3.36.12 \(\int \frac {6-2 x^3+(3 x^2-2 x^3) \log (2)+x^2 \log (7)}{x^2} \, dx\)

Optimal. Leaf size=29 \[ -1-3 x+(3-x) \left (x+\frac {-2+x^2 \log (2)}{x}-\log (7)\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 19, normalized size of antiderivative = 0.66, number of steps used = 2, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {14} \begin {gather*} -\left (x^2 (1+\log (2))\right )-\frac {6}{x}+x \log (56) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6 - 2*x^3 + (3*x^2 - 2*x^3)*Log[2] + x^2*Log[7])/x^2,x]

[Out]

-6/x - x^2*(1 + Log[2]) + x*Log[56]

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 {6}{x^2}-2 x (1+\log (2))+\log (7) \left (1+\frac {\log (8)}{\log (7)}\right )\right ) \, dx\\ &=-\frac {6}{x}-x^2 (1+\log (2))+x \log (56)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.72 \begin {gather*} -\frac {6}{x}-\frac {1}{2} x^2 (2+\log (4))+x \log (56) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 - 2*x^3 + (3*x^2 - 2*x^3)*Log[2] + x^2*Log[7])/x^2,x]

[Out]

-6/x - (x^2*(2 + Log[4]))/2 + x*Log[56]

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fricas [A]  time = 0.66, size = 29, normalized size = 1.00 \begin {gather*} -\frac {x^{3} - x^{2} \log \relax (7) + {\left (x^{3} - 3 \, x^{2}\right )} \log \relax (2) + 6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*log(7)+(-2*x^3+3*x^2)*log(2)-2*x^3+6)/x^2,x, algorithm="fricas")

[Out]

-(x^3 - x^2*log(7) + (x^3 - 3*x^2)*log(2) + 6)/x

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giac [A]  time = 0.21, size = 27, normalized size = 0.93 \begin {gather*} -x^{2} \log \relax (2) - x^{2} + x \log \relax (7) + 3 \, x \log \relax (2) - \frac {6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*log(7)+(-2*x^3+3*x^2)*log(2)-2*x^3+6)/x^2,x, algorithm="giac")

[Out]

-x^2*log(2) - x^2 + x*log(7) + 3*x*log(2) - 6/x

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maple [A]  time = 0.04, size = 28, normalized size = 0.97

method	result	size
default	\(-x^{2} \ln \relax (2)+3 x \ln \relax (2)+x \ln \relax (7)-x^{2}-\frac {6}{x}\)	\(28\)
norman	\(\frac {-6+\left (-1-\ln \relax (2)\right ) x^{3}+\left (3 \ln \relax (2)+\ln \relax (7)\right ) x^{2}}{x}\)	\(28\)
risch	\(-x^{2} \ln \relax (2)+3 x \ln \relax (2)+x \ln \relax (7)-x^{2}-\frac {6}{x}\)	\(28\)
gosper	\(-\frac {x^{3} \ln \relax (2)-3 x^{2} \ln \relax (2)-x^{2} \ln \relax (7)+x^{3}+6}{x}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*ln(7)+(-2*x^3+3*x^2)*ln(2)-2*x^3+6)/x^2,x,method=_RETURNVERBOSE)

[Out]

-x^2*ln(2)+3*x*ln(2)+x*ln(7)-x^2-6/x

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maxima [A]  time = 0.47, size = 24, normalized size = 0.83 \begin {gather*} -x^{2} {\left (\log \relax (2) + 1\right )} + x {\left (\log \relax (7) + 3 \, \log \relax (2)\right )} - \frac {6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*log(7)+(-2*x^3+3*x^2)*log(2)-2*x^3+6)/x^2,x, algorithm="maxima")

[Out]

-x^2*(log(2) + 1) + x*(log(7) + 3*log(2)) - 6/x

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mupad [B]  time = 0.06, size = 21, normalized size = 0.72 \begin {gather*} x\,\ln \left (56\right )-x^2\,\left (\frac {\ln \relax (4)}{2}+1\right )-\frac {6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2)*(3*x^2 - 2*x^3) + x^2*log(7) - 2*x^3 + 6)/x^2,x)

[Out]

x*log(56) - x^2*(log(4)/2 + 1) - 6/x

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sympy [A]  time = 0.09, size = 24, normalized size = 0.83 \begin {gather*} - x^{2} \left (\log {\relax (2 )} + 1\right ) - x \left (- 3 \log {\relax (2 )} - \log {\relax (7 )}\right ) - \frac {6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2*ln(7)+(-2*x**3+3*x**2)*ln(2)-2*x**3+6)/x**2,x)

[Out]

-x**2*(log(2) + 1) - x*(-3*log(2) - log(7)) - 6/x

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