3.49.86 \(\int \frac {(-\frac {1}{-3 e^4+3 e^5+3 x})^{x/4} (-4 e^4+4 e^5+4 x-x^2+(-e^4 x+e^5 x+x^2) \log (-\frac {1}{-3 e^4+3 e^5+3 x}))}{-e^4+e^5+x} \, dx\)

Optimal. Leaf size=30 \[ 4\ 3^{-x/4} \left (\frac {1}{e^4-e^5-x}\right )^{x/4} x \]

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Rubi [F]  time = 1.05, 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 (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (-4 e^4+4 e^5+4 x-x^2+\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{-e^4+e^5+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Log[(3*(1 - E)*E^4 - 3*x)^(-1)]*Defer[Int][x*(-(-3*E^4 + 3*E^5 + 3*x)^(-1))^(x/4), x] - Defer[Int][Defer[Int][
((3*E^4 - 3*E^5 - 3*x)^(-1))^(x/4)*x, x]/(E^4 - E^5 - x), x] + 4*(4 - E^4 + E^5)*Defer[Subst][Defer[Int][(-(-3
*E^4 + 3*E^5 + 12*x)^(-1))^x, x], x, x/4] - 16*Defer[Subst][Defer[Int][x*(-(-3*E^4 + 3*E^5 + 12*x)^(-1))^x, x]
, x, x/4] - 4*(1 - E)^2*E^8*Defer[Subst][Defer[Int][(-(-3*E^4 + 3*E^5 + 12*x)^(-1))^x/(-E^4 + E^5 + 4*x), x],
x, x/4]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (4 (1-e) e^4-4 x+x^2-\left (-e^4 x+e^5 x+x^2\right ) \log \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )\right )}{e^4-e^5-x} \, dx\\ &=\int \left (\frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (4 (1-e) e^4-4 x+x^2\right )}{e^4-e^5-x}+x \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \log \left (\frac {1}{3 e^4-3 e^5-3 x}\right )\right ) \, dx\\ &=\int \frac {\left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \left (4 (1-e) e^4-4 x+x^2\right )}{e^4-e^5-x} \, dx+\int x \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \log \left (\frac {1}{3 e^4-3 e^5-3 x}\right ) \, dx\\ &=4 \operatorname {Subst}\left (\int \frac {\left (-\frac {1}{-3 e^4+3 e^5+12 x}\right )^x \left (4 (1-e) e^4-16 x+16 x^2\right )}{e^4-e^5-4 x} \, dx,x,\frac {x}{4}\right )+\log \left (\frac {1}{3 e^4-3 e^5-3 x}\right ) \int x \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \, dx-\int \frac {\int \left (\frac {1}{3 e^4-3 e^5-3 x}\right )^{x/4} x \, dx}{e^4-e^5-x} \, dx\\ &=4 \operatorname {Subst}\left (\int \left (4 \left (1+\frac {1}{4} (-1+e) e^4\right ) \left (-\frac {1}{-3 e^4+3 e^5+12 x}\right )^x-4 x \left (-\frac {1}{-3 e^4+3 e^5+12 x}\right )^x-\frac {(-1+e)^2 e^8 \left (-\frac {1}{-3 e^4+3 e^5+12 x}\right )^x}{-e^4+e^5+4 x}\right ) \, dx,x,\frac {x}{4}\right )+\log \left (\frac {1}{3 e^4-3 e^5-3 x}\right ) \int x \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \, dx-\int \frac {\int \left (\frac {1}{3 e^4-3 e^5-3 x}\right )^{x/4} x \, dx}{e^4-e^5-x} \, dx\\ &=-\left (16 \operatorname {Subst}\left (\int x \left (-\frac {1}{-3 e^4+3 e^5+12 x}\right )^x \, dx,x,\frac {x}{4}\right )\right )-\left (4 (1-e)^2 e^8\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {1}{-3 e^4+3 e^5+12 x}\right )^x}{-e^4+e^5+4 x} \, dx,x,\frac {x}{4}\right )+\left (16 \left (1+\frac {1}{4} (-1+e) e^4\right )\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{-3 e^4+3 e^5+12 x}\right )^x \, dx,x,\frac {x}{4}\right )+\log \left (\frac {1}{3 e^4-3 e^5-3 x}\right ) \int x \left (-\frac {1}{-3 e^4+3 e^5+3 x}\right )^{x/4} \, dx-\int \frac {\int \left (\frac {1}{3 e^4-3 e^5-3 x}\right )^{x/4} x \, dx}{e^4-e^5-x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 25, normalized size = 0.83 \begin {gather*} 4 \left (\frac {1}{3 e^4-3 e^5-3 x}\right )^{x/4} x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*((3*E^4 - 3*E^5 - 3*x)^(-1))^(x/4)*x

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fricas [A]  time = 0.52, size = 19, normalized size = 0.63 \begin {gather*} 4 \, x \left (-\frac {1}{3 \, {\left (x + e^{5} - e^{4}\right )}}\right )^{\frac {1}{4} \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x*(-1/3/(x + e^5 - e^4))^(1/4*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (x^{2} - {\left (x^{2} + x e^{5} - x e^{4}\right )} \log \left (-\frac {1}{3 \, {\left (x + e^{5} - e^{4}\right )}}\right ) - 4 \, x - 4 \, e^{5} + 4 \, e^{4}\right )} \left (-\frac {1}{3 \, {\left (x + e^{5} - e^{4}\right )}}\right )^{\frac {1}{4} \, x}}{x + e^{5} - e^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.26, size = 24, normalized size = 0.80

method	result	size
risch	\(4 x \left (-\frac {1}{3 \,{\mathrm e}^{5}-3 \,{\mathrm e}^{4}+3 x}\right )^{\frac {x}{4}}\)	\(24\)
default	\(4 x \,{\mathrm e}^{\frac {x \ln \left (-\frac {1}{3 \,{\mathrm e}^{5}-3 \,{\mathrm e}^{4}+3 x}\right )}{4}}\)	\(25\)
norman	\(4 x \,{\mathrm e}^{\frac {x \ln \left (-\frac {1}{3 \,{\mathrm e}^{5}-3 \,{\mathrm e}^{4}+3 x}\right )}{4}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x*(-1/(3*exp(5)-3*exp(4)+3*x))^(1/4*x)

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maxima [A]  time = 0.49, size = 24, normalized size = 0.80 \begin {gather*} 4 \, x e^{\left (-\frac {1}{4} \, x \log \relax (3) - \frac {1}{4} \, x \log \left (-x - e^{5} + e^{4}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.06, size = 23, normalized size = 0.77 \begin {gather*} 4\,x\,{\left (-\frac {1}{3\,x-3\,{\mathrm {e}}^4+3\,{\mathrm {e}}^5}\right )}^{x/4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x*(-1/(3*x - 3*exp(4) + 3*exp(5)))^(x/4)

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sympy [A]  time = 0.79, size = 26, normalized size = 0.87 \begin {gather*} 4 x e^{\frac {x \log {\left (- \frac {1}{3 x - 3 e^{4} + 3 e^{5}} \right )}}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x*exp(x*log(-1/(3*x - 3*exp(4) + 3*exp(5)))/4)

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