3.21.40 \(\int \frac {-400+100 x-200 x^2-121 x^3-37 x^4-48 x^5+3 x^6-6 x^7}{-400 x-200 x^3+48 x^4-25 x^5+24 x^6+3 x^8} \, dx\)

Optimal. Leaf size=29 \[ 4-\frac {x}{4+x^2}+\log \left (\frac {\log (2)}{-\frac {25}{3 x}+x^2}\right ) \]

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Rubi [A]  time = 0.16, antiderivative size = 23, normalized size of antiderivative = 0.79, number of steps used = 6, number of rules used = 4, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {2074, 199, 203, 260} \begin {gather*} -\log \left (25-3 x^3\right )-\frac {x}{x^2+4}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-400 + 100*x - 200*x^2 - 121*x^3 - 37*x^4 - 48*x^5 + 3*x^6 - 6*x^7)/(-400*x - 200*x^3 + 48*x^4 - 25*x^5 +
24*x^6 + 3*x^8),x]

[Out]

-(x/(4 + x^2)) + Log[x] - Log[25 - 3*x^3]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 {8}{\left (4+x^2\right )^2}+\frac {1}{4+x^2}-\frac {9 x^2}{-25+3 x^3}\right ) \, dx\\ &=\log (x)-8 \int \frac {1}{\left (4+x^2\right )^2} \, dx-9 \int \frac {x^2}{-25+3 x^3} \, dx+\int \frac {1}{4+x^2} \, dx\\ &=-\frac {x}{4+x^2}+\frac {1}{2} \tan ^{-1}\left (\frac {x}{2}\right )+\log (x)-\log \left (25-3 x^3\right )-\int \frac {1}{4+x^2} \, dx\\ &=-\frac {x}{4+x^2}+\log (x)-\log \left (25-3 x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 0.79 \begin {gather*} -\frac {x}{4+x^2}+\log (x)-\log \left (25-3 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-400 + 100*x - 200*x^2 - 121*x^3 - 37*x^4 - 48*x^5 + 3*x^6 - 6*x^7)/(-400*x - 200*x^3 + 48*x^4 - 25
*x^5 + 24*x^6 + 3*x^8),x]

[Out]

-(x/(4 + x^2)) + Log[x] - Log[25 - 3*x^3]

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fricas [A]  time = 0.78, size = 34, normalized size = 1.17 \begin {gather*} -\frac {{\left (x^{2} + 4\right )} \log \left (3 \, x^{3} - 25\right ) - {\left (x^{2} + 4\right )} \log \relax (x) + x}{x^{2} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^7+3*x^6-48*x^5-37*x^4-121*x^3-200*x^2+100*x-400)/(3*x^8+24*x^6-25*x^5+48*x^4-200*x^3-400*x),x,
algorithm="fricas")

[Out]

-((x^2 + 4)*log(3*x^3 - 25) - (x^2 + 4)*log(x) + x)/(x^2 + 4)

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giac [A]  time = 0.34, size = 25, normalized size = 0.86 \begin {gather*} -\frac {x}{x^{2} + 4} - \log \left ({\left | 3 \, x^{3} - 25 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^7+3*x^6-48*x^5-37*x^4-121*x^3-200*x^2+100*x-400)/(3*x^8+24*x^6-25*x^5+48*x^4-200*x^3-400*x),x,
algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 24, normalized size = 0.83

method	result	size
default	\(-\frac {x}{x^{2}+4}-\ln \left (3 x^{3}-25\right )+\ln \relax (x )\)	\(24\)
norman	\(-\frac {x}{x^{2}+4}-\ln \left (3 x^{3}-25\right )+\ln \relax (x )\)	\(24\)
risch	\(-\frac {x}{x^{2}+4}-\ln \left (3 x^{3}-25\right )+\ln \relax (x )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x^7+3*x^6-48*x^5-37*x^4-121*x^3-200*x^2+100*x-400)/(3*x^8+24*x^6-25*x^5+48*x^4-200*x^3-400*x),x,method
=_RETURNVERBOSE)

[Out]

-x/(x^2+4)-ln(3*x^3-25)+ln(x)

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maxima [A]  time = 0.57, size = 23, normalized size = 0.79 \begin {gather*} -\frac {x}{x^{2} + 4} - \log \left (3 \, x^{3} - 25\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^7+3*x^6-48*x^5-37*x^4-121*x^3-200*x^2+100*x-400)/(3*x^8+24*x^6-25*x^5+48*x^4-200*x^3-400*x),x,
algorithm="maxima")

[Out]

-x/(x^2 + 4) - log(3*x^3 - 25) + log(x)

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mupad [B]  time = 0.09, size = 21, normalized size = 0.72 \begin {gather*} \ln \relax (x)-\ln \left (x^3-\frac {25}{3}\right )-\frac {x}{x^2+4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((200*x^2 - 100*x + 121*x^3 + 37*x^4 + 48*x^5 - 3*x^6 + 6*x^7 + 400)/(400*x + 200*x^3 - 48*x^4 + 25*x^5 - 2
4*x^6 - 3*x^8),x)

[Out]

log(x) - log(x^3 - 25/3) - x/(x^2 + 4)

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sympy [A]  time = 0.15, size = 17, normalized size = 0.59 \begin {gather*} - \frac {x}{x^{2} + 4} + \log {\relax (x )} - \log {\left (3 x^{3} - 25 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x**7+3*x**6-48*x**5-37*x**4-121*x**3-200*x**2+100*x-400)/(3*x**8+24*x**6-25*x**5+48*x**4-200*x**
3-400*x),x)

[Out]

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

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