3.73.10 \(\int \frac {5^{\frac {9}{20 x^2-2 x^3+(-40+4 x) \log (5)}} ((-180 x+27 x^2) \log (5)-18 \log ^2(5))}{200 x^4-40 x^5+2 x^6+(-800 x^2+160 x^3-8 x^4) \log (5)+(800-160 x+8 x^2) \log ^2(5)} \, dx\)

Optimal. Leaf size=24 \[ e^{\frac {9}{2 (-10+x) \left (2-\frac {x^2}{\log (5)}\right )}} \]

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Rubi [F]  time = 2.36, 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 {5^{\frac {9}{20 x^2-2 x^3+(-40+4 x) \log (5)}} \left (\left (-180 x+27 x^2\right ) \log (5)-18 \log ^2(5)\right )}{200 x^4-40 x^5+2 x^6+\left (-800 x^2+160 x^3-8 x^4\right ) \log (5)+\left (800-160 x+8 x^2\right ) \log ^2(5)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5^(9/(20*x^2 - 2*x^3 + (-40 + 4*x)*Log[5]))*((-180*x + 27*x^2)*Log[5] - 18*Log[5]^2))/(200*x^4 - 40*x^5 +
2*x^6 + (-800*x^2 + 160*x^3 - 8*x^4)*Log[5] + (800 - 160*x + 8*x^2)*Log[5]^2),x]

[Out]

(9*Log[5]*Defer[Int][1/(5^(9/(2*(-10 + x)*(x^2 - 2*Log[5])))*(-10 + x)^2), x])/(4*(50 - Log[5])) - (9*Log[5]*D
efer[Int][(5^(1 - 9/(2*(-10 + x)*(x^2 - 2*Log[5])))*x)/(x^2 - 2*Log[5])^2, x])/(50 - Log[5]) - (9*Log[5]^2*Def
er[Int][1/(5^(9/(2*(-10 + x)*(x^2 - 2*Log[5])))*(-x + Sqrt[Log[25]])^2), x])/(4*(50 - Log[5])*Log[25]) - (9*Lo
g[5]^2*Defer[Int][1/(5^(9/(2*(-10 + x)*(x^2 - 2*Log[5])))*(-x + Sqrt[Log[25]])), x])/(4*(50 - Log[5])*Log[25]^
(3/2)) + (9*Sqrt[Log[25]]*Defer[Int][1/(5^(9/(2*(-10 + x)*(x^2 - 2*Log[5])))*(-x + Sqrt[Log[25]])), x])/(16*(5
0 - Log[5])) - (9*Log[5]^2*Defer[Int][1/(5^(9/(2*(-10 + x)*(x^2 - 2*Log[5])))*(x + Sqrt[Log[25]])^2), x])/(4*(
50 - Log[5])*Log[25]) - (9*Log[5]^2*Defer[Int][1/(5^(9/(2*(-10 + x)*(x^2 - 2*Log[5])))*(x + Sqrt[Log[25]])), x
])/(4*(50 - Log[5])*Log[25]^(3/2)) + (9*Sqrt[Log[25]]*Defer[Int][1/(5^(9/(2*(-10 + x)*(x^2 - 2*Log[5])))*(x +
Sqrt[Log[25]])), x])/(16*(50 - Log[5]))

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9\ 5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} \left (-20 x+3 x^2-2 \log (5)\right ) \log (5)}{2 (10-x)^2 \left (x^2-2 \log (5)\right )^2} \, dx\\ &=\frac {1}{2} (9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} \left (-20 x+3 x^2-2 \log (5)\right )}{(10-x)^2 \left (x^2-2 \log (5)\right )^2} \, dx\\ &=\frac {1}{2} (9 \log (5)) \int \left (-\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{2 (-10+x)^2 (-50+\log (5))}+\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{2 \left (x^2-2 \log (5)\right ) (-50+\log (5))}+\frac {2\ 5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} (5 x+\log (5))}{\left (x^2-2 \log (5)\right )^2 (-50+\log (5))}\right ) \, dx\\ &=\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{(-10+x)^2} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{x^2-2 \log (5)} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} (5 x+\log (5))}{\left (x^2-2 \log (5)\right )^2} \, dx}{50-\log (5)}\\ &=\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{(-10+x)^2} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \left (-\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} \sqrt {\log (25)}}{4 \log (5) \left (-x+\sqrt {\log (25)}\right )}-\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} \sqrt {\log (25)}}{4 \log (5) \left (x+\sqrt {\log (25)}\right )}\right ) \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \left (\frac {5^{1-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} x}{\left (x^2-2 \log (5)\right )^2}+\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} \log (5)}{\left (x^2-2 \log (5)\right )^2}\right ) \, dx}{50-\log (5)}\\ &=\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{(-10+x)^2} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \frac {5^{1-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} x}{\left (x^2-2 \log (5)\right )^2} \, dx}{50-\log (5)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{\left (x^2-2 \log (5)\right )^2} \, dx}{50-\log (5)}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{-x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}\\ &=\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{(-10+x)^2} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \frac {5^{1-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} x}{\left (x^2-2 \log (5)\right )^2} \, dx}{50-\log (5)}-\frac {\left (9 \log ^2(5)\right ) \int \left (\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{4 \left (-x+\sqrt {\log (25)}\right )^2 \log (25)}+\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{4 \left (x+\sqrt {\log (25)}\right )^2 \log (25)}+\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{2 \log (25) \left (-x^2+\log (25)\right )}\right ) \, dx}{50-\log (5)}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{-x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}\\ &=\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{(-10+x)^2} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \frac {5^{1-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} x}{\left (x^2-2 \log (5)\right )^2} \, dx}{50-\log (5)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{\left (-x+\sqrt {\log (25)}\right )^2} \, dx}{4 (50-\log (5)) \log (25)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{\left (x+\sqrt {\log (25)}\right )^2} \, dx}{4 (50-\log (5)) \log (25)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{-x^2+\log (25)} \, dx}{2 (50-\log (5)) \log (25)}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{-x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}\\ &=\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{(-10+x)^2} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \frac {5^{1-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} x}{\left (x^2-2 \log (5)\right )^2} \, dx}{50-\log (5)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{\left (-x+\sqrt {\log (25)}\right )^2} \, dx}{4 (50-\log (5)) \log (25)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{\left (x+\sqrt {\log (25)}\right )^2} \, dx}{4 (50-\log (5)) \log (25)}-\frac {\left (9 \log ^2(5)\right ) \int \left (\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{2 \left (-x+\sqrt {\log (25)}\right ) \sqrt {\log (25)}}+\frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{2 \left (x+\sqrt {\log (25)}\right ) \sqrt {\log (25)}}\right ) \, dx}{2 (50-\log (5)) \log (25)}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{-x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}\\ &=\frac {(9 \log (5)) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{(-10+x)^2} \, dx}{4 (50-\log (5))}-\frac {(9 \log (5)) \int \frac {5^{1-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} x}{\left (x^2-2 \log (5)\right )^2} \, dx}{50-\log (5)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{-x+\sqrt {\log (25)}} \, dx}{4 (50-\log (5)) \log ^{\frac {3}{2}}(25)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{x+\sqrt {\log (25)}} \, dx}{4 (50-\log (5)) \log ^{\frac {3}{2}}(25)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{\left (-x+\sqrt {\log (25)}\right )^2} \, dx}{4 (50-\log (5)) \log (25)}-\frac {\left (9 \log ^2(5)\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{\left (x+\sqrt {\log (25)}\right )^2} \, dx}{4 (50-\log (5)) \log (25)}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{-x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}+\frac {\left (9 \sqrt {\log (25)}\right ) \int \frac {5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}}}{x+\sqrt {\log (25)}} \, dx}{16 (50-\log (5))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.40, size = 21, normalized size = 0.88 \begin {gather*} 5^{-\frac {9}{2 (-10+x) \left (x^2-2 \log (5)\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5^(9/(20*x^2 - 2*x^3 + (-40 + 4*x)*Log[5]))*((-180*x + 27*x^2)*Log[5] - 18*Log[5]^2))/(200*x^4 - 40
*x^5 + 2*x^6 + (-800*x^2 + 160*x^3 - 8*x^4)*Log[5] + (800 - 160*x + 8*x^2)*Log[5]^2),x]

[Out]

5^(-9/(2*(-10 + x)*(x^2 - 2*Log[5])))

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fricas [A]  time = 0.49, size = 24, normalized size = 1.00 \begin {gather*} \frac {1}{5^{\frac {9}{2 \, {\left (x^{3} - 10 \, x^{2} - 2 \, {\left (x - 10\right )} \log \relax (5)\right )}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*log(5)^2+(27*x^2-180*x)*log(5))*exp(9*log(5)/((4*x-40)*log(5)-2*x^3+20*x^2))/((8*x^2-160*x+800)
*log(5)^2+(-8*x^4+160*x^3-800*x^2)*log(5)+2*x^6-40*x^5+200*x^4),x, algorithm="fricas")

[Out]

1/(5^(9/2/(x^3 - 10*x^2 - 2*(x - 10)*log(5))))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {9 \, {\left ({\left (3 \, x^{2} - 20 \, x\right )} \log \relax (5) - 2 \, \log \relax (5)^{2}\right )}}{2 \, {\left (x^{6} - 20 \, x^{5} + 100 \, x^{4} + 4 \, {\left (x^{2} - 20 \, x + 100\right )} \log \relax (5)^{2} - 4 \, {\left (x^{4} - 20 \, x^{3} + 100 \, x^{2}\right )} \log \relax (5)\right )} 5^{\frac {9}{2 \, {\left (x^{3} - 10 \, x^{2} - 2 \, {\left (x - 10\right )} \log \relax (5)\right )}}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*log(5)^2+(27*x^2-180*x)*log(5))*exp(9*log(5)/((4*x-40)*log(5)-2*x^3+20*x^2))/((8*x^2-160*x+800)
*log(5)^2+(-8*x^4+160*x^3-800*x^2)*log(5)+2*x^6-40*x^5+200*x^4),x, algorithm="giac")

[Out]

integrate(9/2*((3*x^2 - 20*x)*log(5) - 2*log(5)^2)/((x^6 - 20*x^5 + 100*x^4 + 4*(x^2 - 20*x + 100)*log(5)^2 -
4*(x^4 - 20*x^3 + 100*x^2)*log(5))*5^(9/2/(x^3 - 10*x^2 - 2*(x - 10)*log(5)))), x)

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maple [A]  time = 0.18, size = 22, normalized size = 0.92

method	result	size
risch	\(5^{\frac {9}{2 \left (x -10\right ) \left (-x^{2}+2 \ln \relax (5)\right )}}\)	\(22\)
gosper	\({\mathrm e}^{\frac {9 \ln \relax (5)}{2 \left (-x^{3}+2 x \ln \relax (5)+10 x^{2}-20 \ln \relax (5)\right )}}\)	\(28\)
norman	\(\frac {10 x^{2} {\mathrm e}^{\frac {9 \ln \relax (5)}{\left (4 x -40\right ) \ln \relax (5)-2 x^{3}+20 x^{2}}}-x^{3} {\mathrm e}^{\frac {9 \ln \relax (5)}{\left (4 x -40\right ) \ln \relax (5)-2 x^{3}+20 x^{2}}}-20 \ln \relax (5) {\mathrm e}^{\frac {9 \ln \relax (5)}{\left (4 x -40\right ) \ln \relax (5)-2 x^{3}+20 x^{2}}}+2 x \ln \relax (5) {\mathrm e}^{\frac {9 \ln \relax (5)}{\left (4 x -40\right ) \ln \relax (5)-2 x^{3}+20 x^{2}}}}{\left (x -10\right ) \left (-x^{2}+2 \ln \relax (5)\right )}\)	\(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-18*ln(5)^2+(27*x^2-180*x)*ln(5))*exp(9*ln(5)/((4*x-40)*ln(5)-2*x^3+20*x^2))/((8*x^2-160*x+800)*ln(5)^2+(
-8*x^4+160*x^3-800*x^2)*ln(5)+2*x^6-40*x^5+200*x^4),x,method=_RETURNVERBOSE)

[Out]

5^(9/2/(x-10)/(-x^2+2*ln(5)))

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maxima [B]  time = 0.56, size = 71, normalized size = 2.96 \begin {gather*} e^{\left (-\frac {9 \, x \log \relax (5)}{4 \, {\left (x^{2} {\left (\log \relax (5) - 50\right )} - 2 \, \log \relax (5)^{2} + 100 \, \log \relax (5)\right )}} - \frac {45 \, \log \relax (5)}{2 \, {\left (x^{2} {\left (\log \relax (5) - 50\right )} - 2 \, \log \relax (5)^{2} + 100 \, \log \relax (5)\right )}} + \frac {9 \, \log \relax (5)}{4 \, {\left (x {\left (\log \relax (5) - 50\right )} - 10 \, \log \relax (5) + 500\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*log(5)^2+(27*x^2-180*x)*log(5))*exp(9*log(5)/((4*x-40)*log(5)-2*x^3+20*x^2))/((8*x^2-160*x+800)
*log(5)^2+(-8*x^4+160*x^3-800*x^2)*log(5)+2*x^6-40*x^5+200*x^4),x, algorithm="maxima")

[Out]

e^(-9/4*x*log(5)/(x^2*(log(5) - 50) - 2*log(5)^2 + 100*log(5)) - 45/2*log(5)/(x^2*(log(5) - 50) - 2*log(5)^2 +
100*log(5)) + 9/4*log(5)/(x*(log(5) - 50) - 10*log(5) + 500))

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mupad [B]  time = 4.94, size = 28, normalized size = 1.17 \begin {gather*} \frac {1}{5^{\frac {9}{2\,x^3-20\,x^2-4\,\ln \relax (5)\,x+40\,\ln \relax (5)}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((9*log(5))/(log(5)*(4*x - 40) + 20*x^2 - 2*x^3))*(log(5)*(180*x - 27*x^2) + 18*log(5)^2))/(log(5)^2*
(8*x^2 - 160*x + 800) - log(5)*(800*x^2 - 160*x^3 + 8*x^4) + 200*x^4 - 40*x^5 + 2*x^6),x)

[Out]

1/5^(9/(40*log(5) - 4*x*log(5) - 20*x^2 + 2*x^3))

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sympy [A]  time = 1.15, size = 24, normalized size = 1.00 \begin {gather*} e^{\frac {9 \log {\relax (5 )}}{- 2 x^{3} + 20 x^{2} + \left (4 x - 40\right ) \log {\relax (5 )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*ln(5)**2+(27*x**2-180*x)*ln(5))*exp(9*ln(5)/((4*x-40)*ln(5)-2*x**3+20*x**2))/((8*x**2-160*x+800
)*ln(5)**2+(-8*x**4+160*x**3-800*x**2)*ln(5)+2*x**6-40*x**5+200*x**4),x)

[Out]

exp(9*log(5)/(-2*x**3 + 20*x**2 + (4*x - 40)*log(5)))

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