3.94.92 \(\int \frac {98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} (49 x+154 x^2+121 x^3)+(98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} (7+71 x+154 x^2+121 x^3)) \log (x)}{196 x+616 x^2+484 x^3+e^{\frac {2}{7 x+11 x^2}} (49 x+154 x^2+121 x^3)+e^{\frac {1}{7 x+11 x^2}} (196 x+616 x^2+484 x^3)} \, dx\)

Optimal. Leaf size=21 \[ \frac {x \log (x)}{2+e^{\frac {1}{x (7+11 x)}}} \]

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Rubi [F]  time = 9.89, 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 {98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+\left (98 x+308 x^2+242 x^3+e^{\frac {1}{7 x+11 x^2}} \left (7+71 x+154 x^2+121 x^3\right )\right ) \log (x)}{196 x+616 x^2+484 x^3+e^{\frac {2}{7 x+11 x^2}} \left (49 x+154 x^2+121 x^3\right )+e^{\frac {1}{7 x+11 x^2}} \left (196 x+616 x^2+484 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(98*x + 308*x^2 + 242*x^3 + E^(7*x + 11*x^2)^(-1)*(49*x + 154*x^2 + 121*x^3) + (98*x + 308*x^2 + 242*x^3 +
E^(7*x + 11*x^2)^(-1)*(7 + 71*x + 154*x^2 + 121*x^3))*Log[x])/(196*x + 616*x^2 + 484*x^3 + E^(2/(7*x + 11*x^2
))*(49*x + 154*x^2 + 121*x^3) + E^(7*x + 11*x^2)^(-1)*(196*x + 616*x^2 + 484*x^3)),x]

[Out]

Defer[Int][(2 + E^(7*x + 11*x^2)^(-1))^(-1), x] + Log[x]*Defer[Int][(2 + E^(7*x + 11*x^2)^(-1))^(-1), x] - (2*
Log[x]*Defer[Int][1/((2 + E^(7*x + 11*x^2)^(-1))^2*x), x])/7 + (Log[x]*Defer[Int][1/((2 + E^(7*x + 11*x^2)^(-1
))*x), x])/7 - 22*Log[x]*Defer[Int][1/((2 + E^(7*x + 11*x^2)^(-1))^2*(7 + 11*x)^2), x] + 11*Log[x]*Defer[Int][
1/((2 + E^(7*x + 11*x^2)^(-1))*(7 + 11*x)^2), x] + (22*Log[x]*Defer[Int][1/((2 + E^(7*x + 11*x^2)^(-1))^2*(7 +
11*x)), x])/7 - (11*Log[x]*Defer[Int][1/((2 + E^(7*x + 11*x^2)^(-1))*(7 + 11*x)), x])/7 - Defer[Int][Defer[In
t][(2 + E^(7*x + 11*x^2)^(-1))^(-1), x]/x, x] + (2*Defer[Int][Defer[Int][1/((2 + E^(7*x + 11*x^2)^(-1))^2*x),
x]/x, x])/7 + 22*Defer[Int][Defer[Int][1/((2 + E^(7*x + 11*x^2)^(-1))^2*(7 + 11*x)^2), x]/x, x] - 11*Defer[Int
][Defer[Int][1/((2 + E^(7*x + 11*x^2)^(-1))*(7 + 11*x)^2), x]/x, x] - (22*Defer[Int][Defer[Int][1/((2 + E^(7*x
+ 11*x^2)^(-1))^2*(7 + 11*x)), x]/x, x])/7 + (11*Defer[Int][Defer[Int][1/((2 + E^(7*x + 11*x^2)^(-1))*(7 + 11
*x)), x]/x, x])/7 - Defer[Int][Defer[Int][(2*x + E^(7*x + 11*x^2)^(-1)*x)^(-1), x]/x, x]/7

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2+e^{\frac {1}{7 x+11 x^2}}+2 \log (x)+\frac {e^{\frac {1}{7 x+11 x^2}} \left (7+71 x+154 x^2+121 x^3\right ) \log (x)}{x (7+11 x)^2}}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2} \, dx\\ &=\int \left (-\frac {2 (7+22 x) \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 x (7+11 x)^2}+\frac {49 x+154 x^2+121 x^3+7 \log (x)+71 x \log (x)+154 x^2 \log (x)+121 x^3 \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) x (7+11 x)^2}\right ) \, dx\\ &=-\left (2 \int \frac {(7+22 x) \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 x (7+11 x)^2} \, dx\right )+\int \frac {49 x+154 x^2+121 x^3+7 \log (x)+71 x \log (x)+154 x^2 \log (x)+121 x^3 \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) x (7+11 x)^2} \, dx\\ &=2 \int \frac {\int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 x} \, dx+77 \int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 (7+11 x)^2} \, dx-11 \int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 (7+11 x)} \, dx}{7 x} \, dx-\frac {1}{7} (2 \log (x)) \int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 x} \, dx+\frac {1}{7} (22 \log (x)) \int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 (7+11 x)} \, dx-(22 \log (x)) \int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 (7+11 x)^2} \, dx+\int \left (\frac {49 x+154 x^2+121 x^3+7 \log (x)+71 x \log (x)+154 x^2 \log (x)+121 x^3 \log (x)}{49 \left (2+e^{\frac {1}{7 x+11 x^2}}\right ) x}-\frac {11 \left (49 x+154 x^2+121 x^3+7 \log (x)+71 x \log (x)+154 x^2 \log (x)+121 x^3 \log (x)\right )}{7 \left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)^2}-\frac {11 \left (49 x+154 x^2+121 x^3+7 \log (x)+71 x \log (x)+154 x^2 \log (x)+121 x^3 \log (x)\right )}{49 \left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)}\right ) \, dx\\ &=\frac {1}{49} \int \frac {49 x+154 x^2+121 x^3+7 \log (x)+71 x \log (x)+154 x^2 \log (x)+121 x^3 \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) x} \, dx-\frac {11}{49} \int \frac {49 x+154 x^2+121 x^3+7 \log (x)+71 x \log (x)+154 x^2 \log (x)+121 x^3 \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)} \, dx+\frac {2}{7} \int \frac {\int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 x} \, dx+77 \int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 (7+11 x)^2} \, dx-11 \int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 (7+11 x)} \, dx}{x} \, dx-\frac {11}{7} \int \frac {49 x+154 x^2+121 x^3+7 \log (x)+71 x \log (x)+154 x^2 \log (x)+121 x^3 \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)^2} \, dx-\frac {1}{7} (2 \log (x)) \int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 x} \, dx+\frac {1}{7} (22 \log (x)) \int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 (7+11 x)} \, dx-(22 \log (x)) \int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 (7+11 x)^2} \, dx\\ &=\frac {1}{49} \int \left (\frac {49}{2+e^{\frac {1}{7 x+11 x^2}}}+\frac {154 x}{2+e^{\frac {1}{7 x+11 x^2}}}+\frac {121 x^2}{2+e^{\frac {1}{7 x+11 x^2}}}+\frac {71 \log (x)}{2+e^{\frac {1}{7 x+11 x^2}}}+\frac {7 \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) x}+\frac {154 x \log (x)}{2+e^{\frac {1}{7 x+11 x^2}}}+\frac {121 x^2 \log (x)}{2+e^{\frac {1}{7 x+11 x^2}}}\right ) \, dx-\frac {11}{49} \int \left (\frac {49 x}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)}+\frac {154 x^2}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)}+\frac {121 x^3}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)}+\frac {7 \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)}+\frac {71 x \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)}+\frac {154 x^2 \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)}+\frac {121 x^3 \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)}\right ) \, dx+\frac {2}{7} \int \left (\frac {\int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 x} \, dx+77 \int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 (7+11 x)^2} \, dx}{x}-\frac {11 \int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 (7+11 x)} \, dx}{x}\right ) \, dx-\frac {11}{7} \int \left (\frac {49 x}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)^2}+\frac {154 x^2}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)^2}+\frac {121 x^3}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)^2}+\frac {7 \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)^2}+\frac {71 x \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)^2}+\frac {154 x^2 \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)^2}+\frac {121 x^3 \log (x)}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right ) (7+11 x)^2}\right ) \, dx-\frac {1}{7} (2 \log (x)) \int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 x} \, dx+\frac {1}{7} (22 \log (x)) \int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 (7+11 x)} \, dx-(22 \log (x)) \int \frac {1}{\left (2+e^{\frac {1}{7 x+11 x^2}}\right )^2 (7+11 x)^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.58, size = 21, normalized size = 1.00 \begin {gather*} \frac {x \log (x)}{2+e^{\frac {1}{7 x+11 x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(98*x + 308*x^2 + 242*x^3 + E^(7*x + 11*x^2)^(-1)*(49*x + 154*x^2 + 121*x^3) + (98*x + 308*x^2 + 242
*x^3 + E^(7*x + 11*x^2)^(-1)*(7 + 71*x + 154*x^2 + 121*x^3))*Log[x])/(196*x + 616*x^2 + 484*x^3 + E^(2/(7*x +
11*x^2))*(49*x + 154*x^2 + 121*x^3) + E^(7*x + 11*x^2)^(-1)*(196*x + 616*x^2 + 484*x^3)),x]

[Out]

(x*Log[x])/(2 + E^(7*x + 11*x^2)^(-1))

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fricas [A]  time = 0.65, size = 20, normalized size = 0.95 \begin {gather*} \frac {x \log \relax (x)}{e^{\left (\frac {1}{11 \, x^{2} + 7 \, x}\right )} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((121*x^3+154*x^2+71*x+7)*exp(1/(11*x^2+7*x))+242*x^3+308*x^2+98*x)*log(x)+(121*x^3+154*x^2+49*x)*e
xp(1/(11*x^2+7*x))+242*x^3+308*x^2+98*x)/((121*x^3+154*x^2+49*x)*exp(1/(11*x^2+7*x))^2+(484*x^3+616*x^2+196*x)
*exp(1/(11*x^2+7*x))+484*x^3+616*x^2+196*x),x, algorithm="fricas")

[Out]

x*log(x)/(e^(1/(11*x^2 + 7*x)) + 2)

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giac [A]  time = 0.28, size = 20, normalized size = 0.95 \begin {gather*} \frac {x \log \relax (x)}{e^{\left (\frac {1}{11 \, x^{2} + 7 \, x}\right )} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((121*x^3+154*x^2+71*x+7)*exp(1/(11*x^2+7*x))+242*x^3+308*x^2+98*x)*log(x)+(121*x^3+154*x^2+49*x)*e
xp(1/(11*x^2+7*x))+242*x^3+308*x^2+98*x)/((121*x^3+154*x^2+49*x)*exp(1/(11*x^2+7*x))^2+(484*x^3+616*x^2+196*x)
*exp(1/(11*x^2+7*x))+484*x^3+616*x^2+196*x),x, algorithm="giac")

[Out]

x*log(x)/(e^(1/(11*x^2 + 7*x)) + 2)

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maple [A]  time = 0.04, size = 21, normalized size = 1.00

method	result	size
risch	\(\frac {\ln \relax (x ) x}{2+{\mathrm e}^{\frac {1}{x \left (11 x +7\right )}}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((121*x^3+154*x^2+71*x+7)*exp(1/(11*x^2+7*x))+242*x^3+308*x^2+98*x)*ln(x)+(121*x^3+154*x^2+49*x)*exp(1/(1
1*x^2+7*x))+242*x^3+308*x^2+98*x)/((121*x^3+154*x^2+49*x)*exp(1/(11*x^2+7*x))^2+(484*x^3+616*x^2+196*x)*exp(1/
(11*x^2+7*x))+484*x^3+616*x^2+196*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)/(2+exp(1/x/(11*x+7)))*x

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maxima [A]  time = 0.41, size = 35, normalized size = 1.67 \begin {gather*} \frac {x e^{\left (\frac {11}{7 \, {\left (11 \, x + 7\right )}}\right )} \log \relax (x)}{2 \, e^{\left (\frac {11}{7 \, {\left (11 \, x + 7\right )}}\right )} + e^{\left (\frac {1}{7 \, x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((121*x^3+154*x^2+71*x+7)*exp(1/(11*x^2+7*x))+242*x^3+308*x^2+98*x)*log(x)+(121*x^3+154*x^2+49*x)*e
xp(1/(11*x^2+7*x))+242*x^3+308*x^2+98*x)/((121*x^3+154*x^2+49*x)*exp(1/(11*x^2+7*x))^2+(484*x^3+616*x^2+196*x)
*exp(1/(11*x^2+7*x))+484*x^3+616*x^2+196*x),x, algorithm="maxima")

[Out]

x*e^(11/7/(11*x + 7))*log(x)/(2*e^(11/7/(11*x + 7)) + e^(1/7/x))

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mupad [B]  time = 6.74, size = 20, normalized size = 0.95 \begin {gather*} \frac {x\,\ln \relax (x)}{{\mathrm {e}}^{\frac {1}{11\,x^2+7\,x}}+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((98*x + log(x)*(98*x + exp(1/(7*x + 11*x^2))*(71*x + 154*x^2 + 121*x^3 + 7) + 308*x^2 + 242*x^3) + exp(1/(
7*x + 11*x^2))*(49*x + 154*x^2 + 121*x^3) + 308*x^2 + 242*x^3)/(196*x + exp(1/(7*x + 11*x^2))*(196*x + 616*x^2
+ 484*x^3) + exp(2/(7*x + 11*x^2))*(49*x + 154*x^2 + 121*x^3) + 616*x^2 + 484*x^3),x)

[Out]

(x*log(x))/(exp(1/(7*x + 11*x^2)) + 2)

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sympy [A]  time = 0.43, size = 17, normalized size = 0.81 \begin {gather*} \frac {x \log {\relax (x )}}{e^{\frac {1}{11 x^{2} + 7 x}} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((121*x**3+154*x**2+71*x+7)*exp(1/(11*x**2+7*x))+242*x**3+308*x**2+98*x)*ln(x)+(121*x**3+154*x**2+4
9*x)*exp(1/(11*x**2+7*x))+242*x**3+308*x**2+98*x)/((121*x**3+154*x**2+49*x)*exp(1/(11*x**2+7*x))**2+(484*x**3+
616*x**2+196*x)*exp(1/(11*x**2+7*x))+484*x**3+616*x**2+196*x),x)

[Out]

x*log(x)/(exp(1/(11*x**2 + 7*x)) + 2)

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