3.12.66 \(\int \frac {300 x^3+(1000 x^5+375 x^6-125 x^8) \log (2)}{9+(120 x^2+90 x^3-150 x^4+30 x^5) \log (2)+(400 x^4+600 x^5-775 x^6-550 x^7+775 x^8-250 x^9+25 x^{10}) \log ^2(2)} \, dx\)

Optimal. Leaf size=30 \[ \frac {5 x^2}{\frac {3}{5 x^2}-(-4+x) \left (1+x-x^2\right ) \log (2)} \]

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Rubi [F]  time = 1.38, 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 {300 x^3+\left (1000 x^5+375 x^6-125 x^8\right ) \log (2)}{9+\left (120 x^2+90 x^3-150 x^4+30 x^5\right ) \log (2)+\left (400 x^4+600 x^5-775 x^6-550 x^7+775 x^8-250 x^9+25 x^{10}\right ) \log ^2(2)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(300*x^3 + (1000*x^5 + 375*x^6 - 125*x^8)*Log[2])/(9 + (120*x^2 + 90*x^3 - 150*x^4 + 30*x^5)*Log[2] + (400
*x^4 + 600*x^5 - 775*x^6 - 550*x^7 + 775*x^8 - 250*x^9 + 25*x^10)*Log[2]^2),x]

[Out]

(6575*Log[2])/(Log[32]*(3 + 20*x^2*Log[2] + 15*x^3*Log[2] - 25*x^4*Log[2] + x^5*Log[32])) + 5100*Defer[Int][(3
+ 20*x^2*Log[2] + 15*x^3*Log[2] - 25*x^4*Log[2] + x^5*Log[32])^(-2), x] + 25*(57 + (10520*Log[2]^2)/Log[32])*
Defer[Int][x/(3 + 20*x^2*Log[2] + 15*x^3*Log[2] - 25*x^4*Log[2] + x^5*Log[32])^2, x] + 125*(3 + 272*Log[2] + (
2367*Log[2]^2)/Log[32])*Defer[Int][x^2/(3 + 20*x^2*Log[2] + 15*x^3*Log[2] - 25*x^4*Log[2] + x^5*Log[32])^2, x]
+ 125*(3 + 280*Log[2] - (5260*Log[2]^2)/Log[32])*Defer[Int][x^3/(3 + 20*x^2*Log[2] + 15*x^3*Log[2] - 25*x^4*L
og[2] + x^5*Log[32])^2, x] - 1700*Defer[Int][(3 + 20*x^2*Log[2] + 15*x^3*Log[2] - 25*x^4*Log[2] + x^5*Log[32])
^(-1), x] - 475*Defer[Int][x/(3 + 20*x^2*Log[2] + 15*x^3*Log[2] - 25*x^4*Log[2] + x^5*Log[32]), x] - 125*Defer
[Int][x^2/(3 + 20*x^2*Log[2] + 15*x^3*Log[2] - 25*x^4*Log[2] + x^5*Log[32]), x] - 25*Defer[Int][x^3/(3 + 20*x^
2*Log[2] + 15*x^3*Log[2] - 25*x^4*Log[2] + x^5*Log[32]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {25 \left (204+57 x-1315 x^4 \log (2)+5 x^2 (3+272 \log (2))+5 x^3 (3+280 \log (2))\right )}{\left (3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)\right )^2}-\frac {25 \left (68+19 x+5 x^2+x^3\right )}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)}\right ) \, dx\\ &=25 \int \frac {204+57 x-1315 x^4 \log (2)+5 x^2 (3+272 \log (2))+5 x^3 (3+280 \log (2))}{\left (3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)\right )^2} \, dx-25 \int \frac {68+19 x+5 x^2+x^3}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)} \, dx\\ &=\frac {6575 \log (2)}{\log (32) \left (3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)\right )}-25 \int \left (\frac {68}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)}+\frac {19 x}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)}+\frac {5 x^2}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)}+\frac {x^3}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)}\right ) \, dx+\frac {5 \int \frac {1020 \log (32)+5 x \left (10520 \log ^2(2)+57 \log (32)\right )-25 x^3 \left (5260 \log ^2(2)-3 \log (32)-280 \log (2) \log (32)\right )+25 x^2 \left (2367 \log ^2(2)+3 \log (32)+272 \log (2) \log (32)\right )}{\left (3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)\right )^2} \, dx}{\log (32)}\\ &=\frac {6575 \log (2)}{\log (32) \left (3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)\right )}-25 \int \frac {x^3}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)} \, dx-125 \int \frac {x^2}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)} \, dx-475 \int \frac {x}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)} \, dx-1700 \int \frac {1}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)} \, dx+\frac {5 \int \left (\frac {1020 \log (32)}{\left (3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)\right )^2}+\frac {5 x \left (10520 \log ^2(2)+57 \log (32)\right )}{\left (3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)\right )^2}-\frac {25 x^3 \left (5260 \log ^2(2)-3 \log (32)-280 \log (2) \log (32)\right )}{\left (3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)\right )^2}+\frac {25 x^2 \left (2367 \log ^2(2)+3 \log (32)+272 \log (2) \log (32)\right )}{\left (3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)\right )^2}\right ) \, dx}{\log (32)}\\ &=\frac {6575 \log (2)}{\log (32) \left (3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)\right )}-25 \int \frac {x^3}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)} \, dx-125 \int \frac {x^2}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)} \, dx-475 \int \frac {x}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)} \, dx-1700 \int \frac {1}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)} \, dx+5100 \int \frac {1}{\left (3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)\right )^2} \, dx+\left (125 \left (3+280 \log (2)-\frac {5260 \log ^2(2)}{\log (32)}\right )\right ) \int \frac {x^3}{\left (3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)\right )^2} \, dx+\left (125 \left (3+272 \log (2)+\frac {2367 \log ^2(2)}{\log (32)}\right )\right ) \int \frac {x^2}{\left (3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)\right )^2} \, dx+\left (25 \left (57+\frac {10520 \log ^2(2)}{\log (32)}\right )\right ) \int \frac {x}{\left (3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 36, normalized size = 1.20 \begin {gather*} \frac {25 x^4}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(300*x^3 + (1000*x^5 + 375*x^6 - 125*x^8)*Log[2])/(9 + (120*x^2 + 90*x^3 - 150*x^4 + 30*x^5)*Log[2]
+ (400*x^4 + 600*x^5 - 775*x^6 - 550*x^7 + 775*x^8 - 250*x^9 + 25*x^10)*Log[2]^2),x]

[Out]

(25*x^4)/(3 + 20*x^2*Log[2] + 15*x^3*Log[2] - 25*x^4*Log[2] + x^5*Log[32])

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fricas [A]  time = 0.78, size = 32, normalized size = 1.07 \begin {gather*} \frac {25 \, x^{4}}{5 \, {\left (x^{5} - 5 \, x^{4} + 3 \, x^{3} + 4 \, x^{2}\right )} \log \relax (2) + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-125*x^8+375*x^6+1000*x^5)*log(2)+300*x^3)/((25*x^10-250*x^9+775*x^8-550*x^7-775*x^6+600*x^5+400*x
^4)*log(2)^2+(30*x^5-150*x^4+90*x^3+120*x^2)*log(2)+9),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.66, size = 37, normalized size = 1.23 \begin {gather*} \frac {25 \, x^{4}}{5 \, x^{5} \log \relax (2) - 25 \, x^{4} \log \relax (2) + 15 \, x^{3} \log \relax (2) + 20 \, x^{2} \log \relax (2) + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-125*x^8+375*x^6+1000*x^5)*log(2)+300*x^3)/((25*x^10-250*x^9+775*x^8-550*x^7-775*x^6+600*x^5+400*x
^4)*log(2)^2+(30*x^5-150*x^4+90*x^3+120*x^2)*log(2)+9),x, algorithm="giac")

[Out]

25*x^4/(5*x^5*log(2) - 25*x^4*log(2) + 15*x^3*log(2) + 20*x^2*log(2) + 3)

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maple [A]  time = 0.16, size = 37, normalized size = 1.23

method	result	size
default	\(\frac {5 x^{4}}{x^{5} \ln \relax (2)-5 x^{4} \ln \relax (2)+3 x^{3} \ln \relax (2)+4 x^{2} \ln \relax (2)+\frac {3}{5}}\)	\(37\)
risch	\(\frac {5 x^{4}}{x^{5} \ln \relax (2)-5 x^{4} \ln \relax (2)+3 x^{3} \ln \relax (2)+4 x^{2} \ln \relax (2)+\frac {3}{5}}\)	\(37\)
gosper	\(\frac {25 x^{4}}{5 x^{5} \ln \relax (2)-25 x^{4} \ln \relax (2)+15 x^{3} \ln \relax (2)+20 x^{2} \ln \relax (2)+3}\)	\(38\)
norman	\(\frac {25 x^{4}}{5 x^{5} \ln \relax (2)-25 x^{4} \ln \relax (2)+15 x^{3} \ln \relax (2)+20 x^{2} \ln \relax (2)+3}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-125*x^8+375*x^6+1000*x^5)*ln(2)+300*x^3)/((25*x^10-250*x^9+775*x^8-550*x^7-775*x^6+600*x^5+400*x^4)*ln(
2)^2+(30*x^5-150*x^4+90*x^3+120*x^2)*ln(2)+9),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.36, size = 37, normalized size = 1.23 \begin {gather*} \frac {25 \, x^{4}}{5 \, x^{5} \log \relax (2) - 25 \, x^{4} \log \relax (2) + 15 \, x^{3} \log \relax (2) + 20 \, x^{2} \log \relax (2) + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-125*x^8+375*x^6+1000*x^5)*log(2)+300*x^3)/((25*x^10-250*x^9+775*x^8-550*x^7-775*x^6+600*x^5+400*x
^4)*log(2)^2+(30*x^5-150*x^4+90*x^3+120*x^2)*log(2)+9),x, algorithm="maxima")

[Out]

25*x^4/(5*x^5*log(2) - 25*x^4*log(2) + 15*x^3*log(2) + 20*x^2*log(2) + 3)

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mupad [B]  time = 0.32, size = 37, normalized size = 1.23 \begin {gather*} \frac {25\,x^4}{5\,\ln \relax (2)\,x^5-25\,\ln \relax (2)\,x^4+15\,\ln \relax (2)\,x^3+20\,\ln \relax (2)\,x^2+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2)*(1000*x^5 + 375*x^6 - 125*x^8) + 300*x^3)/(log(2)*(120*x^2 + 90*x^3 - 150*x^4 + 30*x^5) + log(2)^2
*(400*x^4 + 600*x^5 - 775*x^6 - 550*x^7 + 775*x^8 - 250*x^9 + 25*x^10) + 9),x)

[Out]

(25*x^4)/(20*x^2*log(2) + 15*x^3*log(2) - 25*x^4*log(2) + 5*x^5*log(2) + 3)

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sympy [A]  time = 3.86, size = 39, normalized size = 1.30 \begin {gather*} \frac {25 x^{4}}{5 x^{5} \log {\relax (2 )} - 25 x^{4} \log {\relax (2 )} + 15 x^{3} \log {\relax (2 )} + 20 x^{2} \log {\relax (2 )} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-125*x**8+375*x**6+1000*x**5)*ln(2)+300*x**3)/((25*x**10-250*x**9+775*x**8-550*x**7-775*x**6+600*x
**5+400*x**4)*ln(2)**2+(30*x**5-150*x**4+90*x**3+120*x**2)*ln(2)+9),x)

[Out]

25*x**4/(5*x**5*log(2) - 25*x**4*log(2) + 15*x**3*log(2) + 20*x**2*log(2) + 3)

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