 30 x^{2} - 73 x - 30}{5 x^{3} + 8 x^{2} + 5 x} + \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x**6+80*x**5+114*x**4+80*x**3+25*x**2)*ln(x)-125*x**5-375*x**4-40*x**3+359*x**2+355*x+150)/((25
*x**6+80*x**5+114*x**4+80*x**3+25*x**2)*ln(x)-150*x**5-605*x**4-884*x**3-605*x**2-150*x),x)

[Out]

x + log((-30*x**2 - 73*x - 30)/(5*x**3 + 8*x**2 + 5*x) + log(x))

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