3.71.5 \(\int \frac {3 e^8+x^4+12 x^7}{x^4} \, dx\)

Optimal. Leaf size=15 \[ -\frac {e^8}{x^3}+x+3 x^4 \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {14} \begin {gather*} 3 x^4-\frac {e^8}{x^3}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3*E^8 + x^4 + 12*x^7)/x^4,x]

[Out]

-(E^8/x^3) + x + 3*x^4

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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {3 e^8}{x^4}+12 x^3\right ) \, dx\\ &=-\frac {e^8}{x^3}+x+3 x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -\frac {e^8}{x^3}+x+3 x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*E^8 + x^4 + 12*x^7)/x^4,x]

[Out]

-(E^8/x^3) + x + 3*x^4

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fricas [A]  time = 0.76, size = 17, normalized size = 1.13 \begin {gather*} \frac {3 \, x^{7} + x^{4} - e^{8}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(4)^2+12*x^7+x^4)/x^4,x, algorithm="fricas")

[Out]

(3*x^7 + x^4 - e^8)/x^3

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giac [A]  time = 0.18, size = 14, normalized size = 0.93 \begin {gather*} 3 \, x^{4} + x - \frac {e^{8}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(4)^2+12*x^7+x^4)/x^4,x, algorithm="giac")

[Out]

3*x^4 + x - e^8/x^3

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maple [A]  time = 0.03, size = 15, normalized size = 1.00

method	result	size
default	\(3 x^{4}+x -\frac {{\mathrm e}^{8}}{x^{3}}\)	\(15\)
risch	\(3 x^{4}+x -\frac {{\mathrm e}^{8}}{x^{3}}\)	\(15\)
norman	\(\frac {3 x^{7}+x^{4}-{\mathrm e}^{8}}{x^{3}}\)	\(20\)
gosper	\(-\frac {-3 x^{7}-x^{4}+{\mathrm e}^{8}}{x^{3}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*exp(4)^2+12*x^7+x^4)/x^4,x,method=_RETURNVERBOSE)

[Out]

3*x^4+x-exp(8)/x^3

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maxima [A]  time = 0.38, size = 14, normalized size = 0.93 \begin {gather*} 3 \, x^{4} + x - \frac {e^{8}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(4)^2+12*x^7+x^4)/x^4,x, algorithm="maxima")

[Out]

3*x^4 + x - e^8/x^3

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mupad [B]  time = 4.05, size = 17, normalized size = 1.13 \begin {gather*} \frac {3\,x^7+x^4-{\mathrm {e}}^8}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*exp(8) + x^4 + 12*x^7)/x^4,x)

[Out]

(x^4 - exp(8) + 3*x^7)/x^3

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sympy [A]  time = 0.09, size = 12, normalized size = 0.80 \begin {gather*} 3 x^{4} + x - \frac {e^{8}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(4)**2+12*x**7+x**4)/x**4,x)

[Out]

3*x**4 + x - exp(8)/x**3

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