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

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

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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 = {} $$\begin{array}{c}\int \frac{3x^4+e^{\frac{9+6x^2+x^4}{x^3}}(-27-27x-6x^2-6x^3+x^5)}{x^4+2x^5+x^6}dx\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{3x^4+e^{\frac{9+6x^2+x^4}{x^3}}(-27-27x-6x^2-6x^3+x^5)}{x^4(1+2x+x^2)}dx\\ & =\int \frac{3x^4+e^{\frac{9+6x^2+x^4}{x^3}}(-27-27x-6x^2-6x^3+x^5)}{x^4(1+x{)}^2}dx\\ & =\int (\frac{3}{(1+x{)}^2}+\frac{e^{\frac{{(3+x^2)}^2}{x^3}}(-27-27x-6x^2-6x^3+x^5)}{x^4(1+x{)}^2})dx\\ & =-\frac{3}{1+x}+\int \frac{e^{\frac{{(3+x^2)}^2}{x^3}}(-27-27x-6x^2-6x^3+x^5)}{x^4(1+x{)}^2}dx\\ & =-\frac{3}{1+x}+\int (-\frac{27e^{\frac{{(3+x^2)}^2}{x^3}}}{x^4}+\frac{27e^{\frac{{(3+x^2)}^2}{x^3}}}{x^3}-\frac{33e^{\frac{{(3+x^2)}^2}{x^3}}}{x^2}+\frac{33e^{\frac{{(3+x^2)}^2}{x^3}}}{x}-\frac{e^{\frac{{(3+x^2)}^2}{x^3}}}{(1+x{)}^2}-\frac{32e^{\frac{{(3+x^2)}^2}{x^3}}}{1+x})dx\\ & =-\frac{3}{1+x}-27\int \frac{e^{\frac{{(3+x^2)}^2}{x^3}}}{x^4}dx+27\int \frac{e^{\frac{{(3+x^2)}^2}{x^3}}}{x^3}dx-32\int \frac{e^{\frac{{(3+x^2)}^2}{x^3}}}{1+x}dx-33\int \frac{e^{\frac{{(3+x^2)}^2}{x^3}}}{x^2}dx+33\int \frac{e^{\frac{{(3+x^2)}^2}{x^3}}}{x}dx-\int \frac{e^{\frac{{(3+x^2)}^2}{x^3}}}{(1+x{)}^2}dx\end{array}\end{array}$$

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

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 $$\begin{array}{c}\frac{e^{\left(\frac{x^4+6x^2+9}{x^3}\right)}-3}{x+1}\end{array}$$

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 $$\begin{array}{c}\frac{e^{\left(\frac{x^4+6x^2+9}{x^3}\right)}-3}{x+1}\end{array}$$

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	$-\frac{3}{x+1}+\frac{{\mathrm{e}}^{\frac{{(x^2+3)}^2}{x^3}}}{x+1}$	$27$
norman	$\frac{-3x^3+{\mathrm{e}}^{\frac{x^4+6x^2+9}{x^3}}{x}^3}{x^3(x+1)}$	$35$

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 $$\begin{array}{c}\frac{e^{(x+\frac{6}{x}+\frac{9}{x^3})}}{x+1}-\frac{3}{x+1}\end{array}$$

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 $$\begin{array}{c}\frac{3x+{\mathrm{e}}^{6/x}{\mathrm{e}}^{\frac{9}{x^3}}{\mathrm{e}}^x}{x+1}\end{array}$$

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 $$\begin{array}{c}\frac{e^{\frac{x^4+6x^2+9}{x^3}}}{x+1}-\frac{3}{x+1}\end{array}$$

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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