3.42.54 \(\int \frac {(200 x-80 x^2+8 x^3) \log (\frac {25}{x^2})+(400 x-160 x^2+16 x^3+(200 x-120 x^2+16 x^3) \log (\frac {25}{x^2})) \log (\frac {12}{x})}{\log ^3(\frac {25}{x^2}) \log ^3(\frac {12}{x})} \, dx\)

Optimal. Leaf size=28 \[ \frac {4 (5-x)^2 x^2}{\log ^2\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )} \]

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Rubi [F]  time = 1.86, 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 {\left (200 x-80 x^2+8 x^3\right ) \log \left (\frac {25}{x^2}\right )+\left (400 x-160 x^2+16 x^3+\left (200 x-120 x^2+16 x^3\right ) \log \left (\frac {25}{x^2}\right )\right ) \log \left (\frac {12}{x}\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((200*x - 80*x^2 + 8*x^3)*Log[25/x^2] + (400*x - 160*x^2 + 16*x^3 + (200*x - 120*x^2 + 16*x^3)*Log[25/x^2]
)*Log[12/x])/(Log[25/x^2]^3*Log[12/x]^3),x]

[Out]

200*Defer[Int][x/(Log[25/x^2]^2*Log[12/x]^3), x] - 80*Defer[Int][x^2/(Log[25/x^2]^2*Log[12/x]^3), x] + 8*Defer
[Int][x^3/(Log[25/x^2]^2*Log[12/x]^3), x] + 400*Defer[Int][x/(Log[25/x^2]^3*Log[12/x]^2), x] - 160*Defer[Int][
x^2/(Log[25/x^2]^3*Log[12/x]^2), x] + 16*Defer[Int][x^3/(Log[25/x^2]^3*Log[12/x]^2), x] + 200*Defer[Int][x/(Lo
g[25/x^2]^2*Log[12/x]^2), x] - 120*Defer[Int][x^2/(Log[25/x^2]^2*Log[12/x]^2), x] + 16*Defer[Int][x^3/(Log[25/
x^2]^2*Log[12/x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 (5-x) x \left (-2 (-5+x) \log \left (\frac {12}{x}\right )-\log \left (\frac {25}{x^2}\right ) \left (-5+x+(-5+2 x) \log \left (\frac {12}{x}\right )\right )\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx\\ &=8 \int \frac {(5-x) x \left (-2 (-5+x) \log \left (\frac {12}{x}\right )-\log \left (\frac {25}{x^2}\right ) \left (-5+x+(-5+2 x) \log \left (\frac {12}{x}\right )\right )\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx\\ &=8 \int \left (\frac {(-5+x)^2 x}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )}+\frac {(-5+x) x \left (-10+2 x-5 \log \left (\frac {25}{x^2}\right )+2 x \log \left (\frac {25}{x^2}\right )\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}\right ) \, dx\\ &=8 \int \frac {(-5+x)^2 x}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx+8 \int \frac {(-5+x) x \left (-10+2 x-5 \log \left (\frac {25}{x^2}\right )+2 x \log \left (\frac {25}{x^2}\right )\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )} \, dx\\ &=8 \int \left (\frac {25 x}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )}-\frac {10 x^2}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )}+\frac {x^3}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )}\right ) \, dx+8 \int \left (-\frac {5 x \left (-10+2 x-5 \log \left (\frac {25}{x^2}\right )+2 x \log \left (\frac {25}{x^2}\right )\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}+\frac {x^2 \left (-10+2 x-5 \log \left (\frac {25}{x^2}\right )+2 x \log \left (\frac {25}{x^2}\right )\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}\right ) \, dx\\ &=8 \int \frac {x^3}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx+8 \int \frac {x^2 \left (-10+2 x-5 \log \left (\frac {25}{x^2}\right )+2 x \log \left (\frac {25}{x^2}\right )\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )} \, dx-40 \int \frac {x \left (-10+2 x-5 \log \left (\frac {25}{x^2}\right )+2 x \log \left (\frac {25}{x^2}\right )\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )} \, dx-80 \int \frac {x^2}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx+200 \int \frac {x}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx\\ &=8 \int \left (-\frac {10 x^2}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}+\frac {2 x^3}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}-\frac {5 x^2}{\log ^2\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}+\frac {2 x^3}{\log ^2\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}\right ) \, dx+8 \int \frac {x^3}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx-40 \int \left (-\frac {10 x}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}+\frac {2 x^2}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}-\frac {5 x}{\log ^2\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}+\frac {2 x^2}{\log ^2\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}\right ) \, dx-80 \int \frac {x^2}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx+200 \int \frac {x}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx\\ &=8 \int \frac {x^3}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx+16 \int \frac {x^3}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )} \, dx+16 \int \frac {x^3}{\log ^2\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )} \, dx-40 \int \frac {x^2}{\log ^2\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )} \, dx-80 \int \frac {x^2}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx-2 \left (80 \int \frac {x^2}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )} \, dx\right )-80 \int \frac {x^2}{\log ^2\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )} \, dx+200 \int \frac {x}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx+200 \int \frac {x}{\log ^2\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )} \, dx+400 \int \frac {x}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 26, normalized size = 0.93 \begin {gather*} \frac {4 (-5+x)^2 x^2}{\log ^2\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((200*x - 80*x^2 + 8*x^3)*Log[25/x^2] + (400*x - 160*x^2 + 16*x^3 + (200*x - 120*x^2 + 16*x^3)*Log[2
5/x^2])*Log[12/x])/(Log[25/x^2]^3*Log[12/x]^3),x]

[Out]

(4*(-5 + x)^2*x^2)/(Log[25/x^2]^2*Log[12/x]^2)

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fricas [A]  time = 0.80, size = 52, normalized size = 1.86 \begin {gather*} \frac {16 \, {\left (x^{4} - 10 \, x^{3} + 25 \, x^{2}\right )}}{\log \left (\frac {144}{25}\right )^{2} \log \left (\frac {25}{x^{2}}\right )^{2} + 2 \, \log \left (\frac {144}{25}\right ) \log \left (\frac {25}{x^{2}}\right )^{3} + \log \left (\frac {25}{x^{2}}\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x^3-120*x^2+200*x)*log(25/x^2)+16*x^3-160*x^2+400*x)*log(12/x)+(8*x^3-80*x^2+200*x)*log(25/x^2
))/log(25/x^2)^3/log(12/x)^3,x, algorithm="fricas")

[Out]

16*(x^4 - 10*x^3 + 25*x^2)/(log(144/25)^2*log(25/x^2)^2 + 2*log(144/25)*log(25/x^2)^3 + log(25/x^2)^4)

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giac [B]  time = 0.28, size = 284, normalized size = 10.14 \begin {gather*} \frac {x^{4} \log \left (12\right )^{2} - 2 \, x^{4} \log \left (12\right ) \log \relax (5) + x^{4} \log \relax (5)^{2} - 10 \, x^{3} \log \left (12\right )^{2} + 20 \, x^{3} \log \left (12\right ) \log \relax (5) - 10 \, x^{3} \log \relax (5)^{2} + 25 \, x^{2} \log \left (12\right )^{2} - 50 \, x^{2} \log \left (12\right ) \log \relax (5) + 25 \, x^{2} \log \relax (5)^{2}}{\log \left (12\right )^{4} \log \relax (5)^{2} - 2 \, \log \left (12\right )^{3} \log \relax (5)^{3} + \log \left (12\right )^{2} \log \relax (5)^{4} - 2 \, \log \left (12\right )^{4} \log \relax (5) \log \relax (x) + 2 \, \log \left (12\right )^{3} \log \relax (5)^{2} \log \relax (x) + 2 \, \log \left (12\right )^{2} \log \relax (5)^{3} \log \relax (x) - 2 \, \log \left (12\right ) \log \relax (5)^{4} \log \relax (x) + \log \left (12\right )^{4} \log \relax (x)^{2} + 2 \, \log \left (12\right )^{3} \log \relax (5) \log \relax (x)^{2} - 6 \, \log \left (12\right )^{2} \log \relax (5)^{2} \log \relax (x)^{2} + 2 \, \log \left (12\right ) \log \relax (5)^{3} \log \relax (x)^{2} + \log \relax (5)^{4} \log \relax (x)^{2} - 2 \, \log \left (12\right )^{3} \log \relax (x)^{3} + 2 \, \log \left (12\right )^{2} \log \relax (5) \log \relax (x)^{3} + 2 \, \log \left (12\right ) \log \relax (5)^{2} \log \relax (x)^{3} - 2 \, \log \relax (5)^{3} \log \relax (x)^{3} + \log \left (12\right )^{2} \log \relax (x)^{4} - 2 \, \log \left (12\right ) \log \relax (5) \log \relax (x)^{4} + \log \relax (5)^{2} \log \relax (x)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x^3-120*x^2+200*x)*log(25/x^2)+16*x^3-160*x^2+400*x)*log(12/x)+(8*x^3-80*x^2+200*x)*log(25/x^2
))/log(25/x^2)^3/log(12/x)^3,x, algorithm="giac")

[Out]

(x^4*log(12)^2 - 2*x^4*log(12)*log(5) + x^4*log(5)^2 - 10*x^3*log(12)^2 + 20*x^3*log(12)*log(5) - 10*x^3*log(5
)^2 + 25*x^2*log(12)^2 - 50*x^2*log(12)*log(5) + 25*x^2*log(5)^2)/(log(12)^4*log(5)^2 - 2*log(12)^3*log(5)^3 +
log(12)^2*log(5)^4 - 2*log(12)^4*log(5)*log(x) + 2*log(12)^3*log(5)^2*log(x) + 2*log(12)^2*log(5)^3*log(x) -
2*log(12)*log(5)^4*log(x) + log(12)^4*log(x)^2 + 2*log(12)^3*log(5)*log(x)^2 - 6*log(12)^2*log(5)^2*log(x)^2 +
2*log(12)*log(5)^3*log(x)^2 + log(5)^4*log(x)^2 - 2*log(12)^3*log(x)^3 + 2*log(12)^2*log(5)*log(x)^3 + 2*log(
12)*log(5)^2*log(x)^3 - 2*log(5)^3*log(x)^3 + log(12)^2*log(x)^4 - 2*log(12)*log(5)*log(x)^4 + log(5)^2*log(x)
^4)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (16 x^{3}-120 x^{2}+200 x \right ) \ln \left (\frac {25}{x^{2}}\right )+16 x^{3}-160 x^{2}+400 x \right ) \ln \left (\frac {12}{x}\right )+\left (8 x^{3}-80 x^{2}+200 x \right ) \ln \left (\frac {25}{x^{2}}\right )}{\ln \left (\frac {25}{x^{2}}\right )^{3} \ln \left (\frac {12}{x}\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((16*x^3-120*x^2+200*x)*ln(25/x^2)+16*x^3-160*x^2+400*x)*ln(12/x)+(8*x^3-80*x^2+200*x)*ln(25/x^2))/ln(25/
x^2)^3/ln(12/x)^3,x)

[Out]

int((((16*x^3-120*x^2+200*x)*ln(25/x^2)+16*x^3-160*x^2+400*x)*ln(12/x)+(8*x^3-80*x^2+200*x)*ln(25/x^2))/ln(25/
x^2)^3/ln(12/x)^3,x)

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maxima [B]  time = 0.53, size = 145, normalized size = 5.18 \begin {gather*} \frac {x^{4} - 10 \, x^{3} + 25 \, x^{2}}{\log \relax (5)^{2} \log \relax (3)^{2} + 4 \, \log \relax (5)^{2} \log \relax (3) \log \relax (2) + 4 \, \log \relax (5)^{2} \log \relax (2)^{2} - 2 \, {\left (\log \relax (5) + \log \relax (3) + 2 \, \log \relax (2)\right )} \log \relax (x)^{3} + \log \relax (x)^{4} + {\left (\log \relax (5)^{2} + 4 \, \log \relax (5) \log \relax (3) + \log \relax (3)^{2} + 4 \, {\left (2 \, \log \relax (5) + \log \relax (3)\right )} \log \relax (2) + 4 \, \log \relax (2)^{2}\right )} \log \relax (x)^{2} - 2 \, {\left (\log \relax (5)^{2} \log \relax (3) + \log \relax (5) \log \relax (3)^{2} + 4 \, \log \relax (5) \log \relax (2)^{2} + 2 \, {\left (\log \relax (5)^{2} + 2 \, \log \relax (5) \log \relax (3)\right )} \log \relax (2)\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x^3-120*x^2+200*x)*log(25/x^2)+16*x^3-160*x^2+400*x)*log(12/x)+(8*x^3-80*x^2+200*x)*log(25/x^2
))/log(25/x^2)^3/log(12/x)^3,x, algorithm="maxima")

[Out]

(x^4 - 10*x^3 + 25*x^2)/(log(5)^2*log(3)^2 + 4*log(5)^2*log(3)*log(2) + 4*log(5)^2*log(2)^2 - 2*(log(5) + log(
3) + 2*log(2))*log(x)^3 + log(x)^4 + (log(5)^2 + 4*log(5)*log(3) + log(3)^2 + 4*(2*log(5) + log(3))*log(2) + 4
*log(2)^2)*log(x)^2 - 2*(log(5)^2*log(3) + log(5)*log(3)^2 + 4*log(5)*log(2)^2 + 2*(log(5)^2 + 2*log(5)*log(3)
)*log(2))*log(x))

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mupad [B]  time = 3.35, size = 70, normalized size = 2.50 \begin {gather*} \frac {4\,x^2\,{\left (x-5\right )}^2}{4\,{\ln \left (\frac {12}{x}\right )}^4-4\,{\ln \left (\frac {12}{x}\right )}^3\,\left (2\,\ln \left (\frac {1}{x}\right )+\ln \left (\frac {144\,x^2}{25}\right )\right )+{\ln \left (\frac {12}{x}\right )}^2\,{\left (2\,\ln \left (\frac {1}{x}\right )+\ln \left (\frac {144\,x^2}{25}\right )\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(12/x)*(400*x + log(25/x^2)*(200*x - 120*x^2 + 16*x^3) - 160*x^2 + 16*x^3) + log(25/x^2)*(200*x - 80*x
^2 + 8*x^3))/(log(12/x)^3*log(25/x^2)^3),x)

[Out]

(4*x^2*(x - 5)^2)/(4*log(12/x)^4 - 4*log(12/x)^3*(2*log(1/x) + log((144*x^2)/25)) + log(12/x)^2*(2*log(1/x) +
log((144*x^2)/25))^2)

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sympy [B]  time = 0.47, size = 71, normalized size = 2.54 \begin {gather*} \frac {16 x^{4} - 160 x^{3} + 400 x^{2}}{\log {\left (\frac {25}{x^{2}} \right )}^{4} + \left (- 4 \log {\relax (5 )} + 4 \log {\left (12 \right )}\right ) \log {\left (\frac {25}{x^{2}} \right )}^{3} + \left (- 8 \log {\relax (5 )} \log {\left (12 \right )} + 4 \log {\relax (5 )}^{2} + 4 \log {\left (12 \right )}^{2}\right ) \log {\left (\frac {25}{x^{2}} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x**3-120*x**2+200*x)*ln(25/x**2)+16*x**3-160*x**2+400*x)*ln(12/x)+(8*x**3-80*x**2+200*x)*ln(25
/x**2))/ln(25/x**2)**3/ln(12/x)**3,x)

[Out]

(16*x**4 - 160*x**3 + 400*x**2)/(log(25/x**2)**4 + (-4*log(5) + 4*log(12))*log(25/x**2)**3 + (-8*log(5)*log(12
) + 4*log(5)**2 + 4*log(12)**2)*log(25/x**2)**2)

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