3.64.73 \(\int \frac {4 x^3+2 x^4+e^x (-100+100 x) \log (\frac {2+x}{x})+e^x (-100+50 x-25 x^3) \log ^2(\frac {2+x}{x})}{4 x^3+2 x^4} \, dx\)

Optimal. Leaf size=27 \[ x+\frac {25 e^x (1-x) \log ^2\left (1+\frac {2}{x}\right )}{2 x^2} \]

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Rubi [F]  time = 4.90, 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 {4 x^3+2 x^4+e^x (-100+100 x) \log \left (\frac {2+x}{x}\right )+e^x \left (-100+50 x-25 x^3\right ) \log ^2\left (\frac {2+x}{x}\right )}{4 x^3+2 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*x^3 + 2*x^4 + E^x*(-100 + 100*x)*Log[(2 + x)/x] + E^x*(-100 + 50*x - 25*x^3)*Log[(2 + x)/x]^2)/(4*x^3 +
2*x^4),x]

[Out]

(-25*E^x)/(4*x^2) + (25*E^x)/x + x - (75*ExpIntegralEi[x])/8 - (125*ExpIntegralEi[2 + x])/(8*E^2) + (25*x*Hype
rgeometricPFQ[{1, 1, 1}, {2, 2, 2}, x])/4 + (25*E^x*Log[1 + 2/x])/(2*x^2) - (25*E^x*Log[1 + 2/x])/x + (25*ExpI
ntegralEi[x]*Log[1 + 2/x])/4 + (75*ExpIntegralEi[2 + x]*Log[1 + 2/x])/(4*E^2) + (25*Log[-x]^2)/8 + (25*EulerGa
mma*Log[x])/4 + (25*(ExpIntegralE[1, -x] + ExpIntegralEi[x])*Log[x])/4 - (25*Defer[Int][ExpIntegralEi[x]/(2 +
x), x])/4 + (75*Defer[Int][ExpIntegralEi[2 + x]/x, x])/(4*E^2) - (75*Defer[Int][ExpIntegralEi[2 + x]/(2 + x),
x])/(4*E^2) - 25*Defer[Int][(E^x*Log[1 + 2/x]^2)/x^3, x] + 25*Defer[Int][(E^x*Log[1 + 2/x]^2)/x^2, x] - (25*De
fer[Int][(E^x*Log[1 + 2/x]^2)/x, x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 x^3+2 x^4+e^x (-100+100 x) \log \left (\frac {2+x}{x}\right )+e^x \left (-100+50 x-25 x^3\right ) \log ^2\left (\frac {2+x}{x}\right )}{x^3 (4+2 x)} \, dx\\ &=\int \left (1+\frac {25 e^x \log \left (1+\frac {2}{x}\right ) \left (-4+4 x-4 \log \left (\frac {2+x}{x}\right )+2 x \log \left (\frac {2+x}{x}\right )-x^3 \log \left (\frac {2+x}{x}\right )\right )}{2 x^3 (2+x)}\right ) \, dx\\ &=x+\frac {25}{2} \int \frac {e^x \log \left (1+\frac {2}{x}\right ) \left (-4+4 x-4 \log \left (\frac {2+x}{x}\right )+2 x \log \left (\frac {2+x}{x}\right )-x^3 \log \left (\frac {2+x}{x}\right )\right )}{x^3 (2+x)} \, dx\\ &=x+\frac {25}{2} \int \frac {e^x \log \left (1+\frac {2}{x}\right ) \left (-4+4 x-\left (4-2 x+x^3\right ) \log \left (\frac {2+x}{x}\right )\right )}{x^3 (2+x)} \, dx\\ &=x+\frac {25}{2} \int \left (\frac {4 e^x (-1+x) \log \left (1+\frac {2}{x}\right )}{x^3 (2+x)}+\frac {e^x \left (-2+2 x-x^2\right ) \log ^2\left (1+\frac {2}{x}\right )}{x^3}\right ) \, dx\\ &=x+\frac {25}{2} \int \frac {e^x \left (-2+2 x-x^2\right ) \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+50 \int \frac {e^x (-1+x) \log \left (1+\frac {2}{x}\right )}{x^3 (2+x)} \, dx\\ &=x+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}+\frac {25}{2} \int \left (-\frac {2 e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3}+\frac {2 e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2}-\frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x}\right ) \, dx-50 \int \frac {2 e^{2+x} (-1+2 x)-e^2 x^2 \text {Ei}(x)-3 x^2 \text {Ei}(2+x)}{4 e^2 x^3 (2+x)} \, dx\\ &=x+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx-\frac {25 \int \frac {2 e^{2+x} (-1+2 x)-e^2 x^2 \text {Ei}(x)-3 x^2 \text {Ei}(2+x)}{x^3 (2+x)} \, dx}{2 e^2}\\ &=x+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx-\frac {25 \int \left (\frac {2 e^{2+x} (-1+2 x)}{x^3 (2+x)}+\frac {-e^2 \text {Ei}(x)-3 \text {Ei}(2+x)}{x (2+x)}\right ) \, dx}{2 e^2}\\ &=x+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx-\frac {25 \int \frac {-e^2 \text {Ei}(x)-3 \text {Ei}(2+x)}{x (2+x)} \, dx}{2 e^2}-\frac {25 \int \frac {e^{2+x} (-1+2 x)}{x^3 (2+x)} \, dx}{e^2}\\ &=x+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx-\frac {25 \int \left (-\frac {e^2 \text {Ei}(x)}{x (2+x)}-\frac {3 \text {Ei}(2+x)}{x (2+x)}\right ) \, dx}{2 e^2}-\frac {25 \int \left (-\frac {e^{2+x}}{2 x^3}+\frac {5 e^{2+x}}{4 x^2}-\frac {5 e^{2+x}}{8 x}+\frac {5 e^{2+x}}{8 (2+x)}\right ) \, dx}{e^2}\\ &=x+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}+\frac {25}{2} \int \frac {\text {Ei}(x)}{x (2+x)} \, dx-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx+\frac {25 \int \frac {e^{2+x}}{x^3} \, dx}{2 e^2}+\frac {125 \int \frac {e^{2+x}}{x} \, dx}{8 e^2}-\frac {125 \int \frac {e^{2+x}}{2+x} \, dx}{8 e^2}-\frac {125 \int \frac {e^{2+x}}{x^2} \, dx}{4 e^2}+\frac {75 \int \frac {\text {Ei}(2+x)}{x (2+x)} \, dx}{2 e^2}\\ &=-\frac {25 e^x}{4 x^2}+\frac {125 e^x}{4 x}+x+\frac {125 \text {Ei}(x)}{8}-\frac {125 \text {Ei}(2+x)}{8 e^2}+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}+\frac {25}{2} \int \left (\frac {\text {Ei}(x)}{2 x}-\frac {\text {Ei}(x)}{2 (2+x)}\right ) \, dx-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx+\frac {25 \int \frac {e^{2+x}}{x^2} \, dx}{4 e^2}-\frac {125 \int \frac {e^{2+x}}{x} \, dx}{4 e^2}+\frac {75 \int \left (\frac {\text {Ei}(2+x)}{2 x}-\frac {\text {Ei}(2+x)}{2 (2+x)}\right ) \, dx}{2 e^2}\\ &=-\frac {25 e^x}{4 x^2}+\frac {25 e^x}{x}+x-\frac {125 \text {Ei}(x)}{8}-\frac {125 \text {Ei}(2+x)}{8 e^2}+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}+\frac {25}{4} \int \frac {\text {Ei}(x)}{x} \, dx-\frac {25}{4} \int \frac {\text {Ei}(x)}{2+x} \, dx-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx+\frac {25 \int \frac {e^{2+x}}{x} \, dx}{4 e^2}+\frac {75 \int \frac {\text {Ei}(2+x)}{x} \, dx}{4 e^2}-\frac {75 \int \frac {\text {Ei}(2+x)}{2+x} \, dx}{4 e^2}\\ &=-\frac {25 e^x}{4 x^2}+\frac {25 e^x}{x}+x-\frac {75 \text {Ei}(x)}{8}-\frac {125 \text {Ei}(2+x)}{8 e^2}+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}+\frac {25}{4} (E_1(-x)+\text {Ei}(x)) \log (x)-\frac {25}{4} \int \frac {E_1(-x)}{x} \, dx-\frac {25}{4} \int \frac {\text {Ei}(x)}{2+x} \, dx-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx+\frac {75 \int \frac {\text {Ei}(2+x)}{x} \, dx}{4 e^2}-\frac {75 \int \frac {\text {Ei}(2+x)}{2+x} \, dx}{4 e^2}\\ &=-\frac {25 e^x}{4 x^2}+\frac {25 e^x}{x}+x-\frac {75 \text {Ei}(x)}{8}-\frac {125 \text {Ei}(2+x)}{8 e^2}+\frac {25}{4} x \, _3F_3(1,1,1;2,2,2;x)+\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{2 x^2}-\frac {25 e^x \log \left (1+\frac {2}{x}\right )}{x}+\frac {25}{4} \text {Ei}(x) \log \left (1+\frac {2}{x}\right )+\frac {75 \text {Ei}(2+x) \log \left (1+\frac {2}{x}\right )}{4 e^2}+\frac {25}{8} \log ^2(-x)+\frac {25}{4} \gamma \log (x)+\frac {25}{4} (E_1(-x)+\text {Ei}(x)) \log (x)-\frac {25}{4} \int \frac {\text {Ei}(x)}{2+x} \, dx-\frac {25}{2} \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x} \, dx-25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^3} \, dx+25 \int \frac {e^x \log ^2\left (1+\frac {2}{x}\right )}{x^2} \, dx+\frac {75 \int \frac {\text {Ei}(2+x)}{x} \, dx}{4 e^2}-\frac {75 \int \frac {\text {Ei}(2+x)}{2+x} \, dx}{4 e^2}\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 23.92, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 x^3+2 x^4+e^x (-100+100 x) \log \left (\frac {2+x}{x}\right )+e^x \left (-100+50 x-25 x^3\right ) \log ^2\left (\frac {2+x}{x}\right )}{4 x^3+2 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(4*x^3 + 2*x^4 + E^x*(-100 + 100*x)*Log[(2 + x)/x] + E^x*(-100 + 50*x - 25*x^3)*Log[(2 + x)/x]^2)/(4
*x^3 + 2*x^4),x]

[Out]

Integrate[(4*x^3 + 2*x^4 + E^x*(-100 + 100*x)*Log[(2 + x)/x] + E^x*(-100 + 50*x - 25*x^3)*Log[(2 + x)/x]^2)/(4
*x^3 + 2*x^4), x]

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fricas [A]  time = 0.64, size = 28, normalized size = 1.04 \begin {gather*} -\frac {25 \, {\left (x - 1\right )} e^{x} \log \left (\frac {x + 2}{x}\right )^{2} - 2 \, x^{3}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x^3+50*x-100)*exp(x)*log((2+x)/x)^2+(100*x-100)*exp(x)*log((2+x)/x)+2*x^4+4*x^3)/(2*x^4+4*x^3)
,x, algorithm="fricas")

[Out]

-1/2*(25*(x - 1)*e^x*log((x + 2)/x)^2 - 2*x^3)/x^2

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giac [A]  time = 0.16, size = 40, normalized size = 1.48 \begin {gather*} -\frac {25 \, x e^{x} \log \left (\frac {x + 2}{x}\right )^{2} - 2 \, x^{3} - 25 \, e^{x} \log \left (\frac {x + 2}{x}\right )^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x^3+50*x-100)*exp(x)*log((2+x)/x)^2+(100*x-100)*exp(x)*log((2+x)/x)+2*x^4+4*x^3)/(2*x^4+4*x^3)
,x, algorithm="giac")

[Out]

-1/2*(25*x*e^x*log((x + 2)/x)^2 - 2*x^3 - 25*e^x*log((x + 2)/x)^2)/x^2

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maple [C]  time = 0.61, size = 1010, normalized size = 37.41

method	result	size
risch	\(-\frac {25 \left (x -1\right ) {\mathrm e}^{x} \ln \left (2+x \right )^{2}}{2 x^{2}}+\frac {25 \,{\mathrm e}^{x} \left (i \pi x \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{3}-i \pi x \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )-i \pi x \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{2} \mathrm {csgn}\left (i \left (2+x \right )\right )+i \pi x \,\mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (2+x \right )\right )-i \pi \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{3}+i \pi \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )+i \pi \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{2} \mathrm {csgn}\left (i \left (2+x \right )\right )-i \pi \,\mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (2+x \right )\right )+2 x \ln \relax (x )-2 \ln \relax (x )\right ) \ln \left (2+x \right )}{2 x^{2}}+\frac {-100 x \,{\mathrm e}^{x} \ln \relax (x )^{2}-100 i {\mathrm e}^{x} \pi x \,\mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (2+x \right )\right ) \ln \relax (x )+8 x^{3}+100 \,{\mathrm e}^{x} \ln \relax (x )^{2}-25 \,{\mathrm e}^{x} \pi ^{2} \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{4} \mathrm {csgn}\left (i \left (2+x \right )\right )^{2}+50 \,{\mathrm e}^{x} \pi ^{2} \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{5} \mathrm {csgn}\left (\frac {i}{x}\right )+50 \,{\mathrm e}^{x} \pi ^{2} \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{5} \mathrm {csgn}\left (i \left (2+x \right )\right )+25 \,{\mathrm e}^{x} \pi ^{2} x \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{6}-25 \,{\mathrm e}^{x} \pi ^{2} \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{4} \mathrm {csgn}\left (\frac {i}{x}\right )^{2}-25 \,{\mathrm e}^{x} \pi ^{2} \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{6}-100 i {\mathrm e}^{x} \pi x \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{3} \ln \relax (x )-100 i {\mathrm e}^{x} \ln \relax (x ) \pi \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )-100 i {\mathrm e}^{x} \ln \relax (x ) \pi \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{2} \mathrm {csgn}\left (i \left (2+x \right )\right )-50 \,{\mathrm e}^{x} \pi ^{2} x \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{3} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i \left (2+x \right )\right )+100 \,{\mathrm e}^{x} \pi ^{2} x \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{4} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (2+x \right )\right )-50 \,{\mathrm e}^{x} \pi ^{2} x \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{3} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (2+x \right )\right )^{2}+25 \,{\mathrm e}^{x} \pi ^{2} x \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i \left (2+x \right )\right )^{2}+25 \,{\mathrm e}^{x} \pi ^{2} x \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{4} \mathrm {csgn}\left (\frac {i}{x}\right )^{2}+25 \,{\mathrm e}^{x} \pi ^{2} x \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{4} \mathrm {csgn}\left (i \left (2+x \right )\right )^{2}-50 \,{\mathrm e}^{x} \pi ^{2} x \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{5} \mathrm {csgn}\left (\frac {i}{x}\right )-50 \,{\mathrm e}^{x} \pi ^{2} x \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{5} \mathrm {csgn}\left (i \left (2+x \right )\right )-25 \,{\mathrm e}^{x} \pi ^{2} \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i \left (2+x \right )\right )^{2}+50 \,{\mathrm e}^{x} \pi ^{2} \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{3} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i \left (2+x \right )\right )+50 \,{\mathrm e}^{x} \pi ^{2} \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{3} \mathrm {csgn}\left (i \left (2+x \right )\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )-100 \,{\mathrm e}^{x} \pi ^{2} \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{4} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (2+x \right )\right )+100 i {\mathrm e}^{x} \ln \relax (x ) \pi \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{3}+100 i {\mathrm e}^{x} \pi x \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \ln \relax (x )+100 i {\mathrm e}^{x} \pi x \mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right )^{2} \mathrm {csgn}\left (i \left (2+x \right )\right ) \ln \relax (x )+100 i {\mathrm e}^{x} \ln \relax (x ) \pi \,\mathrm {csgn}\left (\frac {i \left (2+x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (2+x \right )\right )}{8 x^{2}}\)	\(1010\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-25*x^3+50*x-100)*exp(x)*ln((2+x)/x)^2+(100*x-100)*exp(x)*ln((2+x)/x)+2*x^4+4*x^3)/(2*x^4+4*x^3),x,metho
d=_RETURNVERBOSE)

[Out]

-25/2/x^2*(x-1)*exp(x)*ln(2+x)^2+25/2*exp(x)*(I*Pi*x*csgn(I/x*(2+x))^3-I*Pi*x*csgn(I/x*(2+x))^2*csgn(I/x)-I*Pi
*x*csgn(I/x*(2+x))^2*csgn(I*(2+x))+I*Pi*x*csgn(I/x*(2+x))*csgn(I/x)*csgn(I*(2+x))-I*Pi*csgn(I/x*(2+x))^3+I*Pi*
csgn(I/x*(2+x))^2*csgn(I/x)+I*Pi*csgn(I/x*(2+x))^2*csgn(I*(2+x))-I*Pi*csgn(I/x*(2+x))*csgn(I/x)*csgn(I*(2+x))+
2*x*ln(x)-2*ln(x))/x^2*ln(2+x)+1/8*(-100*x*exp(x)*ln(x)^2-25*exp(x)*Pi^2*csgn(I/x*(2+x))^6+8*x^3+100*exp(x)*ln
(x)^2-25*exp(x)*Pi^2*csgn(I/x*(2+x))^4*csgn(I*(2+x))^2+50*exp(x)*Pi^2*csgn(I/x*(2+x))^5*csgn(I/x)+50*exp(x)*Pi
^2*csgn(I/x*(2+x))^5*csgn(I*(2+x))+25*exp(x)*Pi^2*x*csgn(I/x*(2+x))^6-25*exp(x)*Pi^2*csgn(I/x*(2+x))^4*csgn(I/
x)^2+25*exp(x)*Pi^2*x*csgn(I/x*(2+x))^4*csgn(I/x)^2+25*exp(x)*Pi^2*x*csgn(I/x*(2+x))^4*csgn(I*(2+x))^2-50*exp(
x)*Pi^2*x*csgn(I/x*(2+x))^5*csgn(I/x)-50*exp(x)*Pi^2*x*csgn(I/x*(2+x))^5*csgn(I*(2+x))-25*exp(x)*Pi^2*csgn(I/x
*(2+x))^2*csgn(I/x)^2*csgn(I*(2+x))^2+50*exp(x)*Pi^2*csgn(I/x*(2+x))^3*csgn(I/x)^2*csgn(I*(2+x))+50*exp(x)*Pi^
2*csgn(I/x*(2+x))^3*csgn(I*(2+x))^2*csgn(I/x)-100*exp(x)*Pi^2*csgn(I/x*(2+x))^4*csgn(I/x)*csgn(I*(2+x))+100*I*
exp(x)*ln(x)*Pi*csgn(I/x*(2+x))^3-100*I*exp(x)*Pi*x*csgn(I/x*(2+x))*csgn(I/x)*csgn(I*(2+x))*ln(x)+100*I*exp(x)
*Pi*x*csgn(I/x*(2+x))^2*csgn(I/x)*ln(x)-100*I*exp(x)*Pi*x*csgn(I/x*(2+x))^3*ln(x)-100*I*exp(x)*ln(x)*Pi*csgn(I
/x*(2+x))^2*csgn(I/x)-100*I*exp(x)*ln(x)*Pi*csgn(I/x*(2+x))^2*csgn(I*(2+x))-50*exp(x)*Pi^2*x*csgn(I/x*(2+x))^3
*csgn(I/x)^2*csgn(I*(2+x))+100*exp(x)*Pi^2*x*csgn(I/x*(2+x))^4*csgn(I/x)*csgn(I*(2+x))-50*exp(x)*Pi^2*x*csgn(I
/x*(2+x))^3*csgn(I/x)*csgn(I*(2+x))^2+25*exp(x)*Pi^2*x*csgn(I/x*(2+x))^2*csgn(I/x)^2*csgn(I*(2+x))^2+100*I*exp
(x)*Pi*x*csgn(I/x*(2+x))^2*csgn(I*(2+x))*ln(x)+100*I*exp(x)*ln(x)*Pi*csgn(I/x*(2+x))*csgn(I/x)*csgn(I*(2+x)))/
x^2

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maxima [A]  time = 0.52, size = 43, normalized size = 1.59 \begin {gather*} x - \frac {25 \, {\left ({\left (x - 1\right )} e^{x} \log \left (x + 2\right )^{2} - 2 \, {\left (x - 1\right )} e^{x} \log \left (x + 2\right ) \log \relax (x) + {\left (x - 1\right )} e^{x} \log \relax (x)^{2}\right )}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x^3+50*x-100)*exp(x)*log((2+x)/x)^2+(100*x-100)*exp(x)*log((2+x)/x)+2*x^4+4*x^3)/(2*x^4+4*x^3)
,x, algorithm="maxima")

[Out]

x - 25/2*((x - 1)*e^x*log(x + 2)^2 - 2*(x - 1)*e^x*log(x + 2)*log(x) + (x - 1)*e^x*log(x)^2)/x^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {4\,x^3+2\,x^4-{\mathrm {e}}^x\,{\ln \left (\frac {x+2}{x}\right )}^2\,\left (25\,x^3-50\,x+100\right )+{\mathrm {e}}^x\,\ln \left (\frac {x+2}{x}\right )\,\left (100\,x-100\right )}{2\,x^4+4\,x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3 + 2*x^4 - exp(x)*log((x + 2)/x)^2*(25*x^3 - 50*x + 100) + exp(x)*log((x + 2)/x)*(100*x - 100))/(4*x
^3 + 2*x^4),x)

[Out]

int((4*x^3 + 2*x^4 - exp(x)*log((x + 2)/x)^2*(25*x^3 - 50*x + 100) + exp(x)*log((x + 2)/x)*(100*x - 100))/(4*x
^3 + 2*x^4), x)

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sympy [A]  time = 0.34, size = 31, normalized size = 1.15 \begin {gather*} x + \frac {\left (- 25 x \log {\left (\frac {x + 2}{x} \right )}^{2} + 25 \log {\left (\frac {x + 2}{x} \right )}^{2}\right ) e^{x}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x**3+50*x-100)*exp(x)*ln((2+x)/x)**2+(100*x-100)*exp(x)*ln((2+x)/x)+2*x**4+4*x**3)/(2*x**4+4*x
**3),x)

[Out]

x + (-25*x*log((x + 2)/x)**2 + 25*log((x + 2)/x)**2)*exp(x)/(2*x**2)

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