\(\int \frac {2 x-8 x^3+6 x^5+(8 x-16 x^3-20 x^4+28 x^6) \log (5)+(8 x-40 x^4+32 x^7) \log ^2(5)}{5 \log ^2(5)} \, dx\) [7594]

Optimal result

Integrand size = 63, antiderivative size = 26 \[ \int \frac {2 x-8 x^3+6 x^5+\left (8 x-16 x^3-20 x^4+28 x^6\right ) \log (5)+\left (8 x-40 x^4+32 x^7\right ) \log ^2(5)}{5 \log ^2(5)} \, dx=\frac {1}{5} x^2 \left (-2+2 x^3+\frac {-1+x^2}{\log (5)}\right )^2 \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(99\) vs. \(2(26)=52\).

Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.81, number of steps used = 4, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {12} \[ \int \frac {2 x-8 x^3+6 x^5+\left (8 x-16 x^3-20 x^4+28 x^6\right ) \log (5)+\left (8 x-40 x^4+32 x^7\right ) \log ^2(5)}{5 \log ^2(5)} \, dx=\frac {4 x^8}{5}+\frac {4 x^7}{5 \log (5)}+\frac {x^6}{5 \log ^2(5)}-\frac {8 x^5}{5}-\frac {4 x^5}{5 \log (5)}-\frac {2 x^4}{5 \log ^2(5)}-\frac {4 x^4}{5 \log (5)}+\frac {4 x^2}{5}+\frac {x^2}{5 \log ^2(5)}+\frac {4 x^2}{5 \log (5)} \]

[In]

Int[(2*x - 8*x^3 + 6*x^5 + (8*x - 16*x^3 - 20*x^4 + 28*x^6)*Log[5] + (8*x - 40*x^4 + 32*x^7)*Log[5]^2)/(5*Log[
5]^2),x]

[Out]

(4*x^2)/5 - (8*x^5)/5 + (4*x^8)/5 + x^2/(5*Log[5]^2) - (2*x^4)/(5*Log[5]^2) + x^6/(5*Log[5]^2) + (4*x^2)/(5*Lo
g[5]) - (4*x^4)/(5*Log[5]) - (4*x^5)/(5*Log[5]) + (4*x^7)/(5*Log[5])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (2 x-8 x^3+6 x^5+\left (8 x-16 x^3-20 x^4+28 x^6\right ) \log (5)+\left (8 x-40 x^4+32 x^7\right ) \log ^2(5)\right ) \, dx}{5 \log ^2(5)} \\ & = \frac {x^2}{5 \log ^2(5)}-\frac {2 x^4}{5 \log ^2(5)}+\frac {x^6}{5 \log ^2(5)}+\frac {1}{5} \int \left (8 x-40 x^4+32 x^7\right ) \, dx+\frac {\int \left (8 x-16 x^3-20 x^4+28 x^6\right ) \, dx}{5 \log (5)} \\ & = \frac {4 x^2}{5}-\frac {8 x^5}{5}+\frac {4 x^8}{5}+\frac {x^2}{5 \log ^2(5)}-\frac {2 x^4}{5 \log ^2(5)}+\frac {x^6}{5 \log ^2(5)}+\frac {4 x^2}{5 \log (5)}-\frac {4 x^4}{5 \log (5)}-\frac {4 x^5}{5 \log (5)}+\frac {4 x^7}{5 \log (5)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {2 x-8 x^3+6 x^5+\left (8 x-16 x^3-20 x^4+28 x^6\right ) \log (5)+\left (8 x-40 x^4+32 x^7\right ) \log ^2(5)}{5 \log ^2(5)} \, dx=\frac {x^2 \left (-1+x^2-\log (25)+x^3 \log (25)\right )^2}{5 \log ^2(5)} \]

[In]

Integrate[(2*x - 8*x^3 + 6*x^5 + (8*x - 16*x^3 - 20*x^4 + 28*x^6)*Log[5] + (8*x - 40*x^4 + 32*x^7)*Log[5]^2)/(
5*Log[5]^2),x]

[Out]

(x^2*(-1 + x^2 - Log[25] + x^3*Log[25])^2)/(5*Log[5]^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(24)=48\).

Time = 0.13 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.42

method	result	size
default	\(\frac {4 x^{8} \ln \left (5\right )^{2}+4 x^{7} \ln \left (5\right )+x^{6}+4 \left (-2 \ln \left (5\right )-1\right ) \ln \left (5\right ) x^{5}+\frac {\left (-8 \ln \left (5\right )-4\right ) x^{4}}{2}+\left (-2 \ln \left (5\right )-1\right )^{2} x^{2}}{5 \ln \left (5\right )^{2}}\)	\(63\)
norman	\(\frac {\left (-\frac {8 \ln \left (5\right )}{5}-\frac {4}{5}\right ) x^{5}+\frac {4 x^{7}}{5}+\frac {4 x^{8} \ln \left (5\right )}{5}+\frac {x^{6}}{5 \ln \left (5\right )}-\frac {2 \left (2 \ln \left (5\right )+1\right ) x^{4}}{5 \ln \left (5\right )}+\frac {\left (4 \ln \left (5\right )^{2}+4 \ln \left (5\right )+1\right ) x^{2}}{5 \ln \left (5\right )}}{\ln \left (5\right )}\)	\(74\)
parallelrisch	\(\frac {4 x^{8} \ln \left (5\right )^{2}+4 x^{7} \ln \left (5\right )-8 x^{5} \ln \left (5\right )^{2}-4 x^{5} \ln \left (5\right )+x^{6}-4 x^{4} \ln \left (5\right )+4 x^{2} \ln \left (5\right )^{2}-2 x^{4}+4 x^{2} \ln \left (5\right )+x^{2}}{5 \ln \left (5\right )^{2}}\)	\(74\)
risch	\(\frac {4 x^{8}}{5}+\frac {4 x^{7}}{5 \ln \left (5\right )}+\frac {x^{6}}{5 \ln \left (5\right )^{2}}-\frac {8 x^{5}}{5}-\frac {4 x^{5}}{5 \ln \left (5\right )}-\frac {4 x^{4}}{5 \ln \left (5\right )}-\frac {2 x^{4}}{5 \ln \left (5\right )^{2}}+\frac {4 x^{2}}{5}+\frac {4 x^{2}}{5 \ln \left (5\right )}+\frac {x^{2}}{5 \ln \left (5\right )^{2}}\)	\(80\)
gosper	\(\frac {\left (-1+x \right ) \left (4 x^{5} \ln \left (5\right )^{2}+4 x^{4} \ln \left (5\right )^{2}+4 x^{3} \ln \left (5\right )^{2}+4 x^{4} \ln \left (5\right )-4 x^{2} \ln \left (5\right )^{2}+4 x^{3} \ln \left (5\right )-4 x \ln \left (5\right )^{2}+x^{3}-4 \ln \left (5\right )^{2}-4 x \ln \left (5\right )+x^{2}-4 \ln \left (5\right )-x -1\right ) x^{2}}{5 \ln \left (5\right )^{2}}\)	\(96\)

[In]

int(1/5*((32*x^7-40*x^4+8*x)*ln(5)^2+(28*x^6-20*x^4-16*x^3+8*x)*ln(5)+6*x^5-8*x^3+2*x)/ln(5)^2,x,method=_RETUR
NVERBOSE)

[Out]

1/5/ln(5)^2*(4*x^8*ln(5)^2+4*x^7*ln(5)+x^6+4*(-2*ln(5)-1)*ln(5)*x^5+1/2*(-8*ln(5)-4)*x^4+(-2*ln(5)-1)^2*x^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (24) = 48\).

Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19 \[ \int \frac {2 x-8 x^3+6 x^5+\left (8 x-16 x^3-20 x^4+28 x^6\right ) \log (5)+\left (8 x-40 x^4+32 x^7\right ) \log ^2(5)}{5 \log ^2(5)} \, dx=\frac {x^{6} - 2 \, x^{4} + 4 \, {\left (x^{8} - 2 \, x^{5} + x^{2}\right )} \log \left (5\right )^{2} + x^{2} + 4 \, {\left (x^{7} - x^{5} - x^{4} + x^{2}\right )} \log \left (5\right )}{5 \, \log \left (5\right )^{2}} \]

[In]

integrate(1/5*((32*x^7-40*x^4+8*x)*log(5)^2+(28*x^6-20*x^4-16*x^3+8*x)*log(5)+6*x^5-8*x^3+2*x)/log(5)^2,x, alg
orithm="fricas")

[Out]

1/5*(x^6 - 2*x^4 + 4*(x^8 - 2*x^5 + x^2)*log(5)^2 + x^2 + 4*(x^7 - x^5 - x^4 + x^2)*log(5))/log(5)^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (20) = 40\).

Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.27 \[ \int \frac {2 x-8 x^3+6 x^5+\left (8 x-16 x^3-20 x^4+28 x^6\right ) \log (5)+\left (8 x-40 x^4+32 x^7\right ) \log ^2(5)}{5 \log ^2(5)} \, dx=\frac {4 x^{8}}{5} + \frac {4 x^{7}}{5 \log {\left (5 \right )}} + \frac {x^{6}}{5 \log {\left (5 \right )}^{2}} + \frac {x^{5} \left (- 8 \log {\left (5 \right )} - 4\right )}{5 \log {\left (5 \right )}} + \frac {x^{4} \left (- 4 \log {\left (5 \right )} - 2\right )}{5 \log {\left (5 \right )}^{2}} + \frac {x^{2} \cdot \left (1 + 4 \log {\left (5 \right )} + 4 \log {\left (5 \right )}^{2}\right )}{5 \log {\left (5 \right )}^{2}} \]

[In]

integrate(1/5*((32*x**7-40*x**4+8*x)*ln(5)**2+(28*x**6-20*x**4-16*x**3+8*x)*ln(5)+6*x**5-8*x**3+2*x)/ln(5)**2,
x)

[Out]

4*x**8/5 + 4*x**7/(5*log(5)) + x**6/(5*log(5)**2) + x**5*(-8*log(5) - 4)/(5*log(5)) + x**4*(-4*log(5) - 2)/(5*
log(5)**2) + x**2*(1 + 4*log(5) + 4*log(5)**2)/(5*log(5)**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (24) = 48\).

Time = 0.18 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19 \[ \int \frac {2 x-8 x^3+6 x^5+\left (8 x-16 x^3-20 x^4+28 x^6\right ) \log (5)+\left (8 x-40 x^4+32 x^7\right ) \log ^2(5)}{5 \log ^2(5)} \, dx=\frac {x^{6} - 2 \, x^{4} + 4 \, {\left (x^{8} - 2 \, x^{5} + x^{2}\right )} \log \left (5\right )^{2} + x^{2} + 4 \, {\left (x^{7} - x^{5} - x^{4} + x^{2}\right )} \log \left (5\right )}{5 \, \log \left (5\right )^{2}} \]

[In]

integrate(1/5*((32*x^7-40*x^4+8*x)*log(5)^2+(28*x^6-20*x^4-16*x^3+8*x)*log(5)+6*x^5-8*x^3+2*x)/log(5)^2,x, alg
orithm="maxima")

[Out]

1/5*(x^6 - 2*x^4 + 4*(x^8 - 2*x^5 + x^2)*log(5)^2 + x^2 + 4*(x^7 - x^5 - x^4 + x^2)*log(5))/log(5)^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (24) = 48\).

Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19 \[ \int \frac {2 x-8 x^3+6 x^5+\left (8 x-16 x^3-20 x^4+28 x^6\right ) \log (5)+\left (8 x-40 x^4+32 x^7\right ) \log ^2(5)}{5 \log ^2(5)} \, dx=\frac {x^{6} - 2 \, x^{4} + 4 \, {\left (x^{8} - 2 \, x^{5} + x^{2}\right )} \log \left (5\right )^{2} + x^{2} + 4 \, {\left (x^{7} - x^{5} - x^{4} + x^{2}\right )} \log \left (5\right )}{5 \, \log \left (5\right )^{2}} \]

[In]

integrate(1/5*((32*x^7-40*x^4+8*x)*log(5)^2+(28*x^6-20*x^4-16*x^3+8*x)*log(5)+6*x^5-8*x^3+2*x)/log(5)^2,x, alg
orithm="giac")

[Out]

1/5*(x^6 - 2*x^4 + 4*(x^8 - 2*x^5 + x^2)*log(5)^2 + x^2 + 4*(x^7 - x^5 - x^4 + x^2)*log(5))/log(5)^2

Mupad [B] (verification not implemented)

Time = 14.85 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.08 \[ \int \frac {2 x-8 x^3+6 x^5+\left (8 x-16 x^3-20 x^4+28 x^6\right ) \log (5)+\left (8 x-40 x^4+32 x^7\right ) \log ^2(5)}{5 \log ^2(5)} \, dx=\frac {4\,x^8}{5}+\frac {4\,x^7}{5\,\ln \left (5\right )}+\frac {x^6}{5\,{\ln \left (5\right )}^2}-\frac {\left (20\,\ln \left (5\right )+40\,{\ln \left (5\right )}^2\right )\,x^5}{25\,{\ln \left (5\right )}^2}-\frac {\left (16\,\ln \left (5\right )+8\right )\,x^4}{20\,{\ln \left (5\right )}^2}+\frac {\left (8\,\ln \left (5\right )+8\,{\ln \left (5\right )}^2+2\right )\,x^2}{10\,{\ln \left (5\right )}^2} \]

[In]

int(((2*x)/5 + (log(5)*(8*x - 16*x^3 - 20*x^4 + 28*x^6))/5 + (log(5)^2*(8*x - 40*x^4 + 32*x^7))/5 - (8*x^3)/5
+ (6*x^5)/5)/log(5)^2,x)

[Out]

x^6/(5*log(5)^2) + (4*x^7)/(5*log(5)) + (4*x^8)/5 - (x^4*(16*log(5) + 8))/(20*log(5)^2) + (x^2*(8*log(5) + 8*l
og(5)^2 + 2))/(10*log(5)^2) - (x^5*(20*log(5) + 40*log(5)^2))/(25*log(5)^2)