3.14.57 \(\int \frac {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} (9-30 x+13 x^2+20 x^3+4 x^4)+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} (9 x-30 x^2+13 x^3+20 x^4+4 x^5) \log (x)} \, dx\)

Optimal. Leaf size=31 \[ x+\log \left (x^2 \left (2 e^{-\frac {3}{-x+\frac {3}{5+2 x}}}+\log (x)\right )\right ) \]

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Rubi [F]  time = 115.67, 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 {36-288 x-128 x^2+82 x^3+56 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9-30 x+13 x^2+20 x^3+4 x^4\right )+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (18-51 x-4 x^2+53 x^3+28 x^4+4 x^5\right ) \log (x)}{18 x-60 x^2+26 x^3+40 x^4+8 x^5+e^{\frac {-15-6 x}{-3+5 x+2 x^2}} \left (9 x-30 x^2+13 x^3+20 x^4+4 x^5\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(36 - 288*x - 128*x^2 + 82*x^3 + 56*x^4 + 8*x^5 + E^((-15 - 6*x)/(-3 + 5*x + 2*x^2))*(9 - 30*x + 13*x^2 +
20*x^3 + 4*x^4) + E^((-15 - 6*x)/(-3 + 5*x + 2*x^2))*(18 - 51*x - 4*x^2 + 53*x^3 + 28*x^4 + 4*x^5)*Log[x])/(18
*x - 60*x^2 + 26*x^3 + 40*x^4 + 8*x^5 + E^((-15 - 6*x)/(-3 + 5*x + 2*x^2))*(9*x - 30*x^2 + 13*x^3 + 20*x^4 + 4
*x^5)*Log[x]),x]

[Out]

-36/(7*(1 - 2*x)) + x + 3/(7*(3 + x)) + 2*Log[x] + Defer[Int][1/(x*(2*E^((3*(5 + 2*x))/(-3 + 5*x + 2*x^2)) + L
og[x])), x] + (3*Defer[Int][Log[x]/((3 + x)^2*(2*E^((3*(5 + 2*x))/(-3 + 5*x + 2*x^2)) + Log[x])), x])/7 + (72*
Defer[Int][Log[x]/((-1 + 2*x)^2*(2*E^(15/(-3 + 5*x + 2*x^2) + (6*x)/(-3 + 5*x + 2*x^2)) + Log[x])), x])/7

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-3+5 x+2 x^2\right )^2+2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}} \left (18-144 x-64 x^2+41 x^3+28 x^4+4 x^5\right )+(2+x) \left (-3+5 x+2 x^2\right )^2 \log (x)}{x \left (3-5 x-2 x^2\right )^2 \left (2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}}+\log (x)\right )} \, dx\\ &=\int \left (\frac {18-144 x-64 x^2+41 x^3+28 x^4+4 x^5}{x (3+x)^2 (-1+2 x)^2}+\frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{x (3+x)^2 (-1+2 x)^2 \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}\right ) \, dx\\ &=\int \frac {18-144 x-64 x^2+41 x^3+28 x^4+4 x^5}{x (3+x)^2 (-1+2 x)^2} \, dx+\int \frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{x (3+x)^2 (-1+2 x)^2 \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )} \, dx\\ &=\int \left (1+\frac {2}{x}-\frac {3}{7 (3+x)^2}-\frac {72}{7 (-1+2 x)^2}\right ) \, dx+\int \frac {\left (-3+5 x+2 x^2\right )^2+3 x \left (31+20 x+4 x^2\right ) \log (x)}{(1-2 x)^2 x (3+x)^2 \left (2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}}+\log (x)\right )} \, dx\\ &=-\frac {36}{7 (1-2 x)}+x+\frac {3}{7 (3+x)}+2 \log (x)+\int \left (\frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{9 x \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}-\frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{147 (3+x)^2 \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}-\frac {19 \left (9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)\right )}{3087 (3+x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}+\frac {8 \left (9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)\right )}{49 (-1+2 x)^2 \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}-\frac {72 \left (9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)\right )}{343 (-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}\right ) \, dx\\ &=-\frac {36}{7 (1-2 x)}+x+\frac {3}{7 (3+x)}+2 \log (x)-\frac {19 \int \frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{(3+x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )} \, dx}{3087}-\frac {1}{147} \int \frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{(3+x)^2 \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )} \, dx+\frac {1}{9} \int \frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{x \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )} \, dx+\frac {8}{49} \int \frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{(-1+2 x)^2 \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )} \, dx-\frac {72}{343} \int \frac {9-30 x+13 x^2+20 x^3+4 x^4+93 x \log (x)+60 x^2 \log (x)+12 x^3 \log (x)}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )} \, dx\\ &=-\frac {36}{7 (1-2 x)}+x+\frac {3}{7 (3+x)}+2 \log (x)-\frac {19 \int \frac {\left (-3+5 x+2 x^2\right )^2+3 x \left (31+20 x+4 x^2\right ) \log (x)}{(3+x) \left (2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}}+\log (x)\right )} \, dx}{3087}-\frac {1}{147} \int \frac {\left (-3+5 x+2 x^2\right )^2+3 x \left (31+20 x+4 x^2\right ) \log (x)}{(3+x)^2 \left (2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}}+\log (x)\right )} \, dx+\frac {1}{9} \int \frac {\left (-3+5 x+2 x^2\right )^2+3 x \left (31+20 x+4 x^2\right ) \log (x)}{x \left (2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}}+\log (x)\right )} \, dx+\frac {8}{49} \int \frac {\left (-3+5 x+2 x^2\right )^2+3 x \left (31+20 x+4 x^2\right ) \log (x)}{(1-2 x)^2 \left (2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}}+\log (x)\right )} \, dx-\frac {72}{343} \int \left (\frac {9}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}-\frac {30 x}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}+\frac {13 x^2}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}+\frac {20 x^3}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}+\frac {4 x^4}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}+\frac {93 x \log (x)}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}+\frac {60 x^2 \log (x)}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}+\frac {12 x^3 \log (x)}{(-1+2 x) \left (2 \exp \left (\frac {15}{-3+5 x+2 x^2}+\frac {6 x}{-3+5 x+2 x^2}\right )+\log (x)\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 33, normalized size = 1.06 \begin {gather*} x+2 \log (x)+\log \left (2 e^{\frac {3 (5+2 x)}{-3+5 x+2 x^2}}+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36 - 288*x - 128*x^2 + 82*x^3 + 56*x^4 + 8*x^5 + E^((-15 - 6*x)/(-3 + 5*x + 2*x^2))*(9 - 30*x + 13*
x^2 + 20*x^3 + 4*x^4) + E^((-15 - 6*x)/(-3 + 5*x + 2*x^2))*(18 - 51*x - 4*x^2 + 53*x^3 + 28*x^4 + 4*x^5)*Log[x
])/(18*x - 60*x^2 + 26*x^3 + 40*x^4 + 8*x^5 + E^((-15 - 6*x)/(-3 + 5*x + 2*x^2))*(9*x - 30*x^2 + 13*x^3 + 20*x
^4 + 4*x^5)*Log[x]),x]

[Out]

x + 2*Log[x] + Log[2*E^((3*(5 + 2*x))/(-3 + 5*x + 2*x^2)) + Log[x]]

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fricas [A]  time = 0.62, size = 53, normalized size = 1.71 \begin {gather*} x + \log \left ({\left (e^{\left (-\frac {3 \, {\left (2 \, x + 5\right )}}{2 \, x^{2} + 5 \, x - 3}\right )} \log \relax (x) + 2\right )} e^{\left (\frac {3 \, {\left (2 \, x + 5\right )}}{2 \, x^{2} + 5 \, x - 3}\right )}\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5+28*x^4+53*x^3-4*x^2-51*x+18)*exp((-6*x-15)/(2*x^2+5*x-3))*log(x)+(4*x^4+20*x^3+13*x^2-30*x+9
)*exp((-6*x-15)/(2*x^2+5*x-3))+8*x^5+56*x^4+82*x^3-128*x^2-288*x+36)/((4*x^5+20*x^4+13*x^3-30*x^2+9*x)*exp((-6
*x-15)/(2*x^2+5*x-3))*log(x)+8*x^5+40*x^4+26*x^3-60*x^2+18*x),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 1.45, size = 150, normalized size = 4.84 \begin {gather*} \frac {2 \, x^{3} + 2 \, x^{2} \log \left (e^{\left (-\frac {10 \, x^{2} + 31 \, x}{2 \, x^{2} + 5 \, x - 3} + 5\right )} \log \relax (x) + 2\right ) + 4 \, x^{2} \log \relax (x) + 5 \, x^{2} + 5 \, x \log \left (e^{\left (-\frac {10 \, x^{2} + 31 \, x}{2 \, x^{2} + 5 \, x - 3} + 5\right )} \log \relax (x) + 2\right ) + 10 \, x \log \relax (x) + 3 \, x - 3 \, \log \left (e^{\left (-\frac {10 \, x^{2} + 31 \, x}{2 \, x^{2} + 5 \, x - 3} + 5\right )} \log \relax (x) + 2\right ) - 6 \, \log \relax (x) + 15}{2 \, x^{2} + 5 \, x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5+28*x^4+53*x^3-4*x^2-51*x+18)*exp((-6*x-15)/(2*x^2+5*x-3))*log(x)+(4*x^4+20*x^3+13*x^2-30*x+9
)*exp((-6*x-15)/(2*x^2+5*x-3))+8*x^5+56*x^4+82*x^3-128*x^2-288*x+36)/((4*x^5+20*x^4+13*x^3-30*x^2+9*x)*exp((-6
*x-15)/(2*x^2+5*x-3))*log(x)+8*x^5+40*x^4+26*x^3-60*x^2+18*x),x, algorithm="giac")

[Out]

(2*x^3 + 2*x^2*log(e^(-(10*x^2 + 31*x)/(2*x^2 + 5*x - 3) + 5)*log(x) + 2) + 4*x^2*log(x) + 5*x^2 + 5*x*log(e^(
-(10*x^2 + 31*x)/(2*x^2 + 5*x - 3) + 5)*log(x) + 2) + 10*x*log(x) + 3*x - 3*log(e^(-(10*x^2 + 31*x)/(2*x^2 + 5
*x - 3) + 5)*log(x) + 2) - 6*log(x) + 15)/(2*x^2 + 5*x - 3)

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maple [A]  time = 0.04, size = 33, normalized size = 1.06

method	result	size
risch	\(2 \ln \relax (x )+x +\ln \left (\ln \relax (x )+2 \,{\mathrm e}^{\frac {6 x +15}{\left (3+x \right ) \left (2 x -1\right )}}\right )\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^5+28*x^4+53*x^3-4*x^2-51*x+18)*exp((-6*x-15)/(2*x^2+5*x-3))*ln(x)+(4*x^4+20*x^3+13*x^2-30*x+9)*exp((
-6*x-15)/(2*x^2+5*x-3))+8*x^5+56*x^4+82*x^3-128*x^2-288*x+36)/((4*x^5+20*x^4+13*x^3-30*x^2+9*x)*exp((-6*x-15)/
(2*x^2+5*x-3))*ln(x)+8*x^5+40*x^4+26*x^3-60*x^2+18*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.61, size = 56, normalized size = 1.81 \begin {gather*} \frac {7 \, x^{2} + 21 \, x + 3}{7 \, {\left (x + 3\right )}} + \log \left (\frac {1}{2} \, {\left (2 \, e^{\left (\frac {36}{7 \, {\left (2 \, x - 1\right )}} + \frac {3}{7 \, {\left (x + 3\right )}}\right )} + \log \relax (x)\right )} e^{\left (-\frac {3}{7 \, {\left (x + 3\right )}}\right )}\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5+28*x^4+53*x^3-4*x^2-51*x+18)*exp((-6*x-15)/(2*x^2+5*x-3))*log(x)+(4*x^4+20*x^3+13*x^2-30*x+9
)*exp((-6*x-15)/(2*x^2+5*x-3))+8*x^5+56*x^4+82*x^3-128*x^2-288*x+36)/((4*x^5+20*x^4+13*x^3-30*x^2+9*x)*exp((-6
*x-15)/(2*x^2+5*x-3))*log(x)+8*x^5+40*x^4+26*x^3-60*x^2+18*x),x, algorithm="maxima")

[Out]

1/7*(7*x^2 + 21*x + 3)/(x + 3) + log(1/2*(2*e^(36/7/(2*x - 1) + 3/7/(x + 3)) + log(x))*e^(-3/7/(x + 3))) + 2*l
og(x)

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mupad [B]  time = 1.38, size = 56, normalized size = 1.81 \begin {gather*} x+\ln \left (\ln \relax (x)\right )+\ln \left (\frac {{\mathrm {e}}^{-\frac {6\,x+15}{2\,x^2+5\,x-3}}\,\ln \relax (x)+2}{\ln \relax (x)}\right )+2\,\ln \relax (x)+\frac {3\,x+\frac {15}{2}}{x^2+\frac {5\,x}{2}-\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(6*x + 15)/(5*x + 2*x^2 - 3))*(13*x^2 - 30*x + 20*x^3 + 4*x^4 + 9) - 288*x - 128*x^2 + 82*x^3 + 56*x
^4 + 8*x^5 + exp(-(6*x + 15)/(5*x + 2*x^2 - 3))*log(x)*(53*x^3 - 4*x^2 - 51*x + 28*x^4 + 4*x^5 + 18) + 36)/(18
*x - 60*x^2 + 26*x^3 + 40*x^4 + 8*x^5 + exp(-(6*x + 15)/(5*x + 2*x^2 - 3))*log(x)*(9*x - 30*x^2 + 13*x^3 + 20*
x^4 + 4*x^5)),x)

[Out]

x + log(log(x)) + log((exp(-(6*x + 15)/(5*x + 2*x^2 - 3))*log(x) + 2)/log(x)) + 2*log(x) + (3*x + 15/2)/((5*x)
/2 + x^2 - 3/2)

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sympy [B]  time = 1.39, size = 51, normalized size = 1.65 \begin {gather*} x + \frac {6 x + 15}{2 x^{2} + 5 x - 3} + 2 \log {\relax (x )} + \log {\left (e^{\frac {- 6 x - 15}{2 x^{2} + 5 x - 3}} + \frac {2}{\log {\relax (x )}} \right )} + \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**5+28*x**4+53*x**3-4*x**2-51*x+18)*exp((-6*x-15)/(2*x**2+5*x-3))*ln(x)+(4*x**4+20*x**3+13*x**2
-30*x+9)*exp((-6*x-15)/(2*x**2+5*x-3))+8*x**5+56*x**4+82*x**3-128*x**2-288*x+36)/((4*x**5+20*x**4+13*x**3-30*x
**2+9*x)*exp((-6*x-15)/(2*x**2+5*x-3))*ln(x)+8*x**5+40*x**4+26*x**3-60*x**2+18*x),x)

[Out]

x + (6*x + 15)/(2*x**2 + 5*x - 3) + 2*log(x) + log(exp((-6*x - 15)/(2*x**2 + 5*x - 3)) + 2/log(x)) + log(log(x
))

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