3.96.77 \(\int \frac {-6 e^{25}-3 x^3+e^x x^3}{2 x^3} \, dx\)

Optimal. Leaf size=20 \[ \frac {1}{2} \left (1+e^x+\frac {3 e^{25}}{x^2}-3 x\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {12, 14, 2194} \begin {gather*} \frac {3 e^{25}}{2 x^2}-\frac {3 x}{2}+\frac {e^x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6*E^25 - 3*x^3 + E^x*x^3)/(2*x^3),x]

[Out]

E^x/2 + (3*E^25)/(2*x^2) - (3*x)/2

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 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} &=\frac {1}{2} \int \frac {-6 e^{25}-3 x^3+e^x x^3}{x^3} \, dx\\ &=\frac {1}{2} \int \left (e^x-\frac {3 \left (2 e^{25}+x^3\right )}{x^3}\right ) \, dx\\ &=\frac {\int e^x \, dx}{2}-\frac {3}{2} \int \frac {2 e^{25}+x^3}{x^3} \, dx\\ &=\frac {e^x}{2}-\frac {3}{2} \int \left (1+\frac {2 e^{25}}{x^3}\right ) \, dx\\ &=\frac {e^x}{2}+\frac {3 e^{25}}{2 x^2}-\frac {3 x}{2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 0.95 \begin {gather*} \frac {1}{2} \left (e^x+\frac {3 e^{25}}{x^2}-3 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6*E^25 - 3*x^3 + E^x*x^3)/(2*x^3),x]

[Out]

(E^x + (3*E^25)/x^2 - 3*x)/2

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fricas [A]  time = 0.54, size = 22, normalized size = 1.10 \begin {gather*} -\frac {3 \, x^{3} - x^{2} e^{x} - 3 \, e^{25}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(exp(x)*x^3-6*exp(25)-3*x^3)/x^3,x, algorithm="fricas")

[Out]

-1/2*(3*x^3 - x^2*e^x - 3*e^25)/x^2

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giac [A]  time = 0.13, size = 22, normalized size = 1.10 \begin {gather*} -\frac {3 \, x^{3} - x^{2} e^{x} - 3 \, e^{25}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(exp(x)*x^3-6*exp(25)-3*x^3)/x^3,x, algorithm="giac")

[Out]

-1/2*(3*x^3 - x^2*e^x - 3*e^25)/x^2

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maple [A]  time = 0.03, size = 16, normalized size = 0.80

method	result	size
default	\(-\frac {3 x}{2}+\frac {3 \,{\mathrm e}^{25}}{2 x^{2}}+\frac {{\mathrm e}^{x}}{2}\)	\(16\)
risch	\(-\frac {3 x}{2}+\frac {3 \,{\mathrm e}^{25}}{2 x^{2}}+\frac {{\mathrm e}^{x}}{2}\)	\(16\)
norman	\(\frac {-\frac {3 x^{3}}{2}+\frac {{\mathrm e}^{x} x^{2}}{2}+\frac {3 \,{\mathrm e}^{25}}{2}}{x^{2}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(exp(x)*x^3-6*exp(25)-3*x^3)/x^3,x,method=_RETURNVERBOSE)

[Out]

-3/2*x+3/2*exp(25)/x^2+1/2*exp(x)

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maxima [A]  time = 0.36, size = 15, normalized size = 0.75 \begin {gather*} -\frac {3}{2} \, x + \frac {3 \, e^{25}}{2 \, x^{2}} + \frac {1}{2} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(exp(x)*x^3-6*exp(25)-3*x^3)/x^3,x, algorithm="maxima")

[Out]

-3/2*x + 3/2*e^25/x^2 + 1/2*e^x

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mupad [B]  time = 0.13, size = 15, normalized size = 0.75 \begin {gather*} \frac {{\mathrm {e}}^x}{2}-\frac {3\,x}{2}+\frac {3\,{\mathrm {e}}^{25}}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*exp(25) - (x^3*exp(x))/2 + (3*x^3)/2)/x^3,x)

[Out]

exp(x)/2 - (3*x)/2 + (3*exp(25))/(2*x^2)

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sympy [A]  time = 0.11, size = 19, normalized size = 0.95 \begin {gather*} - \frac {3 x}{2} + \frac {e^{x}}{2} + \frac {3 e^{25}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(exp(x)*x**3-6*exp(25)-3*x**3)/x**3,x)

[Out]

-3*x/2 + exp(x)/2 + 3*exp(25)/(2*x**2)

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