3.3.10 \(\int \frac {15-6 x^2}{350+155 x+297 x^2+62 x^3+56 x^4} \, dx\)

Optimal. Leaf size=19 \[ \log \left (14+\frac {3}{x+\frac {5+x+x^2}{x}}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 23, normalized size of antiderivative = 1.21, number of steps used = 4, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2074, 628} \begin {gather*} \log \left (28 x^2+17 x+70\right )-\log \left (2 x^2+x+5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(15 - 6*x^2)/(350 + 155*x + 297*x^2 + 62*x^3 + 56*x^4),x]

[Out]

-Log[5 + x + 2*x^2] + Log[70 + 17*x + 28*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 {-1-4 x}{5+x+2 x^2}+\frac {17+56 x}{70+17 x+28 x^2}\right ) \, dx\\ &=\int \frac {-1-4 x}{5+x+2 x^2} \, dx+\int \frac {17+56 x}{70+17 x+28 x^2} \, dx\\ &=-\log \left (5+x+2 x^2\right )+\log \left (70+17 x+28 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.21 \begin {gather*} -\log \left (5+x+2 x^2\right )+\log \left (70+17 x+28 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(15 - 6*x^2)/(350 + 155*x + 297*x^2 + 62*x^3 + 56*x^4),x]

[Out]

-Log[5 + x + 2*x^2] + Log[70 + 17*x + 28*x^2]

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fricas [A]  time = 1.09, size = 23, normalized size = 1.21 \begin {gather*} \log \left (28 \, x^{2} + 17 \, x + 70\right ) - \log \left (2 \, x^{2} + x + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2+15)/(56*x^4+62*x^3+297*x^2+155*x+350),x, algorithm="fricas")

[Out]

log(28*x^2 + 17*x + 70) - log(2*x^2 + x + 5)

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giac [A]  time = 0.33, size = 23, normalized size = 1.21 \begin {gather*} \log \left (28 \, x^{2} + 17 \, x + 70\right ) - \log \left (2 \, x^{2} + x + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2+15)/(56*x^4+62*x^3+297*x^2+155*x+350),x, algorithm="giac")

[Out]

log(28*x^2 + 17*x + 70) - log(2*x^2 + x + 5)

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maple [A]  time = 0.03, size = 24, normalized size = 1.26

method	result	size
default	\(\ln \left (28 x^{2}+17 x +70\right )-\ln \left (2 x^{2}+x +5\right )\)	\(24\)
norman	\(\ln \left (28 x^{2}+17 x +70\right )-\ln \left (2 x^{2}+x +5\right )\)	\(24\)
risch	\(\ln \left (28 x^{2}+17 x +70\right )-\ln \left (2 x^{2}+x +5\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x^2+15)/(56*x^4+62*x^3+297*x^2+155*x+350),x,method=_RETURNVERBOSE)

[Out]

ln(28*x^2+17*x+70)-ln(2*x^2+x+5)

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maxima [A]  time = 0.41, size = 23, normalized size = 1.21 \begin {gather*} \log \left (28 \, x^{2} + 17 \, x + 70\right ) - \log \left (2 \, x^{2} + x + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2+15)/(56*x^4+62*x^3+297*x^2+155*x+350),x, algorithm="maxima")

[Out]

log(28*x^2 + 17*x + 70) - log(2*x^2 + x + 5)

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mupad [B]  time = 0.21, size = 26, normalized size = 1.37 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {62\,x^2+577\,x+155}{10130\,x^2+5611\,x+25325}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x^2 - 15)/(155*x + 297*x^2 + 62*x^3 + 56*x^4 + 350),x)

[Out]

2*atanh((577*x + 62*x^2 + 155)/(5611*x + 10130*x^2 + 25325))

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sympy [A]  time = 0.11, size = 24, normalized size = 1.26 \begin {gather*} - \log {\left (x^{2} + \frac {x}{2} + \frac {5}{2} \right )} + \log {\left (x^{2} + \frac {17 x}{28} + \frac {5}{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x**2+15)/(56*x**4+62*x**3+297*x**2+155*x+350),x)

[Out]

-log(x**2 + x/2 + 5/2) + log(x**2 + 17*x/28 + 5/2)

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