3.16.3 \(\int \frac {-30 e^{2/5}+3 x+12 x^3+3 e^x x^3}{x^3} \, dx\)

Optimal. Leaf size=27 \[ 3 \left (4+e^x-\left (4-\frac {1}{x^2}\right ) \left (5 e^{2/5}-x\right )\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 24, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {14, 2194} \begin {gather*} \frac {15 e^{2/5}}{x^2}+12 x+3 e^x-\frac {3}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-30*E^(2/5) + 3*x + 12*x^3 + 3*E^x*x^3)/x^3,x]

[Out]

3*E^x + (15*E^(2/5))/x^2 - 3/x + 12*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} &=\int \left (3 e^x-\frac {3 \left (10 e^{2/5}-x-4 x^3\right )}{x^3}\right ) \, dx\\ &=3 \int e^x \, dx-3 \int \frac {10 e^{2/5}-x-4 x^3}{x^3} \, dx\\ &=3 e^x-3 \int \left (-4+\frac {10 e^{2/5}}{x^3}-\frac {1}{x^2}\right ) \, dx\\ &=3 e^x+\frac {15 e^{2/5}}{x^2}-\frac {3}{x}+12 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.89 \begin {gather*} 3 \left (e^x+\frac {5 e^{2/5}}{x^2}-\frac {1}{x}+4 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-30*E^(2/5) + 3*x + 12*x^3 + 3*E^x*x^3)/x^3,x]

[Out]

3*(E^x + (5*E^(2/5))/x^2 - x^(-1) + 4*x)

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fricas [A]  time = 0.94, size = 24, normalized size = 0.89 \begin {gather*} \frac {3 \, {\left (4 \, x^{3} + x^{2} e^{x} - x + 5 \, e^{\frac {2}{5}}\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(x)*x^3-30*exp(2/5)+12*x^3+3*x)/x^3,x, algorithm="fricas")

[Out]

3*(4*x^3 + x^2*e^x - x + 5*e^(2/5))/x^2

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giac [A]  time = 0.25, size = 24, normalized size = 0.89 \begin {gather*} \frac {3 \, {\left (4 \, x^{3} + x^{2} e^{x} - x + 5 \, e^{\frac {2}{5}}\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(x)*x^3-30*exp(2/5)+12*x^3+3*x)/x^3,x, algorithm="giac")

[Out]

3*(4*x^3 + x^2*e^x - x + 5*e^(2/5))/x^2

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maple [A]  time = 0.03, size = 21, normalized size = 0.78

method	result	size
default	\(12 x -\frac {3}{x}+\frac {15 \,{\mathrm e}^{\frac {2}{5}}}{x^{2}}+3 \,{\mathrm e}^{x}\)	\(21\)
risch	\(12 x +\frac {15 \,{\mathrm e}^{\frac {2}{5}}-3 x}{x^{2}}+3 \,{\mathrm e}^{x}\)	\(21\)
norman	\(\frac {-3 x +12 x^{3}+3 \,{\mathrm e}^{x} x^{2}+15 \,{\mathrm e}^{\frac {2}{5}}}{x^{2}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*exp(x)*x^3-30*exp(2/5)+12*x^3+3*x)/x^3,x,method=_RETURNVERBOSE)

[Out]

12*x-3/x+15*exp(2/5)/x^2+3*exp(x)

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maxima [A]  time = 0.51, size = 20, normalized size = 0.74 \begin {gather*} 12 \, x - \frac {3}{x} + \frac {15 \, e^{\frac {2}{5}}}{x^{2}} + 3 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(x)*x^3-30*exp(2/5)+12*x^3+3*x)/x^3,x, algorithm="maxima")

[Out]

12*x - 3/x + 15*e^(2/5)/x^2 + 3*e^x

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mupad [B]  time = 0.07, size = 21, normalized size = 0.78 \begin {gather*} 12\,x+3\,{\mathrm {e}}^x-\frac {3\,x-15\,{\mathrm {e}}^{2/5}}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x - 30*exp(2/5) + 3*x^3*exp(x) + 12*x^3)/x^3,x)

[Out]

12*x + 3*exp(x) - (3*x - 15*exp(2/5))/x^2

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sympy [A]  time = 0.13, size = 20, normalized size = 0.74 \begin {gather*} 12 x + 3 e^{x} + \frac {- 3 x + 15 e^{\frac {2}{5}}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(x)*x**3-30*exp(2/5)+12*x**3+3*x)/x**3,x)

[Out]

12*x + 3*exp(x) + (-3*x + 15*exp(2/5))/x**2

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