3.92.21 \(\int \frac {900+1757 x+168 x^2+4 x^3}{450 x+43 x^2+x^3} \, dx\)

Optimal. Leaf size=30 \[ x-\log (x)+3 \log \left (\frac {e^{-4+x} x}{\left (9+\frac {x}{2}\right ) (25+x)}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 20, normalized size of antiderivative = 0.67, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1594, 1628} \begin {gather*} 4 x+2 \log (x)-3 \log (x+18)-3 \log (x+25) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(900 + 1757*x + 168*x^2 + 4*x^3)/(450*x + 43*x^2 + x^3),x]

[Out]

4*x + 2*Log[x] - 3*Log[18 + x] - 3*Log[25 + x]

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 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {900+1757 x+168 x^2+4 x^3}{x \left (450+43 x+x^2\right )} \, dx\\ &=\int \left (4+\frac {2}{x}-\frac {3}{18+x}-\frac {3}{25+x}\right ) \, dx\\ &=4 x+2 \log (x)-3 \log (18+x)-3 \log (25+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 0.63 \begin {gather*} 4 x+2 \log (x)-3 \log \left (450+43 x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(900 + 1757*x + 168*x^2 + 4*x^3)/(450*x + 43*x^2 + x^3),x]

[Out]

4*x + 2*Log[x] - 3*Log[450 + 43*x + x^2]

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fricas [A]  time = 0.57, size = 19, normalized size = 0.63 \begin {gather*} 4 \, x - 3 \, \log \left (x^{2} + 43 \, x + 450\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3+168*x^2+1757*x+900)/(x^3+43*x^2+450*x),x, algorithm="fricas")

[Out]

4*x - 3*log(x^2 + 43*x + 450) + 2*log(x)

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giac [A]  time = 0.14, size = 23, normalized size = 0.77 \begin {gather*} 4 \, x - 3 \, \log \left ({\left | x + 25 \right |}\right ) - 3 \, \log \left ({\left | x + 18 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3+168*x^2+1757*x+900)/(x^3+43*x^2+450*x),x, algorithm="giac")

[Out]

4*x - 3*log(abs(x + 25)) - 3*log(abs(x + 18)) + 2*log(abs(x))

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maple [A]  time = 0.04, size = 20, normalized size = 0.67

method	result	size
risch	\(4 x +2 \ln \relax (x )-3 \ln \left (x^{2}+43 x +450\right )\)	\(20\)
default	\(4 x +2 \ln \relax (x )-3 \ln \left (18+x \right )-3 \ln \left (x +25\right )\)	\(21\)
norman	\(4 x +2 \ln \relax (x )-3 \ln \left (18+x \right )-3 \ln \left (x +25\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3+168*x^2+1757*x+900)/(x^3+43*x^2+450*x),x,method=_RETURNVERBOSE)

[Out]

4*x+2*ln(x)-3*ln(x^2+43*x+450)

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maxima [A]  time = 0.37, size = 20, normalized size = 0.67 \begin {gather*} 4 \, x - 3 \, \log \left (x + 25\right ) - 3 \, \log \left (x + 18\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3+168*x^2+1757*x+900)/(x^3+43*x^2+450*x),x, algorithm="maxima")

[Out]

4*x - 3*log(x + 25) - 3*log(x + 18) + 2*log(x)

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mupad [B]  time = 0.07, size = 19, normalized size = 0.63 \begin {gather*} 4\,x-3\,\ln \left (x^2+43\,x+450\right )+2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1757*x + 168*x^2 + 4*x^3 + 900)/(450*x + 43*x^2 + x^3),x)

[Out]

4*x - 3*log(43*x + x^2 + 450) + 2*log(x)

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sympy [A]  time = 0.10, size = 19, normalized size = 0.63 \begin {gather*} 4 x + 2 \log {\relax (x )} - 3 \log {\left (x^{2} + 43 x + 450 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3+168*x**2+1757*x+900)/(x**3+43*x**2+450*x),x)

[Out]

4*x + 2*log(x) - 3*log(x**2 + 43*x + 450)

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