3.64.66 \(\int \frac {1-x^2-e^{1+x} x^2+25 x^6}{x^2} \, dx\)

Optimal. Leaf size=22 \[ -12-e^{1+x}-\frac {1}{x}-x+5 x^5 \]

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Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14, 2194} \begin {gather*} 5 x^5-x-e^{x+1}-\frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x^2 - E^(1 + x)*x^2 + 25*x^6)/x^2,x]

[Out]

-E^(1 + x) - x^(-1) - x + 5*x^5

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-e^{1+x}+\frac {1-x^2+25 x^6}{x^2}\right ) \, dx\\ &=-\int e^{1+x} \, dx+\int \frac {1-x^2+25 x^6}{x^2} \, dx\\ &=-e^{1+x}+\int \left (-1+\frac {1}{x^2}+25 x^4\right ) \, dx\\ &=-e^{1+x}-\frac {1}{x}-x+5 x^5\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.95 \begin {gather*} -e^{1+x}-\frac {1}{x}-x+5 x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^2 - E^(1 + x)*x^2 + 25*x^6)/x^2,x]

[Out]

-E^(1 + x) - x^(-1) - x + 5*x^5

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fricas [A]  time = 0.54, size = 23, normalized size = 1.05 \begin {gather*} \frac {5 \, x^{6} - x^{2} - x e^{\left (x + 1\right )} - 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*exp(x+1)+25*x^6-x^2+1)/x^2,x, algorithm="fricas")

[Out]

(5*x^6 - x^2 - x*e^(x + 1) - 1)/x

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giac [A]  time = 0.15, size = 23, normalized size = 1.05 \begin {gather*} \frac {5 \, x^{6} - x^{2} - x e^{\left (x + 1\right )} - 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*exp(x+1)+25*x^6-x^2+1)/x^2,x, algorithm="giac")

[Out]

(5*x^6 - x^2 - x*e^(x + 1) - 1)/x

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maple [A]  time = 0.15, size = 21, normalized size = 0.95

method	result	size
risch	\(5 x^{5}-x -\frac {1}{x}-{\mathrm e}^{x +1}\)	\(21\)
norman	\(\frac {-1-x^{2}+5 x^{6}-x \,{\mathrm e}^{x +1}}{x}\)	\(24\)
derivativedivides	\(-\frac {1}{x}+24 x +24-50 \left (x +1\right )^{2}+50 \left (x +1\right )^{3}-25 \left (x +1\right )^{4}+5 \left (x +1\right )^{5}-{\mathrm e}^{x +1}\)	\(45\)
default	\(-\frac {1}{x}+24 x +24-50 \left (x +1\right )^{2}+50 \left (x +1\right )^{3}-25 \left (x +1\right )^{4}+5 \left (x +1\right )^{5}-{\mathrm e}^{x +1}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2*exp(x+1)+25*x^6-x^2+1)/x^2,x,method=_RETURNVERBOSE)

[Out]

5*x^5-x-1/x-exp(x+1)

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maxima [A]  time = 0.36, size = 20, normalized size = 0.91 \begin {gather*} 5 \, x^{5} - x - \frac {1}{x} - e^{\left (x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2*exp(x+1)+25*x^6-x^2+1)/x^2,x, algorithm="maxima")

[Out]

5*x^5 - x - 1/x - e^(x + 1)

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mupad [B]  time = 4.12, size = 20, normalized size = 0.91 \begin {gather*} 5\,x^5-{\mathrm {e}}^{x+1}-\frac {1}{x}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2*exp(x + 1) + x^2 - 25*x^6 - 1)/x^2,x)

[Out]

5*x^5 - exp(x + 1) - 1/x - x

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sympy [A]  time = 0.10, size = 14, normalized size = 0.64 \begin {gather*} 5 x^{5} - x - e^{x + 1} - \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2*exp(x+1)+25*x**6-x**2+1)/x**2,x)

[Out]

5*x**5 - x - exp(x + 1) - 1/x

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