3.33.45 \(\int \frac {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} (-27-27 x-6 x^2-6 x^3+x^5)}{x^4+2 x^5+x^6} \, dx\)

Optimal. Leaf size=22 \[ \frac {-2+e^{\frac {\left (3+x^2\right )^2}{x^3}}+x}{1+x} \]

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Rubi [F]  time = 1.16, 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 {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4+2 x^5+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3*x^4 + E^((9 + 6*x^2 + x^4)/x^3)*(-27 - 27*x - 6*x^2 - 6*x^3 + x^5))/(x^4 + 2*x^5 + x^6),x]

[Out]

-3/(1 + x) - 27*Defer[Int][E^((3 + x^2)^2/x^3)/x^4, x] + 27*Defer[Int][E^((3 + x^2)^2/x^3)/x^3, x] - 33*Defer[
Int][E^((3 + x^2)^2/x^3)/x^2, x] + 33*Defer[Int][E^((3 + x^2)^2/x^3)/x, x] - Defer[Int][E^((3 + x^2)^2/x^3)/(1
+ x)^2, x] - 32*Defer[Int][E^((3 + x^2)^2/x^3)/(1 + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4 \left (1+2 x+x^2\right )} \, dx\\ &=\int \frac {3 x^4+e^{\frac {9+6 x^2+x^4}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4 (1+x)^2} \, dx\\ &=\int \left (\frac {3}{(1+x)^2}+\frac {e^{\frac {\left (3+x^2\right )^2}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4 (1+x)^2}\right ) \, dx\\ &=-\frac {3}{1+x}+\int \frac {e^{\frac {\left (3+x^2\right )^2}{x^3}} \left (-27-27 x-6 x^2-6 x^3+x^5\right )}{x^4 (1+x)^2} \, dx\\ &=-\frac {3}{1+x}+\int \left (-\frac {27 e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x^4}+\frac {27 e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x^3}-\frac {33 e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x^2}+\frac {33 e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x}-\frac {e^{\frac {\left (3+x^2\right )^2}{x^3}}}{(1+x)^2}-\frac {32 e^{\frac {\left (3+x^2\right )^2}{x^3}}}{1+x}\right ) \, dx\\ &=-\frac {3}{1+x}-27 \int \frac {e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x^4} \, dx+27 \int \frac {e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x^3} \, dx-32 \int \frac {e^{\frac {\left (3+x^2\right )^2}{x^3}}}{1+x} \, dx-33 \int \frac {e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x^2} \, dx+33 \int \frac {e^{\frac {\left (3+x^2\right )^2}{x^3}}}{x} \, dx-\int \frac {e^{\frac {\left (3+x^2\right )^2}{x^3}}}{(1+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 21, normalized size = 0.95 \begin {gather*} \frac {-3+e^{\frac {\left (3+x^2\right )^2}{x^3}}}{1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x^4 + E^((9 + 6*x^2 + x^4)/x^3)*(-27 - 27*x - 6*x^2 - 6*x^3 + x^5))/(x^4 + 2*x^5 + x^6),x]

[Out]

(-3 + E^((3 + x^2)^2/x^3))/(1 + x)

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fricas [A]  time = 0.66, size = 23, normalized size = 1.05 \begin {gather*} \frac {e^{\left (\frac {x^{4} + 6 \, x^{2} + 9}{x^{3}}\right )} - 3}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5-6*x^3-6*x^2-27*x-27)*exp((x^4+6*x^2+9)/x^3)+3*x^4)/(x^6+2*x^5+x^4),x, algorithm="fricas")

[Out]

(e^((x^4 + 6*x^2 + 9)/x^3) - 3)/(x + 1)

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giac [A]  time = 0.41, size = 23, normalized size = 1.05 \begin {gather*} \frac {e^{\left (\frac {x^{4} + 6 \, x^{2} + 9}{x^{3}}\right )} - 3}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5-6*x^3-6*x^2-27*x-27)*exp((x^4+6*x^2+9)/x^3)+3*x^4)/(x^6+2*x^5+x^4),x, algorithm="giac")

[Out]

(e^((x^4 + 6*x^2 + 9)/x^3) - 3)/(x + 1)

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maple [A]  time = 0.18, size = 27, normalized size = 1.23

method	result	size
risch	\(-\frac {3}{x +1}+\frac {{\mathrm e}^{\frac {\left (x^{2}+3\right )^{2}}{x^{3}}}}{x +1}\)	\(27\)
norman	\(\frac {-3 x^{3}+{\mathrm e}^{\frac {x^{4}+6 x^{2}+9}{x^{3}}} x^{3}}{x^{3} \left (x +1\right )}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^5-6*x^3-6*x^2-27*x-27)*exp((x^4+6*x^2+9)/x^3)+3*x^4)/(x^6+2*x^5+x^4),x,method=_RETURNVERBOSE)

[Out]

-3/(x+1)+1/(x+1)*exp((x^2+3)^2/x^3)

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maxima [A]  time = 0.58, size = 27, normalized size = 1.23 \begin {gather*} \frac {e^{\left (x + \frac {6}{x} + \frac {9}{x^{3}}\right )}}{x + 1} - \frac {3}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^5-6*x^3-6*x^2-27*x-27)*exp((x^4+6*x^2+9)/x^3)+3*x^4)/(x^6+2*x^5+x^4),x, algorithm="maxima")

[Out]

e^(x + 6/x + 9/x^3)/(x + 1) - 3/(x + 1)

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mupad [B]  time = 1.96, size = 25, normalized size = 1.14 \begin {gather*} \frac {3\,x+{\mathrm {e}}^{6/x}\,{\mathrm {e}}^{\frac {9}{x^3}}\,{\mathrm {e}}^x}{x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((6*x^2 + x^4 + 9)/x^3)*(27*x + 6*x^2 + 6*x^3 - x^5 + 27) - 3*x^4)/(x^4 + 2*x^5 + x^6),x)

[Out]

(3*x + exp(6/x)*exp(9/x^3)*exp(x))/(x + 1)

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sympy [A]  time = 0.17, size = 22, normalized size = 1.00 \begin {gather*} \frac {e^{\frac {x^{4} + 6 x^{2} + 9}{x^{3}}}}{x + 1} - \frac {3}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**5-6*x**3-6*x**2-27*x-27)*exp((x**4+6*x**2+9)/x**3)+3*x**4)/(x**6+2*x**5+x**4),x)

[Out]

exp((x**4 + 6*x**2 + 9)/x**3)/(x + 1) - 3/(x + 1)

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