3.46.65 \(\int \frac {(-30+5 x^3) \log (3)+((45+150 x^2+15 x^3) \log (3)+(15+50 x^2+5 x^3) \log (3) \log (\frac {-3-10 x^2-x^3}{2 x^2})) \log (3+\log (\frac {-3-10 x^2-x^3}{2 x^2}))}{9+30 x^2+3 x^3+(3+10 x^2+x^3) \log (\frac {-3-10 x^2-x^3}{2 x^2})} \, dx\)

Optimal. Leaf size=23 \[ 5 x \log (3) \log \left (3+\log \left (\frac {1}{2} \left (-10-\frac {3}{x^2}-x\right )\right )\right ) \]

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Rubi [F]  time = 0.79, 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 (-30+5 x^3\right ) \log (3)+\left (\left (45+150 x^2+15 x^3\right ) \log (3)+\left (15+50 x^2+5 x^3\right ) \log (3) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right ) \log \left (3+\log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )\right )}{9+30 x^2+3 x^3+\left (3+10 x^2+x^3\right ) \log \left (\frac {-3-10 x^2-x^3}{2 x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-30 + 5*x^3)*Log[3] + ((45 + 150*x^2 + 15*x^3)*Log[3] + (15 + 50*x^2 + 5*x^3)*Log[3]*Log[(-3 - 10*x^2 -
x^3)/(2*x^2)])*Log[3 + Log[(-3 - 10*x^2 - x^3)/(2*x^2)]])/(9 + 30*x^2 + 3*x^3 + (3 + 10*x^2 + x^3)*Log[(-3 - 1
0*x^2 - x^3)/(2*x^2)]),x]

[Out]

5*Log[3]*Defer[Int][(3 + Log[-5 - 3/(2*x^2) - x/2])^(-1), x] - 45*Log[3]*Defer[Int][1/((3 + 10*x^2 + x^3)*(3 +
Log[-5 - 3/(2*x^2) - x/2])), x] - 50*Log[3]*Defer[Int][x^2/((3 + 10*x^2 + x^3)*(3 + Log[-5 - 3/(2*x^2) - x/2]
)), x] + 5*Log[3]*Defer[Int][Log[3 + Log[-5 - 3/(2*x^2) - x/2]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \log (3) \left (-6+x^3+\left (3+10 x^2+x^3\right ) \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right ) \log \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right )\right )}{\left (3+10 x^2+x^3\right ) \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right )} \, dx\\ &=(5 \log (3)) \int \frac {-6+x^3+\left (3+10 x^2+x^3\right ) \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right ) \log \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right )}{\left (3+10 x^2+x^3\right ) \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right )} \, dx\\ &=(5 \log (3)) \int \left (\frac {-6+x^3}{\left (3+10 x^2+x^3\right ) \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right )}+\log \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right )\right ) \, dx\\ &=(5 \log (3)) \int \frac {-6+x^3}{\left (3+10 x^2+x^3\right ) \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right )} \, dx+(5 \log (3)) \int \log \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right ) \, dx\\ &=(5 \log (3)) \int \left (\frac {1}{3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )}+\frac {-9-10 x^2}{\left (3+10 x^2+x^3\right ) \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right )}\right ) \, dx+(5 \log (3)) \int \log \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right ) \, dx\\ &=(5 \log (3)) \int \frac {1}{3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )} \, dx+(5 \log (3)) \int \frac {-9-10 x^2}{\left (3+10 x^2+x^3\right ) \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right )} \, dx+(5 \log (3)) \int \log \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right ) \, dx\\ &=(5 \log (3)) \int \frac {1}{3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )} \, dx+(5 \log (3)) \int \left (-\frac {9}{\left (3+10 x^2+x^3\right ) \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right )}-\frac {10 x^2}{\left (3+10 x^2+x^3\right ) \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right )}\right ) \, dx+(5 \log (3)) \int \log \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right ) \, dx\\ &=(5 \log (3)) \int \frac {1}{3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )} \, dx+(5 \log (3)) \int \log \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right ) \, dx-(45 \log (3)) \int \frac {1}{\left (3+10 x^2+x^3\right ) \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right )} \, dx-(50 \log (3)) \int \frac {x^2}{\left (3+10 x^2+x^3\right ) \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 23, normalized size = 1.00 \begin {gather*} 5 x \log (3) \log \left (3+\log \left (-5-\frac {3}{2 x^2}-\frac {x}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-30 + 5*x^3)*Log[3] + ((45 + 150*x^2 + 15*x^3)*Log[3] + (15 + 50*x^2 + 5*x^3)*Log[3]*Log[(-3 - 10*
x^2 - x^3)/(2*x^2)])*Log[3 + Log[(-3 - 10*x^2 - x^3)/(2*x^2)]])/(9 + 30*x^2 + 3*x^3 + (3 + 10*x^2 + x^3)*Log[(
-3 - 10*x^2 - x^3)/(2*x^2)]),x]

[Out]

5*x*Log[3]*Log[3 + Log[-5 - 3/(2*x^2) - x/2]]

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fricas [A]  time = 0.66, size = 24, normalized size = 1.04 \begin {gather*} 5 \, x \log \relax (3) \log \left (\log \left (-\frac {x^{3} + 10 \, x^{2} + 3}{2 \, x^{2}}\right ) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^3+50*x^2+15)*log(3)*log(1/2*(-x^3-10*x^2-3)/x^2)+(15*x^3+150*x^2+45)*log(3))*log(log(1/2*(-x^
3-10*x^2-3)/x^2)+3)+(5*x^3-30)*log(3))/((x^3+10*x^2+3)*log(1/2*(-x^3-10*x^2-3)/x^2)+3*x^3+30*x^2+9),x, algorit
hm="fricas")

[Out]

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

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giac [A]  time = 0.75, size = 24, normalized size = 1.04 \begin {gather*} 5 \, x \log \relax (3) \log \left (\log \left (-\frac {x^{3} + 10 \, x^{2} + 3}{2 \, x^{2}}\right ) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^3+50*x^2+15)*log(3)*log(1/2*(-x^3-10*x^2-3)/x^2)+(15*x^3+150*x^2+45)*log(3))*log(log(1/2*(-x^
3-10*x^2-3)/x^2)+3)+(5*x^3-30)*log(3))/((x^3+10*x^2+3)*log(1/2*(-x^3-10*x^2-3)/x^2)+3*x^3+30*x^2+9),x, algorit
hm="giac")

[Out]

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

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (5 x^{3}+50 x^{2}+15\right ) \ln \relax (3) \ln \left (\frac {-x^{3}-10 x^{2}-3}{2 x^{2}}\right )+\left (15 x^{3}+150 x^{2}+45\right ) \ln \relax (3)\right ) \ln \left (\ln \left (\frac {-x^{3}-10 x^{2}-3}{2 x^{2}}\right )+3\right )+\left (5 x^{3}-30\right ) \ln \relax (3)}{\left (x^{3}+10 x^{2}+3\right ) \ln \left (\frac {-x^{3}-10 x^{2}-3}{2 x^{2}}\right )+3 x^{3}+30 x^{2}+9}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((5*x^3+50*x^2+15)*ln(3)*ln(1/2*(-x^3-10*x^2-3)/x^2)+(15*x^3+150*x^2+45)*ln(3))*ln(ln(1/2*(-x^3-10*x^2-3)
/x^2)+3)+(5*x^3-30)*ln(3))/((x^3+10*x^2+3)*ln(1/2*(-x^3-10*x^2-3)/x^2)+3*x^3+30*x^2+9),x)

[Out]

int((((5*x^3+50*x^2+15)*ln(3)*ln(1/2*(-x^3-10*x^2-3)/x^2)+(15*x^3+150*x^2+45)*ln(3))*ln(ln(1/2*(-x^3-10*x^2-3)
/x^2)+3)+(5*x^3-30)*ln(3))/((x^3+10*x^2+3)*ln(1/2*(-x^3-10*x^2-3)/x^2)+3*x^3+30*x^2+9),x)

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maxima [A]  time = 0.48, size = 29, normalized size = 1.26 \begin {gather*} 5 \, x \log \relax (3) \log \left (-\log \relax (2) + \log \left (-x^{3} - 10 \, x^{2} - 3\right ) - 2 \, \log \relax (x) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x^3+50*x^2+15)*log(3)*log(1/2*(-x^3-10*x^2-3)/x^2)+(15*x^3+150*x^2+45)*log(3))*log(log(1/2*(-x^
3-10*x^2-3)/x^2)+3)+(5*x^3-30)*log(3))/((x^3+10*x^2+3)*log(1/2*(-x^3-10*x^2-3)/x^2)+3*x^3+30*x^2+9),x, algorit
hm="maxima")

[Out]

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

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mupad [B]  time = 3.53, size = 26, normalized size = 1.13 \begin {gather*} 5\,x\,\ln \left (\ln \left (-\frac {\frac {x^3}{2}+5\,x^2+\frac {3}{2}}{x^2}\right )+3\right )\,\ln \relax (3) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3)*(5*x^3 - 30) + log(log(-(5*x^2 + x^3/2 + 3/2)/x^2) + 3)*(log(3)*(150*x^2 + 15*x^3 + 45) + log(3)*l
og(-(5*x^2 + x^3/2 + 3/2)/x^2)*(50*x^2 + 5*x^3 + 15)))/(log(-(5*x^2 + x^3/2 + 3/2)/x^2)*(10*x^2 + x^3 + 3) + 3
0*x^2 + 3*x^3 + 9),x)

[Out]

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

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sympy [B]  time = 1.16, size = 66, normalized size = 2.87 \begin {gather*} \left (5 x \log {\relax (3 )} + \frac {5 \log {\relax (3 )}}{2}\right ) \log {\left (\log {\left (\frac {- \frac {x^{3}}{2} - 5 x^{2} - \frac {3}{2}}{x^{2}} \right )} + 3 \right )} - \frac {5 \log {\relax (3 )} \log {\left (\log {\left (\frac {- \frac {x^{3}}{2} - 5 x^{2} - \frac {3}{2}}{x^{2}} \right )} + 3 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((5*x**3+50*x**2+15)*ln(3)*ln(1/2*(-x**3-10*x**2-3)/x**2)+(15*x**3+150*x**2+45)*ln(3))*ln(ln(1/2*(-
x**3-10*x**2-3)/x**2)+3)+(5*x**3-30)*ln(3))/((x**3+10*x**2+3)*ln(1/2*(-x**3-10*x**2-3)/x**2)+3*x**3+30*x**2+9)
,x)

[Out]

(5*x*log(3) + 5*log(3)/2)*log(log((-x**3/2 - 5*x**2 - 3/2)/x**2) + 3) - 5*log(3)*log(log((-x**3/2 - 5*x**2 - 3
/2)/x**2) + 3)/2

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