3.63.51 \(\int \frac {-26 x^2+e^x (-3+3 x)}{2 x^2} \, dx\)

Optimal. Leaf size=14 \[ \frac {3 e^x}{2 x}-13 x \]

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Rubi [A]  time = 0.03, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {12, 14, 2197} \begin {gather*} \frac {3 e^x}{2 x}-13 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-26*x^2 + E^x*(-3 + 3*x))/(2*x^2),x]

[Out]

(3*E^x)/(2*x) - 13*x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {-26 x^2+e^x (-3+3 x)}{x^2} \, dx\\ &=\frac {1}{2} \int \left (-26+\frac {3 e^x (-1+x)}{x^2}\right ) \, dx\\ &=-13 x+\frac {3}{2} \int \frac {e^x (-1+x)}{x^2} \, dx\\ &=\frac {3 e^x}{2 x}-13 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {3 e^x}{2 x}-13 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-26*x^2 + E^x*(-3 + 3*x))/(2*x^2),x]

[Out]

(3*E^x)/(2*x) - 13*x

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fricas [A]  time = 0.57, size = 15, normalized size = 1.07 \begin {gather*} -\frac {26 \, x^{2} - 3 \, e^{x}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((3*x-3)*exp(x)-26*x^2)/x^2,x, algorithm="fricas")

[Out]

-1/2*(26*x^2 - 3*e^x)/x

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giac [A]  time = 0.31, size = 15, normalized size = 1.07 \begin {gather*} -\frac {26 \, x^{2} - 3 \, e^{x}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((3*x-3)*exp(x)-26*x^2)/x^2,x, algorithm="giac")

[Out]

-1/2*(26*x^2 - 3*e^x)/x

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maple [A]  time = 0.07, size = 12, normalized size = 0.86

method	result	size
default	\(-13 x +\frac {3 \,{\mathrm e}^{x}}{2 x}\)	\(12\)
risch	\(-13 x +\frac {3 \,{\mathrm e}^{x}}{2 x}\)	\(12\)
norman	\(\frac {-13 x^{2}+\frac {3 \,{\mathrm e}^{x}}{2}}{x}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((3*x-3)*exp(x)-26*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

-13*x+3/2*exp(x)/x

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maxima [C]  time = 0.52, size = 15, normalized size = 1.07 \begin {gather*} -13 \, x + \frac {3}{2} \, {\rm Ei}\relax (x) - \frac {3}{2} \, \Gamma \left (-1, -x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((3*x-3)*exp(x)-26*x^2)/x^2,x, algorithm="maxima")

[Out]

-13*x + 3/2*Ei(x) - 3/2*gamma(-1, -x)

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mupad [B]  time = 4.06, size = 11, normalized size = 0.79 \begin {gather*} \frac {3\,{\mathrm {e}}^x}{2\,x}-13\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x)*(3*x - 3))/2 - 13*x^2)/x^2,x)

[Out]

(3*exp(x))/(2*x) - 13*x

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sympy [A]  time = 0.08, size = 10, normalized size = 0.71 \begin {gather*} - 13 x + \frac {3 e^{x}}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((3*x-3)*exp(x)-26*x**2)/x**2,x)

[Out]

-13*x + 3*exp(x)/(2*x)

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