3.10.43 \(\int \frac {18+2 x-4 x^2+e^{x+x^3+2 x^4+x^5} (-10 x+45 x^2+120 x^3+75 x^4)}{-12+15 e^{x+x^3+2 x^4+x^5}+6 x} \, dx\)

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

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Rubi [F]  time = 1.45, 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 {18+2 x-4 x^2+e^{x+x^3+2 x^4+x^5} \left (-10 x+45 x^2+120 x^3+75 x^4\right )}{-12+15 e^{x+x^3+2 x^4+x^5}+6 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(18 + 2*x - 4*x^2 + E^(x + x^3 + 2*x^4 + x^5)*(-10*x + 45*x^2 + 120*x^3 + 75*x^4))/(-12 + 15*E^(x + x^3 +
2*x^4 + x^5) + 6*x),x]

[Out]

-1/3*x^2 + x^3 + 2*x^4 + x^5 + 6*Defer[Int][(-4 + 5*E^(x + x^3 + 2*x^4 + x^5) + 2*x)^(-1), x] - 2*Defer[Int][x
/(-4 + 5*E^(x + x^3 + 2*x^4 + x^5) + 2*x), x] + 12*Defer[Int][x^2/(-4 + 5*E^(x + x^3 + 2*x^4 + x^5) + 2*x), x]
+ 26*Defer[Int][x^3/(-4 + 5*E^(x + x^3 + 2*x^4 + x^5) + 2*x), x] + 4*Defer[Int][x^4/(-4 + 5*E^(x + x^3 + 2*x^
4 + x^5) + 2*x), x] - 10*Defer[Int][x^5/(-4 + 5*E^(x + x^3 + 2*x^4 + x^5) + 2*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-18-2 x+4 x^2-e^{x+x^3+2 x^4+x^5} \left (-10 x+45 x^2+120 x^3+75 x^4\right )}{3 \left (4-5 e^{x+x^3+2 x^4+x^5}-2 x\right )} \, dx\\ &=\frac {1}{3} \int \frac {-18-2 x+4 x^2-e^{x+x^3+2 x^4+x^5} \left (-10 x+45 x^2+120 x^3+75 x^4\right )}{4-5 e^{x+x^3+2 x^4+x^5}-2 x} \, dx\\ &=\frac {1}{3} \int \left (x \left (-2+9 x+24 x^2+15 x^3\right )-\frac {6 \left (-3+x-6 x^2-13 x^3-2 x^4+5 x^5\right )}{-4+5 e^{x+x^3+2 x^4+x^5}+2 x}\right ) \, dx\\ &=\frac {1}{3} \int x \left (-2+9 x+24 x^2+15 x^3\right ) \, dx-2 \int \frac {-3+x-6 x^2-13 x^3-2 x^4+5 x^5}{-4+5 e^{x+x^3+2 x^4+x^5}+2 x} \, dx\\ &=\frac {1}{3} \int \left (-2 x+9 x^2+24 x^3+15 x^4\right ) \, dx-2 \int \left (-\frac {3}{-4+5 e^{x+x^3+2 x^4+x^5}+2 x}+\frac {x}{-4+5 e^{x+x^3+2 x^4+x^5}+2 x}-\frac {6 x^2}{-4+5 e^{x+x^3+2 x^4+x^5}+2 x}-\frac {13 x^3}{-4+5 e^{x+x^3+2 x^4+x^5}+2 x}-\frac {2 x^4}{-4+5 e^{x+x^3+2 x^4+x^5}+2 x}+\frac {5 x^5}{-4+5 e^{x+x^3+2 x^4+x^5}+2 x}\right ) \, dx\\ &=-\frac {x^2}{3}+x^3+2 x^4+x^5-2 \int \frac {x}{-4+5 e^{x+x^3+2 x^4+x^5}+2 x} \, dx+4 \int \frac {x^4}{-4+5 e^{x+x^3+2 x^4+x^5}+2 x} \, dx+6 \int \frac {1}{-4+5 e^{x+x^3+2 x^4+x^5}+2 x} \, dx-10 \int \frac {x^5}{-4+5 e^{x+x^3+2 x^4+x^5}+2 x} \, dx+12 \int \frac {x^2}{-4+5 e^{x+x^3+2 x^4+x^5}+2 x} \, dx+26 \int \frac {x^3}{-4+5 e^{x+x^3+2 x^4+x^5}+2 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.38, size = 38, normalized size = 1.19 \begin {gather*} \frac {1}{3} \left (-3 x-x^2+3 \log \left (4-5 e^{x+x^3+2 x^4+x^5}-2 x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(18 + 2*x - 4*x^2 + E^(x + x^3 + 2*x^4 + x^5)*(-10*x + 45*x^2 + 120*x^3 + 75*x^4))/(-12 + 15*E^(x +
x^3 + 2*x^4 + x^5) + 6*x),x]

[Out]

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

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fricas [A]  time = 0.79, size = 31, normalized size = 0.97 \begin {gather*} -\frac {1}{3} \, x^{2} - x + \log \left (2 \, x + 5 \, e^{\left (x^{5} + 2 \, x^{4} + x^{3} + x\right )} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*x^4+120*x^3+45*x^2-10*x)*exp(x^5+2*x^4+x^3+x)-4*x^2+2*x+18)/(15*exp(x^5+2*x^4+x^3+x)+6*x-12),x,
algorithm="fricas")

[Out]

-1/3*x^2 - x + log(2*x + 5*e^(x^5 + 2*x^4 + x^3 + x) - 4)

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giac [A]  time = 0.52, size = 31, normalized size = 0.97 \begin {gather*} -\frac {1}{3} \, x^{2} - x + \log \left (2 \, x + 5 \, e^{\left (x^{5} + 2 \, x^{4} + x^{3} + x\right )} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*x^4+120*x^3+45*x^2-10*x)*exp(x^5+2*x^4+x^3+x)-4*x^2+2*x+18)/(15*exp(x^5+2*x^4+x^3+x)+6*x-12),x,
algorithm="giac")

[Out]

-1/3*x^2 - x + log(2*x + 5*e^(x^5 + 2*x^4 + x^3 + x) - 4)

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maple [A]  time = 0.04, size = 32, normalized size = 1.00

method	result	size
norman	\(-\frac {x^{2}}{3}-x +\ln \left (15 \,{\mathrm e}^{x^{5}+2 x^{4}+x^{3}+x}+6 x -12\right )\)	\(32\)
risch	\(-\frac {x^{2}}{3}-x +\ln \left ({\mathrm e}^{x \left (x^{4}+2 x^{3}+x^{2}+1\right )}+\frac {2 x}{5}-\frac {4}{5}\right )\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((75*x^4+120*x^3+45*x^2-10*x)*exp(x^5+2*x^4+x^3+x)-4*x^2+2*x+18)/(15*exp(x^5+2*x^4+x^3+x)+6*x-12),x,method
=_RETURNVERBOSE)

[Out]

-1/3*x^2-x+ln(15*exp(x^5+2*x^4+x^3+x)+6*x-12)

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maxima [A]  time = 0.40, size = 53, normalized size = 1.66 \begin {gather*} 2 \, x^{4} + x^{3} - \frac {1}{3} \, x^{2} + \log \left (\frac {1}{5} \, {\left (2 \, x + 5 \, e^{\left (x^{5} + 2 \, x^{4} + x^{3} + x\right )} - 4\right )} e^{\left (-2 \, x^{4} - x^{3} - x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*x^4+120*x^3+45*x^2-10*x)*exp(x^5+2*x^4+x^3+x)-4*x^2+2*x+18)/(15*exp(x^5+2*x^4+x^3+x)+6*x-12),x,
algorithm="maxima")

[Out]

2*x^4 + x^3 - 1/3*x^2 + log(1/5*(2*x + 5*e^(x^5 + 2*x^4 + x^3 + x) - 4)*e^(-2*x^4 - x^3 - x))

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mupad [B]  time = 0.74, size = 31, normalized size = 0.97 \begin {gather*} \ln \left (x+\frac {5\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{x^5}\,{\mathrm {e}}^{2\,x^4}\,{\mathrm {e}}^x}{2}-2\right )-x-\frac {x^2}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + exp(x + x^3 + 2*x^4 + x^5)*(45*x^2 - 10*x + 120*x^3 + 75*x^4) - 4*x^2 + 18)/(6*x + 15*exp(x + x^3 +
2*x^4 + x^5) - 12),x)

[Out]

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

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sympy [B]  time = 0.24, size = 54, normalized size = 1.69 \begin {gather*} \frac {4 x^{5}}{5} + \frac {8 x^{4}}{5} + \frac {4 x^{3}}{5} - \frac {x^{2}}{3} - \frac {x}{5} + \frac {\log {\left (\frac {2 x}{5} + e^{x^{5} + 2 x^{4} + x^{3} + x} - \frac {4}{5} \right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*x**4+120*x**3+45*x**2-10*x)*exp(x**5+2*x**4+x**3+x)-4*x**2+2*x+18)/(15*exp(x**5+2*x**4+x**3+x)+
6*x-12),x)

[Out]

4*x**5/5 + 8*x**4/5 + 4*x**3/5 - x**2/3 - x/5 + log(2*x/5 + exp(x**5 + 2*x**4 + x**3 + x) - 4/5)/5

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