3.17.4 $\int \frac{1024-448x+8x^2-208x^3+9x^4}{1024x-192x^2+8x^3-16x^4+x^5}dx$

Optimal. Leaf size=26 $$2+e^5+\log \left(x{(-4+\frac{x(4+x^2)}{16-x})}^4\right)$$

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Rubi [A]  time = 0.08, antiderivative size = 24, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 2, integrand size = 45, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.044, Rules used = {2074, 1587} $$\begin{array}{c}4\log (-x^3-8x+64)-4\log (16-x)+\log (x)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(1024 - 448*x + 8*x^2 - 208*x^3 + 9*x^4)/(1024*x - 192*x^2 + 8*x^3 - 16*x^4 + x^5),x]

[Out]

-4*Log[16 - x] + Log[x] + 4*Log[64 - 8*x - x^3]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, 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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (-\frac{4}{-16+x}+\frac{1}{x}+\frac{4(8+3x^2)}{-64+8x+x^3})dx\\ & =-4\log (16-x)+\log (x)+4\int \frac{8+3x^2}{-64+8x+x^3}dx\\ & =-4\log (16-x)+\log (x)+4\log (64-8x-x^3)\end{array}\end{array}$$

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.92 $$\begin{array}{c}-4\log (16-x)+\log (x)+4\log (64-8x-x^3)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(1024 - 448*x + 8*x^2 - 208*x^3 + 9*x^4)/(1024*x - 192*x^2 + 8*x^3 - 16*x^4 + x^5),x]

[Out]

-4*Log[16 - x] + Log[x] + 4*Log[64 - 8*x - x^3]

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fricas [A]  time = 0.81, size = 20, normalized size = 0.77 $$\begin{array}{c}4\log (x^3+8x-64)-4\log (x-16)+\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^4-208*x^3+8*x^2-448*x+1024)/(x^5-16*x^4+8*x^3-192*x^2+1024*x),x, algorithm="fricas")

[Out]

4*log(x^3 + 8*x - 64) - 4*log(x - 16) + log(x)

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giac [A]  time = 0.17, size = 23, normalized size = 0.88 $$\begin{array}{c}4\log \left(|x^3+8x-64|\right)-4\log \left(|x-16|\right)+\log \left(\left|x\right|\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^4-208*x^3+8*x^2-448*x+1024)/(x^5-16*x^4+8*x^3-192*x^2+1024*x),x, algorithm="giac")

[Out]

4*log(abs(x^3 + 8*x - 64)) - 4*log(abs(x - 16)) + log(abs(x))

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maple [A]  time = 0.03, size = 21, normalized size = 0.81

method	result	size
default	$4\ln (x^3+8x-64)+\ln (x)-4\ln (x-16)$	$21$
norman	$4\ln (x^3+8x-64)+\ln (x)-4\ln (x-16)$	$21$
risch	$4\ln (x^3+8x-64)+\ln (x)-4\ln (x-16)$	$21$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x^4-208*x^3+8*x^2-448*x+1024)/(x^5-16*x^4+8*x^3-192*x^2+1024*x),x,method=_RETURNVERBOSE)

[Out]

4*ln(x^3+8*x-64)+ln(x)-4*ln(x-16)

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maxima [A]  time = 0.75, size = 20, normalized size = 0.77 $$\begin{array}{c}4\log (x^3+8x-64)-4\log (x-16)+\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^4-208*x^3+8*x^2-448*x+1024)/(x^5-16*x^4+8*x^3-192*x^2+1024*x),x, algorithm="maxima")

[Out]

4*log(x^3 + 8*x - 64) - 4*log(x - 16) + log(x)

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mupad [B]  time = 1.07, size = 20, normalized size = 0.77 $$\begin{array}{c}4\ln (x^3+8x-64)-4\ln (x-16)+\ln (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^2 - 448*x - 208*x^3 + 9*x^4 + 1024)/(1024*x - 192*x^2 + 8*x^3 - 16*x^4 + x^5),x)

[Out]

4*log(8*x + x^3 - 64) - 4*log(x - 16) + log(x)

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sympy [A]  time = 0.13, size = 20, normalized size = 0.77 $$\begin{array}{c}\log (x)-4\log (x-16)+4\log (x^3+8x-64)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x**4-208*x**3+8*x**2-448*x+1024)/(x**5-16*x**4+8*x**3-192*x**2+1024*x),x)

[Out]

log(x) - 4*log(x - 16) + 4*log(x**3 + 8*x - 64)

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