3.33.36 $\int \frac{-4050+(-180-90x){\log }^2(2)+(48+88x+60x^2+18x^3+2x^4){\log }^4(2)}{(8+12x+6x^2+x^3){\log }^4(2)}dx$

Optimal. Leaf size=18 $$-3+{(3+x+\frac{45}{(2+x){\log }^2(2)})}^2$$

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 29, normalized size of antiderivative = 1.61, number of steps used = 3, number of rules used = 2, integrand size = 57, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.035, Rules used = {12, 2074} $$\begin{array}{c}{x}^2+6x+\frac{2025}{(x+2{)}^2{\log }^4(2)}+\frac{90}{(x+2){\log }^2(2)}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-4050 + (-180 - 90*x)*Log[2]^2 + (48 + 88*x + 60*x^2 + 18*x^3 + 2*x^4)*Log[2]^4)/((8 + 12*x + 6*x^2 + x^3
)*Log[2]^4),x]

[Out]

6*x + x^2 + 2025/((2 + x)^2*Log[2]^4) + 90/((2 + x)*Log[2]^2)

Rule 12

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

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{\int \frac{-4050+(-180-90x){\log }^2(2)+(48+88x+60x^2+18x^3+2x^4){\log }^4(2)}{8+12x+6x^2+x^3}dx}{{\log }^4(2)}\\ & =\frac{\int (-\frac{4050}{(2+x{)}^3}-\frac{90{\log }^2(2)}{(2+x{)}^2}+6{\log }^4(2)+2x{\log }^4(2))dx}{{\log }^4(2)}\\ & =6x+x^2+\frac{2025}{(2+x{)}^2{\log }^4(2)}+\frac{90}{(2+x){\log }^2(2)}\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 29, normalized size = 1.61 $$\begin{array}{c}6x+x^2+\frac{2025}{(2+x{)}^2{\log }^4(2)}+\frac{90}{(2+x){\log }^2(2)}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-4050 + (-180 - 90*x)*Log[2]^2 + (48 + 88*x + 60*x^2 + 18*x^3 + 2*x^4)*Log[2]^4)/((8 + 12*x + 6*x^2
+ x^3)*Log[2]^4),x]

[Out]

6*x + x^2 + 2025/((2 + x)^2*Log[2]^4) + 90/((2 + x)*Log[2]^2)

________________________________________________________________________________________

fricas [B]  time = 0.58, size = 48, normalized size = 2.67 $$\begin{array}{c}\frac{(x^4+10x^3+28x^2+24x)\log (2{)}^4+90(x+2)\log (2{)}^2+2025}{(x^2+4x+4)\log (2{)}^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+18*x^3+60*x^2+88*x+48)*log(2)^4+(-90*x-180)*log(2)^2-4050)/(x^3+6*x^2+12*x+8)/log(2)^4,x, al
gorithm="fricas")

[Out]

((x^4 + 10*x^3 + 28*x^2 + 24*x)*log(2)^4 + 90*(x + 2)*log(2)^2 + 2025)/((x^2 + 4*x + 4)*log(2)^4)

________________________________________________________________________________________

giac [B]  time = 0.22, size = 43, normalized size = 2.39 $$\begin{array}{c}\frac{x^2\log (2{)}^4+6x\log (2{)}^4+\frac{45(2x\log (2{)}^2+4\log (2{)}^2+45)}{{(x+2)}^2}}{\log (2{)}^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+18*x^3+60*x^2+88*x+48)*log(2)^4+(-90*x-180)*log(2)^2-4050)/(x^3+6*x^2+12*x+8)/log(2)^4,x, al
gorithm="giac")

[Out]

(x^2*log(2)^4 + 6*x*log(2)^4 + 45*(2*x*log(2)^2 + 4*log(2)^2 + 45)/(x + 2)^2)/log(2)^4

________________________________________________________________________________________

maple [B]  time = 0.07, size = 38, normalized size = 2.11

method	result	size
risch	$x^2+6x+\frac{90x\ln (2{)}^2+180\ln (2{)}^2+2025}{\ln (2{)}^4(x^2+4x+4)}$	$38$
default	$\frac{x^2\ln (2{)}^4+6x\ln (2{)}^4+\frac{90\ln (2{)}^2}{2+x}+\frac{2025}{{(2+x)}^2}}{\ln (2{)}^4}$	$40$
gosper	$\frac{\ln (2{)}^4{x}^4+10x^3\ln (2{)}^4-88x\ln (2{)}^4-112\ln (2{)}^4+90x\ln (2{)}^2+180\ln (2{)}^2+2025}{\ln (2{)}^4(x^2+4x+4)}$	$61$
norman	$\frac{x^4\ln (2{)}^3-2\ln (2)(44\ln (2{)}^2-45)x+10x^3\ln (2{)}^3-\frac{112\ln (2{)}^4-180\ln (2{)}^2-2025}{\ln (2)}}{{(2+x)}^2\ln (2{)}^3}$	$62$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^4+18*x^3+60*x^2+88*x+48)*ln(2)^4+(-90*x-180)*ln(2)^2-4050)/(x^3+6*x^2+12*x+8)/ln(2)^4,x,method=_RETU
RNVERBOSE)

[Out]

x^2+6*x+1/ln(2)^4*(90*x*ln(2)^2+180*ln(2)^2+2025)/(x^2+4*x+4)

________________________________________________________________________________________

maxima [B]  time = 0.36, size = 48, normalized size = 2.67 $$\begin{array}{c}\frac{x^2\log (2{)}^4+6x\log (2{)}^4+\frac{45(2x\log (2{)}^2+4\log (2{)}^2+45)}{x^2+4x+4}}{\log (2{)}^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+18*x^3+60*x^2+88*x+48)*log(2)^4+(-90*x-180)*log(2)^2-4050)/(x^3+6*x^2+12*x+8)/log(2)^4,x, al
gorithm="maxima")

[Out]

(x^2*log(2)^4 + 6*x*log(2)^4 + 45*(2*x*log(2)^2 + 4*log(2)^2 + 45)/(x^2 + 4*x + 4))/log(2)^4

________________________________________________________________________________________

mupad [B]  time = 0.12, size = 58, normalized size = 3.22 $$\begin{array}{c}\frac{28x^2{\ln (2)}^4+10x^3{\ln (2)}^4+x^4{\ln (2)}^4+90x{\ln (2)}^2+24x{\ln (2)}^4+180{\ln (2)}^2+2025}{{\ln (2)}^4{(x+2)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(2)^2*(90*x + 180) - log(2)^4*(88*x + 60*x^2 + 18*x^3 + 2*x^4 + 48) + 4050)/(log(2)^4*(12*x + 6*x^2 +
x^3 + 8)),x)

[Out]

(28*x^2*log(2)^4 + 10*x^3*log(2)^4 + x^4*log(2)^4 + 90*x*log(2)^2 + 24*x*log(2)^4 + 180*log(2)^2 + 2025)/(log(
2)^4*(x + 2)^2)

________________________________________________________________________________________

sympy [B]  time = 0.30, size = 46, normalized size = 2.56 $$\begin{array}{c}{x}^2+6x+\frac{90x\log {(2)}^2+180\log {(2)}^2+2025}{x^2\log {(2)}^4+4x\log {(2)}^4+4\log {(2)}^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**4+18*x**3+60*x**2+88*x+48)*ln(2)**4+(-90*x-180)*ln(2)**2-4050)/(x**3+6*x**2+12*x+8)/ln(2)**4,
x)

[Out]

x**2 + 6*x + (90*x*log(2)**2 + 180*log(2)**2 + 2025)/(x**2*log(2)**4 + 4*x*log(2)**4 + 4*log(2)**4)

________________________________________________________________________________________