3.98.94 \(\int \frac {2304+5632 x+512 x^2+3 x^4+22 x^5+6 x^6+e^9 (-768-x^4)}{x^4} \, dx\)

Optimal. Leaf size=26 \[ \left (\frac {256}{x^3}-x\right ) \left (-3+e^9-x-x (10+2 x)\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.62, number of steps used = 2, number of rules used = 1, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {14} \begin {gather*} 2 x^3-\frac {256 \left (3-e^9\right )}{x^3}+11 x^2-\frac {2816}{x^2}+\left (3-e^9\right ) x-\frac {512}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2304 + 5632*x + 512*x^2 + 3*x^4 + 22*x^5 + 6*x^6 + E^9*(-768 - x^4))/x^4,x]

[Out]

(-256*(3 - E^9))/x^3 - 2816/x^2 - 512/x + (3 - E^9)*x + 11*x^2 + 2*x^3

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 (3 \left (1-\frac {e^9}{3}\right )-\frac {768 \left (-3+e^9\right )}{x^4}+\frac {5632}{x^3}+\frac {512}{x^2}+22 x+6 x^2\right ) \, dx\\ &=-\frac {256 \left (3-e^9\right )}{x^3}-\frac {2816}{x^2}-\frac {512}{x}+\left (3-e^9\right ) x+11 x^2+2 x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 40, normalized size = 1.54 \begin {gather*} \frac {256 \left (-3+e^9\right )}{x^3}-\frac {2816}{x^2}-\frac {512}{x}+\left (3-e^9\right ) x+11 x^2+2 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2304 + 5632*x + 512*x^2 + 3*x^4 + 22*x^5 + 6*x^6 + E^9*(-768 - x^4))/x^4,x]

[Out]

(256*(-3 + E^9))/x^3 - 2816/x^2 - 512/x + (3 - E^9)*x + 11*x^2 + 2*x^3

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fricas [A]  time = 0.77, size = 38, normalized size = 1.46 \begin {gather*} \frac {2 \, x^{6} + 11 \, x^{5} + 3 \, x^{4} - 512 \, x^{2} - {\left (x^{4} - 256\right )} e^{9} - 2816 \, x - 768}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4-768)*exp(9)+6*x^6+22*x^5+3*x^4+512*x^2+5632*x+2304)/x^4,x, algorithm="fricas")

[Out]

(2*x^6 + 11*x^5 + 3*x^4 - 512*x^2 - (x^4 - 256)*e^9 - 2816*x - 768)/x^3

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giac [A]  time = 0.12, size = 38, normalized size = 1.46 \begin {gather*} 2 \, x^{3} + 11 \, x^{2} - x e^{9} + 3 \, x - \frac {256 \, {\left (2 \, x^{2} + 11 \, x - e^{9} + 3\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4-768)*exp(9)+6*x^6+22*x^5+3*x^4+512*x^2+5632*x+2304)/x^4,x, algorithm="giac")

[Out]

2*x^3 + 11*x^2 - x*e^9 + 3*x - 256*(2*x^2 + 11*x - e^9 + 3)/x^3

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maple [A]  time = 0.04, size = 38, normalized size = 1.46

method	result	size
risch	\(2 x^{3}-x \,{\mathrm e}^{9}+11 x^{2}+3 x +\frac {-512 x^{2}+256 \,{\mathrm e}^{9}-2816 x -768}{x^{3}}\)	\(38\)
norman	\(\frac {\left (-{\mathrm e}^{9}+3\right ) x^{4}-2816 x -512 x^{2}+11 x^{5}+2 x^{6}+256 \,{\mathrm e}^{9}-768}{x^{3}}\)	\(39\)
gosper	\(-\frac {-2 x^{6}+x^{4} {\mathrm e}^{9}-11 x^{5}-3 x^{4}+512 x^{2}-256 \,{\mathrm e}^{9}+2816 x +768}{x^{3}}\)	\(41\)
default	\(2 x^{3}+11 x^{2}-x \,{\mathrm e}^{9}+3 x -\frac {2816}{x^{2}}-\frac {512}{x}-\frac {-768 \,{\mathrm e}^{9}+2304}{3 x^{3}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^4-768)*exp(9)+6*x^6+22*x^5+3*x^4+512*x^2+5632*x+2304)/x^4,x,method=_RETURNVERBOSE)

[Out]

2*x^3-x*exp(9)+11*x^2+3*x+(-512*x^2+256*exp(9)-2816*x-768)/x^3

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maxima [A]  time = 0.35, size = 37, normalized size = 1.42 \begin {gather*} 2 \, x^{3} + 11 \, x^{2} - x {\left (e^{9} - 3\right )} - \frac {256 \, {\left (2 \, x^{2} + 11 \, x - e^{9} + 3\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4-768)*exp(9)+6*x^6+22*x^5+3*x^4+512*x^2+5632*x+2304)/x^4,x, algorithm="maxima")

[Out]

2*x^3 + 11*x^2 - x*(e^9 - 3) - 256*(2*x^2 + 11*x - e^9 + 3)/x^3

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mupad [B]  time = 5.57, size = 37, normalized size = 1.42 \begin {gather*} 11\,x^2-x\,\left ({\mathrm {e}}^9-3\right )-\frac {512\,x^2+2816\,x-256\,{\mathrm {e}}^9+768}{x^3}+2\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5632*x + 512*x^2 + 3*x^4 + 22*x^5 + 6*x^6 - exp(9)*(x^4 + 768) + 2304)/x^4,x)

[Out]

11*x^2 - x*(exp(9) - 3) - (2816*x - 256*exp(9) + 512*x^2 + 768)/x^3 + 2*x^3

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sympy [A]  time = 0.24, size = 34, normalized size = 1.31 \begin {gather*} 2 x^{3} + 11 x^{2} + x \left (3 - e^{9}\right ) + \frac {- 512 x^{2} - 2816 x - 768 + 256 e^{9}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**4-768)*exp(9)+6*x**6+22*x**5+3*x**4+512*x**2+5632*x+2304)/x**4,x)

[Out]

2*x**3 + 11*x**2 + x*(3 - exp(9)) + (-512*x**2 - 2816*x - 768 + 256*exp(9))/x**3

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