3.28.21 \(\int \frac {(-3300+10836 x-10980 x^2+3660 x^3) \log (3)}{-25+75 x-75 x^2+25 x^3} \, dx\)

Optimal. Leaf size=28 \[ 12 \left (x+(5+x)^2-\left (\frac {3}{-5+\frac {5}{x}}+x\right )^2\right ) \log (3) \]

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Rubi [A]  time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.21, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {12, 2074} \begin {gather*} \frac {732}{5} x \log (3)-\frac {144 \log (3)}{25 (1-x)}-\frac {108 \log (3)}{25 (1-x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-3300 + 10836*x - 10980*x^2 + 3660*x^3)*Log[3])/(-25 + 75*x - 75*x^2 + 25*x^3),x]

[Out]

(-108*Log[3])/(25*(1 - x)^2) - (144*Log[3])/(25*(1 - x)) + (732*x*Log[3])/5

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 {gather*} \begin {aligned} \text {integral} &=\log (3) \int \frac {-3300+10836 x-10980 x^2+3660 x^3}{-25+75 x-75 x^2+25 x^3} \, dx\\ &=\log (3) \int \left (\frac {732}{5}+\frac {216}{25 (-1+x)^3}-\frac {144}{25 (-1+x)^2}\right ) \, dx\\ &=-\frac {108 \log (3)}{25 (1-x)^2}-\frac {144 \log (3)}{25 (1-x)}+\frac {732}{5} x \log (3)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.93 \begin {gather*} \frac {12}{25} \left (-\frac {9}{(-1+x)^2}+\frac {12}{-1+x}+305 (-1+x)\right ) \log (3) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-3300 + 10836*x - 10980*x^2 + 3660*x^3)*Log[3])/(-25 + 75*x - 75*x^2 + 25*x^3),x]

[Out]

(12*(-9/(-1 + x)^2 + 12/(-1 + x) + 305*(-1 + x))*Log[3])/25

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fricas [A]  time = 0.48, size = 29, normalized size = 1.04 \begin {gather*} \frac {12 \, {\left (305 \, x^{3} - 610 \, x^{2} + 317 \, x - 21\right )} \log \relax (3)}{25 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3660*x^3-10980*x^2+10836*x-3300)*log(3)/(25*x^3-75*x^2+75*x-25),x, algorithm="fricas")

[Out]

12/25*(305*x^3 - 610*x^2 + 317*x - 21)*log(3)/(x^2 - 2*x + 1)

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giac [A]  time = 0.27, size = 20, normalized size = 0.71 \begin {gather*} \frac {12}{25} \, {\left (305 \, x + \frac {3 \, {\left (4 \, x - 7\right )}}{{\left (x - 1\right )}^{2}}\right )} \log \relax (3) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3660*x^3-10980*x^2+10836*x-3300)*log(3)/(25*x^3-75*x^2+75*x-25),x, algorithm="giac")

[Out]

12/25*(305*x + 3*(4*x - 7)/(x - 1)^2)*log(3)

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maple [A]  time = 0.03, size = 23, normalized size = 0.82

method	result	size
default	\(\frac {12 \ln \relax (3) \left (305 x +\frac {12}{x -1}-\frac {9}{\left (x -1\right )^{2}}\right )}{25}\)	\(23\)
norman	\(\frac {-\frac {10836 x \ln \relax (3)}{25}+\frac {732 x^{3} \ln \relax (3)}{5}+\frac {7068 \ln \relax (3)}{25}}{\left (x -1\right )^{2}}\)	\(24\)
gosper	\(\frac {12 \left (305 x^{3}-903 x +589\right ) \ln \relax (3)}{25 \left (x^{2}-2 x +1\right )}\)	\(25\)
risch	\(\frac {732 x \ln \relax (3)}{5}+\frac {\ln \relax (3) \left (\frac {144 x}{25}-\frac {252}{25}\right )}{x^{2}-2 x +1}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3660*x^3-10980*x^2+10836*x-3300)*ln(3)/(25*x^3-75*x^2+75*x-25),x,method=_RETURNVERBOSE)

[Out]

12/25*ln(3)*(305*x+12/(x-1)-9/(x-1)^2)

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maxima [A]  time = 0.50, size = 25, normalized size = 0.89 \begin {gather*} \frac {12}{25} \, {\left (305 \, x + \frac {3 \, {\left (4 \, x - 7\right )}}{x^{2} - 2 \, x + 1}\right )} \log \relax (3) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3660*x^3-10980*x^2+10836*x-3300)*log(3)/(25*x^3-75*x^2+75*x-25),x, algorithm="maxima")

[Out]

12/25*(305*x + 3*(4*x - 7)/(x^2 - 2*x + 1))*log(3)

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mupad [B]  time = 0.09, size = 23, normalized size = 0.82 \begin {gather*} \frac {732\,x\,\ln \relax (3)}{5}-\frac {\frac {252\,\ln \relax (3)}{25}-\frac {144\,x\,\ln \relax (3)}{25}}{{\left (x-1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3)*(10836*x - 10980*x^2 + 3660*x^3 - 3300))/(75*x - 75*x^2 + 25*x^3 - 25),x)

[Out]

(732*x*log(3))/5 - ((252*log(3))/25 - (144*x*log(3))/25)/(x - 1)^2

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sympy [A]  time = 0.15, size = 29, normalized size = 1.04 \begin {gather*} \frac {732 x \log {\relax (3 )}}{5} + \frac {144 x \log {\relax (3 )} - 252 \log {\relax (3 )}}{25 x^{2} - 50 x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3660*x**3-10980*x**2+10836*x-3300)*ln(3)/(25*x**3-75*x**2+75*x-25),x)

[Out]

732*x*log(3)/5 + (144*x*log(3) - 252*log(3))/(25*x**2 - 50*x + 25)

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