3.6.62 \(\int \frac {5 x^3+(-50+2 x^4) \log (27)}{x^3 \log (27)} \, dx\)

Optimal. Leaf size=22 \[ 2+\left (-\frac {5}{x}+x\right )^2-4 \log (2)+\frac {5 x}{\log (27)} \]

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.18, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {12, 14} \begin {gather*} \frac {25}{x^2}+\frac {x^2 \log (729)}{2 \log (27)}+\frac {5 x}{\log (27)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5*x^3 + (-50 + 2*x^4)*Log[27])/(x^3*Log[27]),x]

[Out]

25/x^2 + (5*x)/Log[27] + (x^2*Log[729])/(2*Log[27])

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} &=\frac {\int \frac {5 x^3+\left (-50+2 x^4\right ) \log (27)}{x^3} \, dx}{\log (27)}\\ &=\frac {\int \left (5-\frac {50 \log (27)}{x^3}+x \log (729)\right ) \, dx}{\log (27)}\\ &=\frac {25}{x^2}+\frac {5 x}{\log (27)}+\frac {x^2 \log (729)}{2 \log (27)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.73 \begin {gather*} \frac {25}{x^2}+x^2+\frac {5 x}{\log (27)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5*x^3 + (-50 + 2*x^4)*Log[27])/(x^3*Log[27]),x]

[Out]

25/x^2 + x^2 + (5*x)/Log[27]

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fricas [A]  time = 0.77, size = 24, normalized size = 1.09 \begin {gather*} \frac {5 \, x^{3} + 3 \, {\left (x^{4} + 25\right )} \log \relax (3)}{3 \, x^{2} \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*(2*x^4-50)*log(3)+5*x^3)/x^3/log(3),x, algorithm="fricas")

[Out]

1/3*(5*x^3 + 3*(x^4 + 25)*log(3))/(x^2*log(3))

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giac [A]  time = 0.41, size = 24, normalized size = 1.09 \begin {gather*} \frac {3 \, x^{2} \log \relax (3) + 5 \, x + \frac {75 \, \log \relax (3)}{x^{2}}}{3 \, \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*(2*x^4-50)*log(3)+5*x^3)/x^3/log(3),x, algorithm="giac")

[Out]

1/3*(3*x^2*log(3) + 5*x + 75*log(3)/x^2)/log(3)

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maple [A]  time = 0.05, size = 17, normalized size = 0.77

method	result	size
risch	\(\frac {5 x}{3 \ln \relax (3)}+x^{2}+\frac {25}{x^{2}}\)	\(17\)
norman	\(\frac {25+x^{4}+\frac {5 x^{3}}{3 \ln \relax (3)}}{x^{2}}\)	\(19\)
default	\(\frac {5 x +3 x^{2} \ln \relax (3)+\frac {75 \ln \relax (3)}{x^{2}}}{3 \ln \relax (3)}\)	\(25\)
gosper	\(\frac {3 x^{4} \ln \relax (3)+5 x^{3}+75 \ln \relax (3)}{3 \ln \relax (3) x^{2}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(3*(2*x^4-50)*ln(3)+5*x^3)/x^3/ln(3),x,method=_RETURNVERBOSE)

[Out]

5/3*x/ln(3)+x^2+25/x^2

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maxima [A]  time = 0.50, size = 24, normalized size = 1.09 \begin {gather*} \frac {3 \, x^{2} \log \relax (3) + 5 \, x + \frac {75 \, \log \relax (3)}{x^{2}}}{3 \, \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*(2*x^4-50)*log(3)+5*x^3)/x^3/log(3),x, algorithm="maxima")

[Out]

1/3*(3*x^2*log(3) + 5*x + 75*log(3)/x^2)/log(3)

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mupad [B]  time = 0.45, size = 16, normalized size = 0.73 \begin {gather*} \frac {5\,x}{3\,\ln \relax (3)}+\frac {25}{x^2}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3)*(2*x^4 - 50) + (5*x^3)/3)/(x^3*log(3)),x)

[Out]

(5*x)/(3*log(3)) + 25/x^2 + x^2

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sympy [A]  time = 0.09, size = 24, normalized size = 1.09 \begin {gather*} \frac {3 x^{2} \log {\relax (3 )} + 5 x + \frac {75 \log {\relax (3 )}}{x^{2}}}{3 \log {\relax (3 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(3*(2*x**4-50)*ln(3)+5*x**3)/x**3/ln(3),x)

[Out]

(3*x**2*log(3) + 5*x + 75*log(3)/x**2)/(3*log(3))

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