\(\int \frac {11264+83968 x+107520 x^2+50688 x^3+8192 x^4+(2304+14592 x+12864 x^2+3072 x^3) \log (5)+(144+840 x+384 x^2) \log ^2(5)+(3+16 x) \log ^3(5)}{32768+49152 x+24576 x^2+4096 x^3+(6144+6144 x+1536 x^2) \log (5)+(384+192 x) \log ^2(5)+8 \log ^3(5)} \, dx\) [6881]

Optimal result

Integrand size = 110, antiderivative size = 27 \[ \int \frac {11264+83968 x+107520 x^2+50688 x^3+8192 x^4+\left (2304+14592 x+12864 x^2+3072 x^3\right ) \log (5)+\left (144+840 x+384 x^2\right ) \log ^2(5)+(3+16 x) \log ^3(5)}{32768+49152 x+24576 x^2+4096 x^3+\left (6144+6144 x+1536 x^2\right ) \log (5)+(384+192 x) \log ^2(5)+8 \log ^3(5)} \, dx=x^2+\frac {3}{8} \left (x+\frac {4}{3 \left (4+2 x+\frac {\log (5)}{4}\right )^2}\right ) \]

[Out]

1/2/(1/4*ln(5)+2*x+4)^2+3/8*x+x^2

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {2099} \[ \int \frac {11264+83968 x+107520 x^2+50688 x^3+8192 x^4+\left (2304+14592 x+12864 x^2+3072 x^3\right ) \log (5)+\left (144+840 x+384 x^2\right ) \log ^2(5)+(3+16 x) \log ^3(5)}{32768+49152 x+24576 x^2+4096 x^3+\left (6144+6144 x+1536 x^2\right ) \log (5)+(384+192 x) \log ^2(5)+8 \log ^3(5)} \, dx=x^2+\frac {3 x}{8}+\frac {8}{(8 x+16+\log (5))^2} \]

[In]

Int[(11264 + 83968*x + 107520*x^2 + 50688*x^3 + 8192*x^4 + (2304 + 14592*x + 12864*x^2 + 3072*x^3)*Log[5] + (1
44 + 840*x + 384*x^2)*Log[5]^2 + (3 + 16*x)*Log[5]^3)/(32768 + 49152*x + 24576*x^2 + 4096*x^3 + (6144 + 6144*x
+ 1536*x^2)*Log[5] + (384 + 192*x)*Log[5]^2 + 8*Log[5]^3),x]

[Out]

(3*x)/8 + x^2 + 8/(16 + 8*x + Log[5])^2

Rule 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{8}+2 x-\frac {128}{(16+8 x+\log (5))^3}\right ) \, dx \\ & = \frac {3 x}{8}+x^2+\frac {8}{(16+8 x+\log (5))^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(27)=54\).

Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.52 \[ \int \frac {11264+83968 x+107520 x^2+50688 x^3+8192 x^4+\left (2304+14592 x+12864 x^2+3072 x^3\right ) \log (5)+\left (144+840 x+384 x^2\right ) \log ^2(5)+(3+16 x) \log ^3(5)}{32768+49152 x+24576 x^2+4096 x^3+\left (6144+6144 x+1536 x^2\right ) \log (5)+(384+192 x) \log ^2(5)+8 \log ^3(5)} \, dx=\frac {4096 x^4+9 \log ^4(5)+576 x^2 (16+\log (5))+\log ^3(5) (419-5 \log (25))-3840 \log (5) (-7+\log (25))-8 x (16+\log (5))^2 (23+\log (25))+512 x^3 (35+\log (25))-48 \log ^2(5) (-131+5 \log (25))-512 (103+40 \log (25))}{64 (16+8 x+\log (5))^2} \]

[In]

Integrate[(11264 + 83968*x + 107520*x^2 + 50688*x^3 + 8192*x^4 + (2304 + 14592*x + 12864*x^2 + 3072*x^3)*Log[5
] + (144 + 840*x + 384*x^2)*Log[5]^2 + (3 + 16*x)*Log[5]^3)/(32768 + 49152*x + 24576*x^2 + 4096*x^3 + (6144 +
6144*x + 1536*x^2)*Log[5] + (384 + 192*x)*Log[5]^2 + 8*Log[5]^3),x]

[Out]

(4096*x^4 + 9*Log[5]^4 + 576*x^2*(16 + Log[5]) + Log[5]^3*(419 - 5*Log[25]) - 3840*Log[5]*(-7 + Log[25]) - 8*x
*(16 + Log[5])^2*(23 + Log[25]) + 512*x^3*(35 + Log[25]) - 48*Log[5]^2*(-131 + 5*Log[25]) - 512*(103 + 40*Log[
25]))/(64*(16 + 8*x + Log[5])^2)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70

method	result	size
default	\(x^{2}+\frac {3 x}{8}+\frac {8}{\left (\ln \left (5\right )+8 x +16\right )^{2}}\)	\(19\)
risch	\(x^{2}+\frac {3 x}{8}+\frac {8}{\ln \left (5\right )^{2}+16 x \ln \left (5\right )+64 x^{2}+32 \ln \left (5\right )+256 x +256}\)	\(35\)
norman	\(\frac {\left (16 \ln \left (5\right )+280\right ) x^{3}+\left (-\frac {\ln \left (5\right )^{3}}{4}-\frac {105 \ln \left (5\right )^{2}}{8}-1312-228 \ln \left (5\right )\right ) x +64 x^{4}-\frac {\ln \left (5\right )^{4}}{64}-\frac {35 \ln \left (5\right )^{3}}{32}-\frac {57 \ln \left (5\right )^{2}}{2}-328 \ln \left (5\right )-1400}{\left (\ln \left (5\right )+8 x +16\right )^{2}}\)	\(70\)
gosper	\(-\frac {\ln \left (5\right )^{4}+16 \ln \left (5\right )^{3} x -1024 x^{3} \ln \left (5\right )-4096 x^{4}+70 \ln \left (5\right )^{3}+840 x \ln \left (5\right )^{2}-17920 x^{3}+1824 \ln \left (5\right )^{2}+14592 x \ln \left (5\right )+20992 \ln \left (5\right )+83968 x +89600}{64 \left (\ln \left (5\right )^{2}+16 x \ln \left (5\right )+64 x^{2}+32 \ln \left (5\right )+256 x +256\right )}\)	\(89\)
parallelrisch	\(-\frac {8 \ln \left (5\right )^{4}+128 \ln \left (5\right )^{3} x -8192 x^{3} \ln \left (5\right )-32768 x^{4}+560 \ln \left (5\right )^{3}+6720 x \ln \left (5\right )^{2}[phone] x^{3}+14592 \ln \left (5\right )^{2}+116736 x \ln \left (5\right )+167936 \ln \left (5\right )+671744 x}{512 \left (\ln \left (5\right )^{2}+16 x \ln \left (5\right )+64 x^{2}+32 \ln \left (5\right )+256 x +256\right )}\)	\(91\)

[In]

int(((16*x+3)*ln(5)^3+(384*x^2+840*x+144)*ln(5)^2+(3072*x^3+12864*x^2+14592*x+2304)*ln(5)+8192*x^4+50688*x^3+1
07520*x^2+83968*x+11264)/(8*ln(5)^3+(192*x+384)*ln(5)^2+(1536*x^2+6144*x+6144)*ln(5)+4096*x^3+24576*x^2+49152*
x+32768),x,method=_RETURNVERBOSE)

[Out]

x^2+3/8*x+8/(ln(5)+8*x+16)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (18) = 36\).

Time = 0.24 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.85 \[ \int \frac {11264+83968 x+107520 x^2+50688 x^3+8192 x^4+\left (2304+14592 x+12864 x^2+3072 x^3\right ) \log (5)+\left (144+840 x+384 x^2\right ) \log ^2(5)+(3+16 x) \log ^3(5)}{32768+49152 x+24576 x^2+4096 x^3+\left (6144+6144 x+1536 x^2\right ) \log (5)+(384+192 x) \log ^2(5)+8 \log ^3(5)} \, dx=\frac {512 \, x^{4} + 2240 \, x^{3} + {\left (8 \, x^{2} + 3 \, x\right )} \log \left (5\right )^{2} + 2816 \, x^{2} + 16 \, {\left (8 \, x^{3} + 19 \, x^{2} + 6 \, x\right )} \log \left (5\right ) + 768 \, x + 64}{8 \, {\left (64 \, x^{2} + 16 \, {\left (x + 2\right )} \log \left (5\right ) + \log \left (5\right )^{2} + 256 \, x + 256\right )}} \]

[In]

integrate(((16*x+3)*log(5)^3+(384*x^2+840*x+144)*log(5)^2+(3072*x^3+12864*x^2+14592*x+2304)*log(5)+8192*x^4+50
688*x^3+107520*x^2+83968*x+11264)/(8*log(5)^3+(192*x+384)*log(5)^2+(1536*x^2+6144*x+6144)*log(5)+4096*x^3+2457
6*x^2+49152*x+32768),x, algorithm="fricas")

[Out]

1/8*(512*x^4 + 2240*x^3 + (8*x^2 + 3*x)*log(5)^2 + 2816*x^2 + 16*(8*x^3 + 19*x^2 + 6*x)*log(5) + 768*x + 64)/(
64*x^2 + 16*(x + 2)*log(5) + log(5)^2 + 256*x + 256)

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {11264+83968 x+107520 x^2+50688 x^3+8192 x^4+\left (2304+14592 x+12864 x^2+3072 x^3\right ) \log (5)+\left (144+840 x+384 x^2\right ) \log ^2(5)+(3+16 x) \log ^3(5)}{32768+49152 x+24576 x^2+4096 x^3+\left (6144+6144 x+1536 x^2\right ) \log (5)+(384+192 x) \log ^2(5)+8 \log ^3(5)} \, dx=x^{2} + \frac {3 x}{8} + \frac {8}{64 x^{2} + x \left (16 \log {\left (5 \right )} + 256\right ) + \log {\left (5 \right )}^{2} + 32 \log {\left (5 \right )} + 256} \]

[In]

integrate(((16*x+3)*ln(5)**3+(384*x**2+840*x+144)*ln(5)**2+(3072*x**3+12864*x**2+14592*x+2304)*ln(5)+8192*x**4
+50688*x**3+107520*x**2+83968*x+11264)/(8*ln(5)**3+(192*x+384)*ln(5)**2+(1536*x**2+6144*x+6144)*ln(5)+4096*x**
3+24576*x**2+49152*x+32768),x)

[Out]

x**2 + 3*x/8 + 8/(64*x**2 + x*(16*log(5) + 256) + log(5)**2 + 32*log(5) + 256)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {11264+83968 x+107520 x^2+50688 x^3+8192 x^4+\left (2304+14592 x+12864 x^2+3072 x^3\right ) \log (5)+\left (144+840 x+384 x^2\right ) \log ^2(5)+(3+16 x) \log ^3(5)}{32768+49152 x+24576 x^2+4096 x^3+\left (6144+6144 x+1536 x^2\right ) \log (5)+(384+192 x) \log ^2(5)+8 \log ^3(5)} \, dx=x^{2} + \frac {3}{8} \, x + \frac {8}{64 \, x^{2} + 16 \, x {\left (\log \left (5\right ) + 16\right )} + \log \left (5\right )^{2} + 32 \, \log \left (5\right ) + 256} \]

[In]

integrate(((16*x+3)*log(5)^3+(384*x^2+840*x+144)*log(5)^2+(3072*x^3+12864*x^2+14592*x+2304)*log(5)+8192*x^4+50
688*x^3+107520*x^2+83968*x+11264)/(8*log(5)^3+(192*x+384)*log(5)^2+(1536*x^2+6144*x+6144)*log(5)+4096*x^3+2457
6*x^2+49152*x+32768),x, algorithm="maxima")

[Out]

x^2 + 3/8*x + 8/(64*x^2 + 16*x*(log(5) + 16) + log(5)^2 + 32*log(5) + 256)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \frac {11264+83968 x+107520 x^2+50688 x^3+8192 x^4+\left (2304+14592 x+12864 x^2+3072 x^3\right ) \log (5)+\left (144+840 x+384 x^2\right ) \log ^2(5)+(3+16 x) \log ^3(5)}{32768+49152 x+24576 x^2+4096 x^3+\left (6144+6144 x+1536 x^2\right ) \log (5)+(384+192 x) \log ^2(5)+8 \log ^3(5)} \, dx=x^{2} + \frac {3}{8} \, x + \frac {8}{{\left (8 \, x + \log \left (5\right ) + 16\right )}^{2}} \]

[In]

integrate(((16*x+3)*log(5)^3+(384*x^2+840*x+144)*log(5)^2+(3072*x^3+12864*x^2+14592*x+2304)*log(5)+8192*x^4+50
688*x^3+107520*x^2+83968*x+11264)/(8*log(5)^3+(192*x+384)*log(5)^2+(1536*x^2+6144*x+6144)*log(5)+4096*x^3+2457
6*x^2+49152*x+32768),x, algorithm="giac")

[Out]

x^2 + 3/8*x + 8/(8*x + log(5) + 16)^2

Mupad [B] (verification not implemented)

Time = 11.82 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {11264+83968 x+107520 x^2+50688 x^3+8192 x^4+\left (2304+14592 x+12864 x^2+3072 x^3\right ) \log (5)+\left (144+840 x+384 x^2\right ) \log ^2(5)+(3+16 x) \log ^3(5)}{32768+49152 x+24576 x^2+4096 x^3+\left (6144+6144 x+1536 x^2\right ) \log (5)+(384+192 x) \log ^2(5)+8 \log ^3(5)} \, dx=\frac {3\,x}{8}+\frac {8}{64\,x^2+\left (16\,\ln \left (5\right )+256\right )\,x+32\,\ln \left (5\right )+{\ln \left (5\right )}^2+256}+x^2 \]

[In]

int((83968*x + log(5)^3*(16*x + 3) + log(5)*(14592*x + 12864*x^2 + 3072*x^3 + 2304) + log(5)^2*(840*x + 384*x^
2 + 144) + 107520*x^2 + 50688*x^3 + 8192*x^4 + 11264)/(49152*x + log(5)*(6144*x + 1536*x^2 + 6144) + log(5)^2*
(192*x + 384) + 8*log(5)^3 + 24576*x^2 + 4096*x^3 + 32768),x)

[Out]

(3*x)/8 + 8/(32*log(5) + x*(16*log(5) + 256) + log(5)^2 + 64*x^2 + 256) + x^2