3.68.83 \(\int \frac {-9 x^2+9 e^x x^2+e^{\frac {1+x^3}{x}} (-9+18 x^3)}{x^2} \, dx\)

Optimal. Leaf size=19 \[ 9 \left (-3+e^x+e^{x \left (\frac {1}{x^2}+x\right )}-x\right ) \]

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Rubi [A]  time = 0.12, antiderivative size = 20, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {14, 2194, 6706} \begin {gather*} 9 e^{x^2+\frac {1}{x}}-9 x+9 e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9*x^2 + 9*E^x*x^2 + E^((1 + x^3)/x)*(-9 + 18*x^3))/x^2,x]

[Out]

9*E^x + 9*E^(x^(-1) + x^2) - 9*x

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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 18, normalized size = 0.95 \begin {gather*} 9 \left (e^x+e^{\frac {1}{x}+x^2}-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9*x^2 + 9*E^x*x^2 + E^((1 + x^3)/x)*(-9 + 18*x^3))/x^2,x]

[Out]

9*(E^x + E^(x^(-1) + x^2) - x)

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fricas [A]  time = 0.69, size = 20, normalized size = 1.05 \begin {gather*} -9 \, x + 9 \, e^{x} + 9 \, e^{\left (\frac {x^{3} + 1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^3-9)*exp((x^3+1)/x)+9*exp(x)*x^2-9*x^2)/x^2,x, algorithm="fricas")

[Out]

-9*x + 9*e^x + 9*e^((x^3 + 1)/x)

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giac [A]  time = 0.25, size = 20, normalized size = 1.05 \begin {gather*} -9 \, x + 9 \, e^{x} + 9 \, e^{\left (\frac {x^{3} + 1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^3-9)*exp((x^3+1)/x)+9*exp(x)*x^2-9*x^2)/x^2,x, algorithm="giac")

[Out]

-9*x + 9*e^x + 9*e^((x^3 + 1)/x)

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maple [A]  time = 0.09, size = 27, normalized size = 1.42

method	result	size
risch	\(-9 x +9 \,{\mathrm e}^{x}+9 \,{\mathrm e}^{\frac {\left (x +1\right ) \left (x^{2}-x +1\right )}{x}}\)	\(27\)
norman	\(\frac {-9 x^{2}+9 \,{\mathrm e}^{x} x +9 \,{\mathrm e}^{\frac {x^{3}+1}{x}} x}{x}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((18*x^3-9)*exp((x^3+1)/x)+9*exp(x)*x^2-9*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

-9*x+9*exp(x)+9*exp((x+1)*(x^2-x+1)/x)

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maxima [A]  time = 1.38, size = 18, normalized size = 0.95 \begin {gather*} -9 \, x + 9 \, e^{\left (x^{2} + \frac {1}{x}\right )} + 9 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^3-9)*exp((x^3+1)/x)+9*exp(x)*x^2-9*x^2)/x^2,x, algorithm="maxima")

[Out]

-9*x + 9*e^(x^2 + 1/x) + 9*e^x

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mupad [B]  time = 4.23, size = 18, normalized size = 0.95 \begin {gather*} 9\,{\mathrm {e}}^{\frac {1}{x}+x^2}-9\,x+9\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x^2*exp(x) + exp((x^3 + 1)/x)*(18*x^3 - 9) - 9*x^2)/x^2,x)

[Out]

9*exp(1/x + x^2) - 9*x + 9*exp(x)

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sympy [A]  time = 0.19, size = 17, normalized size = 0.89 \begin {gather*} - 9 x + 9 e^{x} + 9 e^{\frac {x^{3} + 1}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x**3-9)*exp((x**3+1)/x)+9*exp(x)*x**2-9*x**2)/x**2,x)

[Out]

-9*x + 9*exp(x) + 9*exp((x**3 + 1)/x)

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