3.19.81 \(\int \frac {-1528-320 x+(152+32 x) \log (3)}{6859+4332 x+912 x^2+64 x^3} \, dx\) [1881]

Optimal. Leaf size=20 \[ \left (10+\frac {1}{4-x+5 (3+x)}-\log (3)\right )^2 \]

[Out]

(1/(4*x+19)-ln(3)+10)^2

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Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.15, number of steps used = 2, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {2099} \begin {gather*} \frac {1}{(4 x+19)^2}+\frac {2 (10-\log (3))}{4 x+19} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1528 - 320*x + (152 + 32*x)*Log[3])/(6859 + 4332*x + 912*x^2 + 64*x^3),x]

[Out]

(19 + 4*x)^(-2) + (2*(10 - Log[3]))/(19 + 4*x)

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 {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {8}{(19+4 x)^3}+\frac {8 (-10+\log (3))}{(19+4 x)^2}\right ) \, dx\\ &=\frac {1}{(19+4 x)^2}+\frac {2 (10-\log (3))}{19+4 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 21, normalized size = 1.05 \begin {gather*} \frac {381-8 x (-10+\log (3))-38 \log (3)}{(19+4 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1528 - 320*x + (152 + 32*x)*Log[3])/(6859 + 4332*x + 912*x^2 + 64*x^3),x]

[Out]

(381 - 8*x*(-10 + Log[3]) - 38*Log[3])/(19 + 4*x)^2

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Maple [A]
time = 0.07, size = 24, normalized size = 1.20

method	result	size
norman	\(\frac {\left (80-8 \ln \left (3\right )\right ) x +381-38 \ln \left (3\right )}{\left (4 x +19\right )^{2}}\)	\(23\)
default	\(-\frac {8 \left (\frac {\ln \left (3\right )}{4}-\frac {5}{2}\right )}{4 x +19}+\frac {1}{\left (4 x +19\right )^{2}}\)	\(24\)
risch	\(\frac {\left (-\frac {\ln \left (3\right )}{2}+5\right ) x +\frac {381}{16}-\frac {19 \ln \left (3\right )}{8}}{x^{2}+\frac {19}{2} x +\frac {361}{16}}\)	\(26\)
gosper	\(-\frac {8 x \ln \left (3\right )+38 \ln \left (3\right )-80 x -381}{16 x^{2}+152 x +361}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((32*x+152)*ln(3)-320*x-1528)/(64*x^3+912*x^2+4332*x+6859),x,method=_RETURNVERBOSE)

[Out]

-8*(1/4*ln(3)-5/2)/(4*x+19)+1/(4*x+19)^2

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Maxima [A]
time = 0.27, size = 27, normalized size = 1.35 \begin {gather*} -\frac {8 \, x {\left (\log \left (3\right ) - 10\right )} + 38 \, \log \left (3\right ) - 381}{16 \, x^{2} + 152 \, x + 361} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x+152)*log(3)-320*x-1528)/(64*x^3+912*x^2+4332*x+6859),x, algorithm="maxima")

[Out]

-(8*x*(log(3) - 10) + 38*log(3) - 381)/(16*x^2 + 152*x + 361)

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Fricas [A]
time = 0.33, size = 28, normalized size = 1.40 \begin {gather*} -\frac {2 \, {\left (4 \, x + 19\right )} \log \left (3\right ) - 80 \, x - 381}{16 \, x^{2} + 152 \, x + 361} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x+152)*log(3)-320*x-1528)/(64*x^3+912*x^2+4332*x+6859),x, algorithm="fricas")

[Out]

-(2*(4*x + 19)*log(3) - 80*x - 381)/(16*x^2 + 152*x + 361)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).
time = 0.10, size = 26, normalized size = 1.30 \begin {gather*} - \frac {x \left (-80 + 8 \log {\left (3 \right )}\right ) - 381 + 38 \log {\left (3 \right )}}{16 x^{2} + 152 x + 361} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x+152)*ln(3)-320*x-1528)/(64*x**3+912*x**2+4332*x+6859),x)

[Out]

-(x*(-80 + 8*log(3)) - 381 + 38*log(3))/(16*x**2 + 152*x + 361)

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Giac [A]
time = 0.39, size = 23, normalized size = 1.15 \begin {gather*} -\frac {8 \, x \log \left (3\right ) - 80 \, x + 38 \, \log \left (3\right ) - 381}{{\left (4 \, x + 19\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x+152)*log(3)-320*x-1528)/(64*x^3+912*x^2+4332*x+6859),x, algorithm="giac")

[Out]

-(8*x*log(3) - 80*x + 38*log(3) - 381)/(4*x + 19)^2

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Mupad [B]
time = 1.15, size = 23, normalized size = 1.15 \begin {gather*} \frac {1}{{\left (4\,x+19\right )}^2}-\frac {2\,\ln \left (3\right )-20}{4\,x+19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(320*x - log(3)*(32*x + 152) + 1528)/(4332*x + 912*x^2 + 64*x^3 + 6859),x)

[Out]

1/(4*x + 19)^2 - (2*log(3) - 20)/(4*x + 19)

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