3.34.18 \(\int \frac {-120 x-6 x^3+(-120 x-6 x^2+6 x^3) \log (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2})}{e (-20-x+x^2) \log ^2(\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2})} \, dx\)

Optimal. Leaf size=26 \[ \frac {3 x^2}{e \log \left (\left (-4-x+\frac {5 (4+x)}{x}\right )^2\right )} \]

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Rubi [F]  time = 0.75, 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 {-120 x-6 x^3+\left (-120 x-6 x^2+6 x^3\right ) \log \left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )}{e \left (-20-x+x^2\right ) \log ^2\left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-120*x - 6*x^3 + (-120*x - 6*x^2 + 6*x^3)*Log[(400 + 40*x - 39*x^2 - 2*x^3 + x^4)/x^2])/(E*(-20 - x + x^2
)*Log[(400 + 40*x - 39*x^2 - 2*x^3 + x^4)/x^2]^2),x]

[Out]

(-6*Defer[Int][Log[(20 + x - x^2)^2/x^2]^(-2), x])/E - (150*Defer[Int][1/((-5 + x)*Log[(20 + x - x^2)^2/x^2]^2
), x])/E - (6*Defer[Int][x/Log[(20 + x - x^2)^2/x^2]^2, x])/E - (96*Defer[Int][1/((4 + x)*Log[(20 + x - x^2)^2
/x^2]^2), x])/E + (6*Defer[Int][x/Log[(20 + x - x^2)^2/x^2], x])/E

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-120 x-6 x^3+\left (-120 x-6 x^2+6 x^3\right ) \log \left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )}{\left (-20-x+x^2\right ) \log ^2\left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )} \, dx}{e}\\ &=\frac {\int \frac {6 x \left (20+x^2-\left (-20-x+x^2\right ) \log \left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )\right )}{\left (20+x-x^2\right ) \log ^2\left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )} \, dx}{e}\\ &=\frac {6 \int \frac {x \left (20+x^2-\left (-20-x+x^2\right ) \log \left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )\right )}{\left (20+x-x^2\right ) \log ^2\left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )} \, dx}{e}\\ &=\frac {6 \int \left (-\frac {x \left (20+x^2\right )}{(-5+x) (4+x) \log ^2\left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )}+\frac {x}{\log \left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )}\right ) \, dx}{e}\\ &=-\frac {6 \int \frac {x \left (20+x^2\right )}{(-5+x) (4+x) \log ^2\left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )} \, dx}{e}+\frac {6 \int \frac {x}{\log \left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )} \, dx}{e}\\ &=-\frac {6 \int \left (\frac {1}{\log ^2\left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )}+\frac {25}{(-5+x) \log ^2\left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )}+\frac {x}{\log ^2\left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )}+\frac {16}{(4+x) \log ^2\left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )}\right ) \, dx}{e}+\frac {6 \int \frac {x}{\log \left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )} \, dx}{e}\\ &=-\frac {6 \int \frac {1}{\log ^2\left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )} \, dx}{e}-\frac {6 \int \frac {x}{\log ^2\left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )} \, dx}{e}+\frac {6 \int \frac {x}{\log \left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )} \, dx}{e}-\frac {96 \int \frac {1}{(4+x) \log ^2\left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )} \, dx}{e}-\frac {150 \int \frac {1}{(-5+x) \log ^2\left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )} \, dx}{e}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 25, normalized size = 0.96 \begin {gather*} \frac {3 x^2}{e \log \left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-120*x - 6*x^3 + (-120*x - 6*x^2 + 6*x^3)*Log[(400 + 40*x - 39*x^2 - 2*x^3 + x^4)/x^2])/(E*(-20 - x
+ x^2)*Log[(400 + 40*x - 39*x^2 - 2*x^3 + x^4)/x^2]^2),x]

[Out]

(3*x^2)/(E*Log[(20 + x - x^2)^2/x^2])

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fricas [A]  time = 0.64, size = 32, normalized size = 1.23 \begin {gather*} \frac {3 \, x^{2} e^{\left (-1\right )}}{\log \left (\frac {x^{4} - 2 \, x^{3} - 39 \, x^{2} + 40 \, x + 400}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^3-6*x^2-120*x)*log((x^4-2*x^3-39*x^2+40*x+400)/x^2)-6*x^3-120*x)/(x^2-x-20)/exp(1)/log((x^4-2*
x^3-39*x^2+40*x+400)/x^2)^2,x, algorithm="fricas")

[Out]

3*x^2*e^(-1)/log((x^4 - 2*x^3 - 39*x^2 + 40*x + 400)/x^2)

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giac [A]  time = 0.38, size = 32, normalized size = 1.23 \begin {gather*} \frac {3 \, x^{2} e^{\left (-1\right )}}{\log \left (\frac {x^{4} - 2 \, x^{3} - 39 \, x^{2} + 40 \, x + 400}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^3-6*x^2-120*x)*log((x^4-2*x^3-39*x^2+40*x+400)/x^2)-6*x^3-120*x)/(x^2-x-20)/exp(1)/log((x^4-2*
x^3-39*x^2+40*x+400)/x^2)^2,x, algorithm="giac")

[Out]

3*x^2*e^(-1)/log((x^4 - 2*x^3 - 39*x^2 + 40*x + 400)/x^2)

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maple [A]  time = 0.52, size = 33, normalized size = 1.27

method	result	size
risch	\(\frac {3 x^{2} {\mathrm e}^{-1}}{\ln \left (\frac {x^{4}-2 x^{3}-39 x^{2}+40 x +400}{x^{2}}\right )}\)	\(33\)
norman	\(\frac {3 x^{2} {\mathrm e}^{-1}}{\ln \left (\frac {x^{4}-2 x^{3}-39 x^{2}+40 x +400}{x^{2}}\right )}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x^3-6*x^2-120*x)*ln((x^4-2*x^3-39*x^2+40*x+400)/x^2)-6*x^3-120*x)/(x^2-x-20)/exp(1)/ln((x^4-2*x^3-39*x
^2+40*x+400)/x^2)^2,x,method=_RETURNVERBOSE)

[Out]

3*x^2*exp(-1)/ln((x^4-2*x^3-39*x^2+40*x+400)/x^2)

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maxima [A]  time = 0.56, size = 22, normalized size = 0.85 \begin {gather*} \frac {3 \, x^{2} e^{\left (-1\right )}}{2 \, {\left (\log \left (x + 4\right ) + \log \left (x - 5\right ) - \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^3-6*x^2-120*x)*log((x^4-2*x^3-39*x^2+40*x+400)/x^2)-6*x^3-120*x)/(x^2-x-20)/exp(1)/log((x^4-2*
x^3-39*x^2+40*x+400)/x^2)^2,x, algorithm="maxima")

[Out]

3/2*x^2*e^(-1)/(log(x + 4) + log(x - 5) - log(x))

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mupad [B]  time = 2.20, size = 111, normalized size = 4.27 \begin {gather*} \frac {3\,x^2\,{\mathrm {e}}^{-1}+\frac {3\,x^2\,\ln \left (\frac {x^4-2\,x^3-39\,x^2+40\,x+400}{x^2}\right )\,{\mathrm {e}}^{-1}\,\left (-x^2+x+20\right )}{x^2+20}}{\ln \left (\frac {x^4-2\,x^3-39\,x^2+40\,x+400}{x^2}\right )}-3\,x\,{\mathrm {e}}^{-1}+3\,x^2\,{\mathrm {e}}^{-1}+\frac {60\,x+2400}{\mathrm {e}\,x^2+20\,\mathrm {e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-1)*(120*x + log((40*x - 39*x^2 - 2*x^3 + x^4 + 400)/x^2)*(120*x + 6*x^2 - 6*x^3) + 6*x^3))/(log((40*
x - 39*x^2 - 2*x^3 + x^4 + 400)/x^2)^2*(x - x^2 + 20)),x)

[Out]

(3*x^2*exp(-1) + (3*x^2*log((40*x - 39*x^2 - 2*x^3 + x^4 + 400)/x^2)*exp(-1)*(x - x^2 + 20))/(x^2 + 20))/log((
40*x - 39*x^2 - 2*x^3 + x^4 + 400)/x^2) - 3*x*exp(-1) + 3*x^2*exp(-1) + (60*x + 2400)/(20*exp(1) + x^2*exp(1))

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sympy [A]  time = 0.17, size = 31, normalized size = 1.19 \begin {gather*} \frac {3 x^{2}}{e \log {\left (\frac {x^{4} - 2 x^{3} - 39 x^{2} + 40 x + 400}{x^{2}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x**3-6*x**2-120*x)*ln((x**4-2*x**3-39*x**2+40*x+400)/x**2)-6*x**3-120*x)/(x**2-x-20)/exp(1)/ln((
x**4-2*x**3-39*x**2+40*x+400)/x**2)**2,x)

[Out]

3*x**2*exp(-1)/log((x**4 - 2*x**3 - 39*x**2 + 40*x + 400)/x**2)

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