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

Optimal. Leaf size=22 \[ \frac {\log ^4\left (-\frac {627}{x}+x+4 x^2\right )}{81 e^{12}} \]

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Rubi [F]  time = 91.75, 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 {\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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 22, normalized size = 1.00 \begin {gather*} \frac {\log ^4\left (-\frac {627}{x}+x+4 x^2\right )}{81 e^{12}} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.63, size = 21, normalized size = 0.95 \begin {gather*} \frac {1}{81} \, e^{\left (-12\right )} \log \left (\frac {4 \, x^{3} + x^{2} - 627}{x}\right )^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.12, size = 24, normalized size = 1.09

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

Verification of antiderivative is not currently implemented for this CAS.

[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(3)^4*ln((4*x^3+x^2-627)/x)^4

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maxima [B]  time = 0.39, size = 236, normalized size = 10.73 \begin {gather*} \frac {4}{81} \, {\left (\log \left (4 \, x^{3} + x^{2} - 627\right ) - \log \relax (x)\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 \relax (x) + 6 \, \log \left (4 \, x^{3} + x^{2} - 627\right )^{2} \log \relax (x)^{2} - 4 \, \log \left (4 \, x^{3} + x^{2} - 627\right ) \log \relax (x)^{3} + \log \relax (x)^{4} + 6 \, {\left (\log \left (4 \, x^{3} + x^{2} - 627\right )^{2} - 2 \, \log \left (4 \, x^{3} + x^{2} - 627\right ) \log \relax (x) + \log \relax (x)^{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 \relax (x) + 3 \, \log \left (4 \, x^{3} + x^{2} - 627\right ) \log \relax (x)^{2} - \log \relax (x)^{3}\right )} \log \left (\frac {4 \, x^{3} + x^{2} - 627}{x}\right )\right )} e^{\left (-12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 1.81, size = 21, normalized size = 0.95 \begin {gather*} \frac {{\ln \left (\frac {4\,x^3+x^2-627}{x}\right )}^4\,{\mathrm {e}}^{-12}}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.20, size = 19, normalized size = 0.86 \begin {gather*} \frac {\log {\left (\frac {4 x^{3} + x^{2} - 627}{x} \right )}^{4}}{81 e^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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