3.37.16 \(\int \frac {-4624+680 x+876 x^2+96 x^3}{-1156 x-119 x^2+83 x^3+12 x^4} \, dx\)

Optimal. Leaf size=23 \[ 4 \left (\log (x)+\log \left (-4+x+\frac {x^2}{1+x+2 (8+x)}\right )\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 26, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2074, 628} \begin {gather*} 4 \log \left (-4 x^2-5 x+68\right )+4 \log (x)-4 \log (3 x+17) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4624 + 680*x + 876*x^2 + 96*x^3)/(-1156*x - 119*x^2 + 83*x^3 + 12*x^4),x]

[Out]

4*Log[x] - 4*Log[17 + 3*x] + 4*Log[68 - 5*x - 4*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 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 {4}{x}-\frac {12}{17+3 x}+\frac {4 (5+8 x)}{-68+5 x+4 x^2}\right ) \, dx\\ &=4 \log (x)-4 \log (17+3 x)+4 \int \frac {5+8 x}{-68+5 x+4 x^2} \, dx\\ &=4 \log (x)-4 \log (17+3 x)+4 \log \left (68-5 x-4 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 1.13 \begin {gather*} 4 \log (x)-4 \log (17+3 x)+4 \log \left (68-5 x-4 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4624 + 680*x + 876*x^2 + 96*x^3)/(-1156*x - 119*x^2 + 83*x^3 + 12*x^4),x]

[Out]

4*Log[x] - 4*Log[17 + 3*x] + 4*Log[68 - 5*x - 4*x^2]

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fricas [A]  time = 0.71, size = 26, normalized size = 1.13 \begin {gather*} 4 \, \log \left (4 \, x^{3} + 5 \, x^{2} - 68 \, x\right ) - 4 \, \log \left (3 \, x + 17\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((96*x^3+876*x^2+680*x-4624)/(12*x^4+83*x^3-119*x^2-1156*x),x, algorithm="fricas")

[Out]

4*log(4*x^3 + 5*x^2 - 68*x) - 4*log(3*x + 17)

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giac [A]  time = 0.14, size = 29, normalized size = 1.26 \begin {gather*} 4 \, \log \left ({\left | 4 \, x^{2} + 5 \, x - 68 \right |}\right ) - 4 \, \log \left ({\left | 3 \, x + 17 \right |}\right ) + 4 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((96*x^3+876*x^2+680*x-4624)/(12*x^4+83*x^3-119*x^2-1156*x),x, algorithm="giac")

[Out]

4*log(abs(4*x^2 + 5*x - 68)) - 4*log(abs(3*x + 17)) + 4*log(abs(x))

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maple [A]  time = 0.03, size = 27, normalized size = 1.17

method	result	size
default	\(-4 \ln \left (17+3 x \right )+4 \ln \relax (x )+4 \ln \left (4 x^{2}+5 x -68\right )\)	\(27\)
norman	\(-4 \ln \left (17+3 x \right )+4 \ln \relax (x )+4 \ln \left (4 x^{2}+5 x -68\right )\)	\(27\)
risch	\(-4 \ln \left (17+3 x \right )+4 \ln \left (4 x^{3}+5 x^{2}-68 x \right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((96*x^3+876*x^2+680*x-4624)/(12*x^4+83*x^3-119*x^2-1156*x),x,method=_RETURNVERBOSE)

[Out]

-4*ln(17+3*x)+4*ln(x)+4*ln(4*x^2+5*x-68)

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maxima [A]  time = 0.40, size = 26, normalized size = 1.13 \begin {gather*} 4 \, \log \left (4 \, x^{2} + 5 \, x - 68\right ) - 4 \, \log \left (3 \, x + 17\right ) + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((96*x^3+876*x^2+680*x-4624)/(12*x^4+83*x^3-119*x^2-1156*x),x, algorithm="maxima")

[Out]

4*log(4*x^2 + 5*x - 68) - 4*log(3*x + 17) + 4*log(x)

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mupad [B]  time = 2.28, size = 22, normalized size = 0.96 \begin {gather*} 4\,\ln \left (x\,\left (4\,x^2+5\,x-68\right )\right )-4\,\ln \left (x+\frac {17}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(680*x + 876*x^2 + 96*x^3 - 4624)/(1156*x + 119*x^2 - 83*x^3 - 12*x^4),x)

[Out]

4*log(x*(5*x + 4*x^2 - 68)) - 4*log(x + 17/3)

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sympy [A]  time = 0.11, size = 24, normalized size = 1.04 \begin {gather*} - 4 \log {\left (3 x + 17 \right )} + 4 \log {\left (4 x^{3} + 5 x^{2} - 68 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((96*x**3+876*x**2+680*x-4624)/(12*x**4+83*x**3-119*x**2-1156*x),x)

[Out]

-4*log(3*x + 17) + 4*log(4*x**3 + 5*x**2 - 68*x)

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