3.69.81 $\int \frac{11264+83968x+107520x^2+50688x^3+8192x^4+(2304+14592x+12864x^2+3072x^3)\log (5)+(144+840x+384x^2){\log }^2(5)+(3+16x){\log }^3(5)}{32768+49152x+24576x^2+4096x^3+(6144+6144x+1536x^2)\log (5)+(384+192x){\log }^2(5)+8{\log }^3(5)}dx$

Optimal. Leaf size=27 $$x^2+\frac{3}{8}(x+\frac{4}{3{(4+2x+\frac{\log (5)}{4})}^2})$$

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Rubi [A]  time = 0.19, antiderivative size = 20, normalized size of antiderivative = 0.74, number of steps used = 2, number of rules used = 1, integrand size = 110, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.009, Rules used = {2074} $$\begin{array}{c}{x}^2+\frac{3x}{8}+\frac{8}{(8x+16+\log (5){)}^2}\end{array}$$

Antiderivative was successfully verified.

[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 2074

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (\frac{3}{8}+2x-\frac{128}{(16+8x+\log (5){)}^3})dx\\ & =\frac{3x}{8}+x^2+\frac{8}{(16+8x+\log (5){)}^2}\end{array}\end{array}$$

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Mathematica [B]  time = 0.06, size = 95, normalized size = 3.52 $$\begin{array}{c}\frac{4096x^4+9{\log }^4(5)+576x^2(16+\log (5))+{\log }^3(5)(419-5\log (25))-3840\log (5)(-7+\log (25))-8x(16+\log (5){)}^2(23+\log (25))+512x^3(35+\log (25))-48{\log }^2(5)(-131+5\log (25))-512(103+40\log (25))}{64(16+8x+\log (5){)}^2}\end{array}$$

Antiderivative was successfully verified.

[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)

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fricas [B]  time = 0.57, size = 77, normalized size = 2.85 $$\begin{array}{c}\frac{512x^4+2240x^3+(8x^2+3x)\log (5{)}^2+2816x^2+16(8x^3+19x^2+6x)\log (5)+768x+64}{8(64x^2+16(x+2)\log (5)+\log (5{)}^2+256x+256)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.13, size = 18, normalized size = 0.67 $$\begin{array}{c}{x}^2+\frac{3}{8}x+\frac{8}{{(8x+\log (5)+16)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.07, size = 19, normalized size = 0.70

method	result	size
default	$x^2+\frac{3x}{8}+\frac{8}{{(\ln (5)+8x+16)}^2}$	$19$
risch	$x^2+\frac{3x}{8}+\frac{8}{\ln (5{)}^2+16x\ln (5)+64x^2+32\ln (5)+256x+256}$	$35$
norman	$\frac{(16\ln (5)+280)x^3+(-\frac{\ln (5{)}^3}{4}-\frac{105\ln (5{)}^2}{8}-1312-228\ln (5))x+64x^4-\frac{\ln (5{)}^4}{64}-\frac{35\ln (5{)}^3}{32}-\frac{57\ln (5{)}^2}{2}-328\ln (5)-1400}{{(\ln (5)+8x+16)}^2}$	$70$
gosper	$-\frac{\ln (5{)}^4+16\ln (5{)}^{3}x-1024x^3\ln (5)-4096x^4+70\ln (5{)}^3+840x\ln (5{)}^2-17920x^3+1824\ln (5{)}^2+14592x\ln (5)+20992\ln (5)+83968x+89600}{64(\ln (5{)}^2+16x\ln (5)+64x^2+32\ln (5)+256x+256)}$	$89$

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [A]  time = 0.42, size = 33, normalized size = 1.22 $$\begin{array}{c}{x}^2+\frac{3}{8}x+\frac{8}{64x^2+16x(\log (5)+16)+\log (5{)}^2+32\log (5)+256}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 4.15, size = 34, normalized size = 1.26 $$\begin{array}{c}\frac{3x}{8}+\frac{8}{64x^2+(16\ln (5)+256)x+32\ln (5)+{\ln (5)}^2+256}+x^2\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.39, size = 34, normalized size = 1.26 $$\begin{array}{c}{x}^2+\frac{3x}{8}+\frac{8}{64x^2+x(16\log (5)+256)+\log {(5)}^2+32\log (5)+256}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[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)

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