3.85.92 \(\int \frac {-10+2 x^2+e^{-4 x^2+x^3} (-68 x+30 x^2-26 x^3+9 x^4+8 x^5-3 x^6)+e^{-4 x^2+x^3} (40 x-15 x^2-8 x^3+3 x^4) \log (5-x^2)}{-5+x^2} \, dx\)

Optimal. Leaf size=30 \[ 2 x-e^{(-4+x) x^2} \left (2+x^2-\log \left (5-x^2\right )\right ) \]

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Rubi [F]  time = 4.63, 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 {-10+2 x^2+e^{-4 x^2+x^3} \left (-68 x+30 x^2-26 x^3+9 x^4+8 x^5-3 x^6\right )+e^{-4 x^2+x^3} \left (40 x-15 x^2-8 x^3+3 x^4\right ) \log \left (5-x^2\right )}{-5+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-10 + 2*x^2 + E^(-4*x^2 + x^3)*(-68*x + 30*x^2 - 26*x^3 + 9*x^4 + 8*x^5 - 3*x^6) + E^(-4*x^2 + x^3)*(40*x
- 15*x^2 - 8*x^3 + 3*x^4)*Log[5 - x^2])/(-5 + x^2),x]

[Out]

2*x + E^(-4*x^2 + x^3)*Log[5 - x^2] + 14*Defer[Int][E^(-4*x^2 + x^3)*x, x] - 6*Defer[Int][E^(-4*x^2 + x^3)*x^2
, x] + 8*Defer[Int][E^(-4*x^2 + x^3)*x^3, x] - 3*Defer[Int][E^(-4*x^2 + x^3)*x^4, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2+\frac {e^{-4 x^2+x^3} x \left (68-30 x+26 x^2-9 x^3-8 x^4+3 x^5-40 \log \left (5-x^2\right )+15 x \log \left (5-x^2\right )+8 x^2 \log \left (5-x^2\right )-3 x^3 \log \left (5-x^2\right )\right )}{5-x^2}\right ) \, dx\\ &=2 x+\int \frac {e^{-4 x^2+x^3} x \left (68-30 x+26 x^2-9 x^3-8 x^4+3 x^5-40 \log \left (5-x^2\right )+15 x \log \left (5-x^2\right )+8 x^2 \log \left (5-x^2\right )-3 x^3 \log \left (5-x^2\right )\right )}{5-x^2} \, dx\\ &=2 x+\int \frac {e^{-4 x^2+x^3} x \left (68-30 x+26 x^2-9 x^3-8 x^4+3 x^5+\left (-40+15 x+8 x^2-3 x^3\right ) \log \left (5-x^2\right )\right )}{5-x^2} \, dx\\ &=2 x+\int \left (-\frac {e^{-4 x^2+x^3} x \left (68-30 x+26 x^2-9 x^3-8 x^4+3 x^5\right )}{-5+x^2}+e^{-4 x^2+x^3} x (-8+3 x) \log \left (5-x^2\right )\right ) \, dx\\ &=2 x-\int \frac {e^{-4 x^2+x^3} x \left (68-30 x+26 x^2-9 x^3-8 x^4+3 x^5\right )}{-5+x^2} \, dx+\int e^{-4 x^2+x^3} x (-8+3 x) \log \left (5-x^2\right ) \, dx\\ &=2 x+e^{-4 x^2+x^3} \log \left (5-x^2\right )-\int -\frac {2 e^{-4 x^2+x^3} x}{5-x^2} \, dx-\int \left (-14 e^{-4 x^2+x^3} x+6 e^{-4 x^2+x^3} x^2-8 e^{-4 x^2+x^3} x^3+3 e^{-4 x^2+x^3} x^4-\frac {2 e^{-4 x^2+x^3} x}{-5+x^2}\right ) \, dx\\ &=2 x+e^{-4 x^2+x^3} \log \left (5-x^2\right )+2 \int \frac {e^{-4 x^2+x^3} x}{5-x^2} \, dx+2 \int \frac {e^{-4 x^2+x^3} x}{-5+x^2} \, dx-3 \int e^{-4 x^2+x^3} x^4 \, dx-6 \int e^{-4 x^2+x^3} x^2 \, dx+8 \int e^{-4 x^2+x^3} x^3 \, dx+14 \int e^{-4 x^2+x^3} x \, dx\\ &=2 x+e^{-4 x^2+x^3} \log \left (5-x^2\right )+2 \int \left (\frac {e^{-4 x^2+x^3}}{2 \left (\sqrt {5}-x\right )}-\frac {e^{-4 x^2+x^3}}{2 \left (\sqrt {5}+x\right )}\right ) \, dx+2 \int \left (-\frac {e^{-4 x^2+x^3}}{2 \left (\sqrt {5}-x\right )}+\frac {e^{-4 x^2+x^3}}{2 \left (\sqrt {5}+x\right )}\right ) \, dx-3 \int e^{-4 x^2+x^3} x^4 \, dx-6 \int e^{-4 x^2+x^3} x^2 \, dx+8 \int e^{-4 x^2+x^3} x^3 \, dx+14 \int e^{-4 x^2+x^3} x \, dx\\ &=2 x+e^{-4 x^2+x^3} \log \left (5-x^2\right )-3 \int e^{-4 x^2+x^3} x^4 \, dx-6 \int e^{-4 x^2+x^3} x^2 \, dx+8 \int e^{-4 x^2+x^3} x^3 \, dx+14 \int e^{-4 x^2+x^3} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 38, normalized size = 1.27 \begin {gather*} 2 x-e^{(-4+x) x^2} \left (2+x^2\right )+e^{(-4+x) x^2} \log \left (5-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10 + 2*x^2 + E^(-4*x^2 + x^3)*(-68*x + 30*x^2 - 26*x^3 + 9*x^4 + 8*x^5 - 3*x^6) + E^(-4*x^2 + x^3)
*(40*x - 15*x^2 - 8*x^3 + 3*x^4)*Log[5 - x^2])/(-5 + x^2),x]

[Out]

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

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fricas [A]  time = 0.97, size = 40, normalized size = 1.33 \begin {gather*} -{\left (x^{2} + 2\right )} e^{\left (x^{3} - 4 \, x^{2}\right )} + e^{\left (x^{3} - 4 \, x^{2}\right )} \log \left (-x^{2} + 5\right ) + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^4-8*x^3-15*x^2+40*x)*exp(x^3-4*x^2)*log(-x^2+5)+(-3*x^6+8*x^5+9*x^4-26*x^3+30*x^2-68*x)*exp(x^
3-4*x^2)+2*x^2-10)/(x^2-5),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.20, size = 50, normalized size = 1.67 \begin {gather*} -x^{2} e^{\left (x^{3} - 4 \, x^{2}\right )} + e^{\left (x^{3} - 4 \, x^{2}\right )} \log \left (-x^{2} + 5\right ) + 2 \, x - 2 \, e^{\left (x^{3} - 4 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^4-8*x^3-15*x^2+40*x)*exp(x^3-4*x^2)*log(-x^2+5)+(-3*x^6+8*x^5+9*x^4-26*x^3+30*x^2-68*x)*exp(x^
3-4*x^2)+2*x^2-10)/(x^2-5),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.37, size = 45, normalized size = 1.50

method	result	size
risch	\(-{\mathrm e}^{\left (x -4\right ) x^{2}} x^{2}+{\mathrm e}^{\left (x -4\right ) x^{2}} \ln \left (-x^{2}+5\right )+2 x -2 \,{\mathrm e}^{\left (x -4\right ) x^{2}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^4-8*x^3-15*x^2+40*x)*exp(x^3-4*x^2)*ln(-x^2+5)+(-3*x^6+8*x^5+9*x^4-26*x^3+30*x^2-68*x)*exp(x^3-4*x^2
)+2*x^2-10)/(x^2-5),x,method=_RETURNVERBOSE)

[Out]

-exp((x-4)*x^2)*x^2+exp((x-4)*x^2)*ln(-x^2+5)+2*x-2*exp((x-4)*x^2)

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maxima [A]  time = 0.68, size = 37, normalized size = 1.23 \begin {gather*} -{\left ({\left (x^{2} + 2\right )} e^{\left (x^{3}\right )} - e^{\left (x^{3}\right )} \log \left (-x^{2} + 5\right )\right )} e^{\left (-4 \, x^{2}\right )} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^4-8*x^3-15*x^2+40*x)*exp(x^3-4*x^2)*log(-x^2+5)+(-3*x^6+8*x^5+9*x^4-26*x^3+30*x^2-68*x)*exp(x^
3-4*x^2)+2*x^2-10)/(x^2-5),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{x^3-4\,x^2}\,\left (3\,x^6-8\,x^5-9\,x^4+26\,x^3-30\,x^2+68\,x\right )-2\,x^2-\ln \left (5-x^2\right )\,{\mathrm {e}}^{x^3-4\,x^2}\,\left (3\,x^4-8\,x^3-15\,x^2+40\,x\right )+10}{x^2-5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x^3 - 4*x^2)*(68*x - 30*x^2 + 26*x^3 - 9*x^4 - 8*x^5 + 3*x^6) - 2*x^2 - log(5 - x^2)*exp(x^3 - 4*x^2
)*(40*x - 15*x^2 - 8*x^3 + 3*x^4) + 10)/(x^2 - 5),x)

[Out]

int(-(exp(x^3 - 4*x^2)*(68*x - 30*x^2 + 26*x^3 - 9*x^4 - 8*x^5 + 3*x^6) - 2*x^2 - log(5 - x^2)*exp(x^3 - 4*x^2
)*(40*x - 15*x^2 - 8*x^3 + 3*x^4) + 10)/(x^2 - 5), x)

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sympy [A]  time = 0.51, size = 24, normalized size = 0.80 \begin {gather*} 2 x + \left (- x^{2} + \log {\left (5 - x^{2} \right )} - 2\right ) e^{x^{3} - 4 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**4-8*x**3-15*x**2+40*x)*exp(x**3-4*x**2)*ln(-x**2+5)+(-3*x**6+8*x**5+9*x**4-26*x**3+30*x**2-68
*x)*exp(x**3-4*x**2)+2*x**2-10)/(x**2-5),x)

[Out]

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

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