3.23.19 \(\int \frac {-20-2 x-30 x^2}{-10 x+x^2+5 x^3} \, dx\)

Optimal. Leaf size=22 \[ \log \left (\frac {x^2}{16 \left (2-\frac {x}{5}-x^2\right )^4}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 18, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1594, 1628, 628} \begin {gather*} 2 \log (x)-4 \log \left (-5 x^2-x+10\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-20 - 2*x - 30*x^2)/(-10*x + x^2 + 5*x^3),x]

[Out]

2*Log[x] - 4*Log[10 - x - 5*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 {-20-2 x-30 x^2}{x \left (-10+x+5 x^2\right )} \, dx\\ &=\int \left (\frac {2}{x}-\frac {4 (1+10 x)}{-10+x+5 x^2}\right ) \, dx\\ &=2 \log (x)-4 \int \frac {1+10 x}{-10+x+5 x^2} \, dx\\ &=2 \log (x)-4 \log \left (10-x-5 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.91 \begin {gather*} -2 \left (-\log (x)+2 \log \left (10-x-5 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20 - 2*x - 30*x^2)/(-10*x + x^2 + 5*x^3),x]

[Out]

-2*(-Log[x] + 2*Log[10 - x - 5*x^2])

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fricas [A]  time = 0.54, size = 16, normalized size = 0.73 \begin {gather*} -4 \, \log \left (5 \, x^{2} + x - 10\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*x^2-2*x-20)/(5*x^3+x^2-10*x),x, algorithm="fricas")

[Out]

-4*log(5*x^2 + x - 10) + 2*log(x)

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giac [A]  time = 0.29, size = 18, normalized size = 0.82 \begin {gather*} -4 \, \log \left ({\left | 5 \, x^{2} + x - 10 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*x^2-2*x-20)/(5*x^3+x^2-10*x),x, algorithm="giac")

[Out]

-4*log(abs(5*x^2 + x - 10)) + 2*log(abs(x))

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maple [A]  time = 0.02, size = 17, normalized size = 0.77

method	result	size
default	\(2 \ln \relax (x )-4 \ln \left (5 x^{2}+x -10\right )\)	\(17\)
norman	\(2 \ln \relax (x )-4 \ln \left (5 x^{2}+x -10\right )\)	\(17\)
risch	\(2 \ln \relax (x )-4 \ln \left (5 x^{2}+x -10\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-30*x^2-2*x-20)/(5*x^3+x^2-10*x),x,method=_RETURNVERBOSE)

[Out]

2*ln(x)-4*ln(5*x^2+x-10)

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maxima [A]  time = 0.51, size = 16, normalized size = 0.73 \begin {gather*} -4 \, \log \left (5 \, x^{2} + x - 10\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*x^2-2*x-20)/(5*x^3+x^2-10*x),x, algorithm="maxima")

[Out]

-4*log(5*x^2 + x - 10) + 2*log(x)

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mupad [B]  time = 1.27, size = 16, normalized size = 0.73 \begin {gather*} 2\,\ln \relax (x)-4\,\ln \left (5\,x^2+x-10\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + 30*x^2 + 20)/(x^2 - 10*x + 5*x^3),x)

[Out]

2*log(x) - 4*log(x + 5*x^2 - 10)

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sympy [A]  time = 0.10, size = 15, normalized size = 0.68 \begin {gather*} 2 \log {\relax (x )} - 4 \log {\left (5 x^{2} + x - 10 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*x**2-2*x-20)/(5*x**3+x**2-10*x),x)

[Out]

2*log(x) - 4*log(5*x**2 + x - 10)

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