3.64.29 \(\int \frac {e^{\frac {6 x+(-2-x) \log (\frac {x}{-10+8 x^2})}{12+6 x}} (20-40 x+21 x^2+64 x^3+4 x^4) \log (3)}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx\)

Optimal. Leaf size=32 \[ \frac {\sqrt [6]{2} e^{\frac {x}{2+x}} \log (3)}{\sqrt [6]{\frac {x}{-5+4 x^2}}} \]

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Rubi [F]  time = 13.65, 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 {6 x+(-2-x) \log \left (\frac {x}{-10+8 x^2}\right )}{12+6 x}\right ) \left (20-40 x+21 x^2+64 x^3+4 x^4\right ) \log (3)}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((6*x + (-2 - x)*Log[x/(-10 + 8*x^2)])/(12 + 6*x))*(20 - 40*x + 21*x^2 + 64*x^3 + 4*x^4)*Log[3])/(-120*
x - 120*x^2 + 66*x^3 + 96*x^4 + 24*x^5),x]

[Out]

(-32*(-2)^(5/6)*(-(x/(5 - 4*x^2)))^(5/6)*(-5 + 4*x^2)^(5/6)*Log[3]*Defer[Subst][Defer[Int][E^(x^6/(2 + x^6))/(
((-2)^(1/6) - x)*(-10 + 8*x^12)^(5/6)), x], x, x^(1/6)])/x^(5/6) + (10*(-(x/(5 - 4*x^2)))^(5/6)*(-5 + 4*x^2)^(
5/6)*Log[3]*Defer[Subst][Defer[Int][E^(x^6/(2 + x^6))/(x^2*(-10 + 8*x^12)^(5/6)), x], x, x^(1/6)])/x^(5/6) + (
96*(-(x/(5 - 4*x^2)))^(5/6)*(-5 + 4*x^2)^(5/6)*Log[3]*Defer[Subst][Defer[Int][(E^(x^6/(2 + x^6))*x^4)/(-10 + 8
*x^12)^(5/6), x], x, x^(1/6)])/x^(5/6) + (8*(-(x/(5 - 4*x^2)))^(5/6)*(-5 + 4*x^2)^(5/6)*Log[3]*Defer[Subst][De
fer[Int][(E^(x^6/(2 + x^6))*x^10)/(-10 + 8*x^12)^(5/6), x], x, x^(1/6)])/x^(5/6) - (32*(-2)^(5/6)*(-(x/(5 - 4*
x^2)))^(5/6)*(-5 + 4*x^2)^(5/6)*Log[3]*Defer[Subst][Defer[Int][E^(x^6/(2 + x^6))/(((-2)^(1/6) + x)*(-10 + 8*x^
12)^(5/6)), x], x, x^(1/6)])/x^(5/6) + ((32*I)*2^(5/6)*(-(x/(5 - 4*x^2)))^(5/6)*(-5 + 4*x^2)^(5/6)*Log[3]*Defe
r[Subst][Defer[Int][E^(x^6/(2 + x^6))/(((-2)^(1/6) - (-1)^(1/3)*x)*(-10 + 8*x^12)^(5/6)), x], x, x^(1/6)])/x^(
5/6) + ((32*I)*2^(5/6)*(-(x/(5 - 4*x^2)))^(5/6)*(-5 + 4*x^2)^(5/6)*Log[3]*Defer[Subst][Defer[Int][E^(x^6/(2 +
x^6))/(((-2)^(1/6) + (-1)^(1/3)*x)*(-10 + 8*x^12)^(5/6)), x], x, x^(1/6)])/x^(5/6) - (32*(-1)^(1/6)*2^(5/6)*(-
(x/(5 - 4*x^2)))^(5/6)*(-5 + 4*x^2)^(5/6)*Log[3]*Defer[Subst][Defer[Int][E^(x^6/(2 + x^6))/(((-2)^(1/6) - (-1)
^(2/3)*x)*(-10 + 8*x^12)^(5/6)), x], x, x^(1/6)])/x^(5/6) - (32*(-1)^(1/6)*2^(5/6)*(-(x/(5 - 4*x^2)))^(5/6)*(-
5 + 4*x^2)^(5/6)*Log[3]*Defer[Subst][Defer[Int][E^(x^6/(2 + x^6))/(((-2)^(1/6) + (-1)^(2/3)*x)*(-10 + 8*x^12)^
(5/6)), x], x, x^(1/6)])/x^(5/6) - ((33*I)*Sqrt[2]*(-(x/(5 - 4*x^2)))^(5/6)*(-5 + 4*x^2)^(5/6)*Log[3]*Defer[Su
bst][Defer[Int][(E^(x^6/(2 + x^6))*x)/((I*Sqrt[2] - x^3)^2*(-10 + 8*x^12)^(5/6)), x], x, x^(1/6)])/x^(5/6) + (
(33*I)*Sqrt[2]*(-(x/(5 - 4*x^2)))^(5/6)*(-5 + 4*x^2)^(5/6)*Log[3]*Defer[Subst][Defer[Int][(E^(x^6/(2 + x^6))*x
)/((I*Sqrt[2] + x^3)^2*(-10 + 8*x^12)^(5/6)), x], x, x^(1/6)])/x^(5/6)

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log (3) \int \frac {\exp \left (\frac {6 x+(-2-x) \log \left (\frac {x}{-10+8 x^2}\right )}{12+6 x}\right ) \left (20-40 x+21 x^2+64 x^3+4 x^4\right )}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx\\ &=\log (3) \int \frac {e^{\frac {x}{2+x}} \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (20-40 x+21 x^2+64 x^3+4 x^4\right )}{3 x^2 (2+x)^2} \, dx\\ &=\frac {1}{3} \log (3) \int \frac {e^{\frac {x}{2+x}} \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (20-40 x+21 x^2+64 x^3+4 x^4\right )}{x^2 (2+x)^2} \, dx\\ &=\frac {\left (\left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \int \frac {e^{\frac {x}{2+x}} \left (20-40 x+21 x^2+64 x^3+4 x^4\right )}{x^{7/6} (2+x)^2 \left (-10+8 x^2\right )^{5/6}} \, dx}{3 x^{5/6}}\\ &=\frac {\left (2 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} \left (20-40 x^6+21 x^{12}+64 x^{18}+4 x^{24}\right )}{x^2 \left (2+x^6\right )^2 \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}\\ &=\frac {\left (2 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \left (\frac {5 e^{\frac {x^6}{2+x^6}}}{x^2 \left (-10+8 x^{12}\right )^{5/6}}+\frac {48 e^{\frac {x^6}{2+x^6}} x^4}{\left (-10+8 x^{12}\right )^{5/6}}+\frac {4 e^{\frac {x^6}{2+x^6}} x^{10}}{\left (-10+8 x^{12}\right )^{5/6}}+\frac {132 e^{\frac {x^6}{2+x^6}} x^4}{\left (2+x^6\right )^2 \left (-10+8 x^{12}\right )^{5/6}}-\frac {192 e^{\frac {x^6}{2+x^6}} x^4}{\left (2+x^6\right ) \left (-10+8 x^{12}\right )^{5/6}}\right ) \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}\\ &=\frac {\left (8 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^{10}}{\left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (10 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}}}{x^2 \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (96 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^4}{\left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (264 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^4}{\left (2+x^6\right )^2 \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}-\frac {\left (384 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^4}{\left (2+x^6\right ) \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}\\ &=\frac {\left (8 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^{10}}{\left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (10 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}}}{x^2 \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (96 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^4}{\left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (264 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \left (-\frac {i e^{\frac {x^6}{2+x^6}} x}{4 \sqrt {2} \left (i \sqrt {2}-x^3\right )^2 \left (-10+8 x^{12}\right )^{5/6}}+\frac {i e^{\frac {x^6}{2+x^6}} x}{4 \sqrt {2} \left (i \sqrt {2}+x^3\right )^2 \left (-10+8 x^{12}\right )^{5/6}}\right ) \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}-\frac {\left (384 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \left (\frac {e^{\frac {x^6}{2+x^6}} x}{2 \left (-i \sqrt {2}+x^3\right ) \left (-10+8 x^{12}\right )^{5/6}}+\frac {e^{\frac {x^6}{2+x^6}} x}{2 \left (i \sqrt {2}+x^3\right ) \left (-10+8 x^{12}\right )^{5/6}}\right ) \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}\\ &=\frac {\left (8 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^{10}}{\left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (10 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}}}{x^2 \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (96 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^4}{\left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}-\frac {\left (192 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x}{\left (-i \sqrt {2}+x^3\right ) \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}-\frac {\left (192 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x}{\left (i \sqrt {2}+x^3\right ) \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}-\frac {\left (33 i \sqrt {2} \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x}{\left (i \sqrt {2}-x^3\right )^2 \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (33 i \sqrt {2} \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x}{\left (i \sqrt {2}+x^3\right )^2 \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 2.01, size = 84, normalized size = 2.62 \begin {gather*} \left (\int \frac {e^{\frac {6 x+(-2-x) \log \left (\frac {x}{-10+8 x^2}\right )}{12+6 x}} \left (20-40 x+21 x^2+64 x^3+4 x^4\right )}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx\right ) \log (3) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((6*x + (-2 - x)*Log[x/(-10 + 8*x^2)])/(12 + 6*x))*(20 - 40*x + 21*x^2 + 64*x^3 + 4*x^4)*Log[3])/
(-120*x - 120*x^2 + 66*x^3 + 96*x^4 + 24*x^5),x]

[Out]

Integrate[(E^((6*x + (-2 - x)*Log[x/(-10 + 8*x^2)])/(12 + 6*x))*(20 - 40*x + 21*x^2 + 64*x^3 + 4*x^4))/(-120*x
- 120*x^2 + 66*x^3 + 96*x^4 + 24*x^5), x]*Log[3]

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fricas [A]  time = 0.71, size = 32, normalized size = 1.00 \begin {gather*} e^{\left (-\frac {{\left (x + 2\right )} \log \left (\frac {x}{2 \, {\left (4 \, x^{2} - 5\right )}}\right ) - 6 \, x}{6 \, {\left (x + 2\right )}}\right )} \log \relax (3) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4+64*x^3+21*x^2-40*x+20)*log(3)*exp(((-x-2)*log(x/(8*x^2-10))+6*x)/(6*x+12))/(24*x^5+96*x^4+66*
x^3-120*x^2-120*x),x, algorithm="fricas")

[Out]

e^(-1/6*((x + 2)*log(1/2*x/(4*x^2 - 5)) - 6*x)/(x + 2))*log(3)

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giac [A]  time = 0.23, size = 53, normalized size = 1.66 \begin {gather*} e^{\left (-\frac {x \log \left (\frac {x}{2 \, {\left (4 \, x^{2} - 5\right )}}\right )}{6 \, {\left (x + 2\right )}} + \frac {x}{x + 2} - \frac {\log \left (\frac {x}{2 \, {\left (4 \, x^{2} - 5\right )}}\right )}{3 \, {\left (x + 2\right )}}\right )} \log \relax (3) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4+64*x^3+21*x^2-40*x+20)*log(3)*exp(((-x-2)*log(x/(8*x^2-10))+6*x)/(6*x+12))/(24*x^5+96*x^4+66*
x^3-120*x^2-120*x),x, algorithm="giac")

[Out]

e^(-1/6*x*log(1/2*x/(4*x^2 - 5))/(x + 2) + x/(x + 2) - 1/3*log(1/2*x/(4*x^2 - 5))/(x + 2))*log(3)

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maple [A]  time = 0.15, size = 44, normalized size = 1.38

method	result	size
risch	\(\ln \relax (3) {\mathrm e}^{-\frac {\ln \left (\frac {x}{8 x^{2}-10}\right ) x +2 \ln \left (\frac {x}{8 x^{2}-10}\right )-6 x}{6 \left (2+x \right )}}\)	\(44\)
gosper	\(\ln \relax (3) {\mathrm e}^{-\frac {\ln \left (\frac {x}{8 x^{2}-10}\right ) x +2 \ln \left (\frac {x}{8 x^{2}-10}\right )-6 x}{6 \left (2+x \right )}}\)	\(46\)
norman	\(\frac {x \ln \relax (3) {\mathrm e}^{\frac {\left (-x -2\right ) \ln \left (\frac {x}{8 x^{2}-10}\right )+6 x}{6 x +12}}+2 \ln \relax (3) {\mathrm e}^{\frac {\left (-x -2\right ) \ln \left (\frac {x}{8 x^{2}-10}\right )+6 x}{6 x +12}}}{2+x}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^4+64*x^3+21*x^2-40*x+20)*ln(3)*exp(((-x-2)*ln(x/(8*x^2-10))+6*x)/(6*x+12))/(24*x^5+96*x^4+66*x^3-120*
x^2-120*x),x,method=_RETURNVERBOSE)

[Out]

ln(3)*exp(-1/6*(ln(x/(8*x^2-10))*x+2*ln(x/(8*x^2-10))-6*x)/(2+x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{6} \, \int \frac {{\left (4 \, x^{4} + 64 \, x^{3} + 21 \, x^{2} - 40 \, x + 20\right )} e^{\left (-\frac {{\left (x + 2\right )} \log \left (\frac {x}{2 \, {\left (4 \, x^{2} - 5\right )}}\right ) - 6 \, x}{6 \, {\left (x + 2\right )}}\right )}}{4 \, x^{5} + 16 \, x^{4} + 11 \, x^{3} - 20 \, x^{2} - 20 \, x}\,{d x} \log \relax (3) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4+64*x^3+21*x^2-40*x+20)*log(3)*exp(((-x-2)*log(x/(8*x^2-10))+6*x)/(6*x+12))/(24*x^5+96*x^4+66*
x^3-120*x^2-120*x),x, algorithm="maxima")

[Out]

1/6*integrate((4*x^4 + 64*x^3 + 21*x^2 - 40*x + 20)*e^(-1/6*((x + 2)*log(1/2*x/(4*x^2 - 5)) - 6*x)/(x + 2))/(4
*x^5 + 16*x^4 + 11*x^3 - 20*x^2 - 20*x), x)*log(3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{\frac {6\,x-\ln \left (\frac {x}{8\,x^2-10}\right )\,\left (x+2\right )}{6\,x+12}}\,\ln \relax (3)\,\left (4\,x^4+64\,x^3+21\,x^2-40\,x+20\right )}{24\,x^5+96\,x^4+66\,x^3-120\,x^2-120\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((6*x - log(x/(8*x^2 - 10))*(x + 2))/(6*x + 12))*log(3)*(21*x^2 - 40*x + 64*x^3 + 4*x^4 + 20))/(66*x^3
- 120*x^2 - 120*x + 96*x^4 + 24*x^5),x)

[Out]

int((exp((6*x - log(x/(8*x^2 - 10))*(x + 2))/(6*x + 12))*log(3)*(21*x^2 - 40*x + 64*x^3 + 4*x^4 + 20))/(66*x^3
- 120*x^2 - 120*x + 96*x^4 + 24*x^5), x)

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sympy [A]  time = 0.91, size = 27, normalized size = 0.84 \begin {gather*} e^{\frac {6 x + \left (- x - 2\right ) \log {\left (\frac {x}{8 x^{2} - 10} \right )}}{6 x + 12}} \log {\relax (3 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**4+64*x**3+21*x**2-40*x+20)*ln(3)*exp(((-x-2)*ln(x/(8*x**2-10))+6*x)/(6*x+12))/(24*x**5+96*x**4
+66*x**3-120*x**2-120*x),x)

[Out]

exp((6*x + (-x - 2)*log(x/(8*x**2 - 10)))/(6*x + 12))*log(3)

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