3.99.84 \(\int \frac {-20+4 e^{4 x} x^2+8 x^3}{x^2} \, dx\)

Optimal. Leaf size=16 \[ e^{4 x}+4 x \left (\frac {5}{x^2}+x\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {14, 2194} \begin {gather*} 4 x^2+e^{4 x}+\frac {20}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-20 + 4*E^(4*x)*x^2 + 8*x^3)/x^2,x]

[Out]

E^(4*x) + 20/x + 4*x^2

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 (4 e^{4 x}+\frac {4 \left (-5+2 x^3\right )}{x^2}\right ) \, dx\\ &=4 \int e^{4 x} \, dx+4 \int \frac {-5+2 x^3}{x^2} \, dx\\ &=e^{4 x}+4 \int \left (-\frac {5}{x^2}+2 x\right ) \, dx\\ &=e^{4 x}+\frac {20}{x}+4 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.25 \begin {gather*} 4 \left (\frac {e^{4 x}}{4}+\frac {5}{x}+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20 + 4*E^(4*x)*x^2 + 8*x^3)/x^2,x]

[Out]

4*(E^(4*x)/4 + 5/x + x^2)

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fricas [A]  time = 0.78, size = 17, normalized size = 1.06 \begin {gather*} \frac {4 \, x^{3} + x e^{\left (4 \, x\right )} + 20}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*exp(x)^4+8*x^3-20)/x^2,x, algorithm="fricas")

[Out]

(4*x^3 + x*e^(4*x) + 20)/x

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giac [A]  time = 0.12, size = 17, normalized size = 1.06 \begin {gather*} \frac {4 \, x^{3} + x e^{\left (4 \, x\right )} + 20}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*exp(x)^4+8*x^3-20)/x^2,x, algorithm="giac")

[Out]

(4*x^3 + x*e^(4*x) + 20)/x

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maple [A]  time = 0.02, size = 16, normalized size = 1.00

method	result	size
default	\({\mathrm e}^{4 x}+4 x^{2}+\frac {20}{x}\)	\(16\)
risch	\({\mathrm e}^{4 x}+4 x^{2}+\frac {20}{x}\)	\(16\)
norman	\(\frac {20+x \,{\mathrm e}^{4 x}+4 x^{3}}{x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2*exp(x)^4+8*x^3-20)/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(x)^4+4*x^2+20/x

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maxima [A]  time = 0.35, size = 15, normalized size = 0.94 \begin {gather*} 4 \, x^{2} + \frac {20}{x} + e^{\left (4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*exp(x)^4+8*x^3-20)/x^2,x, algorithm="maxima")

[Out]

4*x^2 + 20/x + e^(4*x)

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mupad [B]  time = 5.57, size = 15, normalized size = 0.94 \begin {gather*} {\mathrm {e}}^{4\,x}+\frac {20}{x}+4\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2*exp(4*x) + 8*x^3 - 20)/x^2,x)

[Out]

exp(4*x) + 20/x + 4*x^2

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sympy [A]  time = 0.10, size = 12, normalized size = 0.75 \begin {gather*} 4 x^{2} + e^{4 x} + \frac {20}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**2*exp(x)**4+8*x**3-20)/x**2,x)

[Out]

4*x**2 + exp(4*x) + 20/x

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