3.87.46 \(\int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} (-1+2 x+x^2+x^3)+e^x (-49-149 x+53 x^2+2 x^3+2 x^4)+(49 x+48 x^3+2 x^4+3 x^5+e^{2 x} (-x+2 x^2+3 x^3)+e^x (49 x+96 x^2-45 x^3+6 x^4)) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx\)

Optimal. Leaf size=26 \[ \left (1+x+\frac {49}{e^x+x}\right ) \left (3+x \left (-\frac {1}{x}+x\right ) \log (x)\right ) \]

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Rubi [F]  time = 2.73, 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 {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-196*x - x^2 + 51*x^3 + x^4 + x^5 + E^(2*x)*(-1 + 2*x + x^2 + x^3) + E^x*(-49 - 149*x + 53*x^2 + 2*x^3 +
2*x^4) + (49*x + 48*x^3 + 2*x^4 + 3*x^5 + E^(2*x)*(-x + 2*x^2 + 3*x^3) + E^x*(49*x + 96*x^2 - 45*x^3 + 6*x^4))
*Log[x])/(E^(2*x)*x + 2*E^x*x^2 + x^3),x]

[Out]

3*x + 147/(E^x + x) - Log[x] - x*Log[x] + x^2*Log[x] + x^3*Log[x] - (49*Log[x])/(E^x + x) - 49*Log[x]*Defer[In
t][x^2/(E^x + x)^2, x] + 49*Log[x]*Defer[Int][x^3/(E^x + x)^2, x] + 49*Defer[Int][x/(E^x + x), x] + 98*Log[x]*
Defer[Int][x/(E^x + x), x] - 49*Log[x]*Defer[Int][x^2/(E^x + x), x] + 49*Defer[Int][Defer[Int][x^2/(E^x + x)^2
, x]/x, x] - 49*Defer[Int][Defer[Int][x^3/(E^x + x)^2, x]/x, x] - 98*Defer[Int][Defer[Int][x/(E^x + x), x]/x,
x] + 49*Defer[Int][Defer[Int][x^2/(E^x + x), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{x \left (e^x+x\right )^2} \, dx\\ &=\int \left (\frac {49 (-1+x) \left (3-\log (x)+x^2 \log (x)\right )}{\left (e^x+x\right )^2}-\frac {49 \left (1+3 x-x^2-x \log (x)-2 x^2 \log (x)+x^3 \log (x)\right )}{x \left (e^x+x\right )}+\frac {-1+2 x+x^2+x^3-x \log (x)+2 x^2 \log (x)+3 x^3 \log (x)}{x}\right ) \, dx\\ &=49 \int \frac {(-1+x) \left (3-\log (x)+x^2 \log (x)\right )}{\left (e^x+x\right )^2} \, dx-49 \int \frac {1+3 x-x^2-x \log (x)-2 x^2 \log (x)+x^3 \log (x)}{x \left (e^x+x\right )} \, dx+\int \frac {-1+2 x+x^2+x^3-x \log (x)+2 x^2 \log (x)+3 x^3 \log (x)}{x} \, dx\\ &=-\left (49 \int \left (\frac {3}{e^x+x}+\frac {1}{x \left (e^x+x\right )}-\frac {x}{e^x+x}-\frac {\log (x)}{e^x+x}-\frac {2 x \log (x)}{e^x+x}+\frac {x^2 \log (x)}{e^x+x}\right ) \, dx\right )+49 \int \left (-\frac {3-\log (x)+x^2 \log (x)}{\left (e^x+x\right )^2}+\frac {x \left (3-\log (x)+x^2 \log (x)\right )}{\left (e^x+x\right )^2}\right ) \, dx+\int \left (\frac {-1+2 x+x^2+x^3}{x}+\left (-1+2 x+3 x^2\right ) \log (x)\right ) \, dx\\ &=-\left (49 \int \frac {1}{x \left (e^x+x\right )} \, dx\right )+49 \int \frac {x}{e^x+x} \, dx+49 \int \frac {\log (x)}{e^x+x} \, dx-49 \int \frac {x^2 \log (x)}{e^x+x} \, dx-49 \int \frac {3-\log (x)+x^2 \log (x)}{\left (e^x+x\right )^2} \, dx+49 \int \frac {x \left (3-\log (x)+x^2 \log (x)\right )}{\left (e^x+x\right )^2} \, dx+98 \int \frac {x \log (x)}{e^x+x} \, dx-147 \int \frac {1}{e^x+x} \, dx+\int \frac {-1+2 x+x^2+x^3}{x} \, dx+\int \left (-1+2 x+3 x^2\right ) \log (x) \, dx\\ &=-\left (49 \int \frac {1}{x \left (e^x+x\right )} \, dx\right )+49 \int \frac {x}{e^x+x} \, dx-49 \int \left (\frac {3}{\left (e^x+x\right )^2}-\frac {\log (x)}{\left (e^x+x\right )^2}+\frac {x^2 \log (x)}{\left (e^x+x\right )^2}\right ) \, dx+49 \int \left (\frac {3 x}{\left (e^x+x\right )^2}-\frac {x \log (x)}{\left (e^x+x\right )^2}+\frac {x^3 \log (x)}{\left (e^x+x\right )^2}\right ) \, dx-49 \int \frac {\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x} \, dx-98 \int \frac {\int \frac {x}{e^x+x} \, dx}{x} \, dx-147 \int \frac {1}{e^x+x} \, dx+(49 \log (x)) \int \frac {1}{e^x+x} \, dx-(49 \log (x)) \int \frac {x^2}{e^x+x} \, dx+(98 \log (x)) \int \frac {x}{e^x+x} \, dx+\int \left (2-\frac {1}{x}+x+x^2\right ) \, dx+\int \left (-\log (x)+2 x \log (x)+3 x^2 \log (x)\right ) \, dx\\ &=2 x+\frac {x^2}{2}+\frac {x^3}{3}-\log (x)+2 \int x \log (x) \, dx+3 \int x^2 \log (x) \, dx-49 \int \frac {1}{x \left (e^x+x\right )} \, dx+49 \int \frac {x}{e^x+x} \, dx+49 \int \frac {\log (x)}{\left (e^x+x\right )^2} \, dx-49 \int \frac {x \log (x)}{\left (e^x+x\right )^2} \, dx-49 \int \frac {x^2 \log (x)}{\left (e^x+x\right )^2} \, dx+49 \int \frac {x^3 \log (x)}{\left (e^x+x\right )^2} \, dx-49 \int \frac {\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x} \, dx-98 \int \frac {\int \frac {x}{e^x+x} \, dx}{x} \, dx-147 \int \frac {1}{\left (e^x+x\right )^2} \, dx+147 \int \frac {x}{\left (e^x+x\right )^2} \, dx-147 \int \frac {1}{e^x+x} \, dx+(49 \log (x)) \int \frac {1}{e^x+x} \, dx-(49 \log (x)) \int \frac {x^2}{e^x+x} \, dx+(98 \log (x)) \int \frac {x}{e^x+x} \, dx-\int \log (x) \, dx\\ &=3 x+\frac {147}{e^x+x}-\log (x)-x \log (x)+x^2 \log (x)+x^3 \log (x)-\frac {49 \log (x)}{e^x+x}-49 \int \frac {1}{x \left (e^x+x\right )} \, dx+49 \int \frac {x}{e^x+x} \, dx-49 \int \frac {\int \frac {1}{\left (e^x+x\right )^2} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {x^3}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \frac {\frac {1}{e^x+x}+\int \frac {1}{\left (e^x+x\right )^2} \, dx+\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x} \, dx-98 \int \frac {\int \frac {x}{e^x+x} \, dx}{x} \, dx-(49 \log (x)) \int \frac {x^2}{\left (e^x+x\right )^2} \, dx+(49 \log (x)) \int \frac {x^3}{\left (e^x+x\right )^2} \, dx-(49 \log (x)) \int \frac {x^2}{e^x+x} \, dx+(98 \log (x)) \int \frac {x}{e^x+x} \, dx\\ &=3 x+\frac {147}{e^x+x}-\log (x)-x \log (x)+x^2 \log (x)+x^3 \log (x)-\frac {49 \log (x)}{e^x+x}-49 \int \frac {1}{x \left (e^x+x\right )} \, dx+49 \int \frac {x}{e^x+x} \, dx-49 \int \frac {\int \frac {1}{\left (e^x+x\right )^2} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {x^3}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \left (\frac {1}{x \left (e^x+x\right )}+\frac {\int \frac {1}{\left (e^x+x\right )^2} \, dx+\int \frac {1}{e^x+x} \, dx}{x}\right ) \, dx+49 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x} \, dx-98 \int \frac {\int \frac {x}{e^x+x} \, dx}{x} \, dx-(49 \log (x)) \int \frac {x^2}{\left (e^x+x\right )^2} \, dx+(49 \log (x)) \int \frac {x^3}{\left (e^x+x\right )^2} \, dx-(49 \log (x)) \int \frac {x^2}{e^x+x} \, dx+(98 \log (x)) \int \frac {x}{e^x+x} \, dx\\ &=3 x+\frac {147}{e^x+x}-\log (x)-x \log (x)+x^2 \log (x)+x^3 \log (x)-\frac {49 \log (x)}{e^x+x}+49 \int \frac {x}{e^x+x} \, dx-49 \int \frac {\int \frac {1}{\left (e^x+x\right )^2} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {x^3}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \frac {\int \frac {1}{\left (e^x+x\right )^2} \, dx+\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x} \, dx-98 \int \frac {\int \frac {x}{e^x+x} \, dx}{x} \, dx-(49 \log (x)) \int \frac {x^2}{\left (e^x+x\right )^2} \, dx+(49 \log (x)) \int \frac {x^3}{\left (e^x+x\right )^2} \, dx-(49 \log (x)) \int \frac {x^2}{e^x+x} \, dx+(98 \log (x)) \int \frac {x}{e^x+x} \, dx\\ &=3 x+\frac {147}{e^x+x}-\log (x)-x \log (x)+x^2 \log (x)+x^3 \log (x)-\frac {49 \log (x)}{e^x+x}+49 \int \frac {x}{e^x+x} \, dx-49 \int \frac {\int \frac {1}{\left (e^x+x\right )^2} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {x^3}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {1}{e^x+x} \, dx}{x} \, dx+49 \int \left (\frac {\int \frac {1}{\left (e^x+x\right )^2} \, dx}{x}+\frac {\int \frac {1}{e^x+x} \, dx}{x}\right ) \, dx+49 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x} \, dx-98 \int \frac {\int \frac {x}{e^x+x} \, dx}{x} \, dx-(49 \log (x)) \int \frac {x^2}{\left (e^x+x\right )^2} \, dx+(49 \log (x)) \int \frac {x^3}{\left (e^x+x\right )^2} \, dx-(49 \log (x)) \int \frac {x^2}{e^x+x} \, dx+(98 \log (x)) \int \frac {x}{e^x+x} \, dx\\ &=3 x+\frac {147}{e^x+x}-\log (x)-x \log (x)+x^2 \log (x)+x^3 \log (x)-\frac {49 \log (x)}{e^x+x}+49 \int \frac {x}{e^x+x} \, dx+49 \int \frac {\int \frac {x^2}{\left (e^x+x\right )^2} \, dx}{x} \, dx-49 \int \frac {\int \frac {x^3}{\left (e^x+x\right )^2} \, dx}{x} \, dx+49 \int \frac {\int \frac {x^2}{e^x+x} \, dx}{x} \, dx-98 \int \frac {\int \frac {x}{e^x+x} \, dx}{x} \, dx-(49 \log (x)) \int \frac {x^2}{\left (e^x+x\right )^2} \, dx+(49 \log (x)) \int \frac {x^3}{\left (e^x+x\right )^2} \, dx-(49 \log (x)) \int \frac {x^2}{e^x+x} \, dx+(98 \log (x)) \int \frac {x}{e^x+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 42, normalized size = 1.62 \begin {gather*} \frac {3 \left (49+e^x x+x^2\right )+\left (-1+x^2\right ) \left (49+x+x^2+e^x (1+x)\right ) \log (x)}{e^x+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-196*x - x^2 + 51*x^3 + x^4 + x^5 + E^(2*x)*(-1 + 2*x + x^2 + x^3) + E^x*(-49 - 149*x + 53*x^2 + 2*
x^3 + 2*x^4) + (49*x + 48*x^3 + 2*x^4 + 3*x^5 + E^(2*x)*(-x + 2*x^2 + 3*x^3) + E^x*(49*x + 96*x^2 - 45*x^3 + 6
*x^4))*Log[x])/(E^(2*x)*x + 2*E^x*x^2 + x^3),x]

[Out]

(3*(49 + E^x*x + x^2) + (-1 + x^2)*(49 + x + x^2 + E^x*(1 + x))*Log[x])/(E^x + x)

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fricas [B]  time = 0.66, size = 52, normalized size = 2.00 \begin {gather*} \frac {3 \, x^{2} + 3 \, x e^{x} + {\left (x^{4} + x^{3} + 48 \, x^{2} + {\left (x^{3} + x^{2} - x - 1\right )} e^{x} - x - 49\right )} \log \relax (x) + 147}{x + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^3+2*x^2-x)*exp(x)^2+(6*x^4-45*x^3+96*x^2+49*x)*exp(x)+3*x^5+2*x^4+48*x^3+49*x)*log(x)+(x^3+x^
2+2*x-1)*exp(x)^2+(2*x^4+2*x^3+53*x^2-149*x-49)*exp(x)+x^5+x^4+51*x^3-x^2-196*x)/(x*exp(x)^2+2*exp(x)*x^2+x^3)
,x, algorithm="fricas")

[Out]

(3*x^2 + 3*x*e^x + (x^4 + x^3 + 48*x^2 + (x^3 + x^2 - x - 1)*e^x - x - 49)*log(x) + 147)/(x + e^x)

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giac [B]  time = 0.15, size = 76, normalized size = 2.92 \begin {gather*} \frac {x^{4} \log \relax (x) + x^{3} e^{x} \log \relax (x) + x^{3} \log \relax (x) + x^{2} e^{x} \log \relax (x) + 48 \, x^{2} \log \relax (x) - x e^{x} \log \relax (x) + 3 \, x^{2} + 3 \, x e^{x} - x \log \relax (x) - e^{x} \log \relax (x) - 49 \, \log \relax (x) + 147}{x + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^3+2*x^2-x)*exp(x)^2+(6*x^4-45*x^3+96*x^2+49*x)*exp(x)+3*x^5+2*x^4+48*x^3+49*x)*log(x)+(x^3+x^
2+2*x-1)*exp(x)^2+(2*x^4+2*x^3+53*x^2-149*x-49)*exp(x)+x^5+x^4+51*x^3-x^2-196*x)/(x*exp(x)^2+2*exp(x)*x^2+x^3)
,x, algorithm="giac")

[Out]

(x^4*log(x) + x^3*e^x*log(x) + x^3*log(x) + x^2*e^x*log(x) + 48*x^2*log(x) - x*e^x*log(x) + 3*x^2 + 3*x*e^x -
x*log(x) - e^x*log(x) - 49*log(x) + 147)/(x + e^x)

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maple [B]  time = 0.05, size = 70, normalized size = 2.69

method	result	size
risch	\(\frac {\left (x^{4}+{\mathrm e}^{x} x^{3}+x^{3}+{\mathrm e}^{x} x^{2}+48 x^{2}-{\mathrm e}^{x} x -49\right ) \ln \relax (x )}{{\mathrm e}^{x}+x}-\frac {x \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )-3 x^{2}-3 \,{\mathrm e}^{x} x -147}{{\mathrm e}^{x}+x}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3*x^3+2*x^2-x)*exp(x)^2+(6*x^4-45*x^3+96*x^2+49*x)*exp(x)+3*x^5+2*x^4+48*x^3+49*x)*ln(x)+(x^3+x^2+2*x-1
)*exp(x)^2+(2*x^4+2*x^3+53*x^2-149*x-49)*exp(x)+x^5+x^4+51*x^3-x^2-196*x)/(x*exp(x)^2+2*exp(x)*x^2+x^3),x,meth
od=_RETURNVERBOSE)

[Out]

(x^4+exp(x)*x^3+x^3+exp(x)*x^2+48*x^2-exp(x)*x-49)/(exp(x)+x)*ln(x)-(x*ln(x)+exp(x)*ln(x)-3*x^2-3*exp(x)*x-147
)/(exp(x)+x)

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maxima [B]  time = 0.44, size = 54, normalized size = 2.08 \begin {gather*} \frac {3 \, x^{2} + {\left ({\left (x^{3} + x^{2} - x - 1\right )} \log \relax (x) + 3 \, x\right )} e^{x} + {\left (x^{4} + x^{3} + 48 \, x^{2} - x - 49\right )} \log \relax (x) + 147}{x + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^3+2*x^2-x)*exp(x)^2+(6*x^4-45*x^3+96*x^2+49*x)*exp(x)+3*x^5+2*x^4+48*x^3+49*x)*log(x)+(x^3+x^
2+2*x-1)*exp(x)^2+(2*x^4+2*x^3+53*x^2-149*x-49)*exp(x)+x^5+x^4+51*x^3-x^2-196*x)/(x*exp(x)^2+2*exp(x)*x^2+x^3)
,x, algorithm="maxima")

[Out]

(3*x^2 + ((x^3 + x^2 - x - 1)*log(x) + 3*x)*e^x + (x^4 + x^3 + 48*x^2 - x - 49)*log(x) + 147)/(x + e^x)

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mupad [B]  time = 5.50, size = 57, normalized size = 2.19 \begin {gather*} 3\,x-\ln \relax (x)+\frac {147}{x+{\mathrm {e}}^x}+\frac {\ln \relax (x)\,\left ({\mathrm {e}}^x\,\left (x^3+x^2-x\right )+x\,\left (x^3+x^2-x\right )+49\,x^2-49\right )}{x+{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(53*x^2 - 149*x + 2*x^3 + 2*x^4 - 49) - 196*x + log(x)*(49*x + exp(x)*(49*x + 96*x^2 - 45*x^3 + 6*
x^4) + exp(2*x)*(2*x^2 - x + 3*x^3) + 48*x^3 + 2*x^4 + 3*x^5) + exp(2*x)*(2*x + x^2 + x^3 - 1) - x^2 + 51*x^3
+ x^4 + x^5)/(x*exp(2*x) + 2*x^2*exp(x) + x^3),x)

[Out]

3*x - log(x) + 147/(x + exp(x)) + (log(x)*(exp(x)*(x^2 - x + x^3) + x*(x^2 - x + x^3) + 49*x^2 - 49))/(x + exp
(x))

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sympy [A]  time = 0.42, size = 37, normalized size = 1.42 \begin {gather*} 3 x + \left (x^{3} + x^{2} - x\right ) \log {\relax (x )} - \log {\relax (x )} + \frac {49 x^{2} \log {\relax (x )} - 49 \log {\relax (x )} + 147}{x + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x**3+2*x**2-x)*exp(x)**2+(6*x**4-45*x**3+96*x**2+49*x)*exp(x)+3*x**5+2*x**4+48*x**3+49*x)*ln(x)
+(x**3+x**2+2*x-1)*exp(x)**2+(2*x**4+2*x**3+53*x**2-149*x-49)*exp(x)+x**5+x**4+51*x**3-x**2-196*x)/(x*exp(x)**
2+2*exp(x)*x**2+x**3),x)

[Out]

3*x + (x**3 + x**2 - x)*log(x) - log(x) + (49*x**2*log(x) - 49*log(x) + 147)/(x + exp(x))

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