3.56.71 \(\int \frac {5-10 x^2+(10 x^2+10 x^3) \log (3)}{x^2} \, dx\)

Optimal. Leaf size=20 \[ 5 \left (4-\frac {1}{x}-2 x+(1+x)^2 \log (3)\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.10, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {14} \begin {gather*} 5 x^2 \log (3)-\frac {5}{x}-10 x (1-\log (3)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 - 10*x^2 + (10*x^2 + 10*x^3)*Log[3])/x^2,x]

[Out]

-5/x - 10*x*(1 - Log[3]) + 5*x^2*Log[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 (\frac {5}{x^2}+10 (-1+\log (3))+10 x \log (3)\right ) \, dx\\ &=-\frac {5}{x}-10 x (1-\log (3))+5 x^2 \log (3)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 24, normalized size = 1.20 \begin {gather*} 5 \left (-\frac {1}{x}-2 x+x \log (9)+\frac {1}{2} x^2 \log (9)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - 10*x^2 + (10*x^2 + 10*x^3)*Log[3])/x^2,x]

[Out]

5*(-x^(-1) - 2*x + x*Log[9] + (x^2*Log[9])/2)

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 25, normalized size = 1.25 \begin {gather*} -\frac {5 \, {\left (2 \, x^{2} - {\left (x^{3} + 2 \, x^{2}\right )} \log \relax (3) + 1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^3+10*x^2)*log(3)-10*x^2+5)/x^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.39, size = 21, normalized size = 1.05 \begin {gather*} 5 \, x^{2} \log \relax (3) + 10 \, x \log \relax (3) - 10 \, x - \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^3+10*x^2)*log(3)-10*x^2+5)/x^2,x, algorithm="giac")

[Out]

5*x^2*log(3) + 10*x*log(3) - 10*x - 5/x

________________________________________________________________________________________

maple [A]  time = 0.05, size = 22, normalized size = 1.10

method	result	size
default	\(5 x^{2} \ln \relax (3)+10 x \ln \relax (3)-10 x -\frac {5}{x}\)	\(22\)
risch	\(5 x^{2} \ln \relax (3)+10 x \ln \relax (3)-10 x -\frac {5}{x}\)	\(22\)
norman	\(\frac {-5+\left (10 \ln \relax (3)-10\right ) x^{2}+5 x^{3} \ln \relax (3)}{x}\)	\(24\)
gosper	\(\frac {5 x^{3} \ln \relax (3)+10 x^{2} \ln \relax (3)-10 x^{2}-5}{x}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x^3+10*x^2)*ln(3)-10*x^2+5)/x^2,x,method=_RETURNVERBOSE)

[Out]

5*x^2*ln(3)+10*x*ln(3)-10*x-5/x

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 20, normalized size = 1.00 \begin {gather*} 5 \, x^{2} \log \relax (3) + 10 \, x {\left (\log \relax (3) - 1\right )} - \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^3+10*x^2)*log(3)-10*x^2+5)/x^2,x, algorithm="maxima")

[Out]

5*x^2*log(3) + 10*x*(log(3) - 1) - 5/x

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 21, normalized size = 1.05 \begin {gather*} x\,\left (10\,\ln \relax (3)-10\right )+5\,x^2\,\ln \relax (3)-\frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3)*(10*x^2 + 10*x^3) - 10*x^2 + 5)/x^2,x)

[Out]

x*(10*log(3) - 10) + 5*x^2*log(3) - 5/x

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 19, normalized size = 0.95 \begin {gather*} 5 x^{2} \log {\relax (3 )} + x \left (-10 + 10 \log {\relax (3 )}\right ) - \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x**3+10*x**2)*ln(3)-10*x**2+5)/x**2,x)

[Out]

5*x**2*log(3) + x*(-10 + 10*log(3)) - 5/x

________________________________________________________________________________________