3.87.41 \(\int \frac {-3-3 x^2+e^{3+9 x+6 x^2+x^3} (-1+9 x+12 x^2+3 x^3)}{x^2} \, dx\)

Optimal. Leaf size=25 \[ 2+\frac {3+e^{3+x (3+x)^2}-x}{x}-3 x \]

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Rubi [A]  time = 0.09, antiderivative size = 50, normalized size of antiderivative = 2.00, number of steps used = 5, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {14, 2288} \begin {gather*} \frac {e^{x^3+6 x^2+9 x+3} \left (x^3+4 x^2+3 x\right )}{\left (x^2+4 x+3\right ) x^2}-3 x+\frac {3}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 - 3*x^2 + E^(3 + 9*x + 6*x^2 + x^3)*(-1 + 9*x + 12*x^2 + 3*x^3))/x^2,x]

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {3 \left (1+x^2\right )}{x^2}+\frac {e^{3+9 x+6 x^2+x^3} \left (-1+9 x+12 x^2+3 x^3\right )}{x^2}\right ) \, dx\\ &=-\left (3 \int \frac {1+x^2}{x^2} \, dx\right )+\int \frac {e^{3+9 x+6 x^2+x^3} \left (-1+9 x+12 x^2+3 x^3\right )}{x^2} \, dx\\ &=\frac {e^{3+9 x+6 x^2+x^3} \left (3 x+4 x^2+x^3\right )}{x^2 \left (3+4 x+x^2\right )}-3 \int \left (1+\frac {1}{x^2}\right ) \, dx\\ &=\frac {3}{x}-3 x+\frac {e^{3+9 x+6 x^2+x^3} \left (3 x+4 x^2+x^3\right )}{x^2 \left (3+4 x+x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 26, normalized size = 1.04 \begin {gather*} \frac {3+e^{3+9 x+6 x^2+x^3}-3 x^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 - 3*x^2 + E^(3 + 9*x + 6*x^2 + x^3)*(-1 + 9*x + 12*x^2 + 3*x^3))/x^2,x]

[Out]

(3 + E^(3 + 9*x + 6*x^2 + x^3) - 3*x^2)/x

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fricas [A]  time = 0.58, size = 28, normalized size = 1.12 \begin {gather*} -\frac {3 \, x^{2} - e^{\left (x^{3} + 6 \, x^{2} + 9 \, x + 3\right )} - 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3+12*x^2+9*x-1)*exp(x^3+6*x^2+9*x+3)-3*x^2-3)/x^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.14, size = 28, normalized size = 1.12 \begin {gather*} -\frac {3 \, x^{2} - e^{\left (x^{3} + 6 \, x^{2} + 9 \, x + 3\right )} - 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3+12*x^2+9*x-1)*exp(x^3+6*x^2+9*x+3)-3*x^2-3)/x^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 26, normalized size = 1.04

method	result	size
norman	\(\frac {3-3 x^{2}+{\mathrm e}^{x^{3}+6 x^{2}+9 x +3}}{x}\)	\(26\)
risch	\(\frac {3}{x}-3 x +\frac {{\mathrm e}^{x^{3}+6 x^{2}+9 x +3}}{x}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^3+12*x^2+9*x-1)*exp(x^3+6*x^2+9*x+3)-3*x^2-3)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.44, size = 27, normalized size = 1.08 \begin {gather*} -3 \, x + \frac {e^{\left (x^{3} + 6 \, x^{2} + 9 \, x + 3\right )}}{x} + \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3+12*x^2+9*x-1)*exp(x^3+6*x^2+9*x+3)-3*x^2-3)/x^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.31, size = 29, normalized size = 1.16 \begin {gather*} \frac {3}{x}-3\,x+\frac {{\mathrm {e}}^{9\,x}\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^3\,{\mathrm {e}}^{6\,x^2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x^2 - exp(9*x + 6*x^2 + x^3 + 3)*(9*x + 12*x^2 + 3*x^3 - 1) + 3)/x^2,x)

[Out]

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

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sympy [A]  time = 0.12, size = 22, normalized size = 0.88 \begin {gather*} - 3 x + \frac {e^{x^{3} + 6 x^{2} + 9 x + 3}}{x} + \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**3+12*x**2+9*x-1)*exp(x**3+6*x**2+9*x+3)-3*x**2-3)/x**2,x)

[Out]

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

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