3.7.92 \(\int \frac {(-14623232+5483712 x^2-1827904 x^3+171366 x^4) \log ^2(2)+x^2 \log (5)}{x^2} \, dx\)

Optimal. Leaf size=21 \[ x \left (\frac {57122 (4-x)^4 \log ^2(2)}{x^2}+\log (5)\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.86, number of steps used = 2, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {14} \begin {gather*} 57122 x^3 \log ^2(2)-913952 x^2 \log ^2(2)+x \left (5483712 \log ^2(2)+\log (5)\right )+\frac {14623232 \log ^2(2)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-14623232 + 5483712*x^2 - 1827904*x^3 + 171366*x^4)*Log[2]^2 + x^2*Log[5])/x^2,x]

[Out]

(14623232*Log[2]^2)/x - 913952*x^2*Log[2]^2 + 57122*x^3*Log[2]^2 + x*(5483712*Log[2]^2 + Log[5])

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 {14623232 \log ^2(2)}{x^2}-1827904 x \log ^2(2)+171366 x^2 \log ^2(2)+5483712 \log ^2(2) \left (1+\frac {\log (5)}{5483712 \log ^2(2)}\right )\right ) \, dx\\ &=\frac {14623232 \log ^2(2)}{x}-913952 x^2 \log ^2(2)+57122 x^3 \log ^2(2)+x \left (5483712 \log ^2(2)+\log (5)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 39, normalized size = 1.86 \begin {gather*} \frac {14623232 \log ^2(2)}{x}+5483712 x \log ^2(2)-913952 x^2 \log ^2(2)+57122 x^3 \log ^2(2)+x \log (5) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-14623232 + 5483712*x^2 - 1827904*x^3 + 171366*x^4)*Log[2]^2 + x^2*Log[5])/x^2,x]

[Out]

(14623232*Log[2]^2)/x + 5483712*x*Log[2]^2 - 913952*x^2*Log[2]^2 + 57122*x^3*Log[2]^2 + x*Log[5]

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fricas [A]  time = 0.53, size = 32, normalized size = 1.52 \begin {gather*} \frac {x^{2} \log \relax (5) + 57122 \, {\left (x^{4} - 16 \, x^{3} + 96 \, x^{2} + 256\right )} \log \relax (2)^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*log(5)+(171366*x^4-1827904*x^3+5483712*x^2-14623232)*log(2)^2)/x^2,x, algorithm="fricas")

[Out]

(x^2*log(5) + 57122*(x^4 - 16*x^3 + 96*x^2 + 256)*log(2)^2)/x

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giac [B]  time = 0.38, size = 39, normalized size = 1.86 \begin {gather*} 57122 \, x^{3} \log \relax (2)^{2} - 913952 \, x^{2} \log \relax (2)^{2} + 5483712 \, x \log \relax (2)^{2} + x \log \relax (5) + \frac {14623232 \, \log \relax (2)^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*log(5)+(171366*x^4-1827904*x^3+5483712*x^2-14623232)*log(2)^2)/x^2,x, algorithm="giac")

[Out]

57122*x^3*log(2)^2 - 913952*x^2*log(2)^2 + 5483712*x*log(2)^2 + x*log(5) + 14623232*log(2)^2/x

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maple [A]  time = 0.04, size = 40, normalized size = 1.90

method	result	size
default	\(57122 x^{3} \ln \relax (2)^{2}-913952 x^{2} \ln \relax (2)^{2}+5483712 x \ln \relax (2)^{2}+x \ln \relax (5)+\frac {14623232 \ln \relax (2)^{2}}{x}\)	\(40\)
risch	\(57122 x^{3} \ln \relax (2)^{2}-913952 x^{2} \ln \relax (2)^{2}+5483712 x \ln \relax (2)^{2}+x \ln \relax (5)+\frac {14623232 \ln \relax (2)^{2}}{x}\)	\(40\)
norman	\(\frac {57122 x^{4} \ln \relax (2)^{2}-913952 x^{3} \ln \relax (2)^{2}+\left (5483712 \ln \relax (2)^{2}+\ln \relax (5)\right ) x^{2}[phone] \ln \relax (2)^{2}}{x}\)	\(43\)
gosper	\(\frac {57122 x^{4} \ln \relax (2)^{2}-913952 x^{3} \ln \relax (2)^{2}+5483712 x^{2} \ln \relax (2)^{2}+x^{2} \ln \relax (5)[phone] \ln \relax (2)^{2}}{x}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*ln(5)+(171366*x^4-1827904*x^3+5483712*x^2-14623232)*ln(2)^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

57122*x^3*ln(2)^2-913952*x^2*ln(2)^2+5483712*x*ln(2)^2+x*ln(5)[phone]*ln(2)^2/x

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maxima [B]  time = 0.65, size = 39, normalized size = 1.86 \begin {gather*} 57122 \, x^{3} \log \relax (2)^{2} - 913952 \, x^{2} \log \relax (2)^{2} + {\left (5483712 \, \log \relax (2)^{2} + \log \relax (5)\right )} x + \frac {14623232 \, \log \relax (2)^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*log(5)+(171366*x^4-1827904*x^3+5483712*x^2-14623232)*log(2)^2)/x^2,x, algorithm="maxima")

[Out]

57122*x^3*log(2)^2 - 913952*x^2*log(2)^2 + (5483712*log(2)^2 + log(5))*x + 14623232*log(2)^2/x

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mupad [B]  time = 0.53, size = 39, normalized size = 1.86 \begin {gather*} \frac {14623232\,{\ln \relax (2)}^2}{x}-913952\,x^2\,{\ln \relax (2)}^2+57122\,x^3\,{\ln \relax (2)}^2+x\,\left (\ln \relax (5)+5483712\,{\ln \relax (2)}^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2)^2*(5483712*x^2 - 1827904*x^3 + 171366*x^4 - 14623232) + x^2*log(5))/x^2,x)

[Out]

(14623232*log(2)^2)/x - 913952*x^2*log(2)^2 + 57122*x^3*log(2)^2 + x*(log(5) + 5483712*log(2)^2)

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sympy [A]  time = 0.10, size = 39, normalized size = 1.86 \begin {gather*} 57122 x^{3} \log {\relax (2 )}^{2} - 913952 x^{2} \log {\relax (2 )}^{2} + x \left (\log {\relax (5 )} + 5483712 \log {\relax (2 )}^{2}\right ) + \frac {14623232 \log {\relax (2 )}^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2*ln(5)+(171366*x**4-1827904*x**3+5483712*x**2-14623232)*ln(2)**2)/x**2,x)

[Out]

57122*x**3*log(2)**2 - 913952*x**2*log(2)**2 + x*(log(5) + 5483712*log(2)**2) + 14623232*log(2)**2/x

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