3.65.42 \(\int \frac {12 x^4+24 x^5-4 e x^5+4 \log (2)}{3 x^2} \, dx\)

Optimal. Leaf size=26 \[ -1+\frac {1}{3} (4+x-(-5+e) x) \left (x^3-\frac {\log (2)}{x}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {6, 12, 14} \begin {gather*} \frac {1}{3} (6-e) x^4+\frac {4 x^3}{3}-\frac {\log (16)}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12*x^4 + 24*x^5 - 4*E*x^5 + 4*Log[2])/(3*x^2),x]

[Out]

(4*x^3)/3 + ((6 - E)*x^4)/3 - Log[16]/(3*x)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {12 x^4+(24-4 e) x^5+4 \log (2)}{3 x^2} \, dx\\ &=\frac {1}{3} \int \frac {12 x^4+(24-4 e) x^5+4 \log (2)}{x^2} \, dx\\ &=\frac {1}{3} \int \left (12 x^2-4 (-6+e) x^3+\frac {\log (16)}{x^2}\right ) \, dx\\ &=\frac {4 x^3}{3}+\frac {1}{3} (6-e) x^4-\frac {\log (16)}{3 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.04 \begin {gather*} \frac {4}{3} \left (x^3+\frac {1}{4} (6-e) x^4-\frac {\log (2)}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12*x^4 + 24*x^5 - 4*E*x^5 + 4*Log[2])/(3*x^2),x]

[Out]

(4*(x^3 + ((6 - E)*x^4)/4 - Log[2]/x))/3

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fricas [A]  time = 0.70, size = 26, normalized size = 1.00 \begin {gather*} -\frac {x^{5} e - 6 \, x^{5} - 4 \, x^{4} + 4 \, \log \relax (2)}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*log(2)-4*x^5*exp(1)+24*x^5+12*x^4)/x^2,x, algorithm="fricas")

[Out]

-1/3*(x^5*e - 6*x^5 - 4*x^4 + 4*log(2))/x

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giac [A]  time = 0.15, size = 25, normalized size = 0.96 \begin {gather*} -\frac {1}{3} \, x^{4} e + 2 \, x^{4} + \frac {4}{3} \, x^{3} - \frac {4 \, \log \relax (2)}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*log(2)-4*x^5*exp(1)+24*x^5+12*x^4)/x^2,x, algorithm="giac")

[Out]

-1/3*x^4*e + 2*x^4 + 4/3*x^3 - 4/3*log(2)/x

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maple [A]  time = 0.05, size = 25, normalized size = 0.96

method	result	size
norman	\(\frac {\left (-\frac {{\mathrm e}}{3}+2\right ) x^{5}+\frac {4 x^{4}}{3}-\frac {4 \ln \relax (2)}{3}}{x}\)	\(25\)
default	\(-\frac {x^{4} {\mathrm e}}{3}+2 x^{4}+\frac {4 x^{3}}{3}-\frac {4 \ln \relax (2)}{3 x}\)	\(26\)
risch	\(-\frac {x^{4} {\mathrm e}}{3}+2 x^{4}+\frac {4 x^{3}}{3}-\frac {4 \ln \relax (2)}{3 x}\)	\(26\)
gosper	\(-\frac {x^{5} {\mathrm e}-6 x^{5}-4 x^{4}+4 \ln \relax (2)}{3 x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(4*ln(2)-4*x^5*exp(1)+24*x^5+12*x^4)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.37, size = 22, normalized size = 0.85 \begin {gather*} -\frac {1}{3} \, x^{4} {\left (e - 6\right )} + \frac {4}{3} \, x^{3} - \frac {4 \, \log \relax (2)}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*log(2)-4*x^5*exp(1)+24*x^5+12*x^4)/x^2,x, algorithm="maxima")

[Out]

-1/3*x^4*(e - 6) + 4/3*x^3 - 4/3*log(2)/x

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mupad [B]  time = 0.05, size = 24, normalized size = 0.92 \begin {gather*} \frac {4\,x^3}{3}-\frac {4\,\ln \relax (2)}{3\,x}-x^4\,\left (\frac {\mathrm {e}}{3}-2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*log(2))/3 - (4*x^5*exp(1))/3 + 4*x^4 + 8*x^5)/x^2,x)

[Out]

(4*x^3)/3 - (4*log(2))/(3*x) - x^4*(exp(1)/3 - 2)

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sympy [A]  time = 0.09, size = 24, normalized size = 0.92 \begin {gather*} \frac {x^{4} \left (6 - e\right )}{3} + \frac {4 x^{3}}{3} - \frac {4 \log {\relax (2 )}}{3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*ln(2)-4*x**5*exp(1)+24*x**5+12*x**4)/x**2,x)

[Out]

x**4*(6 - E)/3 + 4*x**3/3 - 4*log(2)/(3*x)

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