3.27.46 \(\int \frac {42 x+32 x^2+5 x^3+e^x (7 x-x^2-x^3)+(18+6 x+e^x (3-2 x-x^2)) \log (\frac {3+x}{5})}{3 x^4+x^5+(6 x^3+2 x^4) \log (\frac {3+x}{5})+(3 x^2+x^3) \log ^2(\frac {3+x}{5})} \, dx\)

Optimal. Leaf size=28 \[ \frac {-1-e^x-5 (1+x)}{x \left (x+\log \left (\frac {3+x}{5}\right )\right )} \]

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Rubi [F]  time = 4.37, 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 {42 x+32 x^2+5 x^3+e^x \left (7 x-x^2-x^3\right )+\left (18+6 x+e^x \left (3-2 x-x^2\right )\right ) \log \left (\frac {3+x}{5}\right )}{3 x^4+x^5+\left (6 x^3+2 x^4\right ) \log \left (\frac {3+x}{5}\right )+\left (3 x^2+x^3\right ) \log ^2\left (\frac {3+x}{5}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(42*x + 32*x^2 + 5*x^3 + E^x*(7*x - x^2 - x^3) + (18 + 6*x + E^x*(3 - 2*x - x^2))*Log[(3 + x)/5])/(3*x^4 +
x^5 + (6*x^3 + 2*x^4)*Log[(3 + x)/5] + (3*x^2 + x^3)*Log[(3 + x)/5]^2),x]

[Out]

5*Defer[Int][(x + Log[(3 + x)/5])^(-2), x] + 14*Defer[Int][1/(x*(x + Log[(3 + x)/5])^2), x] + (4*Defer[Int][E^
x/(x*(x + Log[(3 + x)/5])^2), x])/3 + 3*Defer[Int][1/((3 + x)*(x + Log[(3 + x)/5])^2), x] - Defer[Int][E^x/((3
+ x)*(x + Log[(3 + x)/5])^2), x]/3 + 6*Defer[Int][Log[3/5 + x/5]/(x^2*(x + Log[(3 + x)/5])^2), x] + Defer[Int
][E^x/(x^2*(x + Log[(3 + x)/5])), x] - Defer[Int][E^x/(x*(x + Log[(3 + x)/5])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x \left (-42-32 x-5 x^2+e^x \left (-7+x+x^2\right )\right )-\left (-6+e^x (-1+x)\right ) (3+x) \log \left (\frac {3+x}{5}\right )}{x^2 (3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx\\ &=\int \left (\frac {32}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2}+\frac {42}{x (3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2}+\frac {5 x}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2}+\frac {6 \log \left (\frac {3}{5}+\frac {x}{5}\right )}{x^2 \left (x+\log \left (\frac {3+x}{5}\right )\right )^2}-\frac {e^x \left (-7 x+x^2+x^3-3 \log \left (\frac {3+x}{5}\right )+2 x \log \left (\frac {3+x}{5}\right )+x^2 \log \left (\frac {3+x}{5}\right )\right )}{x^2 (3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2}\right ) \, dx\\ &=5 \int \frac {x}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+6 \int \frac {\log \left (\frac {3}{5}+\frac {x}{5}\right )}{x^2 \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+32 \int \frac {1}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+42 \int \frac {1}{x (3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx-\int \frac {e^x \left (-7 x+x^2+x^3-3 \log \left (\frac {3+x}{5}\right )+2 x \log \left (\frac {3+x}{5}\right )+x^2 \log \left (\frac {3+x}{5}\right )\right )}{x^2 (3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx\\ &=5 \int \left (\frac {1}{\left (x+\log \left (\frac {3+x}{5}\right )\right )^2}-\frac {3}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2}\right ) \, dx+6 \int \frac {\log \left (\frac {3}{5}+\frac {x}{5}\right )}{x^2 \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+32 \int \frac {1}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+42 \int \left (\frac {1}{3 x \left (x+\log \left (\frac {3+x}{5}\right )\right )^2}-\frac {1}{3 (3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2}\right ) \, dx-\int \frac {e^x \left (x \left (-7+x+x^2\right )+\left (-3+2 x+x^2\right ) \log \left (\frac {3+x}{5}\right )\right )}{x^2 (3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx\\ &=5 \int \frac {1}{\left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+6 \int \frac {\log \left (\frac {3}{5}+\frac {x}{5}\right )}{x^2 \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+14 \int \frac {1}{x \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx-14 \int \frac {1}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx-15 \int \frac {1}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+32 \int \frac {1}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx-\int \left (\frac {e^x (-4-x)}{x (3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2}+\frac {e^x (-1+x)}{x^2 \left (x+\log \left (\frac {3+x}{5}\right )\right )}\right ) \, dx\\ &=5 \int \frac {1}{\left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+6 \int \frac {\log \left (\frac {3}{5}+\frac {x}{5}\right )}{x^2 \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+14 \int \frac {1}{x \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx-14 \int \frac {1}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx-15 \int \frac {1}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+32 \int \frac {1}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx-\int \frac {e^x (-4-x)}{x (3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx-\int \frac {e^x (-1+x)}{x^2 \left (x+\log \left (\frac {3+x}{5}\right )\right )} \, dx\\ &=5 \int \frac {1}{\left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+6 \int \frac {\log \left (\frac {3}{5}+\frac {x}{5}\right )}{x^2 \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+14 \int \frac {1}{x \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx-14 \int \frac {1}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx-15 \int \frac {1}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+32 \int \frac {1}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx-\int \left (-\frac {4 e^x}{3 x \left (x+\log \left (\frac {3+x}{5}\right )\right )^2}+\frac {e^x}{3 (3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2}\right ) \, dx-\int \left (-\frac {e^x}{x^2 \left (x+\log \left (\frac {3+x}{5}\right )\right )}+\frac {e^x}{x \left (x+\log \left (\frac {3+x}{5}\right )\right )}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {e^x}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx\right )+\frac {4}{3} \int \frac {e^x}{x \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+5 \int \frac {1}{\left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+6 \int \frac {\log \left (\frac {3}{5}+\frac {x}{5}\right )}{x^2 \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+14 \int \frac {1}{x \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx-14 \int \frac {1}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx-15 \int \frac {1}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+32 \int \frac {1}{(3+x) \left (x+\log \left (\frac {3+x}{5}\right )\right )^2} \, dx+\int \frac {e^x}{x^2 \left (x+\log \left (\frac {3+x}{5}\right )\right )} \, dx-\int \frac {e^x}{x \left (x+\log \left (\frac {3+x}{5}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.66, size = 25, normalized size = 0.89 \begin {gather*} -\frac {6+e^x+5 x}{x \left (x+\log \left (\frac {3+x}{5}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(42*x + 32*x^2 + 5*x^3 + E^x*(7*x - x^2 - x^3) + (18 + 6*x + E^x*(3 - 2*x - x^2))*Log[(3 + x)/5])/(3
*x^4 + x^5 + (6*x^3 + 2*x^4)*Log[(3 + x)/5] + (3*x^2 + x^3)*Log[(3 + x)/5]^2),x]

[Out]

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

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fricas [A]  time = 0.57, size = 23, normalized size = 0.82 \begin {gather*} -\frac {5 \, x + e^{x} + 6}{x^{2} + x \log \left (\frac {1}{5} \, x + \frac {3}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-2*x+3)*exp(x)+18+6*x)*log(3/5+1/5*x)+(-x^3-x^2+7*x)*exp(x)+5*x^3+32*x^2+42*x)/((x^3+3*x^2)*l
og(3/5+1/5*x)^2+(2*x^4+6*x^3)*log(3/5+1/5*x)+x^5+3*x^4),x, algorithm="fricas")

[Out]

-(5*x + e^x + 6)/(x^2 + x*log(1/5*x + 3/5))

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giac [A]  time = 0.36, size = 23, normalized size = 0.82 \begin {gather*} -\frac {5 \, x + e^{x} + 6}{x^{2} + x \log \left (\frac {1}{5} \, x + \frac {3}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-2*x+3)*exp(x)+18+6*x)*log(3/5+1/5*x)+(-x^3-x^2+7*x)*exp(x)+5*x^3+32*x^2+42*x)/((x^3+3*x^2)*l
og(3/5+1/5*x)^2+(2*x^4+6*x^3)*log(3/5+1/5*x)+x^5+3*x^4),x, algorithm="giac")

[Out]

-(5*x + e^x + 6)/(x^2 + x*log(1/5*x + 3/5))

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maple [A]  time = 0.22, size = 23, normalized size = 0.82

method	result	size
risch	\(-\frac {5 x +{\mathrm e}^{x}+6}{x \left (\ln \left (\frac {3}{5}+\frac {x}{5}\right )+x \right )}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^2-2*x+3)*exp(x)+18+6*x)*ln(3/5+1/5*x)+(-x^3-x^2+7*x)*exp(x)+5*x^3+32*x^2+42*x)/((x^3+3*x^2)*ln(3/5+1
/5*x)^2+(2*x^4+6*x^3)*ln(3/5+1/5*x)+x^5+3*x^4),x,method=_RETURNVERBOSE)

[Out]

-(5*x+exp(x)+6)/x/(ln(3/5+1/5*x)+x)

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maxima [A]  time = 0.67, size = 26, normalized size = 0.93 \begin {gather*} -\frac {5 \, x + e^{x} + 6}{x^{2} - x \log \relax (5) + x \log \left (x + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-2*x+3)*exp(x)+18+6*x)*log(3/5+1/5*x)+(-x^3-x^2+7*x)*exp(x)+5*x^3+32*x^2+42*x)/((x^3+3*x^2)*l
og(3/5+1/5*x)^2+(2*x^4+6*x^3)*log(3/5+1/5*x)+x^5+3*x^4),x, algorithm="maxima")

[Out]

-(5*x + e^x + 6)/(x^2 - x*log(5) + x*log(x + 3))

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mupad [B]  time = 1.89, size = 22, normalized size = 0.79 \begin {gather*} -\frac {5\,x+{\mathrm {e}}^x+6}{x\,\left (x+\ln \left (\frac {x}{5}+\frac {3}{5}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((42*x - exp(x)*(x^2 - 7*x + x^3) + log(x/5 + 3/5)*(6*x - exp(x)*(2*x + x^2 - 3) + 18) + 32*x^2 + 5*x^3)/(l
og(x/5 + 3/5)*(6*x^3 + 2*x^4) + log(x/5 + 3/5)^2*(3*x^2 + x^3) + 3*x^4 + x^5),x)

[Out]

-(5*x + exp(x) + 6)/(x*(x + log(x/5 + 3/5)))

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sympy [A]  time = 0.37, size = 36, normalized size = 1.29 \begin {gather*} \frac {- 5 x - 6}{x^{2} + x \log {\left (\frac {x}{5} + \frac {3}{5} \right )}} - \frac {e^{x}}{x^{2} + x \log {\left (\frac {x}{5} + \frac {3}{5} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2-2*x+3)*exp(x)+18+6*x)*ln(3/5+1/5*x)+(-x**3-x**2+7*x)*exp(x)+5*x**3+32*x**2+42*x)/((x**3+3*x
**2)*ln(3/5+1/5*x)**2+(2*x**4+6*x**3)*ln(3/5+1/5*x)+x**5+3*x**4),x)

[Out]

(-5*x - 6)/(x**2 + x*log(x/5 + 3/5)) - exp(x)/(x**2 + x*log(x/5 + 3/5))

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