3.8.86 \(\int \frac {243 (e^{5-x} (3 x+4 x^2)-3 \log (5)+e^{5-x} (-15 x-25 x^2+20 x^3) \log (x))}{(e^{5-x} (3 x+4 x^2)-3 \log (5))^5 (e^{5-x} (3 x^2+4 x^3)-3 x \log (5))} \, dx\)

Optimal. Leaf size=27 \[ \frac {\log (x)}{\left (e^{5-x} \left (x+\frac {4 x^2}{3}\right )-\log (5)\right )^5} \]

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Rubi [F]  time = 4.46, 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 {243 \left (e^{5-x} \left (3 x+4 x^2\right )-3 \log (5)+e^{5-x} \left (-15 x-25 x^2+20 x^3\right ) \log (x)\right )}{\left (e^{5-x} \left (3 x+4 x^2\right )-3 \log (5)\right )^5 \left (e^{5-x} \left (3 x^2+4 x^3\right )-3 x \log (5)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(243*(E^(5 - x)*(3*x + 4*x^2) - 3*Log[5] + E^(5 - x)*(-15*x - 25*x^2 + 20*x^3)*Log[x]))/((E^(5 - x)*(3*x +
4*x^2) - 3*Log[5])^5*(E^(5 - x)*(3*x^2 + 4*x^3) - 3*x*Log[5])),x]

[Out]

-3645*Log[x]*Defer[Int][E^(5 + 5*x)/(3*E^5*x + 4*E^5*x^2 - 3*E^x*Log[5])^6, x] - 6075*Log[x]*Defer[Int][(E^(5
+ 5*x)*x)/(3*E^5*x + 4*E^5*x^2 - 3*E^x*Log[5])^6, x] + 4860*Log[x]*Defer[Int][(E^(5 + 5*x)*x^2)/(3*E^5*x + 4*E
^5*x^2 - 3*E^x*Log[5])^6, x] + 243*Defer[Int][E^(5*x)/(x*(3*E^5*x + 4*E^5*x^2 - 3*E^x*Log[5])^5), x] + 3645*De
fer[Int][Defer[Int][E^(5 + 5*x)/(E^5*x*(3 + 4*x) - 3*E^x*Log[5])^6, x]/x, x] + 6075*Defer[Int][Defer[Int][(E^(
5 + 5*x)*x)/(E^5*x*(3 + 4*x) - 3*E^x*Log[5])^6, x]/x, x] - 4860*Defer[Int][Defer[Int][(E^(5 + 5*x)*x^2)/(E^5*x
*(3 + 4*x) - 3*E^x*Log[5])^6, x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=243 \int \frac {e^{5-x} \left (3 x+4 x^2\right )-3 \log (5)+e^{5-x} \left (-15 x-25 x^2+20 x^3\right ) \log (x)}{\left (e^{5-x} \left (3 x+4 x^2\right )-3 \log (5)\right )^5 \left (e^{5-x} \left (3 x^2+4 x^3\right )-3 x \log (5)\right )} \, dx\\ &=243 \int \frac {e^{5 x} \left (e^5 x (3+4 x)-3 e^x \log (5)+5 e^5 x \left (-3-5 x+4 x^2\right ) \log (x)\right )}{x \left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx\\ &=243 \int \left (\frac {e^{5 x}}{x \left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^5}+\frac {5 e^{5+5 x} \left (-3-5 x+4 x^2\right ) \log (x)}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6}\right ) \, dx\\ &=243 \int \frac {e^{5 x}}{x \left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^5} \, dx+1215 \int \frac {e^{5+5 x} \left (-3-5 x+4 x^2\right ) \log (x)}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx\\ &=243 \int \frac {e^{5 x}}{x \left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^5} \, dx-1215 \int \frac {-3 \int \frac {e^{5+5 x}}{\left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx-5 \int \frac {e^{5+5 x} x}{\left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx+4 \int \frac {e^{5+5 x} x^2}{\left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx}{x} \, dx-(3645 \log (x)) \int \frac {e^{5+5 x}}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx+(4860 \log (x)) \int \frac {e^{5+5 x} x^2}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx-(6075 \log (x)) \int \frac {e^{5+5 x} x}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx\\ &=243 \int \frac {e^{5 x}}{x \left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^5} \, dx-1215 \int \left (\frac {-3 \int \frac {e^{5+5 x}}{\left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx-5 \int \frac {e^{5+5 x} x}{\left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx}{x}+\frac {4 \int \frac {e^{5+5 x} x^2}{\left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx}{x}\right ) \, dx-(3645 \log (x)) \int \frac {e^{5+5 x}}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx+(4860 \log (x)) \int \frac {e^{5+5 x} x^2}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx-(6075 \log (x)) \int \frac {e^{5+5 x} x}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx\\ &=243 \int \frac {e^{5 x}}{x \left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^5} \, dx-1215 \int \frac {-3 \int \frac {e^{5+5 x}}{\left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx-5 \int \frac {e^{5+5 x} x}{\left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx}{x} \, dx-4860 \int \frac {\int \frac {e^{5+5 x} x^2}{\left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx}{x} \, dx-(3645 \log (x)) \int \frac {e^{5+5 x}}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx+(4860 \log (x)) \int \frac {e^{5+5 x} x^2}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx-(6075 \log (x)) \int \frac {e^{5+5 x} x}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx\\ &=243 \int \frac {e^{5 x}}{x \left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^5} \, dx-1215 \int \left (-\frac {3 \int \frac {e^{5+5 x}}{\left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx}{x}-\frac {5 \int \frac {e^{5+5 x} x}{\left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx}{x}\right ) \, dx-4860 \int \frac {\int \frac {e^{5+5 x} x^2}{\left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx}{x} \, dx-(3645 \log (x)) \int \frac {e^{5+5 x}}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx+(4860 \log (x)) \int \frac {e^{5+5 x} x^2}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx-(6075 \log (x)) \int \frac {e^{5+5 x} x}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx\\ &=243 \int \frac {e^{5 x}}{x \left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^5} \, dx+3645 \int \frac {\int \frac {e^{5+5 x}}{\left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx}{x} \, dx-4860 \int \frac {\int \frac {e^{5+5 x} x^2}{\left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx}{x} \, dx+6075 \int \frac {\int \frac {e^{5+5 x} x}{\left (e^5 x (3+4 x)-3 e^x \log (5)\right )^6} \, dx}{x} \, dx-(3645 \log (x)) \int \frac {e^{5+5 x}}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx+(4860 \log (x)) \int \frac {e^{5+5 x} x^2}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx-(6075 \log (x)) \int \frac {e^{5+5 x} x}{\left (3 e^5 x+4 e^5 x^2-3 e^x \log (5)\right )^6} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.00, size = 30, normalized size = 1.11 \begin {gather*} -\frac {243 e^{5 x} \log (x)}{\left (-e^5 x (3+4 x)+3 e^x \log (5)\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(243*(E^(5 - x)*(3*x + 4*x^2) - 3*Log[5] + E^(5 - x)*(-15*x - 25*x^2 + 20*x^3)*Log[x]))/((E^(5 - x)*
(3*x + 4*x^2) - 3*Log[5])^5*(E^(5 - x)*(3*x^2 + 4*x^3) - 3*x*Log[5])),x]

[Out]

(-243*E^(5*x)*Log[x])/(-(E^5*x*(3 + 4*x)) + 3*E^x*Log[5])^5

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fricas [B]  time = 0.78, size = 169, normalized size = 6.26 \begin {gather*} \frac {243 \, \log \relax (x)}{405 \, {\left (4 \, x^{2} + 3 \, x\right )} e^{\left (-x + 5\right )} \log \relax (5)^{4} - 270 \, {\left (16 \, x^{4} + 24 \, x^{3} + 9 \, x^{2}\right )} e^{\left (-2 \, x + 10\right )} \log \relax (5)^{3} - 243 \, \log \relax (5)^{5} + 90 \, {\left (64 \, x^{6} + 144 \, x^{5} + 108 \, x^{4} + 27 \, x^{3}\right )} e^{\left (-3 \, x + 15\right )} \log \relax (5)^{2} - 15 \, {\left (256 \, x^{8} + 768 \, x^{7} + 864 \, x^{6} + 432 \, x^{5} + 81 \, x^{4}\right )} e^{\left (-4 \, x + 20\right )} \log \relax (5) + {\left (1024 \, x^{10} + 3840 \, x^{9} + 5760 \, x^{8} + 4320 \, x^{7} + 1620 \, x^{6} + 243 \, x^{5}\right )} e^{\left (-5 \, x + 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^3-25*x^2-15*x)*exp(5-x)*log(x)+(4*x^2+3*x)*exp(5-x)-3*log(5))/((4*x^3+3*x^2)*exp(5-x)-3*x*log
(5))/(1/3*(4*x^2+3*x)*exp(5-x)-log(5))^5,x, algorithm="fricas")

[Out]

243*log(x)/(405*(4*x^2 + 3*x)*e^(-x + 5)*log(5)^4 - 270*(16*x^4 + 24*x^3 + 9*x^2)*e^(-2*x + 10)*log(5)^3 - 243
*log(5)^5 + 90*(64*x^6 + 144*x^5 + 108*x^4 + 27*x^3)*e^(-3*x + 15)*log(5)^2 - 15*(256*x^8 + 768*x^7 + 864*x^6
+ 432*x^5 + 81*x^4)*e^(-4*x + 20)*log(5) + (1024*x^10 + 3840*x^9 + 5760*x^8 + 4320*x^7 + 1620*x^6 + 243*x^5)*e
^(-5*x + 25))

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giac [B]  time = 1.00, size = 277, normalized size = 10.26 \begin {gather*} \frac {243 \, \log \relax (x)}{1024 \, x^{10} e^{\left (-5 \, x + 25\right )} + 3840 \, x^{9} e^{\left (-5 \, x + 25\right )} - 3840 \, x^{8} e^{\left (-4 \, x + 20\right )} \log \relax (5) + 5760 \, x^{8} e^{\left (-5 \, x + 25\right )} - 11520 \, x^{7} e^{\left (-4 \, x + 20\right )} \log \relax (5) + 5760 \, x^{6} e^{\left (-3 \, x + 15\right )} \log \relax (5)^{2} + 4320 \, x^{7} e^{\left (-5 \, x + 25\right )} - 12960 \, x^{6} e^{\left (-4 \, x + 20\right )} \log \relax (5) + 12960 \, x^{5} e^{\left (-3 \, x + 15\right )} \log \relax (5)^{2} - 4320 \, x^{4} e^{\left (-2 \, x + 10\right )} \log \relax (5)^{3} + 1620 \, x^{6} e^{\left (-5 \, x + 25\right )} - 6480 \, x^{5} e^{\left (-4 \, x + 20\right )} \log \relax (5) + 9720 \, x^{4} e^{\left (-3 \, x + 15\right )} \log \relax (5)^{2} - 6480 \, x^{3} e^{\left (-2 \, x + 10\right )} \log \relax (5)^{3} + 1620 \, x^{2} e^{\left (-x + 5\right )} \log \relax (5)^{4} + 243 \, x^{5} e^{\left (-5 \, x + 25\right )} - 1215 \, x^{4} e^{\left (-4 \, x + 20\right )} \log \relax (5) + 2430 \, x^{3} e^{\left (-3 \, x + 15\right )} \log \relax (5)^{2} - 2430 \, x^{2} e^{\left (-2 \, x + 10\right )} \log \relax (5)^{3} + 1215 \, x e^{\left (-x + 5\right )} \log \relax (5)^{4} - 243 \, \log \relax (5)^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^3-25*x^2-15*x)*exp(5-x)*log(x)+(4*x^2+3*x)*exp(5-x)-3*log(5))/((4*x^3+3*x^2)*exp(5-x)-3*x*log
(5))/(1/3*(4*x^2+3*x)*exp(5-x)-log(5))^5,x, algorithm="giac")

[Out]

243*log(x)/(1024*x^10*e^(-5*x + 25) + 3840*x^9*e^(-5*x + 25) - 3840*x^8*e^(-4*x + 20)*log(5) + 5760*x^8*e^(-5*
x + 25) - 11520*x^7*e^(-4*x + 20)*log(5) + 5760*x^6*e^(-3*x + 15)*log(5)^2 + 4320*x^7*e^(-5*x + 25) - 12960*x^
6*e^(-4*x + 20)*log(5) + 12960*x^5*e^(-3*x + 15)*log(5)^2 - 4320*x^4*e^(-2*x + 10)*log(5)^3 + 1620*x^6*e^(-5*x
+ 25) - 6480*x^5*e^(-4*x + 20)*log(5) + 9720*x^4*e^(-3*x + 15)*log(5)^2 - 6480*x^3*e^(-2*x + 10)*log(5)^3 + 1
620*x^2*e^(-x + 5)*log(5)^4 + 243*x^5*e^(-5*x + 25) - 1215*x^4*e^(-4*x + 20)*log(5) + 2430*x^3*e^(-3*x + 15)*l
og(5)^2 - 2430*x^2*e^(-2*x + 10)*log(5)^3 + 1215*x*e^(-x + 5)*log(5)^4 - 243*log(5)^5)

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maple [A]  time = 0.03, size = 32, normalized size = 1.19

method	result	size
risch	\(-\frac {243 \ln \relax (x )}{\left (-4 \,{\mathrm e}^{5-x} x^{2}-3 x \,{\mathrm e}^{5-x}+3 \ln \relax (5)\right )^{5}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*x^3-25*x^2-15*x)*exp(5-x)*ln(x)+(4*x^2+3*x)*exp(5-x)-3*ln(5))/((4*x^3+3*x^2)*exp(5-x)-3*x*ln(5))/(1/3
*(4*x^2+3*x)*exp(5-x)-ln(5))^5,x,method=_RETURNVERBOSE)

[Out]

-243/(-4*exp(5-x)*x^2-3*x*exp(5-x)+3*ln(5))^5*ln(x)

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maxima [B]  time = 1.66, size = 490, normalized size = 18.15 \begin {gather*} \frac {405 \, {\left (4 \, x^{2} e^{5} \log \relax (5)^{4} + 3 \, x e^{5} \log \relax (5)^{4}\right )} e^{\left (4 \, x\right )} \log \relax (x) - 270 \, {\left (16 \, x^{4} e^{10} \log \relax (5)^{3} + 24 \, x^{3} e^{10} \log \relax (5)^{3} + 9 \, x^{2} e^{10} \log \relax (5)^{3}\right )} e^{\left (3 \, x\right )} \log \relax (x) + 90 \, {\left (64 \, x^{6} e^{15} \log \relax (5)^{2} + 144 \, x^{5} e^{15} \log \relax (5)^{2} + 108 \, x^{4} e^{15} \log \relax (5)^{2} + 27 \, x^{3} e^{15} \log \relax (5)^{2}\right )} e^{\left (2 \, x\right )} \log \relax (x) - 15 \, {\left (256 \, x^{8} e^{20} \log \relax (5) + 768 \, x^{7} e^{20} \log \relax (5) + 864 \, x^{6} e^{20} \log \relax (5) + 432 \, x^{5} e^{20} \log \relax (5) + 81 \, x^{4} e^{20} \log \relax (5)\right )} e^{x} \log \relax (x) + {\left (1024 \, x^{10} e^{25} + 3840 \, x^{9} e^{25} + 5760 \, x^{8} e^{25} + 4320 \, x^{7} e^{25} + 1620 \, x^{6} e^{25} + 243 \, x^{5} e^{25}\right )} \log \relax (x)}{1024 \, x^{10} e^{25} \log \relax (5)^{5} + 3840 \, x^{9} e^{25} \log \relax (5)^{5} + 5760 \, x^{8} e^{25} \log \relax (5)^{5} + 4320 \, x^{7} e^{25} \log \relax (5)^{5} + 1620 \, x^{6} e^{25} \log \relax (5)^{5} + 243 \, x^{5} e^{25} \log \relax (5)^{5} - 243 \, e^{\left (5 \, x\right )} \log \relax (5)^{10} + 405 \, {\left (4 \, x^{2} e^{5} \log \relax (5)^{9} + 3 \, x e^{5} \log \relax (5)^{9}\right )} e^{\left (4 \, x\right )} - 270 \, {\left (16 \, x^{4} e^{10} \log \relax (5)^{8} + 24 \, x^{3} e^{10} \log \relax (5)^{8} + 9 \, x^{2} e^{10} \log \relax (5)^{8}\right )} e^{\left (3 \, x\right )} + 90 \, {\left (64 \, x^{6} e^{15} \log \relax (5)^{7} + 144 \, x^{5} e^{15} \log \relax (5)^{7} + 108 \, x^{4} e^{15} \log \relax (5)^{7} + 27 \, x^{3} e^{15} \log \relax (5)^{7}\right )} e^{\left (2 \, x\right )} - 15 \, {\left (256 \, x^{8} e^{20} \log \relax (5)^{6} + 768 \, x^{7} e^{20} \log \relax (5)^{6} + 864 \, x^{6} e^{20} \log \relax (5)^{6} + 432 \, x^{5} e^{20} \log \relax (5)^{6} + 81 \, x^{4} e^{20} \log \relax (5)^{6}\right )} e^{x}} - \frac {\log \relax (x)}{\log \relax (5)^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^3-25*x^2-15*x)*exp(5-x)*log(x)+(4*x^2+3*x)*exp(5-x)-3*log(5))/((4*x^3+3*x^2)*exp(5-x)-3*x*log
(5))/(1/3*(4*x^2+3*x)*exp(5-x)-log(5))^5,x, algorithm="maxima")

[Out]

(405*(4*x^2*e^5*log(5)^4 + 3*x*e^5*log(5)^4)*e^(4*x)*log(x) - 270*(16*x^4*e^10*log(5)^3 + 24*x^3*e^10*log(5)^3
+ 9*x^2*e^10*log(5)^3)*e^(3*x)*log(x) + 90*(64*x^6*e^15*log(5)^2 + 144*x^5*e^15*log(5)^2 + 108*x^4*e^15*log(5
)^2 + 27*x^3*e^15*log(5)^2)*e^(2*x)*log(x) - 15*(256*x^8*e^20*log(5) + 768*x^7*e^20*log(5) + 864*x^6*e^20*log(
5) + 432*x^5*e^20*log(5) + 81*x^4*e^20*log(5))*e^x*log(x) + (1024*x^10*e^25 + 3840*x^9*e^25 + 5760*x^8*e^25 +
4320*x^7*e^25 + 1620*x^6*e^25 + 243*x^5*e^25)*log(x))/(1024*x^10*e^25*log(5)^5 + 3840*x^9*e^25*log(5)^5 + 5760
*x^8*e^25*log(5)^5 + 4320*x^7*e^25*log(5)^5 + 1620*x^6*e^25*log(5)^5 + 243*x^5*e^25*log(5)^5 - 243*e^(5*x)*log
(5)^10 + 405*(4*x^2*e^5*log(5)^9 + 3*x*e^5*log(5)^9)*e^(4*x) - 270*(16*x^4*e^10*log(5)^8 + 24*x^3*e^10*log(5)^
8 + 9*x^2*e^10*log(5)^8)*e^(3*x) + 90*(64*x^6*e^15*log(5)^7 + 144*x^5*e^15*log(5)^7 + 108*x^4*e^15*log(5)^7 +
27*x^3*e^15*log(5)^7)*e^(2*x) - 15*(256*x^8*e^20*log(5)^6 + 768*x^7*e^20*log(5)^6 + 864*x^6*e^20*log(5)^6 + 43
2*x^5*e^20*log(5)^6 + 81*x^4*e^20*log(5)^6)*e^x) - log(x)/log(5)^5

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {3\,\ln \relax (5)-{\mathrm {e}}^{5-x}\,\left (4\,x^2+3\,x\right )+{\mathrm {e}}^{5-x}\,\ln \relax (x)\,\left (-20\,x^3+25\,x^2+15\,x\right )}{{\left (\ln \relax (5)-\frac {{\mathrm {e}}^{5-x}\,\left (4\,x^2+3\,x\right )}{3}\right )}^5\,\left (3\,x\,\ln \relax (5)-{\mathrm {e}}^{5-x}\,\left (4\,x^3+3\,x^2\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*log(5) - exp(5 - x)*(3*x + 4*x^2) + exp(5 - x)*log(x)*(15*x + 25*x^2 - 20*x^3))/((log(5) - (exp(5 - x)
*(3*x + 4*x^2))/3)^5*(3*x*log(5) - exp(5 - x)*(3*x^2 + 4*x^3))),x)

[Out]

int(-(3*log(5) - exp(5 - x)*(3*x + 4*x^2) + exp(5 - x)*log(x)*(15*x + 25*x^2 - 20*x^3))/((log(5) - (exp(5 - x)
*(3*x + 4*x^2))/3)^5*(3*x*log(5) - exp(5 - x)*(3*x^2 + 4*x^3))), x)

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sympy [B]  time = 1.64, size = 209, normalized size = 7.74 \begin {gather*} \frac {243 \log {\relax (x )}}{\left (1620 x^{2} \log {\relax (5 )}^{4} + 1215 x \log {\relax (5 )}^{4}\right ) e^{5 - x} + \left (- 4320 x^{4} \log {\relax (5 )}^{3} - 6480 x^{3} \log {\relax (5 )}^{3} - 2430 x^{2} \log {\relax (5 )}^{3}\right ) e^{10 - 2 x} + \left (5760 x^{6} \log {\relax (5 )}^{2} + 12960 x^{5} \log {\relax (5 )}^{2} + 9720 x^{4} \log {\relax (5 )}^{2} + 2430 x^{3} \log {\relax (5 )}^{2}\right ) e^{15 - 3 x} + \left (- 3840 x^{8} \log {\relax (5 )} - 11520 x^{7} \log {\relax (5 )} - 12960 x^{6} \log {\relax (5 )} - 6480 x^{5} \log {\relax (5 )} - 1215 x^{4} \log {\relax (5 )}\right ) e^{20 - 4 x} + \left (1024 x^{10} + 3840 x^{9} + 5760 x^{8} + 4320 x^{7} + 1620 x^{6} + 243 x^{5}\right ) e^{25 - 5 x} - 243 \log {\relax (5 )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x**3-25*x**2-15*x)*exp(5-x)*ln(x)+(4*x**2+3*x)*exp(5-x)-3*ln(5))/((4*x**3+3*x**2)*exp(5-x)-3*x*
ln(5))/(1/3*(4*x**2+3*x)*exp(5-x)-ln(5))**5,x)

[Out]

243*log(x)/((1620*x**2*log(5)**4 + 1215*x*log(5)**4)*exp(5 - x) + (-4320*x**4*log(5)**3 - 6480*x**3*log(5)**3
- 2430*x**2*log(5)**3)*exp(10 - 2*x) + (5760*x**6*log(5)**2 + 12960*x**5*log(5)**2 + 9720*x**4*log(5)**2 + 243
0*x**3*log(5)**2)*exp(15 - 3*x) + (-3840*x**8*log(5) - 11520*x**7*log(5) - 12960*x**6*log(5) - 6480*x**5*log(5
) - 1215*x**4*log(5))*exp(20 - 4*x) + (1024*x**10 + 3840*x**9 + 5760*x**8 + 4320*x**7 + 1620*x**6 + 243*x**5)*
exp(25 - 5*x) - 243*log(5)**5)

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