\(\int \frac {e^x (2 x+2 x^2+(-8 x^4-2 x^5) \log ^2(5))+e^x (-2 x-x^2+(2 x^4+x^5) \log ^2(5)) \log (e^x (-x+x^4 \log ^2(5)))}{(-1+x^3 \log ^2(5)) \log ^3(e^x (-x+x^4 \log ^2(5)))} \, dx\) [8605]

Optimal result

Integrand size = 106, antiderivative size = 26 \[ \int \frac {e^x \left (2 x+2 x^2+\left (-8 x^4-2 x^5\right ) \log ^2(5)\right )+e^x \left (-2 x-x^2+\left (2 x^4+x^5\right ) \log ^2(5)\right ) \log \left (e^x \left (-x+x^4 \log ^2(5)\right )\right )}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x \left (-x+x^4 \log ^2(5)\right )\right )} \, dx=\frac {e^x x^2}{\log ^2\left (e^x \left (-x+x^4 \log ^2(5)\right )\right )} \]

[Out]

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

Rubi [F]

\[ \int \frac {e^x \left (2 x+2 x^2+\left (-8 x^4-2 x^5\right ) \log ^2(5)\right )+e^x \left (-2 x-x^2+\left (2 x^4+x^5\right ) \log ^2(5)\right ) \log \left (e^x \left (-x+x^4 \log ^2(5)\right )\right )}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x \left (-x+x^4 \log ^2(5)\right )\right )} \, dx=\int \frac {e^x \left (2 x+2 x^2+\left (-8 x^4-2 x^5\right ) \log ^2(5)\right )+e^x \left (-2 x-x^2+\left (2 x^4+x^5\right ) \log ^2(5)\right ) \log \left (e^x \left (-x+x^4 \log ^2(5)\right )\right )}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x \left (-x+x^4 \log ^2(5)\right )\right )} \, dx \]

[In]

Int[(E^x*(2*x + 2*x^2 + (-8*x^4 - 2*x^5)*Log[5]^2) + E^x*(-2*x - x^2 + (2*x^4 + x^5)*Log[5]^2)*Log[E^x*(-x + x
^4*Log[5]^2)])/((-1 + x^3*Log[5]^2)*Log[E^x*(-x + x^4*Log[5]^2)]^3),x]

[Out]

-8*Defer[Int][(E^x*x)/Log[E^x*x*(-1 + x^3*Log[5]^2)]^3, x] - 2*Defer[Int][(E^x*x^2)/Log[E^x*x*(-1 + x^3*Log[5]
^2)]^3, x] - (2*(-1)^(1/3)*Defer[Int][E^x/((1 - x*(-Log[5])^(2/3))*Log[E^x*x*(-1 + x^3*Log[5]^2)]^3), x])/Log[
5]^(2/3) + (2*Defer[Int][E^x/((1 - x*Log[5]^(2/3))*Log[E^x*x*(-1 + x^3*Log[5]^2)]^3), x])/Log[5]^(2/3) + 2*(-L
og[5]^(-1))^(2/3)*Defer[Int][E^x/((1 + (-1)^(1/3)*x*Log[5]^(2/3))*Log[E^x*x*(-1 + x^3*Log[5]^2)]^3), x] + 2*De
fer[Int][(E^x*x)/Log[E^x*x*(-1 + x^3*Log[5]^2)]^2, x] + Defer[Int][(E^x*x^2)/Log[E^x*x*(-1 + x^3*Log[5]^2)]^2,
x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 e^x x}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}+\frac {2 e^x x^2}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}-\frac {8 e^x x^4 \log ^2(5)}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}-\frac {2 e^x x^5 \log ^2(5)}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}-\frac {2 e^x x}{\left (-1+x^3 \log ^2(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}-\frac {e^x x^2}{\left (-1+x^3 \log ^2(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}+\frac {2 e^x x^4 \log ^2(5)}{\left (-1+x^3 \log ^2(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}+\frac {e^x x^5 \log ^2(5)}{\left (-1+x^3 \log ^2(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}\right ) \, dx \\ & = 2 \int \frac {e^x x}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx+2 \int \frac {e^x x^2}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx-2 \int \frac {e^x x}{\left (-1+x^3 \log ^2(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx+\log ^2(5) \int \frac {e^x x^5}{\left (-1+x^3 \log ^2(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx-\left (2 \log ^2(5)\right ) \int \frac {e^x x^5}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx+\left (2 \log ^2(5)\right ) \int \frac {e^x x^4}{\left (-1+x^3 \log ^2(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx-\left (8 \log ^2(5)\right ) \int \frac {e^x x^4}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx-\int \frac {e^x x^2}{\left (-1+x^3 \log ^2(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx \\ & = 2 \int \left (-\frac {e^x}{3 \left (1-x \log ^{\frac {2}{3}}(5)\right ) \log ^{\frac {4}{3}}(5) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}-\frac {e^x}{3 \left (-\sqrt [3]{-1}-x \log ^{\frac {2}{3}}(5)\right ) \log ^{\frac {4}{3}}(5) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}-\frac {e^x}{3 \left ((-1)^{2/3}-x \log ^{\frac {2}{3}}(5)\right ) \log ^{\frac {4}{3}}(5) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}\right ) \, dx+2 \int \left (-\frac {e^x \left (-\frac {1}{\log (5)}\right )^{2/3}}{3 \left (1+\sqrt [3]{-1} x \log ^{\frac {2}{3}}(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}+\frac {\sqrt [3]{-1} e^x}{3 \left (1-x (-\log (5))^{2/3}\right ) \log ^{\frac {2}{3}}(5) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}-\frac {e^x}{3 \left (1-x \log ^{\frac {2}{3}}(5)\right ) \log ^{\frac {2}{3}}(5) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}\right ) \, dx-2 \int \left (-\frac {e^x \left (-\frac {1}{\log (5)}\right )^{2/3}}{3 \left (1+\sqrt [3]{-1} x \log ^{\frac {2}{3}}(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}+\frac {\sqrt [3]{-1} e^x}{3 \left (1-x (-\log (5))^{2/3}\right ) \log ^{\frac {2}{3}}(5) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}-\frac {e^x}{3 \left (1-x \log ^{\frac {2}{3}}(5)\right ) \log ^{\frac {2}{3}}(5) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}\right ) \, dx+\log ^2(5) \int \left (\frac {e^x x^2}{\log ^2(5) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}+\frac {e^x x^2}{\log ^2(5) \left (-1+x^3 \log ^2(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}\right ) \, dx-\left (2 \log ^2(5)\right ) \int \left (\frac {e^x x^2}{\log ^2(5) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}+\frac {e^x x^2}{\log ^2(5) \left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}\right ) \, dx+\left (2 \log ^2(5)\right ) \int \left (\frac {e^x x}{\log ^2(5) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}+\frac {e^x x}{\log ^2(5) \left (-1+x^3 \log ^2(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}\right ) \, dx-\left (8 \log ^2(5)\right ) \int \left (\frac {e^x x}{\log ^2(5) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}+\frac {e^x x}{\log ^2(5) \left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}\right ) \, dx-\int \left (-\frac {e^x}{3 \left (1-x \log ^{\frac {2}{3}}(5)\right ) \log ^{\frac {4}{3}}(5) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}-\frac {e^x}{3 \left (-\sqrt [3]{-1}-x \log ^{\frac {2}{3}}(5)\right ) \log ^{\frac {4}{3}}(5) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}-\frac {e^x}{3 \left ((-1)^{2/3}-x \log ^{\frac {2}{3}}(5)\right ) \log ^{\frac {4}{3}}(5) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )}\right ) \, dx \\ & = -\left (2 \int \frac {e^x x^2}{\log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx\right )-2 \int \frac {e^x x^2}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx+2 \int \frac {e^x x}{\log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx+2 \int \frac {e^x x}{\left (-1+x^3 \log ^2(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx-8 \int \frac {e^x x}{\log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx-8 \int \frac {e^x x}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx-\frac {1}{3} \left (2 \left (-\frac {1}{\log (5)}\right )^{2/3}\right ) \int \frac {e^x}{\left (1+\sqrt [3]{-1} x \log ^{\frac {2}{3}}(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx+\frac {1}{3} \left (2 \left (-\frac {1}{\log (5)}\right )^{2/3}\right ) \int \frac {e^x}{\left (1+\sqrt [3]{-1} x \log ^{\frac {2}{3}}(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx+\frac {\int \frac {e^x}{\left (1-x \log ^{\frac {2}{3}}(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx}{3 \log ^{\frac {4}{3}}(5)}+\frac {\int \frac {e^x}{\left (-\sqrt [3]{-1}-x \log ^{\frac {2}{3}}(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx}{3 \log ^{\frac {4}{3}}(5)}+\frac {\int \frac {e^x}{\left ((-1)^{2/3}-x \log ^{\frac {2}{3}}(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx}{3 \log ^{\frac {4}{3}}(5)}-\frac {2 \int \frac {e^x}{\left (1-x \log ^{\frac {2}{3}}(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx}{3 \log ^{\frac {4}{3}}(5)}-\frac {2 \int \frac {e^x}{\left (-\sqrt [3]{-1}-x \log ^{\frac {2}{3}}(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx}{3 \log ^{\frac {4}{3}}(5)}-\frac {2 \int \frac {e^x}{\left ((-1)^{2/3}-x \log ^{\frac {2}{3}}(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx}{3 \log ^{\frac {4}{3}}(5)}-\frac {2 \int \frac {e^x}{\left (1-x \log ^{\frac {2}{3}}(5)\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx}{3 \log ^{\frac {2}{3}}(5)}+\frac {2 \int \frac {e^x}{\left (1-x \log ^{\frac {2}{3}}(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx}{3 \log ^{\frac {2}{3}}(5)}+\frac {\left (2 \sqrt [3]{-1}\right ) \int \frac {e^x}{\left (1-x (-\log (5))^{2/3}\right ) \log ^3\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx}{3 \log ^{\frac {2}{3}}(5)}-\frac {\left (2 \sqrt [3]{-1}\right ) \int \frac {e^x}{\left (1-x (-\log (5))^{2/3}\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx}{3 \log ^{\frac {2}{3}}(5)}+\int \frac {e^x x^2}{\log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx+\int \frac {e^x x^2}{\left (-1+x^3 \log ^2(5)\right ) \log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {e^x \left (2 x+2 x^2+\left (-8 x^4-2 x^5\right ) \log ^2(5)\right )+e^x \left (-2 x-x^2+\left (2 x^4+x^5\right ) \log ^2(5)\right ) \log \left (e^x \left (-x+x^4 \log ^2(5)\right )\right )}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x \left (-x+x^4 \log ^2(5)\right )\right )} \, dx=\frac {e^x x^2}{\log ^2\left (e^x x \left (-1+x^3 \log ^2(5)\right )\right )} \]

[In]

Integrate[(E^x*(2*x + 2*x^2 + (-8*x^4 - 2*x^5)*Log[5]^2) + E^x*(-2*x - x^2 + (2*x^4 + x^5)*Log[5]^2)*Log[E^x*(
-x + x^4*Log[5]^2)])/((-1 + x^3*Log[5]^2)*Log[E^x*(-x + x^4*Log[5]^2)]^3),x]

[Out]

(E^x*x^2)/Log[E^x*x*(-1 + x^3*Log[5]^2)]^2

Maple [A] (verified)

Time = 4.75 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{x} x^{2}}{{\ln \left (\left (x^{4} \ln \left (5\right )^{2}-x \right ) {\mathrm e}^{x}\right )}^{2}}\)	\(25\)
risch	\(-\frac {4 \,{\mathrm e}^{x} x^{2}}{{\left (\pi \,\operatorname {csgn}\left (i \left (x^{3} \ln \left (5\right )^{2}-1\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{x} \left (x^{3} \ln \left (5\right )^{2}-1\right )\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (x^{3} \ln \left (5\right )^{2}-1\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{x} \left (x^{3} \ln \left (5\right )^{2}-1\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )-\pi \operatorname {csgn}\left (i {\mathrm e}^{x} \left (x^{3} \ln \left (5\right )^{2}-1\right )\right )^{3}+\pi \operatorname {csgn}\left (i {\mathrm e}^{x} \left (x^{3} \ln \left (5\right )^{2}-1\right )\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )+\pi \,\operatorname {csgn}\left (i {\mathrm e}^{x} \left (x^{3} \ln \left (5\right )^{2}-1\right )\right ) {\operatorname {csgn}\left (i x \left (x^{3} \ln \left (5\right )^{2}-1\right ) {\mathrm e}^{x}\right )}^{2}-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{x} \left (x^{3} \ln \left (5\right )^{2}-1\right )\right ) \operatorname {csgn}\left (i x \left (x^{3} \ln \left (5\right )^{2}-1\right ) {\mathrm e}^{x}\right ) \operatorname {csgn}\left (i x \right )-\pi {\operatorname {csgn}\left (i x \left (x^{3} \ln \left (5\right )^{2}-1\right ) {\mathrm e}^{x}\right )}^{3}+\pi {\operatorname {csgn}\left (i x \left (x^{3} \ln \left (5\right )^{2}-1\right ) {\mathrm e}^{x}\right )}^{2} \operatorname {csgn}\left (i x \right )-2 i \ln \left (x \right )-2 i \ln \left (x^{3} \ln \left (5\right )^{2}-1\right )-2 i \ln \left ({\mathrm e}^{x}\right )\right )}^{2}}\)	\(282\)

[In]

int((((x^5+2*x^4)*ln(5)^2-x^2-2*x)*exp(x)*ln((x^4*ln(5)^2-x)*exp(x))+((-2*x^5-8*x^4)*ln(5)^2+2*x^2+2*x)*exp(x)
)/(x^3*ln(5)^2-1)/ln((x^4*ln(5)^2-x)*exp(x))^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {e^x \left (2 x+2 x^2+\left (-8 x^4-2 x^5\right ) \log ^2(5)\right )+e^x \left (-2 x-x^2+\left (2 x^4+x^5\right ) \log ^2(5)\right ) \log \left (e^x \left (-x+x^4 \log ^2(5)\right )\right )}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x \left (-x+x^4 \log ^2(5)\right )\right )} \, dx=\frac {x^{2} e^{x}}{\log \left ({\left (x^{4} \log \left (5\right )^{2} - x\right )} e^{x}\right )^{2}} \]

[In]

integrate((((x^5+2*x^4)*log(5)^2-x^2-2*x)*exp(x)*log((x^4*log(5)^2-x)*exp(x))+((-2*x^5-8*x^4)*log(5)^2+2*x^2+2
*x)*exp(x))/(x^3*log(5)^2-1)/log((x^4*log(5)^2-x)*exp(x))^3,x, algorithm="fricas")

[Out]

x^2*e^x/log((x^4*log(5)^2 - x)*e^x)^2

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {e^x \left (2 x+2 x^2+\left (-8 x^4-2 x^5\right ) \log ^2(5)\right )+e^x \left (-2 x-x^2+\left (2 x^4+x^5\right ) \log ^2(5)\right ) \log \left (e^x \left (-x+x^4 \log ^2(5)\right )\right )}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x \left (-x+x^4 \log ^2(5)\right )\right )} \, dx=\frac {x^{2} e^{x}}{\log {\left (\left (x^{4} \log {\left (5 \right )}^{2} - x\right ) e^{x} \right )}^{2}} \]

[In]

integrate((((x**5+2*x**4)*ln(5)**2-x**2-2*x)*exp(x)*ln((x**4*ln(5)**2-x)*exp(x))+((-2*x**5-8*x**4)*ln(5)**2+2*
x**2+2*x)*exp(x))/(x**3*ln(5)**2-1)/ln((x**4*ln(5)**2-x)*exp(x))**3,x)

[Out]

x**2*exp(x)/log((x**4*log(5)**2 - x)*exp(x))**2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).

Time = 0.36 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {e^x \left (2 x+2 x^2+\left (-8 x^4-2 x^5\right ) \log ^2(5)\right )+e^x \left (-2 x-x^2+\left (2 x^4+x^5\right ) \log ^2(5)\right ) \log \left (e^x \left (-x+x^4 \log ^2(5)\right )\right )}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x \left (-x+x^4 \log ^2(5)\right )\right )} \, dx=\frac {x^{2} e^{x}}{x^{2} + 2 \, {\left (x + \log \left (x\right )\right )} \log \left (x^{3} \log \left (5\right )^{2} - 1\right ) + \log \left (x^{3} \log \left (5\right )^{2} - 1\right )^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} \]

[In]

integrate((((x^5+2*x^4)*log(5)^2-x^2-2*x)*exp(x)*log((x^4*log(5)^2-x)*exp(x))+((-2*x^5-8*x^4)*log(5)^2+2*x^2+2
*x)*exp(x))/(x^3*log(5)^2-1)/log((x^4*log(5)^2-x)*exp(x))^3,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {e^x \left (2 x+2 x^2+\left (-8 x^4-2 x^5\right ) \log ^2(5)\right )+e^x \left (-2 x-x^2+\left (2 x^4+x^5\right ) \log ^2(5)\right ) \log \left (e^x \left (-x+x^4 \log ^2(5)\right )\right )}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x \left (-x+x^4 \log ^2(5)\right )\right )} \, dx=\frac {x^{2} e^{x}}{x^{2} + 2 \, x \log \left (x^{4} \log \left (5\right )^{2} - x\right ) + \log \left (x^{4} \log \left (5\right )^{2} - x\right )^{2}} \]

[In]

integrate((((x^5+2*x^4)*log(5)^2-x^2-2*x)*exp(x)*log((x^4*log(5)^2-x)*exp(x))+((-2*x^5-8*x^4)*log(5)^2+2*x^2+2
*x)*exp(x))/(x^3*log(5)^2-1)/log((x^4*log(5)^2-x)*exp(x))^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 15.50 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {e^x \left (2 x+2 x^2+\left (-8 x^4-2 x^5\right ) \log ^2(5)\right )+e^x \left (-2 x-x^2+\left (2 x^4+x^5\right ) \log ^2(5)\right ) \log \left (e^x \left (-x+x^4 \log ^2(5)\right )\right )}{\left (-1+x^3 \log ^2(5)\right ) \log ^3\left (e^x \left (-x+x^4 \log ^2(5)\right )\right )} \, dx=\frac {x^2\,{\mathrm {e}}^x}{{\ln \left (-{\mathrm {e}}^x\,\left (x-x^4\,{\ln \left (5\right )}^2\right )\right )}^2} \]

[In]

int((exp(x)*(2*x + 2*x^2 - log(5)^2*(8*x^4 + 2*x^5)) - exp(x)*log(-exp(x)*(x - x^4*log(5)^2))*(2*x - log(5)^2*
(2*x^4 + x^5) + x^2))/(log(-exp(x)*(x - x^4*log(5)^2))^3*(x^3*log(5)^2 - 1)),x)

[Out]

(x^2*exp(x))/log(-exp(x)*(x - x^4*log(5)^2))^2