\(\int \frac {(2508+4 x^2+32 x^3) \log ^3(\frac {-627+x^2+4 x^3}{x})}{e^{12} (-50787 x+81 x^3+324 x^4)} \, dx\) [2547]

Optimal result

Integrand size = 49, antiderivative size = 22 \[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\frac {\log ^4\left (-\frac {627}{x}+x+4 x^2\right )}{81 e^{12}} \]

[Out]

1/81*ln(x+4*x^2-627/x)^4/exp(3)^4

Rubi [F]

\[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx \]

[In]

Int[((2508 + 4*x^2 + 32*x^3)*Log[(-627 + x^2 + 4*x^3)/x]^3)/(E^12*(-50787*x + 81*x^3 + 324*x^4)),x]

[Out]

(-4*Log[x]*Log[-((627 - x^2 - 4*x^3)/x)]^3)/(81*E^12) - (4*Defer[Int][(Log[x]*Log[(-627 + x^2 + 4*x^3)/x]^2)/x
, x])/(27*E^12) + (8*Defer[Int][(x*Log[x]*Log[(-627 + x^2 + 4*x^3)/x]^2)/(-627 + x^2 + 4*x^3), x])/(27*E^12) +
(16*Defer[Int][(x^2*Log[x]*Log[(-627 + x^2 + 4*x^3)/x]^2)/(-627 + x^2 + 4*x^3), x])/(9*E^12) + (8*Defer[Int][
(x*Log[(-627 + x^2 + 4*x^3)/x]^3)/(-627 + x^2 + 4*x^3), x])/(81*E^12) + (16*Defer[Int][(x^2*Log[(-627 + x^2 +
4*x^3)/x]^3)/(-627 + x^2 + 4*x^3), x])/(27*E^12)

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{-50787 x+81 x^3+324 x^4} \, dx}{e^{12}} \\ & = \frac {\int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{x \left (-50787+81 x^2+324 x^3\right )} \, dx}{e^{12}} \\ & = \frac {\int \left (-\frac {4 \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{81 x}+\frac {8 x (1+6 x) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{81 \left (-627+x^2+4 x^3\right )}\right ) \, dx}{e^{12}} \\ & = -\frac {4 \int \frac {\log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{x} \, dx}{81 e^{12}}+\frac {8 \int \frac {x (1+6 x) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3} \, dx}{81 e^{12}} \\ & = -\frac {4 \log (x) \log ^3\left (-\frac {627-x^2-4 x^3}{x}\right )}{81 e^{12}}+\frac {8 \int \left (\frac {x \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3}+\frac {6 x^2 \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3}\right ) \, dx}{81 e^{12}}+\frac {4 \int \frac {x \left (\frac {2 x+12 x^2}{x}-\frac {-627+x^2+4 x^3}{x^2}\right ) \log (x) \log ^2\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3} \, dx}{27 e^{12}} \\ & = -\frac {4 \log (x) \log ^3\left (-\frac {627-x^2-4 x^3}{x}\right )}{81 e^{12}}+\frac {8 \int \frac {x \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3} \, dx}{81 e^{12}}+\frac {4 \int \frac {\left (-627-x^2-8 x^3\right ) \log (x) \log ^2\left (\frac {-627+x^2+4 x^3}{x}\right )}{x \left (627-x^2-4 x^3\right )} \, dx}{27 e^{12}}+\frac {16 \int \frac {x^2 \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3} \, dx}{27 e^{12}} \\ & = -\frac {4 \log (x) \log ^3\left (-\frac {627-x^2-4 x^3}{x}\right )}{81 e^{12}}+\frac {8 \int \frac {x \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3} \, dx}{81 e^{12}}+\frac {4 \int \left (-\frac {\log (x) \log ^2\left (\frac {-627+x^2+4 x^3}{x}\right )}{x}+\frac {2 (-1-6 x) x \log (x) \log ^2\left (\frac {-627+x^2+4 x^3}{x}\right )}{627-x^2-4 x^3}\right ) \, dx}{27 e^{12}}+\frac {16 \int \frac {x^2 \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3} \, dx}{27 e^{12}} \\ & = -\frac {4 \log (x) \log ^3\left (-\frac {627-x^2-4 x^3}{x}\right )}{81 e^{12}}+\frac {8 \int \frac {x \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3} \, dx}{81 e^{12}}-\frac {4 \int \frac {\log (x) \log ^2\left (\frac {-627+x^2+4 x^3}{x}\right )}{x} \, dx}{27 e^{12}}+\frac {8 \int \frac {(-1-6 x) x \log (x) \log ^2\left (\frac {-627+x^2+4 x^3}{x}\right )}{627-x^2-4 x^3} \, dx}{27 e^{12}}+\frac {16 \int \frac {x^2 \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3} \, dx}{27 e^{12}} \\ & = -\frac {4 \log (x) \log ^3\left (-\frac {627-x^2-4 x^3}{x}\right )}{81 e^{12}}+\frac {8 \int \frac {x \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3} \, dx}{81 e^{12}}-\frac {4 \int \frac {\log (x) \log ^2\left (\frac {-627+x^2+4 x^3}{x}\right )}{x} \, dx}{27 e^{12}}+\frac {8 \int \left (\frac {x \log (x) \log ^2\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3}+\frac {6 x^2 \log (x) \log ^2\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3}\right ) \, dx}{27 e^{12}}+\frac {16 \int \frac {x^2 \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3} \, dx}{27 e^{12}} \\ & = -\frac {4 \log (x) \log ^3\left (-\frac {627-x^2-4 x^3}{x}\right )}{81 e^{12}}+\frac {8 \int \frac {x \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3} \, dx}{81 e^{12}}-\frac {4 \int \frac {\log (x) \log ^2\left (\frac {-627+x^2+4 x^3}{x}\right )}{x} \, dx}{27 e^{12}}+\frac {8 \int \frac {x \log (x) \log ^2\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3} \, dx}{27 e^{12}}+\frac {16 \int \frac {x^2 \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3} \, dx}{27 e^{12}}+\frac {16 \int \frac {x^2 \log (x) \log ^2\left (\frac {-627+x^2+4 x^3}{x}\right )}{-627+x^2+4 x^3} \, dx}{9 e^{12}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\frac {\log ^4\left (-\frac {627}{x}+x+4 x^2\right )}{81 e^{12}} \]

[In]

Integrate[((2508 + 4*x^2 + 32*x^3)*Log[(-627 + x^2 + 4*x^3)/x]^3)/(E^12*(-50787*x + 81*x^3 + 324*x^4)),x]

[Out]

Log[-627/x + x + 4*x^2]^4/(81*E^12)

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {{\mathrm e}^{-12} \ln \left (\frac {4 x^{3}+x^{2}-627}{x}\right )^{4}}{81}\)	\(22\)
default	\(\frac {{\mathrm e}^{-12} \ln \left (\frac {4 x^{3}+x^{2}-627}{x}\right )^{4}}{81}\)	\(24\)
norman	\(\frac {{\mathrm e}^{-12} \ln \left (\frac {4 x^{3}+x^{2}-627}{x}\right )^{4}}{81}\)	\(24\)

[In]

int((32*x^3+4*x^2+2508)*ln((4*x^3+x^2-627)/x)^3/(324*x^4+81*x^3-50787*x)/exp(3)^4,x,method=_RETURNVERBOSE)

[Out]

1/81*exp(-12)*ln((4*x^3+x^2-627)/x)^4

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\frac {1}{81} \, e^{\left (-12\right )} \log \left (\frac {4 \, x^{3} + x^{2} - 627}{x}\right )^{4} \]

[In]

integrate((32*x^3+4*x^2+2508)*log((4*x^3+x^2-627)/x)^3/(324*x^4+81*x^3-50787*x)/exp(3)^4,x, algorithm="fricas"
)

[Out]

1/81*e^(-12)*log((4*x^3 + x^2 - 627)/x)^4

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\frac {\log {\left (\frac {4 x^{3} + x^{2} - 627}{x} \right )}^{4}}{81 e^{12}} \]

[In]

integrate((32*x**3+4*x**2+2508)*ln((4*x**3+x**2-627)/x)**3/(324*x**4+81*x**3-50787*x)/exp(3)**4,x)

[Out]

exp(-12)*log((4*x**3 + x**2 - 627)/x)**4/81

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (19) = 38\).

Time = 0.22 (sec) , antiderivative size = 236, normalized size of antiderivative = 10.73 \[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\frac {4}{81} \, {\left (\log \left (4 \, x^{3} + x^{2} - 627\right ) - \log \left (x\right )\right )} e^{\left (-12\right )} \log \left (\frac {4 \, x^{3} + x^{2} - 627}{x}\right )^{3} - \frac {1}{81} \, {\left (\log \left (4 \, x^{3} + x^{2} - 627\right )^{4} - 4 \, \log \left (4 \, x^{3} + x^{2} - 627\right )^{3} \log \left (x\right ) + 6 \, \log \left (4 \, x^{3} + x^{2} - 627\right )^{2} \log \left (x\right )^{2} - 4 \, \log \left (4 \, x^{3} + x^{2} - 627\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4} + 6 \, {\left (\log \left (4 \, x^{3} + x^{2} - 627\right )^{2} - 2 \, \log \left (4 \, x^{3} + x^{2} - 627\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )} \log \left (\frac {4 \, x^{3} + x^{2} - 627}{x}\right )^{2} - 4 \, {\left (\log \left (4 \, x^{3} + x^{2} - 627\right )^{3} - 3 \, \log \left (4 \, x^{3} + x^{2} - 627\right )^{2} \log \left (x\right ) + 3 \, \log \left (4 \, x^{3} + x^{2} - 627\right ) \log \left (x\right )^{2} - \log \left (x\right )^{3}\right )} \log \left (\frac {4 \, x^{3} + x^{2} - 627}{x}\right )\right )} e^{\left (-12\right )} \]

[In]

integrate((32*x^3+4*x^2+2508)*log((4*x^3+x^2-627)/x)^3/(324*x^4+81*x^3-50787*x)/exp(3)^4,x, algorithm="maxima"
)

[Out]

4/81*(log(4*x^3 + x^2 - 627) - log(x))*e^(-12)*log((4*x^3 + x^2 - 627)/x)^3 - 1/81*(log(4*x^3 + x^2 - 627)^4 -
4*log(4*x^3 + x^2 - 627)^3*log(x) + 6*log(4*x^3 + x^2 - 627)^2*log(x)^2 - 4*log(4*x^3 + x^2 - 627)*log(x)^3 +
log(x)^4 + 6*(log(4*x^3 + x^2 - 627)^2 - 2*log(4*x^3 + x^2 - 627)*log(x) + log(x)^2)*log((4*x^3 + x^2 - 627)/
x)^2 - 4*(log(4*x^3 + x^2 - 627)^3 - 3*log(4*x^3 + x^2 - 627)^2*log(x) + 3*log(4*x^3 + x^2 - 627)*log(x)^2 - l
og(x)^3)*log((4*x^3 + x^2 - 627)/x))*e^(-12)

Giac [F]

\[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\int { \frac {4 \, {\left (8 \, x^{3} + x^{2} + 627\right )} e^{\left (-12\right )} \log \left (\frac {4 \, x^{3} + x^{2} - 627}{x}\right )^{3}}{81 \, {\left (4 \, x^{4} + x^{3} - 627 \, x\right )}} \,d x } \]

[In]

integrate((32*x^3+4*x^2+2508)*log((4*x^3+x^2-627)/x)^3/(324*x^4+81*x^3-50787*x)/exp(3)^4,x, algorithm="giac")

[Out]

integrate(4/81*(8*x^3 + x^2 + 627)*e^(-12)*log((4*x^3 + x^2 - 627)/x)^3/(4*x^4 + x^3 - 627*x), x)

Mupad [B] (verification not implemented)

Time = 8.69 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {\left (2508+4 x^2+32 x^3\right ) \log ^3\left (\frac {-627+x^2+4 x^3}{x}\right )}{e^{12} \left (-50787 x+81 x^3+324 x^4\right )} \, dx=\frac {{\ln \left (\frac {4\,x^3+x^2-627}{x}\right )}^4\,{\mathrm {e}}^{-12}}{81} \]

[In]

int((log((x^2 + 4*x^3 - 627)/x)^3*exp(-12)*(4*x^2 + 32*x^3 + 2508))/(81*x^3 - 50787*x + 324*x^4),x)

[Out]

(log((x^2 + 4*x^3 - 627)/x)^4*exp(-12))/81