3.65.22 \(\int \frac {24 x+625 (48+6 x)}{4 x^2+x^3+625 (4 x+x^2)} \, dx\)

Optimal. Leaf size=17 \[ 6 \log \left (\frac {x^2}{(4+x) (625+x)}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {1998, 1594, 800} \begin {gather*} 12 \log (x)-6 \log (x+4)-6 \log (x+625) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(24*x + 625*(48 + 6*x))/(4*x^2 + x^3 + 625*(4*x + x^2)),x]

[Out]

12*Log[x] - 6*Log[4 + x] - 6*Log[625 + x]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1998

Int[(u_)^(p_.)*(z_), x_Symbol] :> Int[ExpandToSum[z, x]*ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && BinomialQ[z,
x] && GeneralizedTrinomialQ[u, x] && EqQ[BinomialDegree[z, x] - GeneralizedTrinomialDegree[u, x], 0] &&  !(Bi
nomialMatchQ[z, x] && GeneralizedTrinomialMatchQ[u, x])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {30000+3774 x}{2500 x+629 x^2+x^3} \, dx\\ &=\int \frac {30000+3774 x}{x \left (2500+629 x+x^2\right )} \, dx\\ &=\int \left (\frac {12}{x}-\frac {6}{4+x}-\frac {6}{625+x}\right ) \, dx\\ &=12 \log (x)-6 \log (4+x)-6 \log (625+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.06 \begin {gather*} 6 \left (2 \log (x)-\log \left (2500+629 x+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24*x + 625*(48 + 6*x))/(4*x^2 + x^3 + 625*(4*x + x^2)),x]

[Out]

6*(2*Log[x] - Log[2500 + 629*x + x^2])

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fricas [A]  time = 0.49, size = 16, normalized size = 0.94 \begin {gather*} -6 \, \log \left (x^{2} + 629 \, x + 2500\right ) + 12 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3774*x+30000)/(x^3+629*x^2+2500*x),x, algorithm="fricas")

[Out]

-6*log(x^2 + 629*x + 2500) + 12*log(x)

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giac [A]  time = 0.13, size = 20, normalized size = 1.18 \begin {gather*} -6 \, \log \left ({\left | x + 625 \right |}\right ) - 6 \, \log \left ({\left | x + 4 \right |}\right ) + 12 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3774*x+30000)/(x^3+629*x^2+2500*x),x, algorithm="giac")

[Out]

-6*log(abs(x + 625)) - 6*log(abs(x + 4)) + 12*log(abs(x))

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maple [A]  time = 0.03, size = 17, normalized size = 1.00

method	result	size
risch	\(12 \ln \relax (x )-6 \ln \left (x^{2}+629 x +2500\right )\)	\(17\)
default	\(-6 \ln \left (x +625\right )-6 \ln \left (4+x \right )+12 \ln \relax (x )\)	\(18\)
norman	\(-6 \ln \left (x +625\right )-6 \ln \left (4+x \right )+12 \ln \relax (x )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3774*x+30000)/(x^3+629*x^2+2500*x),x,method=_RETURNVERBOSE)

[Out]

12*ln(x)-6*ln(x^2+629*x+2500)

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maxima [A]  time = 0.36, size = 17, normalized size = 1.00 \begin {gather*} -6 \, \log \left (x + 625\right ) - 6 \, \log \left (x + 4\right ) + 12 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3774*x+30000)/(x^3+629*x^2+2500*x),x, algorithm="maxima")

[Out]

-6*log(x + 625) - 6*log(x + 4) + 12*log(x)

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mupad [B]  time = 4.16, size = 16, normalized size = 0.94 \begin {gather*} 12\,\ln \relax (x)-6\,\ln \left (x^2+629\,x+2500\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3774*x + 30000)/(2500*x + 629*x^2 + x^3),x)

[Out]

12*log(x) - 6*log(629*x + x^2 + 2500)

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sympy [A]  time = 0.09, size = 15, normalized size = 0.88 \begin {gather*} 12 \log {\relax (x )} - 6 \log {\left (x^{2} + 629 x + 2500 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3774*x+30000)/(x**3+629*x**2+2500*x),x)

[Out]

12*log(x) - 6*log(x**2 + 629*x + 2500)

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