3.20.22 \(\int \frac {4 x^4+2 x^5+e^3 (-384 x-64 x^2)+(576+128 x) \log (2)}{x^4} \, dx\)

Optimal. Leaf size=24 \[ x+(3+x) \left (x+\frac {64 \left (e^3-\frac {\log (2)}{x}\right )}{x^2}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.54, number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {14} \begin {gather*} -\frac {192 \log (2)}{x^3}+x^2+\frac {64 \left (3 e^3-\log (2)\right )}{x^2}+4 x+\frac {64 e^3}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*x^4 + 2*x^5 + E^3*(-384*x - 64*x^2) + (576 + 128*x)*Log[2])/x^4,x]

[Out]

(64*E^3)/x + 4*x + x^2 + (64*(3*E^3 - Log[2]))/x^2 - (192*Log[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 (4-\frac {64 e^3}{x^2}+2 x-\frac {128 \left (3 e^3-\log (2)\right )}{x^3}+\frac {576 \log (2)}{x^4}\right ) \, dx\\ &=\frac {64 e^3}{x}+4 x+x^2+\frac {64 \left (3 e^3-\log (2)\right )}{x^2}-\frac {192 \log (2)}{x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.29 \begin {gather*} \frac {4 x^4+x^5+64 e^3 x (3+x)-192 \log (2)-64 x \log (2)}{x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x^4 + 2*x^5 + E^3*(-384*x - 64*x^2) + (576 + 128*x)*Log[2])/x^4,x]

[Out]

(4*x^4 + x^5 + 64*E^3*x*(3 + x) - 192*Log[2] - 64*x*Log[2])/x^3

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fricas [A]  time = 0.64, size = 31, normalized size = 1.29 \begin {gather*} \frac {x^{5} + 4 \, x^{4} + 64 \, {\left (x^{2} + 3 \, x\right )} e^{3} - 64 \, {\left (x + 3\right )} \log \relax (2)}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x+576)*log(2)+(-64*x^2-384*x)*exp(3)+2*x^5+4*x^4)/x^4,x, algorithm="fricas")

[Out]

(x^5 + 4*x^4 + 64*(x^2 + 3*x)*e^3 - 64*(x + 3)*log(2))/x^3

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giac [A]  time = 0.33, size = 33, normalized size = 1.38 \begin {gather*} x^{2} + 4 \, x + \frac {64 \, {\left (x^{2} e^{3} + 3 \, x e^{3} - x \log \relax (2) - 3 \, \log \relax (2)\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x+576)*log(2)+(-64*x^2-384*x)*exp(3)+2*x^5+4*x^4)/x^4,x, algorithm="giac")

[Out]

x^2 + 4*x + 64*(x^2*e^3 + 3*x*e^3 - x*log(2) - 3*log(2))/x^3

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maple [A]  time = 0.05, size = 35, normalized size = 1.46

method	result	size
gosper	\(\frac {x^{5}+4 x^{4}+64 x^{2} {\mathrm e}^{3}+192 x \,{\mathrm e}^{3}-64 x \ln \relax (2)-192 \ln \relax (2)}{x^{3}}\)	\(35\)
risch	\(x^{2}+4 x +\frac {64 x^{2} {\mathrm e}^{3}+\left (192 \,{\mathrm e}^{3}-64 \ln \relax (2)\right ) x -192 \ln \relax (2)}{x^{3}}\)	\(35\)
default	\(4 x +x^{2}+\frac {64 \,{\mathrm e}^{3}}{x}-\frac {-192 \,{\mathrm e}^{3}+64 \ln \relax (2)}{x^{2}}-\frac {192 \ln \relax (2)}{x^{3}}\)	\(36\)
norman	\(\frac {x^{5}+\left (192 \,{\mathrm e}^{3}-64 \ln \relax (2)\right ) x +4 x^{4}+64 x^{2} {\mathrm e}^{3}-192 \ln \relax (2)}{x^{3}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((128*x+576)*ln(2)+(-64*x^2-384*x)*exp(3)+2*x^5+4*x^4)/x^4,x,method=_RETURNVERBOSE)

[Out]

(x^5+4*x^4+64*x^2*exp(3)+192*x*exp(3)-64*x*ln(2)-192*ln(2))/x^3

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maxima [A]  time = 0.57, size = 34, normalized size = 1.42 \begin {gather*} x^{2} + 4 \, x + \frac {64 \, {\left (x^{2} e^{3} + x {\left (3 \, e^{3} - \log \relax (2)\right )} - 3 \, \log \relax (2)\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x+576)*log(2)+(-64*x^2-384*x)*exp(3)+2*x^5+4*x^4)/x^4,x, algorithm="maxima")

[Out]

x^2 + 4*x + 64*(x^2*e^3 + x*(3*e^3 - log(2)) - 3*log(2))/x^3

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mupad [B]  time = 0.07, size = 34, normalized size = 1.42 \begin {gather*} 4\,x+\frac {64\,{\mathrm {e}}^3\,x^2+\left (192\,{\mathrm {e}}^3-64\,\ln \relax (2)\right )\,x-192\,\ln \relax (2)}{x^3}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2)*(128*x + 576) - exp(3)*(384*x + 64*x^2) + 4*x^4 + 2*x^5)/x^4,x)

[Out]

4*x + (64*x^2*exp(3) - 192*log(2) + x*(192*exp(3) - 64*log(2)))/x^3 + x^2

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sympy [A]  time = 0.30, size = 34, normalized size = 1.42 \begin {gather*} x^{2} + 4 x + \frac {64 x^{2} e^{3} + x \left (- 64 \log {\relax (2 )} + 192 e^{3}\right ) - 192 \log {\relax (2 )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x+576)*ln(2)+(-64*x**2-384*x)*exp(3)+2*x**5+4*x**4)/x**4,x)

[Out]

x**2 + 4*x + (64*x**2*exp(3) + x*(-64*log(2) + 192*exp(3)) - 192*log(2))/x**3

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