3.26.65 \(\int \frac {32+8 x+7 x^3+2 e^{x^2} x^4}{-16 x-8 x^2+4 x^3+e^{x^2} x^3+7 x^4} \, dx\)

Optimal. Leaf size=28 \[ \log \left (e^{x^2}-\frac {(4+x)^2}{x^2}+x \left (6+\frac {5+x}{x}\right )\right ) \]

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Rubi [F]  time = 1.15, 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 {32+8 x+7 x^3+2 e^{x^2} x^4}{-16 x-8 x^2+4 x^3+e^{x^2} x^3+7 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(32 + 8*x + 7*x^3 + 2*E^x^2*x^4)/(-16*x - 8*x^2 + 4*x^3 + E^x^2*x^3 + 7*x^4),x]

[Out]

x^2 + 8*Defer[Int][(-16 - 8*x + 4*x^2 + E^x^2*x^2 + 7*x^3)^(-1), x] + 32*Defer[Int][1/(x*(-16 - 8*x + 4*x^2 +
E^x^2*x^2 + 7*x^3)), x] + 32*Defer[Int][x/(-16 - 8*x + 4*x^2 + E^x^2*x^2 + 7*x^3), x] + 23*Defer[Int][x^2/(-16
- 8*x + 4*x^2 + E^x^2*x^2 + 7*x^3), x] - 8*Defer[Int][x^3/(-16 - 8*x + 4*x^2 + E^x^2*x^2 + 7*x^3), x] - 14*De
fer[Int][x^4/(-16 - 8*x + 4*x^2 + E^x^2*x^2 + 7*x^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 x-\frac {-32-8 x-32 x^2-23 x^3+8 x^4+14 x^5}{x \left (-16-8 x+4 x^2+e^{x^2} x^2+7 x^3\right )}\right ) \, dx\\ &=x^2-\int \frac {-32-8 x-32 x^2-23 x^3+8 x^4+14 x^5}{x \left (-16-8 x+4 x^2+e^{x^2} x^2+7 x^3\right )} \, dx\\ &=x^2-\int \left (-\frac {8}{-16-8 x+4 x^2+e^{x^2} x^2+7 x^3}-\frac {32}{x \left (-16-8 x+4 x^2+e^{x^2} x^2+7 x^3\right )}-\frac {32 x}{-16-8 x+4 x^2+e^{x^2} x^2+7 x^3}-\frac {23 x^2}{-16-8 x+4 x^2+e^{x^2} x^2+7 x^3}+\frac {8 x^3}{-16-8 x+4 x^2+e^{x^2} x^2+7 x^3}+\frac {14 x^4}{-16-8 x+4 x^2+e^{x^2} x^2+7 x^3}\right ) \, dx\\ &=x^2+8 \int \frac {1}{-16-8 x+4 x^2+e^{x^2} x^2+7 x^3} \, dx-8 \int \frac {x^3}{-16-8 x+4 x^2+e^{x^2} x^2+7 x^3} \, dx-14 \int \frac {x^4}{-16-8 x+4 x^2+e^{x^2} x^2+7 x^3} \, dx+23 \int \frac {x^2}{-16-8 x+4 x^2+e^{x^2} x^2+7 x^3} \, dx+32 \int \frac {1}{x \left (-16-8 x+4 x^2+e^{x^2} x^2+7 x^3\right )} \, dx+32 \int \frac {x}{-16-8 x+4 x^2+e^{x^2} x^2+7 x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.31, size = 31, normalized size = 1.11 \begin {gather*} -2 \log (x)+\log \left (16+8 x-4 x^2-e^{x^2} x^2-7 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32 + 8*x + 7*x^3 + 2*E^x^2*x^4)/(-16*x - 8*x^2 + 4*x^3 + E^x^2*x^3 + 7*x^4),x]

[Out]

-2*Log[x] + Log[16 + 8*x - 4*x^2 - E^x^2*x^2 - 7*x^3]

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fricas [A]  time = 0.58, size = 28, normalized size = 1.00 \begin {gather*} \log \left (\frac {7 \, x^{3} + x^{2} e^{\left (x^{2}\right )} + 4 \, x^{2} - 8 \, x - 16}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4*exp(x^2)+7*x^3+8*x+32)/(x^3*exp(x^2)+7*x^4+4*x^3-8*x^2-16*x),x, algorithm="fricas")

[Out]

log((7*x^3 + x^2*e^(x^2) + 4*x^2 - 8*x - 16)/x^2)

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giac [A]  time = 0.37, size = 29, normalized size = 1.04 \begin {gather*} \log \left (7 \, x^{3} + x^{2} e^{\left (x^{2}\right )} + 4 \, x^{2} - 8 \, x - 16\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4*exp(x^2)+7*x^3+8*x+32)/(x^3*exp(x^2)+7*x^4+4*x^3-8*x^2-16*x),x, algorithm="giac")

[Out]

log(7*x^3 + x^2*e^(x^2) + 4*x^2 - 8*x - 16) - 2*log(x)

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maple [A]  time = 0.03, size = 26, normalized size = 0.93

method	result	size
risch	\(\ln \left ({\mathrm e}^{x^{2}}+\frac {7 x^{3}+4 x^{2}-8 x -16}{x^{2}}\right )\)	\(26\)
norman	\(-2 \ln \relax (x )+\ln \left (x^{2} {\mathrm e}^{x^{2}}+7 x^{3}+4 x^{2}-8 x -16\right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^4*exp(x^2)+7*x^3+8*x+32)/(x^3*exp(x^2)+7*x^4+4*x^3-8*x^2-16*x),x,method=_RETURNVERBOSE)

[Out]

ln(exp(x^2)+(7*x^3+4*x^2-8*x-16)/x^2)

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maxima [A]  time = 0.40, size = 28, normalized size = 1.00 \begin {gather*} \log \left (\frac {7 \, x^{3} + x^{2} e^{\left (x^{2}\right )} + 4 \, x^{2} - 8 \, x - 16}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4*exp(x^2)+7*x^3+8*x+32)/(x^3*exp(x^2)+7*x^4+4*x^3-8*x^2-16*x),x, algorithm="maxima")

[Out]

log((7*x^3 + x^2*e^(x^2) + 4*x^2 - 8*x - 16)/x^2)

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mupad [B]  time = 1.47, size = 29, normalized size = 1.04 \begin {gather*} \ln \left (x^2\,{\mathrm {e}}^{x^2}-8\,x+4\,x^2+7\,x^3-16\right )-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x + 2*x^4*exp(x^2) + 7*x^3 + 32)/(x^3*exp(x^2) - 16*x - 8*x^2 + 4*x^3 + 7*x^4),x)

[Out]

log(x^2*exp(x^2) - 8*x + 4*x^2 + 7*x^3 - 16) - 2*log(x)

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sympy [A]  time = 0.22, size = 24, normalized size = 0.86 \begin {gather*} \log {\left (e^{x^{2}} + \frac {7 x^{3} + 4 x^{2} - 8 x - 16}{x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**4*exp(x**2)+7*x**3+8*x+32)/(x**3*exp(x**2)+7*x**4+4*x**3-8*x**2-16*x),x)

[Out]

log(exp(x**2) + (7*x**3 + 4*x**2 - 8*x - 16)/x**2)

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