3.83.97 \(\int \frac {6 x+15 e^4 x^3+3 e^9 x^3+(-6-20 e^4 x^2-4 e^9 x^2) \log (5 e^{\frac {3}{5 e^4 x+e^9 x}})+(5 e^4 x+e^9 x) \log ^2(5 e^{\frac {3}{5 e^4 x+e^9 x}})}{5 e^4 x+e^9 x} \, dx\)

Optimal. Leaf size=28 \[ x \left (x-\log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )\right )^2 \]

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Rubi [F]  time = 0.36, 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 {6 x+15 e^4 x^3+3 e^9 x^3+\left (-6-20 e^4 x^2-4 e^9 x^2\right ) \log \left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )+\left (5 e^4 x+e^9 x\right ) \log ^2\left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )}{5 e^4 x+e^9 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6*x + 15*E^4*x^3 + 3*E^9*x^3 + (-6 - 20*E^4*x^2 - 4*E^9*x^2)*Log[5*E^(3/(5*E^4*x + E^9*x))] + (5*E^4*x +
E^9*x)*Log[5*E^(3/(5*E^4*x + E^9*x))]^2)/(5*E^4*x + E^9*x),x]

[Out]

18/(E^8*(5 + E^5)^2*x) + x^3 - 2*x^2*Log[5*E^(3/(E^4*(5 + E^5)*x))] + (18*Log[x])/(E^8*(5 + E^5)^2*x) - (6*Log
[5*E^(3/(E^4*(5 + E^5)*x))]*Log[x])/(E^4*(5 + E^5)) + Defer[Int][Log[5*E^(3/(E^4*(5 + E^5)*x))]^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6 x+15 e^4 x^3+3 e^9 x^3+\left (-6-20 e^4 x^2-4 e^9 x^2\right ) \log \left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )+\left (5 e^4 x+e^9 x\right ) \log ^2\left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )}{\left (5 e^4+e^9\right ) x} \, dx\\ &=\int \frac {6 x+\left (15 e^4+3 e^9\right ) x^3+\left (-6-20 e^4 x^2-4 e^9 x^2\right ) \log \left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )+\left (5 e^4 x+e^9 x\right ) \log ^2\left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )}{\left (5 e^4+e^9\right ) x} \, dx\\ &=\frac {\int \frac {6 x+\left (15 e^4+3 e^9\right ) x^3+\left (-6-20 e^4 x^2-4 e^9 x^2\right ) \log \left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )+\left (5 e^4 x+e^9 x\right ) \log ^2\left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )}{x} \, dx}{e^4 \left (5+e^5\right )}\\ &=\frac {\int \left (3 \left (2+e^4 \left (5+e^5\right ) x^2\right )+\frac {2 \left (-3-2 e^4 \left (5+e^5\right ) x^2\right ) \log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )}{x}+e^4 \left (5+e^5\right ) \log ^2\left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )\right ) \, dx}{e^4 \left (5+e^5\right )}\\ &=\frac {2 \int \frac {\left (-3-2 e^4 \left (5+e^5\right ) x^2\right ) \log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )}{x} \, dx}{e^4 \left (5+e^5\right )}+\frac {3 \int \left (2+e^4 \left (5+e^5\right ) x^2\right ) \, dx}{e^4 \left (5+e^5\right )}+\int \log ^2\left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right ) \, dx\\ &=\frac {6 x}{e^4 \left (5+e^5\right )}+x^3+\frac {2 \int \left (-\frac {3 \log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )}{x}+2 e^4 \left (-5-e^5\right ) x \log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )\right ) \, dx}{e^4 \left (5+e^5\right )}+\int \log ^2\left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right ) \, dx\\ &=\frac {6 x}{e^4 \left (5+e^5\right )}+x^3-4 \int x \log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right ) \, dx-\frac {6 \int \frac {\log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )}{x} \, dx}{e^4 \left (5+e^5\right )}+\int \log ^2\left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right ) \, dx\\ &=\frac {6 x}{e^4 \left (5+e^5\right )}+x^3-2 x^2 \log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )-\frac {6 \log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right ) \log (x)}{e^4 \left (5+e^5\right )}+2 \int -\frac {3}{e^4 \left (5+e^5\right )} \, dx+\frac {6 \int -\frac {3 \log (x)}{e^4 \left (5+e^5\right ) x^2} \, dx}{e^4 \left (5+e^5\right )}+\int \log ^2\left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right ) \, dx\\ &=x^3-2 x^2 \log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )-\frac {6 \log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right ) \log (x)}{e^4 \left (5+e^5\right )}-\frac {18 \int \frac {\log (x)}{x^2} \, dx}{e^8 \left (5+e^5\right )^2}+\int \log ^2\left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right ) \, dx\\ &=\frac {18}{e^8 \left (5+e^5\right )^2 x}+x^3-2 x^2 \log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right )+\frac {18 \log (x)}{e^8 \left (5+e^5\right )^2 x}-\frac {6 \log \left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right ) \log (x)}{e^4 \left (5+e^5\right )}+\int \log ^2\left (5 e^{\frac {3}{e^4 \left (5+e^5\right ) x}}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.06, size = 125, normalized size = 4.46 \begin {gather*} \frac {18+25 e^8 x^4+10 e^{13} x^4+e^{18} x^4-2 e^4 \left (5+e^5\right ) x \left (3+5 e^4 x^2+e^9 x^2\right ) \log \left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )+e^8 \left (5+e^5\right )^2 x^2 \log ^2\left (5 e^{\frac {3}{5 e^4 x+e^9 x}}\right )}{e^8 \left (5+e^5\right )^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6*x + 15*E^4*x^3 + 3*E^9*x^3 + (-6 - 20*E^4*x^2 - 4*E^9*x^2)*Log[5*E^(3/(5*E^4*x + E^9*x))] + (5*E^
4*x + E^9*x)*Log[5*E^(3/(5*E^4*x + E^9*x))]^2)/(5*E^4*x + E^9*x),x]

[Out]

(18 + 25*E^8*x^4 + 10*E^13*x^4 + E^18*x^4 - 2*E^4*(5 + E^5)*x*(3 + 5*E^4*x^2 + E^9*x^2)*Log[5*E^(3/(5*E^4*x +
E^9*x))] + E^8*(5 + E^5)^2*x^2*Log[5*E^(3/(5*E^4*x + E^9*x))]^2)/(E^8*(5 + E^5)^2*x)

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fricas [B]  time = 0.55, size = 105, normalized size = 3.75 \begin {gather*} \frac {x^{4} e^{18} + 10 \, x^{4} e^{13} + 25 \, x^{4} e^{8} - 6 \, x^{2} e^{9} - 30 \, x^{2} e^{4} + {\left (x^{2} e^{18} + 10 \, x^{2} e^{13} + 25 \, x^{2} e^{8}\right )} \log \relax (5)^{2} - 2 \, {\left (x^{3} e^{18} + 10 \, x^{3} e^{13} + 25 \, x^{3} e^{8}\right )} \log \relax (5) + 9}{x e^{18} + 10 \, x e^{13} + 25 \, x e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*exp(4)*exp(5)+5*x*exp(4))*log(5*exp(3/(x*exp(4)*exp(5)+5*x*exp(4))))^2+(-4*x^2*exp(4)*exp(5)-20*
x^2*exp(4)-6)*log(5*exp(3/(x*exp(4)*exp(5)+5*x*exp(4))))+3*x^3*exp(4)*exp(5)+15*x^3*exp(4)+6*x)/(x*exp(4)*exp(
5)+5*x*exp(4)),x, algorithm="fricas")

[Out]

(x^4*e^18 + 10*x^4*e^13 + 25*x^4*e^8 - 6*x^2*e^9 - 30*x^2*e^4 + (x^2*e^18 + 10*x^2*e^13 + 25*x^2*e^8)*log(5)^2
- 2*(x^3*e^18 + 10*x^3*e^13 + 25*x^3*e^8)*log(5) + 9)/(x*e^18 + 10*x*e^13 + 25*x*e^8)

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giac [B]  time = 0.22, size = 253, normalized size = 9.04 \begin {gather*} \frac {x^{3} e^{54} + 30 \, x^{3} e^{49} + 375 \, x^{3} e^{44} + 2500 \, x^{3} e^{39} + 9375 \, x^{3} e^{34} + 18750 \, x^{3} e^{29} + 15625 \, x^{3} e^{24} - 2 \, x^{2} e^{54} \log \relax (5) - 60 \, x^{2} e^{49} \log \relax (5) - 750 \, x^{2} e^{44} \log \relax (5) - 5000 \, x^{2} e^{39} \log \relax (5) - 18750 \, x^{2} e^{34} \log \relax (5) - 37500 \, x^{2} e^{29} \log \relax (5) - 31250 \, x^{2} e^{24} \log \relax (5) + x e^{54} \log \relax (5)^{2} + 30 \, x e^{49} \log \relax (5)^{2} + 375 \, x e^{44} \log \relax (5)^{2} + 2500 \, x e^{39} \log \relax (5)^{2} + 9375 \, x e^{34} \log \relax (5)^{2} + 18750 \, x e^{29} \log \relax (5)^{2} + 15625 \, x e^{24} \log \relax (5)^{2} - 6 \, x e^{45} - 150 \, x e^{40} - 1500 \, x e^{35} - 7500 \, x e^{30} - 18750 \, x e^{25} - 18750 \, x e^{20}}{e^{54} + 30 \, e^{49} + 375 \, e^{44} + 2500 \, e^{39} + 9375 \, e^{34} + 18750 \, e^{29} + 15625 \, e^{24}} + \frac {9}{x {\left (e^{18} + 10 \, e^{13} + 25 \, e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*exp(4)*exp(5)+5*x*exp(4))*log(5*exp(3/(x*exp(4)*exp(5)+5*x*exp(4))))^2+(-4*x^2*exp(4)*exp(5)-20*
x^2*exp(4)-6)*log(5*exp(3/(x*exp(4)*exp(5)+5*x*exp(4))))+3*x^3*exp(4)*exp(5)+15*x^3*exp(4)+6*x)/(x*exp(4)*exp(
5)+5*x*exp(4)),x, algorithm="giac")

[Out]

(x^3*e^54 + 30*x^3*e^49 + 375*x^3*e^44 + 2500*x^3*e^39 + 9375*x^3*e^34 + 18750*x^3*e^29 + 15625*x^3*e^24 - 2*x
^2*e^54*log(5) - 60*x^2*e^49*log(5) - 750*x^2*e^44*log(5) - 5000*x^2*e^39*log(5) - 18750*x^2*e^34*log(5) - 375
00*x^2*e^29*log(5) - 31250*x^2*e^24*log(5) + x*e^54*log(5)^2 + 30*x*e^49*log(5)^2 + 375*x*e^44*log(5)^2 + 2500
*x*e^39*log(5)^2 + 9375*x*e^34*log(5)^2 + 18750*x*e^29*log(5)^2 + 15625*x*e^24*log(5)^2 - 6*x*e^45 - 150*x*e^4
0 - 1500*x*e^35 - 7500*x*e^30 - 18750*x*e^25 - 18750*x*e^20)/(e^54 + 30*e^49 + 375*e^44 + 2500*e^39 + 9375*e^3
4 + 18750*e^29 + 15625*e^24) + 9/(x*(e^18 + 10*e^13 + 25*e^8))

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maple [B]  time = 0.16, size = 66, normalized size = 2.36

method	result	size
risch	\(x \ln \left ({\mathrm e}^{\frac {3}{x \left (5 \,{\mathrm e}^{4}+{\mathrm e}^{9}\right )}}\right )^{2}+\left (2 x \ln \relax (5)-2 x^{2}\right ) \ln \left ({\mathrm e}^{\frac {3}{x \left (5 \,{\mathrm e}^{4}+{\mathrm e}^{9}\right )}}\right )+x \ln \relax (5)^{2}-2 x^{2} \ln \relax (5)+x^{3}\)	\(66\)
default	\(\frac {25 x \left (\ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right )-\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}\right )^{2}}{\left ({\mathrm e}^{5}+5\right )^{2}}+\frac {10 \,{\mathrm e}^{-4} {\mathrm e}^{9} x \left (\ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right )-\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}\right )^{2}}{\left ({\mathrm e}^{5}+5\right )^{2}}+\frac {{\mathrm e}^{-8} {\mathrm e}^{18} x \left (\ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right )-\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}\right )^{2}}{\left ({\mathrm e}^{5}+5\right )^{2}}+\frac {30 \,{\mathrm e}^{-4} \left (\ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right )-\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}\right ) \ln \relax (x )}{\left ({\mathrm e}^{5}+5\right )^{2}}+\frac {6 \,{\mathrm e}^{-8} \left (\ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right )-\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}\right ) \ln \relax (x ) {\mathrm e}^{9}}{\left ({\mathrm e}^{5}+5\right )^{2}}-\frac {9 \,{\mathrm e}^{-8}}{\left ({\mathrm e}^{5}+5\right )^{2} x}+\frac {3 \,{\mathrm e}^{-4} \left (\frac {x^{3} {\mathrm e}^{4} {\mathrm e}^{5}}{3}+\frac {5 x^{3} {\mathrm e}^{4}}{3}+2 x \right )}{{\mathrm e}^{5}+5}-\frac {10 \ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right ) x^{2}}{{\mathrm e}^{5}+5}-\frac {2 \,{\mathrm e}^{-4} {\mathrm e}^{9} \ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right ) x^{2}}{{\mathrm e}^{5}+5}-\frac {6 \,{\mathrm e}^{-4} \ln \relax (x ) \ln \left (5 \,{\mathrm e}^{\frac {3}{x \,{\mathrm e}^{4} {\mathrm e}^{5}+5 x \,{\mathrm e}^{4}}}\right )}{{\mathrm e}^{5}+5}-\frac {30 \,{\mathrm e}^{-4} {\mathrm e}^{5} x}{\left ({\mathrm e}^{5}+5\right )^{3}}-\frac {6 \,{\mathrm e}^{-8} {\mathrm e}^{5} {\mathrm e}^{9} x}{\left ({\mathrm e}^{5}+5\right )^{3}}+\frac {18 \,{\mathrm e}^{-8} {\mathrm e}^{5} \ln \relax (x )}{\left ({\mathrm e}^{5}+5\right )^{3} x}+\frac {18 \,{\mathrm e}^{-8} {\mathrm e}^{5}}{\left ({\mathrm e}^{5}+5\right )^{3} x}-\frac {150 \,{\mathrm e}^{-4} x}{\left ({\mathrm e}^{5}+5\right )^{3}}-\frac {30 \,{\mathrm e}^{-8} {\mathrm e}^{9} x}{\left ({\mathrm e}^{5}+5\right )^{3}}+\frac {90 \,{\mathrm e}^{-8} \ln \relax (x )}{\left ({\mathrm e}^{5}+5\right )^{3} x}+\frac {90 \,{\mathrm e}^{-8}}{\left ({\mathrm e}^{5}+5\right )^{3} x}\)	\(540\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x*exp(4)*exp(5)+5*x*exp(4))*ln(5*exp(3/(x*exp(4)*exp(5)+5*x*exp(4))))^2+(-4*x^2*exp(4)*exp(5)-20*x^2*exp
(4)-6)*ln(5*exp(3/(x*exp(4)*exp(5)+5*x*exp(4))))+3*x^3*exp(4)*exp(5)+15*x^3*exp(4)+6*x)/(x*exp(4)*exp(5)+5*x*e
xp(4)),x,method=_RETURNVERBOSE)

[Out]

x*ln(exp(3/x/(5*exp(4)+exp(9))))^2+(2*x*ln(5)-2*x^2)*ln(exp(3/x/(5*exp(4)+exp(9))))+x*ln(5)^2-2*x^2*ln(5)+x^3

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maxima [B]  time = 0.39, size = 512, normalized size = 18.29 \begin {gather*} \frac {x^{3} e^{9}}{e^{9} + 5 \, e^{4}} + \frac {5 \, x^{3} e^{4}}{e^{9} + 5 \, e^{4}} - \frac {2 \, x^{2} e^{9} \log \left (5 \, e^{\left (\frac {3}{x e^{9} + 5 \, x e^{4}}\right )}\right )}{e^{9} + 5 \, e^{4}} - \frac {10 \, x^{2} e^{4} \log \left (5 \, e^{\left (\frac {3}{x e^{9} + 5 \, x e^{4}}\right )}\right )}{e^{9} + 5 \, e^{4}} + \frac {x e^{9} \log \left (5 \, e^{\left (\frac {3}{x e^{9} + 5 \, x e^{4}}\right )}\right )^{2}}{e^{9} + 5 \, e^{4}} + \frac {5 \, x e^{4} \log \left (5 \, e^{\left (\frac {3}{x e^{9} + 5 \, x e^{4}}\right )}\right )^{2}}{e^{9} + 5 \, e^{4}} - 6 \, {\left (\frac {3 \, {\left (\frac {\log \relax (x)}{{\left (x e^{9} + 5 \, x e^{4}\right )} {\left (e^{9} + 5 \, e^{4}\right )}} + \frac {1}{x {\left (e^{9} + 5 \, e^{4}\right )}^{2}}\right )} {\left (e^{9} + 5 \, e^{4}\right )}}{e^{18} + 10 \, e^{13} + 25 \, e^{8}} - \frac {\log \relax (x) \log \left (5 \, e^{\left (\frac {3}{x e^{9} + 5 \, x e^{4}}\right )}\right )}{e^{18} + 10 \, e^{13} + 25 \, e^{8}}\right )} e^{9} - 30 \, {\left (\frac {3 \, {\left (\frac {\log \relax (x)}{{\left (x e^{9} + 5 \, x e^{4}\right )} {\left (e^{9} + 5 \, e^{4}\right )}} + \frac {1}{x {\left (e^{9} + 5 \, e^{4}\right )}^{2}}\right )} {\left (e^{9} + 5 \, e^{4}\right )}}{e^{18} + 10 \, e^{13} + 25 \, e^{8}} - \frac {\log \relax (x) \log \left (5 \, e^{\left (\frac {3}{x e^{9} + 5 \, x e^{4}}\right )}\right )}{e^{18} + 10 \, e^{13} + 25 \, e^{8}}\right )} e^{4} - \frac {6 \, x e^{9}}{e^{18} + 10 \, e^{13} + 25 \, e^{8}} - \frac {30 \, x e^{4}}{e^{18} + 10 \, e^{13} + 25 \, e^{8}} - \frac {6 \, \log \left (x e^{9} + 5 \, x e^{4}\right ) \log \left (5 \, e^{\left (\frac {3}{x e^{9} + 5 \, x e^{4}}\right )}\right )}{e^{9} + 5 \, e^{4}} + \frac {6 \, x}{e^{9} + 5 \, e^{4}} + \frac {18 \, \log \left (x e^{9} + 5 \, x e^{4}\right )}{{\left (x e^{9} + 5 \, x e^{4}\right )} {\left (e^{9} + 5 \, e^{4}\right )}} + \frac {18}{{\left (x e^{9} + 5 \, x e^{4}\right )} {\left (e^{9} + 5 \, e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*exp(4)*exp(5)+5*x*exp(4))*log(5*exp(3/(x*exp(4)*exp(5)+5*x*exp(4))))^2+(-4*x^2*exp(4)*exp(5)-20*
x^2*exp(4)-6)*log(5*exp(3/(x*exp(4)*exp(5)+5*x*exp(4))))+3*x^3*exp(4)*exp(5)+15*x^3*exp(4)+6*x)/(x*exp(4)*exp(
5)+5*x*exp(4)),x, algorithm="maxima")

[Out]

x^3*e^9/(e^9 + 5*e^4) + 5*x^3*e^4/(e^9 + 5*e^4) - 2*x^2*e^9*log(5*e^(3/(x*e^9 + 5*x*e^4)))/(e^9 + 5*e^4) - 10*
x^2*e^4*log(5*e^(3/(x*e^9 + 5*x*e^4)))/(e^9 + 5*e^4) + x*e^9*log(5*e^(3/(x*e^9 + 5*x*e^4)))^2/(e^9 + 5*e^4) +
5*x*e^4*log(5*e^(3/(x*e^9 + 5*x*e^4)))^2/(e^9 + 5*e^4) - 6*(3*(log(x)/((x*e^9 + 5*x*e^4)*(e^9 + 5*e^4)) + 1/(x
*(e^9 + 5*e^4)^2))*(e^9 + 5*e^4)/(e^18 + 10*e^13 + 25*e^8) - log(x)*log(5*e^(3/(x*e^9 + 5*x*e^4)))/(e^18 + 10*
e^13 + 25*e^8))*e^9 - 30*(3*(log(x)/((x*e^9 + 5*x*e^4)*(e^9 + 5*e^4)) + 1/(x*(e^9 + 5*e^4)^2))*(e^9 + 5*e^4)/(
e^18 + 10*e^13 + 25*e^8) - log(x)*log(5*e^(3/(x*e^9 + 5*x*e^4)))/(e^18 + 10*e^13 + 25*e^8))*e^4 - 6*x*e^9/(e^1
8 + 10*e^13 + 25*e^8) - 30*x*e^4/(e^18 + 10*e^13 + 25*e^8) - 6*log(x*e^9 + 5*x*e^4)*log(5*e^(3/(x*e^9 + 5*x*e^
4)))/(e^9 + 5*e^4) + 6*x/(e^9 + 5*e^4) + 18*log(x*e^9 + 5*x*e^4)/((x*e^9 + 5*x*e^4)*(e^9 + 5*e^4)) + 18/((x*e^
9 + 5*x*e^4)*(e^9 + 5*e^4))

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mupad [B]  time = 5.41, size = 75, normalized size = 2.68 \begin {gather*} \frac {{\mathrm {e}}^8\,{\left ({\mathrm {e}}^5+5\right )}^2\,x^5-2\,{\mathrm {e}}^8\,\ln \relax (5)\,{\left ({\mathrm {e}}^5+5\right )}^2\,x^4+{\mathrm {e}}^4\,\left ({\mathrm {e}}^5+5\right )\,\left (5\,{\mathrm {e}}^4\,{\ln \relax (5)}^2+{\mathrm {e}}^9\,{\ln \relax (5)}^2-6\right )\,x^3+9\,x}{x^2\,\left (25\,{\mathrm {e}}^8+10\,{\mathrm {e}}^{13}+{\mathrm {e}}^{18}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x + log(5*exp(3/(5*x*exp(4) + x*exp(9))))^2*(5*x*exp(4) + x*exp(9)) + 15*x^3*exp(4) + 3*x^3*exp(9) - lo
g(5*exp(3/(5*x*exp(4) + x*exp(9))))*(20*x^2*exp(4) + 4*x^2*exp(9) + 6))/(5*x*exp(4) + x*exp(9)),x)

[Out]

(9*x + x^5*exp(8)*(exp(5) + 5)^2 - 2*x^4*exp(8)*log(5)*(exp(5) + 5)^2 + x^3*exp(4)*(exp(5) + 5)*(5*exp(4)*log(
5)^2 + exp(9)*log(5)^2 - 6))/(x^2*(25*exp(8) + 10*exp(13) + exp(18)))

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sympy [B]  time = 0.66, size = 104, normalized size = 3.71 \begin {gather*} \frac {x^{3} \left (25 e^{8} + 10 e^{13} + e^{18}\right ) + x^{2} \left (- 2 e^{18} \log {\relax (5 )} - 20 e^{13} \log {\relax (5 )} - 50 e^{8} \log {\relax (5 )}\right ) + x \left (- 6 e^{9} - 30 e^{4} + 25 e^{8} \log {\relax (5 )}^{2} + 10 e^{13} \log {\relax (5 )}^{2} + e^{18} \log {\relax (5 )}^{2}\right ) + \frac {9}{x}}{25 e^{8} + 10 e^{13} + e^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*exp(4)*exp(5)+5*x*exp(4))*ln(5*exp(3/(x*exp(4)*exp(5)+5*x*exp(4))))**2+(-4*x**2*exp(4)*exp(5)-20
*x**2*exp(4)-6)*ln(5*exp(3/(x*exp(4)*exp(5)+5*x*exp(4))))+3*x**3*exp(4)*exp(5)+15*x**3*exp(4)+6*x)/(x*exp(4)*e
xp(5)+5*x*exp(4)),x)

[Out]

(x**3*(25*exp(8) + 10*exp(13) + exp(18)) + x**2*(-2*exp(18)*log(5) - 20*exp(13)*log(5) - 50*exp(8)*log(5)) + x
*(-6*exp(9) - 30*exp(4) + 25*exp(8)*log(5)**2 + 10*exp(13)*log(5)**2 + exp(18)*log(5)**2) + 9/x)/(25*exp(8) +
10*exp(13) + exp(18))

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