3.86.57 \(\int \frac {225+91 x+9 x^2}{225 x+90 x^2+9 x^3} \, dx\)

Optimal. Leaf size=13 \[ 5-\frac {1}{9 (5+x)}+\log (x) \]

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Rubi [A]  time = 0.03, antiderivative size = 12, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1594, 27, 12, 893} \begin {gather*} \log (x)-\frac {1}{9 (x+5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(225 + 91*x + 9*x^2)/(225*x + 90*x^2 + 9*x^3),x]

[Out]

-1/9*1/(5 + x) + Log[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {225+91 x+9 x^2}{x \left (225+90 x+9 x^2\right )} \, dx\\ &=\int \frac {225+91 x+9 x^2}{9 x (5+x)^2} \, dx\\ &=\frac {1}{9} \int \frac {225+91 x+9 x^2}{x (5+x)^2} \, dx\\ &=\frac {1}{9} \int \left (\frac {9}{x}+\frac {1}{(5+x)^2}\right ) \, dx\\ &=-\frac {1}{9 (5+x)}+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.23 \begin {gather*} \frac {1}{9} \left (-\frac {1}{5+x}+9 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(225 + 91*x + 9*x^2)/(225*x + 90*x^2 + 9*x^3),x]

[Out]

(-(5 + x)^(-1) + 9*Log[x])/9

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fricas [A]  time = 0.69, size = 16, normalized size = 1.23 \begin {gather*} \frac {9 \, {\left (x + 5\right )} \log \relax (x) - 1}{9 \, {\left (x + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2+91*x+225)/(9*x^3+90*x^2+225*x),x, algorithm="fricas")

[Out]

1/9*(9*(x + 5)*log(x) - 1)/(x + 5)

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giac [A]  time = 0.13, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{9 \, {\left (x + 5\right )}} + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2+91*x+225)/(9*x^3+90*x^2+225*x),x, algorithm="giac")

[Out]

-1/9/(x + 5) + log(abs(x))

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maple [A]  time = 0.03, size = 11, normalized size = 0.85

method	result	size
default	\(\ln \relax (x )-\frac {1}{9 \left (5+x \right )}\)	\(11\)
norman	\(\ln \relax (x )-\frac {1}{9 \left (5+x \right )}\)	\(11\)
risch	\(\ln \relax (x )-\frac {1}{9 \left (5+x \right )}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x^2+91*x+225)/(9*x^3+90*x^2+225*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)-1/9/(5+x)

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maxima [A]  time = 0.35, size = 10, normalized size = 0.77 \begin {gather*} -\frac {1}{9 \, {\left (x + 5\right )}} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2+91*x+225)/(9*x^3+90*x^2+225*x),x, algorithm="maxima")

[Out]

-1/9/(x + 5) + log(x)

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mupad [B]  time = 0.04, size = 12, normalized size = 0.92 \begin {gather*} \ln \relax (x)-\frac {1}{9\,\left (x+5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((91*x + 9*x^2 + 225)/(225*x + 90*x^2 + 9*x^3),x)

[Out]

log(x) - 1/(9*(x + 5))

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sympy [A]  time = 0.13, size = 8, normalized size = 0.62 \begin {gather*} \log {\relax (x )} - \frac {1}{9 x + 45} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x**2+91*x+225)/(9*x**3+90*x**2+225*x),x)

[Out]

log(x) - 1/(9*x + 45)

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