\(\int \frac {(-14623232+5483712 x^2-1827904 x^3+171366 x^4) \log ^2(2)+x^2 \log (5)}{x^2} \, dx\) [692]

Optimal result

Integrand size = 33, antiderivative size = 21 \[ \int \frac {\left (-14623232+5483712 x^2-1827904 x^3+171366 x^4\right ) \log ^2(2)+x^2 \log (5)}{x^2} \, dx=x \left (\frac {57122 (4-x)^4 \log ^2(2)}{x^2}+\log (5)\right ) \]

[Out]

(ln(5)+338*ln(2)^2*(-x+4)^2*(-13*x+52)^2/x^2)*x

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {14} \[ \int \frac {\left (-14623232+5483712 x^2-1827904 x^3+171366 x^4\right ) \log ^2(2)+x^2 \log (5)}{x^2} \, dx=57122 x^3 \log ^2(2)-913952 x^2 \log ^2(2)+x \left (5483712 \log ^2(2)+\log (5)\right )+\frac {14623232 \log ^2(2)}{x} \]

[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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86 \[ \int \frac {\left (-14623232+5483712 x^2-1827904 x^3+171366 x^4\right ) \log ^2(2)+x^2 \log (5)}{x^2} \, dx=\frac {14623232 \log ^2(2)}{x}+5483712 x \log ^2(2)-913952 x^2 \log ^2(2)+57122 x^3 \log ^2(2)+x \log (5) \]

[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]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \frac {\left (-14623232+5483712 x^2-1827904 x^3+171366 x^4\right ) \log ^2(2)+x^2 \log (5)}{x^2} \, dx=\frac {x^{2} \log \left (5\right ) + 57122 \, {\left (x^{4} - 16 \, x^{3} + 96 \, x^{2} + 256\right )} \log \left (2\right )^{2}}{x} \]

[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

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86 \[ \int \frac {\left (-14623232+5483712 x^2-1827904 x^3+171366 x^4\right ) \log ^2(2)+x^2 \log (5)}{x^2} \, dx=57122 x^{3} \log {\left (2 \right )}^{2} - 913952 x^{2} \log {\left (2 \right )}^{2} + x \left (\log {\left (5 \right )} + 5483712 \log {\left (2 \right )}^{2}\right ) + \frac {14623232 \log {\left (2 \right )}^{2}}{x} \]

[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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).

Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86 \[ \int \frac {\left (-14623232+5483712 x^2-1827904 x^3+171366 x^4\right ) \log ^2(2)+x^2 \log (5)}{x^2} \, dx=57122 \, x^{3} \log \left (2\right )^{2} - 913952 \, x^{2} \log \left (2\right )^{2} + {\left (5483712 \, \log \left (2\right )^{2} + \log \left (5\right )\right )} x + \frac {14623232 \, \log \left (2\right )^{2}}{x} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).

Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86 \[ \int \frac {\left (-14623232+5483712 x^2-1827904 x^3+171366 x^4\right ) \log ^2(2)+x^2 \log (5)}{x^2} \, dx=57122 \, x^{3} \log \left (2\right )^{2} - 913952 \, x^{2} \log \left (2\right )^{2} + 5483712 \, x \log \left (2\right )^{2} + x \log \left (5\right ) + \frac {14623232 \, \log \left (2\right )^{2}}{x} \]

[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

Mupad [B] (verification not implemented)

Time = 7.86 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86 \[ \int \frac {\left (-14623232+5483712 x^2-1827904 x^3+171366 x^4\right ) \log ^2(2)+x^2 \log (5)}{x^2} \, dx=\frac {14623232\,{\ln \left (2\right )}^2}{x}-913952\,x^2\,{\ln \left (2\right )}^2+57122\,x^3\,{\ln \left (2\right )}^2+x\,\left (\ln \left (5\right )+5483712\,{\ln \left (2\right )}^2\right ) \]

[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)