3.22.43 \(\int \frac {-8-38 x+21 x^2+(-8-19 x+7 x^2) \log (\frac {1}{2} (8 x+19 x^2-7 x^3))}{-8-19 x+7 x^2} \, dx\)

Optimal. Leaf size=23 \[ x \log \left ((-2+x)^2 x+\frac {9}{2} (3-x) x^2\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.18, antiderivative size = 18, normalized size of antiderivative = 0.78, number of steps used = 13, number of rules used = 5, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {6728, 1657, 632, 31, 2523} \begin {gather*} x \log \left (\frac {1}{2} x \left (-7 x^2+19 x+8\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8 - 38*x + 21*x^2 + (-8 - 19*x + 7*x^2)*Log[(8*x + 19*x^2 - 7*x^3)/2])/(-8 - 19*x + 7*x^2),x]

[Out]

x*Log[(x*(8 + 19*x - 7*x^2))/2]

Rule 31

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

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-8-38 x+21 x^2}{-8-19 x+7 x^2}+\log \left (\frac {1}{2} x \left (8+19 x-7 x^2\right )\right )\right ) \, dx\\ &=\int \frac {-8-38 x+21 x^2}{-8-19 x+7 x^2} \, dx+\int \log \left (\frac {1}{2} x \left (8+19 x-7 x^2\right )\right ) \, dx\\ &=x \log \left (\frac {1}{2} x \left (8+19 x-7 x^2\right )\right )-\int \frac {8+38 x-21 x^2}{8+19 x-7 x^2} \, dx+\int \left (3+\frac {16+19 x}{-8-19 x+7 x^2}\right ) \, dx\\ &=3 x+x \log \left (\frac {1}{2} x \left (8+19 x-7 x^2\right )\right )+\int \frac {16+19 x}{-8-19 x+7 x^2} \, dx-\int \left (3-\frac {16+19 x}{8+19 x-7 x^2}\right ) \, dx\\ &=x \log \left (\frac {1}{2} x \left (8+19 x-7 x^2\right )\right )-\frac {1}{2} \left (-19+3 \sqrt {65}\right ) \int \frac {1}{-\frac {19}{2}+\frac {3 \sqrt {65}}{2}+7 x} \, dx+\frac {1}{2} \left (19+3 \sqrt {65}\right ) \int \frac {1}{-\frac {19}{2}-\frac {3 \sqrt {65}}{2}+7 x} \, dx+\int \frac {16+19 x}{8+19 x-7 x^2} \, dx\\ &=\frac {1}{14} \left (19-3 \sqrt {65}\right ) \log \left (19-3 \sqrt {65}-14 x\right )+\frac {1}{14} \left (19+3 \sqrt {65}\right ) \log \left (19+3 \sqrt {65}-14 x\right )+x \log \left (\frac {1}{2} x \left (8+19 x-7 x^2\right )\right )-\frac {1}{2} \left (-19-3 \sqrt {65}\right ) \int \frac {1}{\frac {19}{2}+\frac {3 \sqrt {65}}{2}-7 x} \, dx+\frac {1}{2} \left (19-3 \sqrt {65}\right ) \int \frac {1}{\frac {19}{2}-\frac {3 \sqrt {65}}{2}-7 x} \, dx\\ &=x \log \left (\frac {1}{2} x \left (8+19 x-7 x^2\right )\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 18, normalized size = 0.78 \begin {gather*} x \log \left (\frac {1}{2} x \left (8+19 x-7 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 - 38*x + 21*x^2 + (-8 - 19*x + 7*x^2)*Log[(8*x + 19*x^2 - 7*x^3)/2])/(-8 - 19*x + 7*x^2),x]

[Out]

x*Log[(x*(8 + 19*x - 7*x^2))/2]

________________________________________________________________________________________

fricas [A]  time = 1.20, size = 17, normalized size = 0.74 \begin {gather*} x \log \left (-\frac {7}{2} \, x^{3} + \frac {19}{2} \, x^{2} + 4 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x^2-19*x-8)*log(-7/2*x^3+19/2*x^2+4*x)+21*x^2-38*x-8)/(7*x^2-19*x-8),x, algorithm="fricas")

[Out]

x*log(-7/2*x^3 + 19/2*x^2 + 4*x)

________________________________________________________________________________________

giac [A]  time = 0.32, size = 17, normalized size = 0.74 \begin {gather*} x \log \left (-\frac {7}{2} \, x^{3} + \frac {19}{2} \, x^{2} + 4 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x^2-19*x-8)*log(-7/2*x^3+19/2*x^2+4*x)+21*x^2-38*x-8)/(7*x^2-19*x-8),x, algorithm="giac")

[Out]

x*log(-7/2*x^3 + 19/2*x^2 + 4*x)

________________________________________________________________________________________

maple [A]  time = 0.49, size = 18, normalized size = 0.78

method	result	size
norman	\(\ln \left (-\frac {7}{2} x^{3}+\frac {19}{2} x^{2}+4 x \right ) x\)	\(18\)
risch	\(\ln \left (-\frac {7}{2} x^{3}+\frac {19}{2} x^{2}+4 x \right ) x\)	\(18\)
default	\(-x \ln \relax (2)+x \ln \left (-7 x^{3}+19 x^{2}+8 x \right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((7*x^2-19*x-8)*ln(-7/2*x^3+19/2*x^2+4*x)+21*x^2-38*x-8)/(7*x^2-19*x-8),x,method=_RETURNVERBOSE)

[Out]

ln(-7/2*x^3+19/2*x^2+4*x)*x

________________________________________________________________________________________

maxima [B]  time = 1.05, size = 46, normalized size = 2.00 \begin {gather*} -x {\left (\log \relax (2) + 3\right )} + \frac {1}{14} \, {\left (14 \, x - 19\right )} \log \left (-7 \, x^{2} + 19 \, x + 8\right ) + x \log \relax (x) + 3 \, x + \frac {19}{14} \, \log \left (7 \, x^{2} - 19 \, x - 8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x^2-19*x-8)*log(-7/2*x^3+19/2*x^2+4*x)+21*x^2-38*x-8)/(7*x^2-19*x-8),x, algorithm="maxima")

[Out]

-x*(log(2) + 3) + 1/14*(14*x - 19)*log(-7*x^2 + 19*x + 8) + x*log(x) + 3*x + 19/14*log(7*x^2 - 19*x - 8)

________________________________________________________________________________________

mupad [B]  time = 1.31, size = 17, normalized size = 0.74 \begin {gather*} x\,\ln \left (-\frac {7\,x^3}{2}+\frac {19\,x^2}{2}+4\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((38*x - 21*x^2 + log(4*x + (19*x^2)/2 - (7*x^3)/2)*(19*x - 7*x^2 + 8) + 8)/(19*x - 7*x^2 + 8),x)

[Out]

x*log(4*x + (19*x^2)/2 - (7*x^3)/2)

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 19, normalized size = 0.83 \begin {gather*} x \log {\left (- \frac {7 x^{3}}{2} + \frac {19 x^{2}}{2} + 4 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x**2-19*x-8)*ln(-7/2*x**3+19/2*x**2+4*x)+21*x**2-38*x-8)/(7*x**2-19*x-8),x)

[Out]

x*log(-7*x**3/2 + 19*x**2/2 + 4*x)

________________________________________________________________________________________