3.34.7 \(\int \frac {48+184 x+187 x^2+90 x^3+27 x^4}{48 x+152 x^2+187 x^3+114 x^4+27 x^5} \, dx\)

Optimal. Leaf size=27 \[ \log \left (\frac {x \left (-2-\frac {16}{3 \left (5+\frac {4}{x}+3 x\right )}\right )}{12 e^5}\right ) \]

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Rubi [A]  time = 0.07, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2074, 628} \begin {gather*} -\log \left (3 x^2+5 x+4\right )+\log \left (9 x^2+23 x+12\right )+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(48 + 184*x + 187*x^2 + 90*x^3 + 27*x^4)/(48*x + 152*x^2 + 187*x^3 + 114*x^4 + 27*x^5),x]

[Out]

Log[x] - Log[4 + 5*x + 3*x^2] + Log[12 + 23*x + 9*x^2]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

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} &=\int \left (\frac {1}{x}+\frac {-5-6 x}{4+5 x+3 x^2}+\frac {23+18 x}{12+23 x+9 x^2}\right ) \, dx\\ &=\log (x)+\int \frac {-5-6 x}{4+5 x+3 x^2} \, dx+\int \frac {23+18 x}{12+23 x+9 x^2} \, dx\\ &=\log (x)-\log \left (4+5 x+3 x^2\right )+\log \left (12+23 x+9 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} \log (x)-\log \left (4+5 x+3 x^2\right )+\log \left (12+23 x+9 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(48 + 184*x + 187*x^2 + 90*x^3 + 27*x^4)/(48*x + 152*x^2 + 187*x^3 + 114*x^4 + 27*x^5),x]

[Out]

Log[x] - Log[4 + 5*x + 3*x^2] + Log[12 + 23*x + 9*x^2]

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fricas [A]  time = 0.50, size = 29, normalized size = 1.07 \begin {gather*} \log \left (9 \, x^{3} + 23 \, x^{2} + 12 \, x\right ) - \log \left (3 \, x^{2} + 5 \, x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*x^4+90*x^3+187*x^2+184*x+48)/(27*x^5+114*x^4+187*x^3+152*x^2+48*x),x, algorithm="fricas")

[Out]

log(9*x^3 + 23*x^2 + 12*x) - log(3*x^2 + 5*x + 4)

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giac [A]  time = 0.18, size = 29, normalized size = 1.07 \begin {gather*} -\log \left (3 \, x^{2} + 5 \, x + 4\right ) + \log \left ({\left | 9 \, x^{2} + 23 \, x + 12 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*x^4+90*x^3+187*x^2+184*x+48)/(27*x^5+114*x^4+187*x^3+152*x^2+48*x),x, algorithm="giac")

[Out]

-log(3*x^2 + 5*x + 4) + log(abs(9*x^2 + 23*x + 12)) + log(abs(x))

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maple [A]  time = 0.03, size = 28, normalized size = 1.04

method	result	size
default	\(-\ln \left (3 x^{2}+5 x +4\right )+\ln \relax (x )+\ln \left (9 x^{2}+23 x +12\right )\)	\(28\)
norman	\(-\ln \left (3 x^{2}+5 x +4\right )+\ln \relax (x )+\ln \left (9 x^{2}+23 x +12\right )\)	\(28\)
risch	\(-\ln \left (3 x^{2}+5 x +4\right )+\ln \left (9 x^{3}+23 x^{2}+12 x \right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((27*x^4+90*x^3+187*x^2+184*x+48)/(27*x^5+114*x^4+187*x^3+152*x^2+48*x),x,method=_RETURNVERBOSE)

[Out]

-ln(3*x^2+5*x+4)+ln(x)+ln(9*x^2+23*x+12)

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maxima [A]  time = 0.82, size = 27, normalized size = 1.00 \begin {gather*} \log \left (9 \, x^{2} + 23 \, x + 12\right ) - \log \left (3 \, x^{2} + 5 \, x + 4\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*x^4+90*x^3+187*x^2+184*x+48)/(27*x^5+114*x^4+187*x^3+152*x^2+48*x),x, algorithm="maxima")

[Out]

log(9*x^2 + 23*x + 12) - log(3*x^2 + 5*x + 4) + log(x)

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mupad [B]  time = 0.15, size = 25, normalized size = 0.93 \begin {gather*} \ln \left (x\,\left (9\,x^2+23\,x+12\right )\right )-\ln \left (x^2+\frac {5\,x}{3}+\frac {4}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((184*x + 187*x^2 + 90*x^3 + 27*x^4 + 48)/(48*x + 152*x^2 + 187*x^3 + 114*x^4 + 27*x^5),x)

[Out]

log(x*(23*x + 9*x^2 + 12)) - log((5*x)/3 + x^2 + 4/3)

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sympy [A]  time = 0.12, size = 26, normalized size = 0.96 \begin {gather*} - \log {\left (3 x^{2} + 5 x + 4 \right )} + \log {\left (9 x^{3} + 23 x^{2} + 12 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*x**4+90*x**3+187*x**2+184*x+48)/(27*x**5+114*x**4+187*x**3+152*x**2+48*x),x)

[Out]

-log(3*x**2 + 5*x + 4) + log(9*x**3 + 23*x**2 + 12*x)

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