3.39.94 \(\int \frac {630+630 x-1311 x^2+101 x^3-2 x^4}{630 x-50 x^2+x^3} \, dx\)

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

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Rubi [A]  time = 0.06, antiderivative size = 20, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {1594, 1628, 628} \begin {gather*} -x^2-\log \left (x^2-50 x+630\right )+x+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(630 + 630*x - 1311*x^2 + 101*x^3 - 2*x^4)/(630*x - 50*x^2 + x^3),x]

[Out]

x - x^2 + Log[x] - Log[630 - 50*x + 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 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 {630+630 x-1311 x^2+101 x^3-2 x^4}{x \left (630-50 x+x^2\right )} \, dx\\ &=\int \left (1+\frac {1}{x}-2 x-\frac {2 (-25+x)}{630-50 x+x^2}\right ) \, dx\\ &=x-x^2+\log (x)-2 \int \frac {-25+x}{630-50 x+x^2} \, dx\\ &=x-x^2+\log (x)-\log \left (630-50 x+x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.95 \begin {gather*} x-x^2+\log (x)-\log \left (630-50 x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(630 + 630*x - 1311*x^2 + 101*x^3 - 2*x^4)/(630*x - 50*x^2 + x^3),x]

[Out]

x - x^2 + Log[x] - Log[630 - 50*x + x^2]

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fricas [A]  time = 0.51, size = 20, normalized size = 0.95 \begin {gather*} -x^{2} + x - \log \left (x^{2} - 50 \, x + 630\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^4+101*x^3-1311*x^2+630*x+630)/(x^3-50*x^2+630*x),x, algorithm="fricas")

[Out]

-x^2 + x - log(x^2 - 50*x + 630) + log(x)

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giac [A]  time = 0.23, size = 21, normalized size = 1.00 \begin {gather*} -x^{2} + x - \log \left (x^{2} - 50 \, x + 630\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^4+101*x^3-1311*x^2+630*x+630)/(x^3-50*x^2+630*x),x, algorithm="giac")

[Out]

-x^2 + x - log(x^2 - 50*x + 630) + log(abs(x))

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

method	result	size
default	\(x -x^{2}+\ln \relax (x )-\ln \left (x^{2}-50 x +630\right )\)	\(21\)
norman	\(x -x^{2}+\ln \relax (x )-\ln \left (x^{2}-50 x +630\right )\)	\(21\)
risch	\(x -x^{2}+\ln \relax (x )-\ln \left (x^{2}-50 x +630\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^4+101*x^3-1311*x^2+630*x+630)/(x^3-50*x^2+630*x),x,method=_RETURNVERBOSE)

[Out]

x-x^2+ln(x)-ln(x^2-50*x+630)

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maxima [A]  time = 0.36, size = 20, normalized size = 0.95 \begin {gather*} -x^{2} + x - \log \left (x^{2} - 50 \, x + 630\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^4+101*x^3-1311*x^2+630*x+630)/(x^3-50*x^2+630*x),x, algorithm="maxima")

[Out]

-x^2 + x - log(x^2 - 50*x + 630) + log(x)

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mupad [B]  time = 2.23, size = 20, normalized size = 0.95 \begin {gather*} x-\ln \left (x^2-50\,x+630\right )+\ln \relax (x)-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((630*x - 1311*x^2 + 101*x^3 - 2*x^4 + 630)/(630*x - 50*x^2 + x^3),x)

[Out]

x - log(x^2 - 50*x + 630) + log(x) - x^2

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sympy [A]  time = 0.11, size = 17, normalized size = 0.81 \begin {gather*} - x^{2} + x + \log {\relax (x )} - \log {\left (x^{2} - 50 x + 630 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**4+101*x**3-1311*x**2+630*x+630)/(x**3-50*x**2+630*x),x)

[Out]

-x**2 + x + log(x) - log(x**2 - 50*x + 630)

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