3.95.92 \(\int \frac {e^x (-105+25 x+25 x^2-25 x^3)+e^x (105+55 x+25 x^2) \log (x)-105 e^x \log ^2(x)}{25 x^4-210 x^2 \log (x)+441 \log ^2(x)} \, dx\)

Optimal. Leaf size=27 \[ \frac {e^x}{-x+\frac {-\frac {21}{5}+x}{1-\frac {x}{\log (x)}}} \]

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Rubi [F]  time = 1.65, 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 {e^x \left (-105+25 x+25 x^2-25 x^3\right )+e^x \left (105+55 x+25 x^2\right ) \log (x)-105 e^x \log ^2(x)}{25 x^4-210 x^2 \log (x)+441 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(-105 + 25*x + 25*x^2 - 25*x^3) + E^x*(105 + 55*x + 25*x^2)*Log[x] - 105*E^x*Log[x]^2)/(25*x^4 - 210*
x^2*Log[x] + 441*Log[x]^2),x]

[Out]

(-5*E^x)/21 - 105*Defer[Int][E^x/(5*x^2 - 21*Log[x])^2, x] + 25*Defer[Int][(E^x*x)/(5*x^2 - 21*Log[x])^2, x] +
50*Defer[Int][(E^x*x^2)/(5*x^2 - 21*Log[x])^2, x] - (250*Defer[Int][(E^x*x^3)/(5*x^2 - 21*Log[x])^2, x])/21 -
5*Defer[Int][E^x/(5*x^2 - 21*Log[x]), x] - (55*Defer[Int][(E^x*x)/(5*x^2 - 21*Log[x]), x])/21 + (25*Defer[Int
][(E^x*x^2)/(5*x^2 - 21*Log[x]), x])/21

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 e^x \left (-21+5 x+5 x^2-5 x^3+21 \log (x)+11 x \log (x)+5 x^2 \log (x)-21 \log ^2(x)\right )}{\left (5 x^2-21 \log (x)\right )^2} \, dx\\ &=5 \int \frac {e^x \left (-21+5 x+5 x^2-5 x^3+21 \log (x)+11 x \log (x)+5 x^2 \log (x)-21 \log ^2(x)\right )}{\left (5 x^2-21 \log (x)\right )^2} \, dx\\ &=5 \int \left (-\frac {e^x}{21}+\frac {e^x \left (-441+105 x+210 x^2-50 x^3\right )}{21 \left (5 x^2-21 \log (x)\right )^2}+\frac {e^x \left (-21-11 x+5 x^2\right )}{21 \left (5 x^2-21 \log (x)\right )}\right ) \, dx\\ &=-\frac {5 \int e^x \, dx}{21}+\frac {5}{21} \int \frac {e^x \left (-441+105 x+210 x^2-50 x^3\right )}{\left (5 x^2-21 \log (x)\right )^2} \, dx+\frac {5}{21} \int \frac {e^x \left (-21-11 x+5 x^2\right )}{5 x^2-21 \log (x)} \, dx\\ &=-\frac {5 e^x}{21}+\frac {5}{21} \int \left (-\frac {441 e^x}{\left (5 x^2-21 \log (x)\right )^2}+\frac {105 e^x x}{\left (5 x^2-21 \log (x)\right )^2}+\frac {210 e^x x^2}{\left (5 x^2-21 \log (x)\right )^2}-\frac {50 e^x x^3}{\left (5 x^2-21 \log (x)\right )^2}\right ) \, dx+\frac {5}{21} \int \left (-\frac {21 e^x}{5 x^2-21 \log (x)}-\frac {11 e^x x}{5 x^2-21 \log (x)}+\frac {5 e^x x^2}{5 x^2-21 \log (x)}\right ) \, dx\\ &=-\frac {5 e^x}{21}+\frac {25}{21} \int \frac {e^x x^2}{5 x^2-21 \log (x)} \, dx-\frac {55}{21} \int \frac {e^x x}{5 x^2-21 \log (x)} \, dx-5 \int \frac {e^x}{5 x^2-21 \log (x)} \, dx-\frac {250}{21} \int \frac {e^x x^3}{\left (5 x^2-21 \log (x)\right )^2} \, dx+25 \int \frac {e^x x}{\left (5 x^2-21 \log (x)\right )^2} \, dx+50 \int \frac {e^x x^2}{\left (5 x^2-21 \log (x)\right )^2} \, dx-105 \int \frac {e^x}{\left (5 x^2-21 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.37, size = 23, normalized size = 0.85 \begin {gather*} -\frac {5 e^x (x-\log (x))}{5 x^2-21 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-105 + 25*x + 25*x^2 - 25*x^3) + E^x*(105 + 55*x + 25*x^2)*Log[x] - 105*E^x*Log[x]^2)/(25*x^4
- 210*x^2*Log[x] + 441*Log[x]^2),x]

[Out]

(-5*E^x*(x - Log[x]))/(5*x^2 - 21*Log[x])

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fricas [A]  time = 0.64, size = 25, normalized size = 0.93 \begin {gather*} -\frac {5 \, {\left (x e^{x} - e^{x} \log \relax (x)\right )}}{5 \, x^{2} - 21 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-105*exp(x)*log(x)^2+(25*x^2+55*x+105)*exp(x)*log(x)+(-25*x^3+25*x^2+25*x-105)*exp(x))/(441*log(x)^
2-210*x^2*log(x)+25*x^4),x, algorithm="fricas")

[Out]

-5*(x*e^x - e^x*log(x))/(5*x^2 - 21*log(x))

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giac [A]  time = 0.21, size = 25, normalized size = 0.93 \begin {gather*} -\frac {5 \, {\left (x e^{x} - e^{x} \log \relax (x)\right )}}{5 \, x^{2} - 21 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-105*exp(x)*log(x)^2+(25*x^2+55*x+105)*exp(x)*log(x)+(-25*x^3+25*x^2+25*x-105)*exp(x))/(441*log(x)^
2-210*x^2*log(x)+25*x^4),x, algorithm="giac")

[Out]

-5*(x*e^x - e^x*log(x))/(5*x^2 - 21*log(x))

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maple [A]  time = 0.02, size = 28, normalized size = 1.04

method	result	size
risch	\(-\frac {5 \,{\mathrm e}^{x}}{21}+\frac {5 \left (5 x -21\right ) x \,{\mathrm e}^{x}}{21 \left (5 x^{2}-21 \ln \relax (x )\right )}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-105*exp(x)*ln(x)^2+(25*x^2+55*x+105)*exp(x)*ln(x)+(-25*x^3+25*x^2+25*x-105)*exp(x))/(441*ln(x)^2-210*x^2
*ln(x)+25*x^4),x,method=_RETURNVERBOSE)

[Out]

-5/21*exp(x)+5/21*(5*x-21)*x*exp(x)/(5*x^2-21*ln(x))

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maxima [A]  time = 0.38, size = 22, normalized size = 0.81 \begin {gather*} -\frac {5 \, {\left (x - \log \relax (x)\right )} e^{x}}{5 \, x^{2} - 21 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-105*exp(x)*log(x)^2+(25*x^2+55*x+105)*exp(x)*log(x)+(-25*x^3+25*x^2+25*x-105)*exp(x))/(441*log(x)^
2-210*x^2*log(x)+25*x^4),x, algorithm="maxima")

[Out]

-5*(x - log(x))*e^x/(5*x^2 - 21*log(x))

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mupad [B]  time = 9.05, size = 22, normalized size = 0.81 \begin {gather*} \frac {5\,{\mathrm {e}}^x\,\left (x-\ln \relax (x)\right )}{21\,\ln \relax (x)-5\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(25*x + 25*x^2 - 25*x^3 - 105) - 105*exp(x)*log(x)^2 + exp(x)*log(x)*(55*x + 25*x^2 + 105))/(441*l
og(x)^2 - 210*x^2*log(x) + 25*x^4),x)

[Out]

(5*exp(x)*(x - log(x)))/(21*log(x) - 5*x^2)

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sympy [A]  time = 0.30, size = 20, normalized size = 0.74 \begin {gather*} \frac {\left (- 5 x + 5 \log {\relax (x )}\right ) e^{x}}{5 x^{2} - 21 \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-105*exp(x)*ln(x)**2+(25*x**2+55*x+105)*exp(x)*ln(x)+(-25*x**3+25*x**2+25*x-105)*exp(x))/(441*ln(x)
**2-210*x**2*ln(x)+25*x**4),x)

[Out]

(-5*x + 5*log(x))*exp(x)/(5*x**2 - 21*log(x))

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