3.78.35 \(\int \frac {-375-525 x-119 x^2+2 x^3}{375 x^2+122 x^3-x^4+(375 x+125 x^2) \log (x)+(75 x+25 x^2) \log (3+x)} \, dx\)

Optimal. Leaf size=21 \[ -\log \left (-\frac {x^2}{25}+5 (x+\log (x))+\log (3+x)\right ) \]

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Rubi [A]  time = 0.23, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6688, 6684} \begin {gather*} -\log ((125-x) x+125 \log (x)+25 \log (x+3)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-375 - 525*x - 119*x^2 + 2*x^3)/(375*x^2 + 122*x^3 - x^4 + (375*x + 125*x^2)*Log[x] + (75*x + 25*x^2)*Log
[3 + x]),x]

[Out]

-Log[(125 - x)*x + 125*Log[x] + 25*Log[3 + x]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {375+525 x+119 x^2-2 x^3}{x (3+x) ((-125+x) x-125 \log (x)-25 \log (3+x))} \, dx\\ &=-\log ((125-x) x+125 \log (x)+25 \log (3+x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.98, size = 20, normalized size = 0.95 \begin {gather*} -\log \left (-125 x+x^2-125 \log (x)-25 \log (3+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-375 - 525*x - 119*x^2 + 2*x^3)/(375*x^2 + 122*x^3 - x^4 + (375*x + 125*x^2)*Log[x] + (75*x + 25*x^
2)*Log[3 + x]),x]

[Out]

-Log[-125*x + x^2 - 125*Log[x] - 25*Log[3 + x]]

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fricas [A]  time = 0.68, size = 22, normalized size = 1.05 \begin {gather*} -\log \left (-x^{2} + 125 \, x + 25 \, \log \left (x + 3\right ) + 125 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3-119*x^2-525*x-375)/((25*x^2+75*x)*log(3+x)+(125*x^2+375*x)*log(x)-x^4+122*x^3+375*x^2),x, alg
orithm="fricas")

[Out]

-log(-x^2 + 125*x + 25*log(x + 3) + 125*log(x))

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giac [A]  time = 0.19, size = 22, normalized size = 1.05 \begin {gather*} -\log \left (-x^{2} + 125 \, x + 25 \, \log \left (x + 3\right ) + 125 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3-119*x^2-525*x-375)/((25*x^2+75*x)*log(3+x)+(125*x^2+375*x)*log(x)-x^4+122*x^3+375*x^2),x, alg
orithm="giac")

[Out]

-log(-x^2 + 125*x + 25*log(x + 3) + 125*log(x))

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maple [A]  time = 0.02, size = 21, normalized size = 1.00

method	result	size
risch	\(-\ln \left (\ln \left (3+x \right )+5 x +5 \ln \relax (x )-\frac {x^{2}}{25}\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^3-119*x^2-525*x-375)/((25*x^2+75*x)*ln(3+x)+(125*x^2+375*x)*ln(x)-x^4+122*x^3+375*x^2),x,method=_RETU
RNVERBOSE)

[Out]

-ln(ln(3+x)+5*x+5*ln(x)-1/25*x^2)

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maxima [A]  time = 0.41, size = 20, normalized size = 0.95 \begin {gather*} -\log \left (-\frac {1}{25} \, x^{2} + 5 \, x + \log \left (x + 3\right ) + 5 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3-119*x^2-525*x-375)/((25*x^2+75*x)*log(3+x)+(125*x^2+375*x)*log(x)-x^4+122*x^3+375*x^2),x, alg
orithm="maxima")

[Out]

-log(-1/25*x^2 + 5*x + log(x + 3) + 5*log(x))

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mupad [B]  time = 5.43, size = 20, normalized size = 0.95 \begin {gather*} -\ln \left (5\,x+\ln \left (x+3\right )+5\,\ln \relax (x)-\frac {x^2}{25}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(525*x + 119*x^2 - 2*x^3 + 375)/(log(x + 3)*(75*x + 25*x^2) + log(x)*(375*x + 125*x^2) + 375*x^2 + 122*x^
3 - x^4),x)

[Out]

-log(5*x + log(x + 3) + 5*log(x) - x^2/25)

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sympy [A]  time = 0.43, size = 20, normalized size = 0.95 \begin {gather*} - \log {\left (- \frac {x^{2}}{25} + 5 x + 5 \log {\relax (x )} + \log {\left (x + 3 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**3-119*x**2-525*x-375)/((25*x**2+75*x)*ln(3+x)+(125*x**2+375*x)*ln(x)-x**4+122*x**3+375*x**2),x
)

[Out]

-log(-x**2/25 + 5*x + 5*log(x) + log(x + 3))

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