3.13.76 \(\int \frac {e^{\frac {3-x+3 \log (\frac {-20-5 x-2 x^2}{6 x})}{\log (\frac {-20-5 x-2 x^2}{6 x})}} (60-20 x-6 x^2+2 x^3+(-20 x-5 x^2-2 x^3) \log (\frac {-20-5 x-2 x^2}{6 x}))}{(20 x+5 x^2+2 x^3) \log ^2(\frac {-20-5 x-2 x^2}{6 x})} \, dx\)

Optimal. Leaf size=31 \[ e^{3+\frac {3-x}{\log \left (-\frac {3}{2}+\frac {1}{3} \left (2-\frac {10}{x}-x\right )\right )}} \]

________________________________________________________________________________________

Rubi [F]  time = 13.47, 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 {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right ) \left (60-20 x-6 x^2+2 x^3+\left (-20 x-5 x^2-2 x^3\right ) \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )\right )}{\left (20 x+5 x^2+2 x^3\right ) \log ^2\left (\frac {-20-5 x-2 x^2}{6 x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((3 - x + 3*Log[(-20 - 5*x - 2*x^2)/(6*x)])/Log[(-20 - 5*x - 2*x^2)/(6*x)])*(60 - 20*x - 6*x^2 + 2*x^3
+ (-20*x - 5*x^2 - 2*x^3)*Log[(-20 - 5*x - 2*x^2)/(6*x)]))/((20*x + 5*x^2 + 2*x^3)*Log[(-20 - 5*x - 2*x^2)/(6*
x)]^2),x]

[Out]

Defer[Int][E^((3 - x + 3*Log[(-20 - 5*x - 2*x^2)/(6*x)])/Log[(-20 - 5*x - 2*x^2)/(6*x)])/Log[(-5 - 20/x - 2*x)
/6]^2, x] - ((44*I)/3)*Sqrt[5/3]*Defer[Int][E^((3 - x + 3*Log[(-20 - 5*x - 2*x^2)/(6*x)])/Log[(-20 - 5*x - 2*x
^2)/(6*x)])/((-5 + (3*I)*Sqrt[15] - 4*x)*Log[(-5 - 20/x - 2*x)/6]^2), x] + 3*Defer[Int][E^((3 - x + 3*Log[(-20
- 5*x - 2*x^2)/(6*x)])/Log[(-20 - 5*x - 2*x^2)/(6*x)])/(x*Log[(-5 - 20/x - 2*x)/6]^2), x] - (17*(9 + I*Sqrt[1
5])*Defer[Int][E^((3 - x + 3*Log[(-20 - 5*x - 2*x^2)/(6*x)])/Log[(-20 - 5*x - 2*x^2)/(6*x)])/((5 - (3*I)*Sqrt[
15] + 4*x)*Log[(-5 - 20/x - 2*x)/6]^2), x])/9 - ((44*I)/3)*Sqrt[5/3]*Defer[Int][E^((3 - x + 3*Log[(-20 - 5*x -
2*x^2)/(6*x)])/Log[(-20 - 5*x - 2*x^2)/(6*x)])/((5 + (3*I)*Sqrt[15] + 4*x)*Log[(-5 - 20/x - 2*x)/6]^2), x] -
(17*(9 - I*Sqrt[15])*Defer[Int][E^((3 - x + 3*Log[(-20 - 5*x - 2*x^2)/(6*x)])/Log[(-20 - 5*x - 2*x^2)/(6*x)])/
((5 + (3*I)*Sqrt[15] + 4*x)*Log[(-5 - 20/x - 2*x)/6]^2), x])/9 - Defer[Int][E^((3 - x + 3*Log[(-20 - 5*x - 2*x
^2)/(6*x)])/Log[(-20 - 5*x - 2*x^2)/(6*x)])/Log[(-5 - 20/x - 2*x)/6], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right ) \left (60-20 x-6 x^2+2 x^3+\left (-20 x-5 x^2-2 x^3\right ) \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )\right )}{x \left (20+5 x+2 x^2\right ) \log ^2\left (\frac {-20-5 x-2 x^2}{6 x}\right )} \, dx\\ &=\int \left (\frac {2 \exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right ) \left (30-10 x-3 x^2+x^3\right )}{x \left (20+5 x+2 x^2\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )}-\frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\log \left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )}\right ) \, dx\\ &=2 \int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right ) \left (30-10 x-3 x^2+x^3\right )}{x \left (20+5 x+2 x^2\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx-\int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\log \left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx\\ &=2 \int \left (\frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{2 \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )}+\frac {3 \exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{2 x \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )}+\frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right ) (-55-17 x)}{2 \left (20+5 x+2 x^2\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )}\right ) \, dx-\int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\log \left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx\\ &=3 \int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{x \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx+\int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx+\int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right ) (-55-17 x)}{\left (20+5 x+2 x^2\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx-\int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\log \left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx\\ &=3 \int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{x \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx+\int \left (-\frac {55 \exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\left (20+5 x+2 x^2\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )}-\frac {17 \exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right ) x}{\left (20+5 x+2 x^2\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )}\right ) \, dx+\int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx-\int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\log \left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx\\ &=3 \int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{x \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx-17 \int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right ) x}{\left (20+5 x+2 x^2\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx-55 \int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\left (20+5 x+2 x^2\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx+\int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx-\int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\log \left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx\\ &=3 \int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{x \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx-17 \int \left (\frac {\left (1+\frac {1}{3} i \sqrt {\frac {5}{3}}\right ) \exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\left (5-3 i \sqrt {15}+4 x\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )}+\frac {\left (1-\frac {1}{3} i \sqrt {\frac {5}{3}}\right ) \exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\left (5+3 i \sqrt {15}+4 x\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )}\right ) \, dx-55 \int \left (\frac {4 i \exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{3 \sqrt {15} \left (-5+3 i \sqrt {15}-4 x\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )}+\frac {4 i \exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{3 \sqrt {15} \left (5+3 i \sqrt {15}+4 x\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )}\right ) \, dx+\int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx-\int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\log \left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx\\ &=3 \int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{x \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx-\frac {1}{3} \left (44 i \sqrt {\frac {5}{3}}\right ) \int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\left (-5+3 i \sqrt {15}-4 x\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx-\frac {1}{3} \left (44 i \sqrt {\frac {5}{3}}\right ) \int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\left (5+3 i \sqrt {15}+4 x\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx-\frac {1}{9} \left (17 \left (9-i \sqrt {15}\right )\right ) \int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\left (5+3 i \sqrt {15}+4 x\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx-\frac {1}{9} \left (17 \left (9+i \sqrt {15}\right )\right ) \int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\left (5-3 i \sqrt {15}+4 x\right ) \log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx+\int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\log ^2\left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx-\int \frac {\exp \left (\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}\right )}{\log \left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 27, normalized size = 0.87 \begin {gather*} e^{3+\frac {3-x}{\log \left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((3 - x + 3*Log[(-20 - 5*x - 2*x^2)/(6*x)])/Log[(-20 - 5*x - 2*x^2)/(6*x)])*(60 - 20*x - 6*x^2 +
2*x^3 + (-20*x - 5*x^2 - 2*x^3)*Log[(-20 - 5*x - 2*x^2)/(6*x)]))/((20*x + 5*x^2 + 2*x^3)*Log[(-20 - 5*x - 2*x^
2)/(6*x)]^2),x]

[Out]

E^(3 + (3 - x)/Log[(-5 - 20/x - 2*x)/6])

________________________________________________________________________________________

fricas [A]  time = 0.80, size = 42, normalized size = 1.35 \begin {gather*} e^{\left (-\frac {x - 3 \, \log \left (-\frac {2 \, x^{2} + 5 \, x + 20}{6 \, x}\right ) - 3}{\log \left (-\frac {2 \, x^{2} + 5 \, x + 20}{6 \, x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-5*x^2-20*x)*log(1/6*(-2*x^2-5*x-20)/x)+2*x^3-6*x^2-20*x+60)*exp((3*log(1/6*(-2*x^2-5*x-20)/
x)+3-x)/log(1/6*(-2*x^2-5*x-20)/x))/(2*x^3+5*x^2+20*x)/log(1/6*(-2*x^2-5*x-20)/x)^2,x, algorithm="fricas")

[Out]

e^(-(x - 3*log(-1/6*(2*x^2 + 5*x + 20)/x) - 3)/log(-1/6*(2*x^2 + 5*x + 20)/x))

________________________________________________________________________________________

giac [A]  time = 10.68, size = 34, normalized size = 1.10 \begin {gather*} e^{\left (-\frac {x}{\log \left (-\frac {1}{3} \, x - \frac {10}{3 \, x} - \frac {5}{6}\right )} + \frac {3}{\log \left (-\frac {1}{3} \, x - \frac {10}{3 \, x} - \frac {5}{6}\right )} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-5*x^2-20*x)*log(1/6*(-2*x^2-5*x-20)/x)+2*x^3-6*x^2-20*x+60)*exp((3*log(1/6*(-2*x^2-5*x-20)/
x)+3-x)/log(1/6*(-2*x^2-5*x-20)/x))/(2*x^3+5*x^2+20*x)/log(1/6*(-2*x^2-5*x-20)/x)^2,x, algorithm="giac")

[Out]

e^(-x/log(-1/3*x - 10/3/x - 5/6) + 3/log(-1/3*x - 10/3/x - 5/6) + 3)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 43, normalized size = 1.39

method	result	size
risch	\({\mathrm e}^{-\frac {-3 \ln \left (\frac {-2 x^{2}-5 x -20}{6 x}\right )-3+x}{\ln \left (\frac {-2 x^{2}-5 x -20}{6 x}\right )}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3-5*x^2-20*x)*ln(1/6*(-2*x^2-5*x-20)/x)+2*x^3-6*x^2-20*x+60)*exp((3*ln(1/6*(-2*x^2-5*x-20)/x)+3-x)/
ln(1/6*(-2*x^2-5*x-20)/x))/(2*x^3+5*x^2+20*x)/ln(1/6*(-2*x^2-5*x-20)/x)^2,x,method=_RETURNVERBOSE)

[Out]

exp(-(-3*ln(1/6*(-2*x^2-5*x-20)/x)-3+x)/ln(1/6*(-2*x^2-5*x-20)/x))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{3} - 6 \, x^{2} - {\left (2 \, x^{3} + 5 \, x^{2} + 20 \, x\right )} \log \left (-\frac {2 \, x^{2} + 5 \, x + 20}{6 \, x}\right ) - 20 \, x + 60\right )} e^{\left (-\frac {x - 3 \, \log \left (-\frac {2 \, x^{2} + 5 \, x + 20}{6 \, x}\right ) - 3}{\log \left (-\frac {2 \, x^{2} + 5 \, x + 20}{6 \, x}\right )}\right )}}{{\left (2 \, x^{3} + 5 \, x^{2} + 20 \, x\right )} \log \left (-\frac {2 \, x^{2} + 5 \, x + 20}{6 \, x}\right )^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-5*x^2-20*x)*log(1/6*(-2*x^2-5*x-20)/x)+2*x^3-6*x^2-20*x+60)*exp((3*log(1/6*(-2*x^2-5*x-20)/
x)+3-x)/log(1/6*(-2*x^2-5*x-20)/x))/(2*x^3+5*x^2+20*x)/log(1/6*(-2*x^2-5*x-20)/x)^2,x, algorithm="maxima")

[Out]

integrate((2*x^3 - 6*x^2 - (2*x^3 + 5*x^2 + 20*x)*log(-1/6*(2*x^2 + 5*x + 20)/x) - 20*x + 60)*e^(-(x - 3*log(-
1/6*(2*x^2 + 5*x + 20)/x) - 3)/log(-1/6*(2*x^2 + 5*x + 20)/x))/((2*x^3 + 5*x^2 + 20*x)*log(-1/6*(2*x^2 + 5*x +
20)/x)^2), x)

________________________________________________________________________________________

mupad [B]  time = 1.39, size = 101, normalized size = 3.26 \begin {gather*} {\mathrm {e}}^{\frac {3}{\ln \left (-\frac {2\,x^2+5\,x+20}{6\,x}\right )}-\frac {x}{\ln \left (-\frac {2\,x^2+5\,x+20}{6\,x}\right )}}\,{\left (-\frac {1}{6\,x}\right )}^{\frac {3}{\ln \left (-\frac {2\,x^2+5\,x+20}{6\,x}\right )}}\,{\left (2\,x^2+5\,x+20\right )}^{\frac {3}{\ln \left (-\frac {2\,x^2+5\,x+20}{6\,x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((3*log(-((5*x)/6 + x^2/3 + 10/3)/x) - x + 3)/log(-((5*x)/6 + x^2/3 + 10/3)/x))*(20*x + log(-((5*x)/6
+ x^2/3 + 10/3)/x)*(20*x + 5*x^2 + 2*x^3) + 6*x^2 - 2*x^3 - 60))/(log(-((5*x)/6 + x^2/3 + 10/3)/x)^2*(20*x +
5*x^2 + 2*x^3)),x)

[Out]

exp(3/log(-(5*x + 2*x^2 + 20)/(6*x)) - x/log(-(5*x + 2*x^2 + 20)/(6*x)))*(-1/(6*x))^(3/log(-(5*x + 2*x^2 + 20)
/(6*x)))*(5*x + 2*x^2 + 20)^(3/log(-(5*x + 2*x^2 + 20)/(6*x)))

________________________________________________________________________________________

sympy [B]  time = 0.63, size = 42, normalized size = 1.35 \begin {gather*} e^{\frac {- x + 3 \log {\left (\frac {- \frac {x^{2}}{3} - \frac {5 x}{6} - \frac {10}{3}}{x} \right )} + 3}{\log {\left (\frac {- \frac {x^{2}}{3} - \frac {5 x}{6} - \frac {10}{3}}{x} \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3-5*x**2-20*x)*ln(1/6*(-2*x**2-5*x-20)/x)+2*x**3-6*x**2-20*x+60)*exp((3*ln(1/6*(-2*x**2-5*x-
20)/x)+3-x)/ln(1/6*(-2*x**2-5*x-20)/x))/(2*x**3+5*x**2+20*x)/ln(1/6*(-2*x**2-5*x-20)/x)**2,x)

[Out]

exp((-x + 3*log((-x**2/3 - 5*x/6 - 10/3)/x) + 3)/log((-x**2/3 - 5*x/6 - 10/3)/x))

________________________________________________________________________________________