3.81.54 \(\int \frac {-250 x^2-25 x^3-125 x^5+(-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6) \log (\frac {37+4 x}{8+x})}{(29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8) \log ^2(\frac {37+4 x}{8+x})} \, dx\)

Optimal. Leaf size=30 \[ \frac {x}{\left (-\frac {1}{5}-\frac {2}{x}-x^2\right ) \log \left (4+\frac {5}{8+x}\right )} \]

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Rubi [F]  time = 1.37, 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 {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-250*x^2 - 25*x^3 - 125*x^5 + (-29600*x - 8380*x^2 - 745*x^3 + 7380*x^4 + 1725*x^5 + 100*x^6)*Log[(37 + 4
*x)/(8 + x)])/((29600 + 12820*x + 2076*x^2 + 29749*x^3 + 9864*x^4 + 1090*x^5 + 7440*x^6 + 1725*x^7 + 100*x^8)*
Log[(37 + 4*x)/(8 + x)]^2),x]

[Out]

-25*Defer[Int][x^2/((8 + x)*(37 + 4*x)*(10 + x + 5*x^3)*Log[(37 + 4*x)/(8 + x)]^2), x] + 5*Defer[Int][(x*(-20
- x + 5*x^3))/((10 + x + 5*x^3)^2*Log[(37 + 4*x)/(8 + x)]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 x \left (-5 x \left (10+x+5 x^3\right )+\left (-5920-1676 x-149 x^2+1476 x^3+345 x^4+20 x^5\right ) \log \left (\frac {37+4 x}{8+x}\right )\right )}{\left (296+69 x+4 x^2\right ) \left (10+x+5 x^3\right )^2 \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx\\ &=5 \int \frac {x \left (-5 x \left (10+x+5 x^3\right )+\left (-5920-1676 x-149 x^2+1476 x^3+345 x^4+20 x^5\right ) \log \left (\frac {37+4 x}{8+x}\right )\right )}{\left (296+69 x+4 x^2\right ) \left (10+x+5 x^3\right )^2 \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx\\ &=5 \int \left (-\frac {5 x^2}{(8+x) (37+4 x) \left (10+x+5 x^3\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )}+\frac {x \left (-20-x+5 x^3\right )}{\left (10+x+5 x^3\right )^2 \log \left (\frac {37+4 x}{8+x}\right )}\right ) \, dx\\ &=5 \int \frac {x \left (-20-x+5 x^3\right )}{\left (10+x+5 x^3\right )^2 \log \left (\frac {37+4 x}{8+x}\right )} \, dx-25 \int \frac {x^2}{(8+x) (37+4 x) \left (10+x+5 x^3\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 29, normalized size = 0.97 \begin {gather*} -\frac {5 x^2}{\left (10+x+5 x^3\right ) \log \left (\frac {37+4 x}{8+x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-250*x^2 - 25*x^3 - 125*x^5 + (-29600*x - 8380*x^2 - 745*x^3 + 7380*x^4 + 1725*x^5 + 100*x^6)*Log[(
37 + 4*x)/(8 + x)])/((29600 + 12820*x + 2076*x^2 + 29749*x^3 + 9864*x^4 + 1090*x^5 + 7440*x^6 + 1725*x^7 + 100
*x^8)*Log[(37 + 4*x)/(8 + x)]^2),x]

[Out]

(-5*x^2)/((10 + x + 5*x^3)*Log[(37 + 4*x)/(8 + x)])

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fricas [A]  time = 0.63, size = 29, normalized size = 0.97 \begin {gather*} -\frac {5 \, x^{2}}{{\left (5 \, x^{3} + x + 10\right )} \log \left (\frac {4 \, x + 37}{x + 8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x^6+1725*x^5+7380*x^4-745*x^3-8380*x^2-29600*x)*log((4*x+37)/(x+8))-125*x^5-25*x^3-250*x^2)/(1
00*x^8+1725*x^7+7440*x^6+1090*x^5+9864*x^4+29749*x^3+2076*x^2+12820*x+29600)/log((4*x+37)/(x+8))^2,x, algorith
m="fricas")

[Out]

-5*x^2/((5*x^3 + x + 10)*log((4*x + 37)/(x + 8)))

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giac [B]  time = 0.30, size = 137, normalized size = 4.57 \begin {gather*} \frac {5 \, {\left (\frac {64 \, {\left (4 \, x + 37\right )}^{3}}{{\left (x + 8\right )}^{3}} - \frac {848 \, {\left (4 \, x + 37\right )}^{2}}{{\left (x + 8\right )}^{2}} + \frac {3737 \, {\left (4 \, x + 37\right )}}{x + 8} - 5476\right )}}{\frac {2558 \, {\left (4 \, x + 37\right )}^{3} \log \left (\frac {4 \, x + 37}{x + 8}\right )}{{\left (x + 8\right )}^{3}} - \frac {35501 \, {\left (4 \, x + 37\right )}^{2} \log \left (\frac {4 \, x + 37}{x + 8}\right )}{{\left (x + 8\right )}^{2}} + \frac {164224 \, {\left (4 \, x + 37\right )} \log \left (\frac {4 \, x + 37}{x + 8}\right )}{x + 8} - 253217 \, \log \left (\frac {4 \, x + 37}{x + 8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x^6+1725*x^5+7380*x^4-745*x^3-8380*x^2-29600*x)*log((4*x+37)/(x+8))-125*x^5-25*x^3-250*x^2)/(1
00*x^8+1725*x^7+7440*x^6+1090*x^5+9864*x^4+29749*x^3+2076*x^2+12820*x+29600)/log((4*x+37)/(x+8))^2,x, algorith
m="giac")

[Out]

5*(64*(4*x + 37)^3/(x + 8)^3 - 848*(4*x + 37)^2/(x + 8)^2 + 3737*(4*x + 37)/(x + 8) - 5476)/(2558*(4*x + 37)^3
*log((4*x + 37)/(x + 8))/(x + 8)^3 - 35501*(4*x + 37)^2*log((4*x + 37)/(x + 8))/(x + 8)^2 + 164224*(4*x + 37)*
log((4*x + 37)/(x + 8))/(x + 8) - 253217*log((4*x + 37)/(x + 8)))

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maple [A]  time = 0.41, size = 30, normalized size = 1.00

method	result	size
risch	\(-\frac {5 x^{2}}{\left (5 x^{3}+x +10\right ) \ln \left (\frac {4 x +37}{x +8}\right )}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((100*x^6+1725*x^5+7380*x^4-745*x^3-8380*x^2-29600*x)*ln((4*x+37)/(x+8))-125*x^5-25*x^3-250*x^2)/(100*x^8+
1725*x^7+7440*x^6+1090*x^5+9864*x^4+29749*x^3+2076*x^2+12820*x+29600)/ln((4*x+37)/(x+8))^2,x,method=_RETURNVER
BOSE)

[Out]

-5*x^2/(5*x^3+x+10)/ln((4*x+37)/(x+8))

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maxima [A]  time = 0.41, size = 37, normalized size = 1.23 \begin {gather*} -\frac {5 \, x^{2}}{{\left (5 \, x^{3} + x + 10\right )} \log \left (4 \, x + 37\right ) - {\left (5 \, x^{3} + x + 10\right )} \log \left (x + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x^6+1725*x^5+7380*x^4-745*x^3-8380*x^2-29600*x)*log((4*x+37)/(x+8))-125*x^5-25*x^3-250*x^2)/(1
00*x^8+1725*x^7+7440*x^6+1090*x^5+9864*x^4+29749*x^3+2076*x^2+12820*x+29600)/log((4*x+37)/(x+8))^2,x, algorith
m="maxima")

[Out]

-5*x^2/((5*x^3 + x + 10)*log(4*x + 37) - (5*x^3 + x + 10)*log(x + 8))

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mupad [B]  time = 8.30, size = 51, normalized size = 1.70 \begin {gather*} \frac {4\,x^3+\frac {4\,x}{5}+8}{5\,x^3+x+10}-\frac {5\,x^2}{\ln \left (\frac {4\,x+37}{x+8}\right )\,\left (5\,x^3+x+10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(250*x^2 + 25*x^3 + 125*x^5 + log((4*x + 37)/(x + 8))*(29600*x + 8380*x^2 + 745*x^3 - 7380*x^4 - 1725*x^5
- 100*x^6))/(log((4*x + 37)/(x + 8))^2*(12820*x + 2076*x^2 + 29749*x^3 + 9864*x^4 + 1090*x^5 + 7440*x^6 + 172
5*x^7 + 100*x^8 + 29600)),x)

[Out]

((4*x)/5 + 4*x^3 + 8)/(x + 5*x^3 + 10) - (5*x^2)/(log((4*x + 37)/(x + 8))*(x + 5*x^3 + 10))

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sympy [A]  time = 0.21, size = 24, normalized size = 0.80 \begin {gather*} - \frac {5 x^{2}}{\left (5 x^{3} + x + 10\right ) \log {\left (\frac {4 x + 37}{x + 8} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x**6+1725*x**5+7380*x**4-745*x**3-8380*x**2-29600*x)*ln((4*x+37)/(x+8))-125*x**5-25*x**3-250*x
**2)/(100*x**8+1725*x**7+7440*x**6+1090*x**5+9864*x**4+29749*x**3+2076*x**2+12820*x+29600)/ln((4*x+37)/(x+8))*
*2,x)

[Out]

-5*x**2/((5*x**3 + x + 10)*log((4*x + 37)/(x + 8)))

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