3.82.40 \(\int \frac {e^5 (45 x-92 x^2-14 x^3+40 x^4+x^5-4 x^6)+(e^5 (-145 x+100 x^2+54 x^3-40 x^4-5 x^5+4 x^6)+e^5 (-145+100 x+54 x^2-40 x^3-5 x^4+4 x^5) \log (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2})) \log (x+\log (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}))}{-145 x^3+100 x^4+54 x^5-40 x^6-5 x^7+4 x^8+(-145 x^2+100 x^3+54 x^4-40 x^5-5 x^6+4 x^7) \log (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2})} \, dx\)

Optimal. Leaf size=28 \[ \frac {e^5 \left (x-\log \left (x+\log \left (5-4 \left (x+\frac {1}{-5+x^2}\right )\right )\right )\right )}{x} \]

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Rubi [F]  time = 3.92, 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 {e^5 \left (45 x-92 x^2-14 x^3+40 x^4+x^5-4 x^6\right )+\left (e^5 \left (-145 x+100 x^2+54 x^3-40 x^4-5 x^5+4 x^6\right )+e^5 \left (-145+100 x+54 x^2-40 x^3-5 x^4+4 x^5\right ) \log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right ) \log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{-145 x^3+100 x^4+54 x^5-40 x^6-5 x^7+4 x^8+\left (-145 x^2+100 x^3+54 x^4-40 x^5-5 x^6+4 x^7\right ) \log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^5*(45*x - 92*x^2 - 14*x^3 + 40*x^4 + x^5 - 4*x^6) + (E^5*(-145*x + 100*x^2 + 54*x^3 - 40*x^4 - 5*x^5 +
4*x^6) + E^5*(-145 + 100*x + 54*x^2 - 40*x^3 - 5*x^4 + 4*x^5)*Log[(-29 + 20*x + 5*x^2 - 4*x^3)/(-5 + x^2)])*Lo
g[x + Log[(-29 + 20*x + 5*x^2 - 4*x^3)/(-5 + x^2)]])/(-145*x^3 + 100*x^4 + 54*x^5 - 40*x^6 - 5*x^7 + 4*x^8 + (
-145*x^2 + 100*x^3 + 54*x^4 - 40*x^5 - 5*x^6 + 4*x^7)*Log[(-29 + 20*x + 5*x^2 - 4*x^3)/(-5 + x^2)]),x]

[Out]

-((E^5*Defer[Int][1/((Sqrt[5] - x)*(x + Log[(-29 + 20*x + 5*x^2 - 4*x^3)/(-5 + x^2)])), x])/Sqrt[5]) - (9*E^5*
Defer[Int][1/(x*(x + Log[(-29 + 20*x + 5*x^2 - 4*x^3)/(-5 + x^2)])), x])/29 - (E^5*Defer[Int][1/((Sqrt[5] + x)
*(x + Log[(-29 + 20*x + 5*x^2 - 4*x^3)/(-5 + x^2)])), x])/Sqrt[5] + (690*E^5*Defer[Int][1/((29 - 20*x - 5*x^2
+ 4*x^3)*(x + Log[(-29 + 20*x + 5*x^2 - 4*x^3)/(-5 + x^2)])), x])/29 - (248*E^5*Defer[Int][x/((29 - 20*x - 5*x
^2 + 4*x^3)*(x + Log[(-29 + 20*x + 5*x^2 - 4*x^3)/(-5 + x^2)])), x])/29 - (80*E^5*Defer[Int][x^2/((29 - 20*x -
5*x^2 + 4*x^3)*(x + Log[(-29 + 20*x + 5*x^2 - 4*x^3)/(-5 + x^2)])), x])/29 + E^5*Defer[Int][Log[x + Log[(-29
+ 20*x + 5*x^2 - 4*x^3)/(-5 + x^2)]]/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^5 \left (\frac {x \left (45-92 x-14 x^2+40 x^3+x^4-4 x^5\right )}{\left (-145+100 x+54 x^2-40 x^3-5 x^4+4 x^5\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}+\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )\right )}{x^2} \, dx\\ &=e^5 \int \frac {\frac {x \left (45-92 x-14 x^2+40 x^3+x^4-4 x^5\right )}{\left (-145+100 x+54 x^2-40 x^3-5 x^4+4 x^5\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}+\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x^2} \, dx\\ &=e^5 \int \left (\frac {45-92 x-14 x^2+40 x^3+x^4-4 x^5}{x \left (-5+x^2\right ) \left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}+\frac {\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x^2}\right ) \, dx\\ &=e^5 \int \frac {45-92 x-14 x^2+40 x^3+x^4-4 x^5}{x \left (-5+x^2\right ) \left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx+e^5 \int \frac {\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x^2} \, dx\\ &=e^5 \int \left (-\frac {9}{29 x \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}+\frac {2}{\left (-5+x^2\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}-\frac {2 \left (-345+124 x+40 x^2\right )}{29 \left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}\right ) \, dx+e^5 \int \frac {\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x^2} \, dx\\ &=-\left (\frac {1}{29} \left (2 e^5\right ) \int \frac {-345+124 x+40 x^2}{\left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx\right )-\frac {1}{29} \left (9 e^5\right ) \int \frac {1}{x \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx+e^5 \int \frac {\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x^2} \, dx+\left (2 e^5\right ) \int \frac {1}{\left (-5+x^2\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx\\ &=-\left (\frac {1}{29} \left (2 e^5\right ) \int \left (-\frac {345}{\left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}+\frac {124 x}{\left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}+\frac {40 x^2}{\left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}\right ) \, dx\right )-\frac {1}{29} \left (9 e^5\right ) \int \frac {1}{x \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx+e^5 \int \frac {\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x^2} \, dx+\left (2 e^5\right ) \int \left (-\frac {1}{2 \sqrt {5} \left (\sqrt {5}-x\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}-\frac {1}{2 \sqrt {5} \left (\sqrt {5}+x\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}\right ) \, dx\\ &=-\left (\frac {1}{29} \left (9 e^5\right ) \int \frac {1}{x \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx\right )+e^5 \int \frac {\log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x^2} \, dx-\frac {1}{29} \left (80 e^5\right ) \int \frac {x^2}{\left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx-\frac {1}{29} \left (248 e^5\right ) \int \frac {x}{\left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx+\frac {1}{29} \left (690 e^5\right ) \int \frac {1}{\left (29-20 x-5 x^2+4 x^3\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx-\frac {e^5 \int \frac {1}{\left (\sqrt {5}-x\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx}{\sqrt {5}}-\frac {e^5 \int \frac {1}{\left (\sqrt {5}+x\right ) \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )} \, dx}{\sqrt {5}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 35, normalized size = 1.25 \begin {gather*} -\frac {e^5 \log \left (x+\log \left (\frac {-29+20 x+5 x^2-4 x^3}{-5+x^2}\right )\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^5*(45*x - 92*x^2 - 14*x^3 + 40*x^4 + x^5 - 4*x^6) + (E^5*(-145*x + 100*x^2 + 54*x^3 - 40*x^4 - 5*
x^5 + 4*x^6) + E^5*(-145 + 100*x + 54*x^2 - 40*x^3 - 5*x^4 + 4*x^5)*Log[(-29 + 20*x + 5*x^2 - 4*x^3)/(-5 + x^2
)])*Log[x + Log[(-29 + 20*x + 5*x^2 - 4*x^3)/(-5 + x^2)]])/(-145*x^3 + 100*x^4 + 54*x^5 - 40*x^6 - 5*x^7 + 4*x
^8 + (-145*x^2 + 100*x^3 + 54*x^4 - 40*x^5 - 5*x^6 + 4*x^7)*Log[(-29 + 20*x + 5*x^2 - 4*x^3)/(-5 + x^2)]),x]

[Out]

-((E^5*Log[x + Log[(-29 + 20*x + 5*x^2 - 4*x^3)/(-5 + x^2)]])/x)

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fricas [A]  time = 1.16, size = 35, normalized size = 1.25 \begin {gather*} -\frac {e^{5} \log \left (x + \log \left (-\frac {4 \, x^{3} - 5 \, x^{2} - 20 \, x + 29}{x^{2} - 5}\right )\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^5-5*x^4-40*x^3+54*x^2+100*x-145)*exp(5)*log((-4*x^3+5*x^2+20*x-29)/(x^2-5))+(4*x^6-5*x^5-40*x
^4+54*x^3+100*x^2-145*x)*exp(5))*log(log((-4*x^3+5*x^2+20*x-29)/(x^2-5))+x)+(-4*x^6+x^5+40*x^4-14*x^3-92*x^2+4
5*x)*exp(5))/((4*x^7-5*x^6-40*x^5+54*x^4+100*x^3-145*x^2)*log((-4*x^3+5*x^2+20*x-29)/(x^2-5))+4*x^8-5*x^7-40*x
^6+54*x^5+100*x^4-145*x^3),x, algorithm="fricas")

[Out]

-e^5*log(x + log(-(4*x^3 - 5*x^2 - 20*x + 29)/(x^2 - 5)))/x

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giac [A]  time = 0.78, size = 35, normalized size = 1.25 \begin {gather*} -\frac {e^{5} \log \left (x + \log \left (-\frac {4 \, x^{3} - 5 \, x^{2} - 20 \, x + 29}{x^{2} - 5}\right )\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^5-5*x^4-40*x^3+54*x^2+100*x-145)*exp(5)*log((-4*x^3+5*x^2+20*x-29)/(x^2-5))+(4*x^6-5*x^5-40*x
^4+54*x^3+100*x^2-145*x)*exp(5))*log(log((-4*x^3+5*x^2+20*x-29)/(x^2-5))+x)+(-4*x^6+x^5+40*x^4-14*x^3-92*x^2+4
5*x)*exp(5))/((4*x^7-5*x^6-40*x^5+54*x^4+100*x^3-145*x^2)*log((-4*x^3+5*x^2+20*x-29)/(x^2-5))+4*x^8-5*x^7-40*x
^6+54*x^5+100*x^4-145*x^3),x, algorithm="giac")

[Out]

-e^5*log(x + log(-(4*x^3 - 5*x^2 - 20*x + 29)/(x^2 - 5)))/x

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maple [F]  time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (4 x^{5}-5 x^{4}-40 x^{3}+54 x^{2}+100 x -145\right ) {\mathrm e}^{5} \ln \left (\frac {-4 x^{3}+5 x^{2}+20 x -29}{x^{2}-5}\right )+\left (4 x^{6}-5 x^{5}-40 x^{4}+54 x^{3}+100 x^{2}-145 x \right ) {\mathrm e}^{5}\right ) \ln \left (\ln \left (\frac {-4 x^{3}+5 x^{2}+20 x -29}{x^{2}-5}\right )+x \right )+\left (-4 x^{6}+x^{5}+40 x^{4}-14 x^{3}-92 x^{2}+45 x \right ) {\mathrm e}^{5}}{\left (4 x^{7}-5 x^{6}-40 x^{5}+54 x^{4}+100 x^{3}-145 x^{2}\right ) \ln \left (\frac {-4 x^{3}+5 x^{2}+20 x -29}{x^{2}-5}\right )+4 x^{8}-5 x^{7}-40 x^{6}+54 x^{5}+100 x^{4}-145 x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x^5-5*x^4-40*x^3+54*x^2+100*x-145)*exp(5)*ln((-4*x^3+5*x^2+20*x-29)/(x^2-5))+(4*x^6-5*x^5-40*x^4+54*x
^3+100*x^2-145*x)*exp(5))*ln(ln((-4*x^3+5*x^2+20*x-29)/(x^2-5))+x)+(-4*x^6+x^5+40*x^4-14*x^3-92*x^2+45*x)*exp(
5))/((4*x^7-5*x^6-40*x^5+54*x^4+100*x^3-145*x^2)*ln((-4*x^3+5*x^2+20*x-29)/(x^2-5))+4*x^8-5*x^7-40*x^6+54*x^5+
100*x^4-145*x^3),x)

[Out]

int((((4*x^5-5*x^4-40*x^3+54*x^2+100*x-145)*exp(5)*ln((-4*x^3+5*x^2+20*x-29)/(x^2-5))+(4*x^6-5*x^5-40*x^4+54*x
^3+100*x^2-145*x)*exp(5))*ln(ln((-4*x^3+5*x^2+20*x-29)/(x^2-5))+x)+(-4*x^6+x^5+40*x^4-14*x^3-92*x^2+45*x)*exp(
5))/((4*x^7-5*x^6-40*x^5+54*x^4+100*x^3-145*x^2)*ln((-4*x^3+5*x^2+20*x-29)/(x^2-5))+4*x^8-5*x^7-40*x^6+54*x^5+
100*x^4-145*x^3),x)

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maxima [A]  time = 0.46, size = 34, normalized size = 1.21 \begin {gather*} -\frac {e^{5} \log \left (x + \log \left (-4 \, x^{3} + 5 \, x^{2} + 20 \, x - 29\right ) - \log \left (x^{2} - 5\right )\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^5-5*x^4-40*x^3+54*x^2+100*x-145)*exp(5)*log((-4*x^3+5*x^2+20*x-29)/(x^2-5))+(4*x^6-5*x^5-40*x
^4+54*x^3+100*x^2-145*x)*exp(5))*log(log((-4*x^3+5*x^2+20*x-29)/(x^2-5))+x)+(-4*x^6+x^5+40*x^4-14*x^3-92*x^2+4
5*x)*exp(5))/((4*x^7-5*x^6-40*x^5+54*x^4+100*x^3-145*x^2)*log((-4*x^3+5*x^2+20*x-29)/(x^2-5))+4*x^8-5*x^7-40*x
^6+54*x^5+100*x^4-145*x^3),x, algorithm="maxima")

[Out]

-e^5*log(x + log(-4*x^3 + 5*x^2 + 20*x - 29) - log(x^2 - 5))/x

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mupad [B]  time = 5.86, size = 34, normalized size = 1.21 \begin {gather*} -\frac {\ln \left (x+\ln \left (\frac {-4\,x^3+5\,x^2+20\,x-29}{x^2-5}\right )\right )\,{\mathrm {e}}^5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(5)*(45*x - 92*x^2 - 14*x^3 + 40*x^4 + x^5 - 4*x^6) - log(x + log((20*x + 5*x^2 - 4*x^3 - 29)/(x^2 -
5)))*(exp(5)*(145*x - 100*x^2 - 54*x^3 + 40*x^4 + 5*x^5 - 4*x^6) - log((20*x + 5*x^2 - 4*x^3 - 29)/(x^2 - 5))*
exp(5)*(100*x + 54*x^2 - 40*x^3 - 5*x^4 + 4*x^5 - 145)))/(log((20*x + 5*x^2 - 4*x^3 - 29)/(x^2 - 5))*(145*x^2
- 100*x^3 - 54*x^4 + 40*x^5 + 5*x^6 - 4*x^7) + 145*x^3 - 100*x^4 - 54*x^5 + 40*x^6 + 5*x^7 - 4*x^8),x)

[Out]

-(log(x + log((20*x + 5*x^2 - 4*x^3 - 29)/(x^2 - 5)))*exp(5))/x

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sympy [A]  time = 1.23, size = 31, normalized size = 1.11 \begin {gather*} - \frac {e^{5} \log {\left (x + \log {\left (\frac {- 4 x^{3} + 5 x^{2} + 20 x - 29}{x^{2} - 5} \right )} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x**5-5*x**4-40*x**3+54*x**2+100*x-145)*exp(5)*ln((-4*x**3+5*x**2+20*x-29)/(x**2-5))+(4*x**6-5*x
**5-40*x**4+54*x**3+100*x**2-145*x)*exp(5))*ln(ln((-4*x**3+5*x**2+20*x-29)/(x**2-5))+x)+(-4*x**6+x**5+40*x**4-
14*x**3-92*x**2+45*x)*exp(5))/((4*x**7-5*x**6-40*x**5+54*x**4+100*x**3-145*x**2)*ln((-4*x**3+5*x**2+20*x-29)/(
x**2-5))+4*x**8-5*x**7-40*x**6+54*x**5+100*x**4-145*x**3),x)

[Out]

-exp(5)*log(x + log((-4*x**3 + 5*x**2 + 20*x - 29)/(x**2 - 5)))/x

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