3.62.87 \(\int \frac {-70 x-5 x^2+10 x^3+e^{-2+5 x+x^2} (-15+14 x-6 x^2+x^3)-5 e^{-2+5 x+x^2} x \log (x)}{15 x-5 x^2+e^{-2+5 x+x^2} (9 x^2-6 x^3+x^4)+e^{-2+5 x+x^2} (-15 x+5 x^2) \log (x)} \, dx\)

Optimal. Leaf size=29 \[ \log \left (\frac {1}{4} \left (x+\frac {5 \left (-e^{2-x (5+x)}+\log (x)\right )}{-3+x}\right )\right ) \]

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Rubi [F]  time = 27.87, 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 {-70 x-5 x^2+10 x^3+e^{-2+5 x+x^2} \left (-15+14 x-6 x^2+x^3\right )-5 e^{-2+5 x+x^2} x \log (x)}{15 x-5 x^2+e^{-2+5 x+x^2} \left (9 x^2-6 x^3+x^4\right )+e^{-2+5 x+x^2} \left (-15 x+5 x^2\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-70*x - 5*x^2 + 10*x^3 + E^(-2 + 5*x + x^2)*(-15 + 14*x - 6*x^2 + x^3) - 5*E^(-2 + 5*x + x^2)*x*Log[x])/(
15*x - 5*x^2 + E^(-2 + 5*x + x^2)*(9*x^2 - 6*x^3 + x^4) + E^(-2 + 5*x + x^2)*(-15*x + 5*x^2)*Log[x]),x]

[Out]

-Log[3 - x] - 3*Defer[Int][(-3*x + x^2 + 5*Log[x])^(-1), x] + 5*Defer[Int][1/(x*(-3*x + x^2 + 5*Log[x])), x] +
2*Defer[Int][x/(-3*x + x^2 + 5*Log[x]), x] - 15*E^2*Defer[Int][1/((-3*x + x^2 + 5*Log[x])*(-5*E^2 - 3*E^(x*(5
+ x))*x + E^(x*(5 + x))*x^2 + 5*E^(x*(5 + x))*Log[x])), x] + 25*E^2*Defer[Int][1/(x*(-3*x + x^2 + 5*Log[x])*(
-5*E^2 - 3*E^(x*(5 + x))*x + E^(x*(5 + x))*x^2 + 5*E^(x*(5 + x))*Log[x])), x] - 65*E^2*Defer[Int][x/((-3*x + x
^2 + 5*Log[x])*(-5*E^2 - 3*E^(x*(5 + x))*x + E^(x*(5 + x))*x^2 + 5*E^(x*(5 + x))*Log[x])), x] - 5*E^2*Defer[In
t][x^2/((-3*x + x^2 + 5*Log[x])*(-5*E^2 - 3*E^(x*(5 + x))*x + E^(x*(5 + x))*x^2 + 5*E^(x*(5 + x))*Log[x])), x]
+ 10*E^2*Defer[Int][x^3/((-3*x + x^2 + 5*Log[x])*(-5*E^2 - 3*E^(x*(5 + x))*x + E^(x*(5 + x))*x^2 + 5*E^(x*(5
+ x))*Log[x])), x] + 125*E^2*Defer[Int][Log[x]/((-3*x + x^2 + 5*Log[x])*(-5*E^2 - 3*E^(x*(5 + x))*x + E^(x*(5
+ x))*x^2 + 5*E^(x*(5 + x))*Log[x])), x] + 50*E^2*Defer[Int][(x*Log[x])/((-3*x + x^2 + 5*Log[x])*(-5*E^2 - 3*E
^(x*(5 + x))*x + E^(x*(5 + x))*x^2 + 5*E^(x*(5 + x))*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^2 \left (-70 x-5 x^2+10 x^3+e^{-2+5 x+x^2} \left (-15+14 x-6 x^2+x^3\right )-5 e^{-2+5 x+x^2} x \log (x)\right )}{(3-x) x \left (5 e^2+3 e^{x (5+x)} x-e^{x (5+x)} x^2-5 e^{x (5+x)} \log (x)\right )} \, dx\\ &=e^2 \int \frac {-70 x-5 x^2+10 x^3+e^{-2+5 x+x^2} \left (-15+14 x-6 x^2+x^3\right )-5 e^{-2+5 x+x^2} x \log (x)}{(3-x) x \left (5 e^2+3 e^{x (5+x)} x-e^{x (5+x)} x^2-5 e^{x (5+x)} \log (x)\right )} \, dx\\ &=e^2 \int \left (-\frac {-15 e^{x (5+x)}-70 e^2 x+14 e^{x (5+x)} x-5 e^2 x^2-6 e^{x (5+x)} x^2+10 e^2 x^3+e^{x (5+x)} x^3-5 e^{x (5+x)} x \log (x)}{3 e^2 (-3+x) \left (5 e^2+3 e^{x (5+x)} x-e^{x (5+x)} x^2-5 e^{x (5+x)} \log (x)\right )}+\frac {-15 e^{x (5+x)}-70 e^2 x+14 e^{x (5+x)} x-5 e^2 x^2-6 e^{x (5+x)} x^2+10 e^2 x^3+e^{x (5+x)} x^3-5 e^{x (5+x)} x \log (x)}{3 e^2 x \left (5 e^2+3 e^{x (5+x)} x-e^{x (5+x)} x^2-5 e^{x (5+x)} \log (x)\right )}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {-15 e^{x (5+x)}-70 e^2 x+14 e^{x (5+x)} x-5 e^2 x^2-6 e^{x (5+x)} x^2+10 e^2 x^3+e^{x (5+x)} x^3-5 e^{x (5+x)} x \log (x)}{(-3+x) \left (5 e^2+3 e^{x (5+x)} x-e^{x (5+x)} x^2-5 e^{x (5+x)} \log (x)\right )} \, dx\right )+\frac {1}{3} \int \frac {-15 e^{x (5+x)}-70 e^2 x+14 e^{x (5+x)} x-5 e^2 x^2-6 e^{x (5+x)} x^2+10 e^2 x^3+e^{x (5+x)} x^3-5 e^{x (5+x)} x \log (x)}{x \left (5 e^2+3 e^{x (5+x)} x-e^{x (5+x)} x^2-5 e^{x (5+x)} \log (x)\right )} \, dx\\ &=\frac {1}{3} \int \frac {-5 e^2 x \left (14+x-2 x^2\right )+e^{x (5+x)} \left (-15+14 x-6 x^2+x^3\right )-5 e^{x (5+x)} x \log (x)}{x \left (5 e^2-e^{x (5+x)} (-3+x) x-5 e^{x (5+x)} \log (x)\right )} \, dx-\frac {1}{3} \int \frac {5 e^2 x \left (14+x-2 x^2\right )-e^{x (5+x)} \left (-15+14 x-6 x^2+x^3\right )+5 e^{x (5+x)} x \log (x)}{(3-x) \left (5 e^2-e^{x (5+x)} (-3+x) x-5 e^{x (5+x)} \log (x)\right )} \, dx\\ &=-\left (\frac {1}{3} \int \left (\frac {15-14 x+6 x^2-x^3+5 x \log (x)}{(-3+x) \left (-3 x+x^2+5 \log (x)\right )}+\frac {5 e^2 \left (5-3 x-13 x^2-x^3+2 x^4+25 x \log (x)+10 x^2 \log (x)\right )}{\left (-3 x+x^2+5 \log (x)\right ) \left (5 e^2+3 e^{x (5+x)} x-e^{x (5+x)} x^2-5 e^{x (5+x)} \log (x)\right )}\right ) \, dx\right )+\frac {1}{3} \int \left (\frac {15-14 x+6 x^2-x^3+5 x \log (x)}{x \left (-3 x+x^2+5 \log (x)\right )}+\frac {5 e^2 (-3+x) \left (5-3 x-13 x^2-x^3+2 x^4+25 x \log (x)+10 x^2 \log (x)\right )}{x \left (-3 x+x^2+5 \log (x)\right ) \left (5 e^2+3 e^{x (5+x)} x-e^{x (5+x)} x^2-5 e^{x (5+x)} \log (x)\right )}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {15-14 x+6 x^2-x^3+5 x \log (x)}{(-3+x) \left (-3 x+x^2+5 \log (x)\right )} \, dx\right )+\frac {1}{3} \int \frac {15-14 x+6 x^2-x^3+5 x \log (x)}{x \left (-3 x+x^2+5 \log (x)\right )} \, dx-\frac {1}{3} \left (5 e^2\right ) \int \frac {5-3 x-13 x^2-x^3+2 x^4+25 x \log (x)+10 x^2 \log (x)}{\left (-3 x+x^2+5 \log (x)\right ) \left (5 e^2+3 e^{x (5+x)} x-e^{x (5+x)} x^2-5 e^{x (5+x)} \log (x)\right )} \, dx+\frac {1}{3} \left (5 e^2\right ) \int \frac {(-3+x) \left (5-3 x-13 x^2-x^3+2 x^4+25 x \log (x)+10 x^2 \log (x)\right )}{x \left (-3 x+x^2+5 \log (x)\right ) \left (5 e^2+3 e^{x (5+x)} x-e^{x (5+x)} x^2-5 e^{x (5+x)} \log (x)\right )} \, dx\\ &=-\left (\frac {1}{3} \int \left (\frac {x}{-3+x}+\frac {-5+3 x-2 x^2}{-3 x+x^2+5 \log (x)}\right ) \, dx\right )+\frac {1}{3} \int \left (1+\frac {15-14 x+9 x^2-2 x^3}{x \left (-3 x+x^2+5 \log (x)\right )}\right ) \, dx-\frac {1}{3} \left (5 e^2\right ) \int \left (-\frac {5}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )}+\frac {3 x}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )}+\frac {13 x^2}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )}+\frac {x^3}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )}-\frac {2 x^4}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )}-\frac {25 x \log (x)}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )}-\frac {10 x^2 \log (x)}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )}\right ) \, dx+\frac {1}{3} \left (5 e^2\right ) \int \left (-\frac {5-3 x-13 x^2-x^3+2 x^4+25 x \log (x)+10 x^2 \log (x)}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )}+\frac {3 \left (5-3 x-13 x^2-x^3+2 x^4+25 x \log (x)+10 x^2 \log (x)\right )}{x \left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )}\right ) \, dx\\ &=\frac {x}{3}-\frac {1}{3} \int \frac {x}{-3+x} \, dx-\frac {1}{3} \int \frac {-5+3 x-2 x^2}{-3 x+x^2+5 \log (x)} \, dx+\frac {1}{3} \int \frac {15-14 x+9 x^2-2 x^3}{x \left (-3 x+x^2+5 \log (x)\right )} \, dx-\frac {1}{3} \left (5 e^2\right ) \int \frac {x^3}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )} \, dx-\frac {1}{3} \left (5 e^2\right ) \int \frac {5-3 x-13 x^2-x^3+2 x^4+25 x \log (x)+10 x^2 \log (x)}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )} \, dx+\frac {1}{3} \left (10 e^2\right ) \int \frac {x^4}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )} \, dx-\left (5 e^2\right ) \int \frac {x}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )} \, dx+\left (5 e^2\right ) \int \frac {5-3 x-13 x^2-x^3+2 x^4+25 x \log (x)+10 x^2 \log (x)}{x \left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )} \, dx+\frac {1}{3} \left (25 e^2\right ) \int \frac {1}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )} \, dx+\frac {1}{3} \left (50 e^2\right ) \int \frac {x^2 \log (x)}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )} \, dx-\frac {1}{3} \left (65 e^2\right ) \int \frac {x^2}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )} \, dx+\frac {1}{3} \left (125 e^2\right ) \int \frac {x \log (x)}{\left (-3 x+x^2+5 \log (x)\right ) \left (-5 e^2-3 e^{x (5+x)} x+e^{x (5+x)} x^2+5 e^{x (5+x)} \log (x)\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 51, normalized size = 1.76 \begin {gather*} -\frac {1}{4} (5+2 x)^2-\log (3-x)+\log \left (5 e^2-e^{x (5+x)} (-3+x) x-5 e^{x (5+x)} \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-70*x - 5*x^2 + 10*x^3 + E^(-2 + 5*x + x^2)*(-15 + 14*x - 6*x^2 + x^3) - 5*E^(-2 + 5*x + x^2)*x*Log
[x])/(15*x - 5*x^2 + E^(-2 + 5*x + x^2)*(9*x^2 - 6*x^3 + x^4) + E^(-2 + 5*x + x^2)*(-15*x + 5*x^2)*Log[x]),x]

[Out]

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

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fricas [A]  time = 1.02, size = 52, normalized size = 1.79 \begin {gather*} \log \left ({\left ({\left (x^{2} - 3 \, x\right )} e^{\left (x^{2} + 5 \, x - 2\right )} + 5 \, e^{\left (x^{2} + 5 \, x - 2\right )} \log \relax (x) - 5\right )} e^{\left (-x^{2} - 5 \, x + 2\right )}\right ) - \log \left (x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x*exp(x^2+5*x-2)*log(x)+(x^3-6*x^2+14*x-15)*exp(x^2+5*x-2)+10*x^3-5*x^2-70*x)/((5*x^2-15*x)*exp(
x^2+5*x-2)*log(x)+(x^4-6*x^3+9*x^2)*exp(x^2+5*x-2)-5*x^2+15*x),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.19, size = 85, normalized size = 2.93 \begin {gather*} -x^{2} - 5 \, x + \log \left (-x^{2} e^{\left (x^{2} + 5 \, x\right )} + 3 \, x e^{\left (x^{2} + 5 \, x\right )} - 5 \, e^{\left (x^{2} + 5 \, x\right )} \log \relax (x) + 5 \, e^{2}\right ) + \log \left (x^{2} - 3 \, x + 5 \, \log \relax (x)\right ) - \log \left (-x^{2} + 3 \, x - 5 \, \log \relax (x)\right ) - \log \left (x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x*exp(x^2+5*x-2)*log(x)+(x^3-6*x^2+14*x-15)*exp(x^2+5*x-2)+10*x^3-5*x^2-70*x)/((5*x^2-15*x)*exp(
x^2+5*x-2)*log(x)+(x^4-6*x^3+9*x^2)*exp(x^2+5*x-2)-5*x^2+15*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.08, size = 52, normalized size = 1.79

method	result	size
risch	\(-\ln \left (x -3\right )+\ln \left (\ln \relax (x )+\frac {\left (x^{2} {\mathrm e}^{x^{2}+5 x -2}-3 \,{\mathrm e}^{x^{2}+5 x -2} x -5\right ) {\mathrm e}^{-x^{2}-5 x +2}}{5}\right )\)	\(52\)
norman	\(-x^{2}-5 x -\ln \left (x -3\right )+\ln \left (x^{2} {\mathrm e}^{x^{2}+5 x -2}-3 \,{\mathrm e}^{x^{2}+5 x -2} x +5 \,{\mathrm e}^{x^{2}+5 x -2} \ln \relax (x )-5\right )\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-5*x*exp(x^2+5*x-2)*ln(x)+(x^3-6*x^2+14*x-15)*exp(x^2+5*x-2)+10*x^3-5*x^2-70*x)/((5*x^2-15*x)*exp(x^2+5*x
-2)*ln(x)+(x^4-6*x^3+9*x^2)*exp(x^2+5*x-2)-5*x^2+15*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.39, size = 68, normalized size = 2.34 \begin {gather*} -x^{2} + \log \left (\frac {1}{5} \, x^{2} - \frac {3}{5} \, x + \log \relax (x)\right ) - \log \left (x - 3\right ) + \log \left (\frac {{\left ({\left (x^{2} - 3 \, x + 5 \, \log \relax (x)\right )} e^{\left (x^{2} + 5 \, x\right )} - 5 \, e^{2}\right )} e^{\left (-5 \, x\right )}}{x^{2} - 3 \, x + 5 \, \log \relax (x)}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x*exp(x^2+5*x-2)*log(x)+(x^3-6*x^2+14*x-15)*exp(x^2+5*x-2)+10*x^3-5*x^2-70*x)/((5*x^2-15*x)*exp(
x^2+5*x-2)*log(x)+(x^4-6*x^3+9*x^2)*exp(x^2+5*x-2)-5*x^2+15*x),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {70\,x-{\mathrm {e}}^{x^2+5\,x-2}\,\left (x^3-6\,x^2+14\,x-15\right )+5\,x^2-10\,x^3+5\,x\,{\mathrm {e}}^{x^2+5\,x-2}\,\ln \relax (x)}{15\,x+{\mathrm {e}}^{x^2+5\,x-2}\,\left (x^4-6\,x^3+9\,x^2\right )-5\,x^2-{\mathrm {e}}^{x^2+5\,x-2}\,\ln \relax (x)\,\left (15\,x-5\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(70*x - exp(5*x + x^2 - 2)*(14*x - 6*x^2 + x^3 - 15) + 5*x^2 - 10*x^3 + 5*x*exp(5*x + x^2 - 2)*log(x))/(1
5*x + exp(5*x + x^2 - 2)*(9*x^2 - 6*x^3 + x^4) - 5*x^2 - exp(5*x + x^2 - 2)*log(x)*(15*x - 5*x^2)),x)

[Out]

int(-(70*x - exp(5*x + x^2 - 2)*(14*x - 6*x^2 + x^3 - 15) + 5*x^2 - 10*x^3 + 5*x*exp(5*x + x^2 - 2)*log(x))/(1
5*x + exp(5*x + x^2 - 2)*(9*x^2 - 6*x^3 + x^4) - 5*x^2 - exp(5*x + x^2 - 2)*log(x)*(15*x - 5*x^2)), x)

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sympy [A]  time = 1.32, size = 51, normalized size = 1.76 \begin {gather*} - x^{2} - 5 x - \log {\left (x - 3 \right )} + \log {\left (e^{x^{2} + 5 x - 2} - \frac {5}{x^{2} - 3 x + 5 \log {\relax (x )}} \right )} + \log {\left (\frac {x^{2}}{5} - \frac {3 x}{5} + \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x*exp(x**2+5*x-2)*ln(x)+(x**3-6*x**2+14*x-15)*exp(x**2+5*x-2)+10*x**3-5*x**2-70*x)/((5*x**2-15*x
)*exp(x**2+5*x-2)*ln(x)+(x**4-6*x**3+9*x**2)*exp(x**2+5*x-2)-5*x**2+15*x),x)

[Out]

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

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