3.51.32 \(\int \frac {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+(-32 x-16 x^2-2 x^3) \log (\frac {4 e^{\frac {14+3 x}{4+x}}}{x})}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+(-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5) \log (\frac {4 e^{\frac {14+3 x}{4+x}}}{x})+(16+8 x+x^2) \log ^2(\frac {4 e^{\frac {14+3 x}{4+x}}}{x})} \, dx\)

Optimal. Leaf size=32 \[ \frac {x^2}{5-x^3-\log \left (\frac {4 e^{3+\frac {2}{4+x}}}{x}\right )} \]

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Rubi [F]  time = 1.42, 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 {144 x+70 x^2+9 x^3+16 x^4+8 x^5+x^6+\left (-32 x-16 x^2-2 x^3\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}{400+200 x+25 x^2-160 x^3-80 x^4-10 x^5+16 x^6+8 x^7+x^8+\left (-160-80 x-10 x^2+32 x^3+16 x^4+2 x^5\right ) \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )+\left (16+8 x+x^2\right ) \log ^2\left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(144*x + 70*x^2 + 9*x^3 + 16*x^4 + 8*x^5 + x^6 + (-32*x - 16*x^2 - 2*x^3)*Log[(4*E^((14 + 3*x)/(4 + x)))/x
])/(400 + 200*x + 25*x^2 - 160*x^3 - 80*x^4 - 10*x^5 + 16*x^6 + 8*x^7 + x^8 + (-160 - 80*x - 10*x^2 + 32*x^3 +
16*x^4 + 2*x^5)*Log[(4*E^((14 + 3*x)/(4 + x)))/x] + (16 + 8*x + x^2)*Log[(4*E^((14 + 3*x)/(4 + x)))/x]^2),x]

[Out]

-2*Defer[Int][(-5 + x^3 + Log[(4*E^((14 + 3*x)/(4 + x)))/x])^(-2), x] - Defer[Int][x/(-5 + x^3 + Log[(4*E^((14
+ 3*x)/(4 + x)))/x])^2, x] + 3*Defer[Int][x^4/(-5 + x^3 + Log[(4*E^((14 + 3*x)/(4 + x)))/x])^2, x] - 32*Defer
[Int][1/((4 + x)^2*(-5 + x^3 + Log[(4*E^((14 + 3*x)/(4 + x)))/x])^2), x] + 16*Defer[Int][1/((4 + x)*(-5 + x^3
+ Log[(4*E^((14 + 3*x)/(4 + x)))/x])^2), x] - 2*Defer[Int][x/(-5 + x^3 + Log[(4*E^((14 + 3*x)/(4 + x)))/x]), x
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (144+70 x+9 x^2+16 x^3+8 x^4+x^5-2 (4+x)^2 \log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )\right )}{(4+x)^2 \left (5-x^3-\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )\right )^2} \, dx\\ &=\int \left (\frac {x \left (-16-10 x-x^2+48 x^3+24 x^4+3 x^5\right )}{(4+x)^2 \left (-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )\right )^2}-\frac {2 x}{-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x}{-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx\right )+\int \frac {x \left (-16-10 x-x^2+48 x^3+24 x^4+3 x^5\right )}{(4+x)^2 \left (-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )\right )^2} \, dx\\ &=-\left (2 \int \frac {x}{-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx\right )+\int \left (-\frac {2}{\left (-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )\right )^2}-\frac {x}{\left (-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )\right )^2}+\frac {3 x^4}{\left (-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )\right )^2}-\frac {32}{(4+x)^2 \left (-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )\right )^2}+\frac {16}{(4+x) \left (-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\left (-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )\right )^2} \, dx\right )-2 \int \frac {x}{-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )} \, dx+3 \int \frac {x^4}{\left (-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )\right )^2} \, dx+16 \int \frac {1}{(4+x) \left (-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )\right )^2} \, dx-32 \int \frac {1}{(4+x)^2 \left (-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )\right )^2} \, dx-\int \frac {x}{\left (-5+x^3+\log \left (\frac {4 e^{\frac {14+3 x}{4+x}}}{x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 29, normalized size = 0.91 \begin {gather*} -\frac {x^2}{-5+x^3+\log \left (\frac {4 e^{3+\frac {2}{4+x}}}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(144*x + 70*x^2 + 9*x^3 + 16*x^4 + 8*x^5 + x^6 + (-32*x - 16*x^2 - 2*x^3)*Log[(4*E^((14 + 3*x)/(4 +
x)))/x])/(400 + 200*x + 25*x^2 - 160*x^3 - 80*x^4 - 10*x^5 + 16*x^6 + 8*x^7 + x^8 + (-160 - 80*x - 10*x^2 + 32
*x^3 + 16*x^4 + 2*x^5)*Log[(4*E^((14 + 3*x)/(4 + x)))/x] + (16 + 8*x + x^2)*Log[(4*E^((14 + 3*x)/(4 + x)))/x]^
2),x]

[Out]

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

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fricas [A]  time = 1.05, size = 30, normalized size = 0.94 \begin {gather*} -\frac {x^{2}}{x^{3} + \log \left (\frac {4 \, e^{\left (\frac {3 \, x + 14}{x + 4}\right )}}{x}\right ) - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-16*x^2-32*x)*log(4*exp((3*x+14)/(4+x))/x)+x^6+8*x^5+16*x^4+9*x^3+70*x^2+144*x)/((x^2+8*x+16
)*log(4*exp((3*x+14)/(4+x))/x)^2+(2*x^5+16*x^4+32*x^3-10*x^2-80*x-160)*log(4*exp((3*x+14)/(4+x))/x)+x^8+8*x^7+
16*x^6-10*x^5-80*x^4-160*x^3+25*x^2+200*x+400),x, algorithm="fricas")

[Out]

-x^2/(x^3 + log(4*e^((3*x + 14)/(x + 4))/x) - 5)

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giac [A]  time = 0.39, size = 44, normalized size = 1.38 \begin {gather*} -\frac {x^{3} + 4 \, x^{2}}{x^{4} + 4 \, x^{3} + 2 \, x \log \relax (2) - x \log \relax (x) - 2 \, x + 8 \, \log \relax (2) - 4 \, \log \relax (x) - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-16*x^2-32*x)*log(4*exp((3*x+14)/(4+x))/x)+x^6+8*x^5+16*x^4+9*x^3+70*x^2+144*x)/((x^2+8*x+16
)*log(4*exp((3*x+14)/(4+x))/x)^2+(2*x^5+16*x^4+32*x^3-10*x^2-80*x-160)*log(4*exp((3*x+14)/(4+x))/x)+x^8+8*x^7+
16*x^6-10*x^5-80*x^4-160*x^3+25*x^2+200*x+400),x, algorithm="giac")

[Out]

-(x^3 + 4*x^2)/(x^4 + 4*x^3 + 2*x*log(2) - x*log(x) - 2*x + 8*log(2) - 4*log(x) - 6)

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maple [C]  time = 0.21, size = 182, normalized size = 5.69

method	result	size
risch	\(-\frac {2 x^{2}}{-i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\frac {3 x +14}{4+x}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )^{2}-10+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{\frac {3 x +14}{4+x}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{\frac {3 x +14}{4+x}}}{x}\right )^{3}+2 x^{3}+4 \ln \relax (2)-2 \ln \relax (x )+2 \ln \left ({\mathrm e}^{\frac {3 x +14}{4+x}}\right )}\)	\(182\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3-16*x^2-32*x)*ln(4*exp((3*x+14)/(4+x))/x)+x^6+8*x^5+16*x^4+9*x^3+70*x^2+144*x)/((x^2+8*x+16)*ln(4*
exp((3*x+14)/(4+x))/x)^2+(2*x^5+16*x^4+32*x^3-10*x^2-80*x-160)*ln(4*exp((3*x+14)/(4+x))/x)+x^8+8*x^7+16*x^6-10
*x^5-80*x^4-160*x^3+25*x^2+200*x+400),x,method=_RETURNVERBOSE)

[Out]

-2*x^2/(-I*Pi*csgn(I/x)*csgn(I*exp((3*x+14)/(4+x)))*csgn(I/x*exp((3*x+14)/(4+x)))+I*Pi*csgn(I/x)*csgn(I/x*exp(
(3*x+14)/(4+x)))^2-10+I*Pi*csgn(I*exp((3*x+14)/(4+x)))*csgn(I/x*exp((3*x+14)/(4+x)))^2-I*Pi*csgn(I/x*exp((3*x+
14)/(4+x)))^3+2*x^3+4*ln(2)-2*ln(x)+2*ln(exp((3*x+14)/(4+x))))

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maxima [A]  time = 0.49, size = 41, normalized size = 1.28 \begin {gather*} -\frac {x^{3} + 4 \, x^{2}}{x^{4} + 4 \, x^{3} + 2 \, x {\left (\log \relax (2) - 1\right )} - {\left (x + 4\right )} \log \relax (x) + 8 \, \log \relax (2) - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-16*x^2-32*x)*log(4*exp((3*x+14)/(4+x))/x)+x^6+8*x^5+16*x^4+9*x^3+70*x^2+144*x)/((x^2+8*x+16
)*log(4*exp((3*x+14)/(4+x))/x)^2+(2*x^5+16*x^4+32*x^3-10*x^2-80*x-160)*log(4*exp((3*x+14)/(4+x))/x)+x^8+8*x^7+
16*x^6-10*x^5-80*x^4-160*x^3+25*x^2+200*x+400),x, algorithm="maxima")

[Out]

-(x^3 + 4*x^2)/(x^4 + 4*x^3 + 2*x*(log(2) - 1) - (x + 4)*log(x) + 8*log(2) - 6)

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mupad [B]  time = 3.74, size = 33, normalized size = 1.03 \begin {gather*} -\frac {x^2}{\ln \left (\frac {4}{x}\right )+\frac {3\,x}{x+4}+\frac {14}{x+4}+x^3-5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((144*x + 70*x^2 + 9*x^3 + 16*x^4 + 8*x^5 + x^6 - log((4*exp((3*x + 14)/(x + 4)))/x)*(32*x + 16*x^2 + 2*x^3
))/(200*x - log((4*exp((3*x + 14)/(x + 4)))/x)*(80*x + 10*x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 160) + log((4*exp((3
*x + 14)/(x + 4)))/x)^2*(8*x + x^2 + 16) + 25*x^2 - 160*x^3 - 80*x^4 - 10*x^5 + 16*x^6 + 8*x^7 + x^8 + 400),x)

[Out]

-x^2/(log(4/x) + (3*x)/(x + 4) + 14/(x + 4) + x^3 - 5)

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sympy [A]  time = 0.48, size = 24, normalized size = 0.75 \begin {gather*} - \frac {x^{2}}{x^{3} + \log {\left (\frac {4 e^{\frac {3 x + 14}{x + 4}}}{x} \right )} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3-16*x**2-32*x)*ln(4*exp((3*x+14)/(4+x))/x)+x**6+8*x**5+16*x**4+9*x**3+70*x**2+144*x)/((x**2
+8*x+16)*ln(4*exp((3*x+14)/(4+x))/x)**2+(2*x**5+16*x**4+32*x**3-10*x**2-80*x-160)*ln(4*exp((3*x+14)/(4+x))/x)+
x**8+8*x**7+16*x**6-10*x**5-80*x**4-160*x**3+25*x**2+200*x+400),x)

[Out]

-x**2/(x**3 + log(4*exp((3*x + 14)/(x + 4))/x) - 5)

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