3.20.2 \(\int \frac {9215-36864 x+27648 x^2+(9215-18432 x+9216 x^2) \log (9215 x-18432 x^2+9216 x^3)}{9215-18432 x+9216 x^2} \, dx\)

Optimal. Leaf size=17 \[ x \log \left (-x+9216 (1-x)^2 x\right ) \]

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Rubi [A]  time = 0.15, antiderivative size = 15, normalized size of antiderivative = 0.88, number of steps used = 10, number of rules used = 6, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {6728, 893, 2523, 1657, 632, 31} \begin {gather*} x \log \left (x \left (9216 x^2-18432 x+9215\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(9215 - 36864*x + 27648*x^2 + (9215 - 18432*x + 9216*x^2)*Log[9215*x - 18432*x^2 + 9216*x^3])/(9215 - 1843
2*x + 9216*x^2),x]

[Out]

x*Log[x*(9215 - 18432*x + 9216*x^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 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

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 {9215-36864 x+27648 x^2}{(-97+96 x) (-95+96 x)}+\log \left (x \left (9215-18432 x+9216 x^2\right )\right )\right ) \, dx\\ &=\int \frac {9215-36864 x+27648 x^2}{(-97+96 x) (-95+96 x)} \, dx+\int \log \left (x \left (9215-18432 x+9216 x^2\right )\right ) \, dx\\ &=x \log \left (x \left (9215-18432 x+9216 x^2\right )\right )-\int \frac {9215-36864 x+27648 x^2}{9215-18432 x+9216 x^2} \, dx+\int \left (3+\frac {97}{-97+96 x}+\frac {95}{-95+96 x}\right ) \, dx\\ &=3 x+\frac {95}{96} \log (95-96 x)+\frac {97}{96} \log (97-96 x)+x \log \left (x \left (9215-18432 x+9216 x^2\right )\right )-\int \left (3-\frac {2 (9215-9216 x)}{9215-18432 x+9216 x^2}\right ) \, dx\\ &=\frac {95}{96} \log (95-96 x)+\frac {97}{96} \log (97-96 x)+x \log \left (x \left (9215-18432 x+9216 x^2\right )\right )+2 \int \frac {9215-9216 x}{9215-18432 x+9216 x^2} \, dx\\ &=\frac {95}{96} \log (95-96 x)+\frac {97}{96} \log (97-96 x)+x \log \left (x \left (9215-18432 x+9216 x^2\right )\right )-9120 \int \frac {1}{-9120+9216 x} \, dx-9312 \int \frac {1}{-9312+9216 x} \, dx\\ &=x \log \left (x \left (9215-18432 x+9216 x^2\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 15, normalized size = 0.88 \begin {gather*} x \log \left (x \left (9215-18432 x+9216 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9215 - 36864*x + 27648*x^2 + (9215 - 18432*x + 9216*x^2)*Log[9215*x - 18432*x^2 + 9216*x^3])/(9215
- 18432*x + 9216*x^2),x]

[Out]

x*Log[x*(9215 - 18432*x + 9216*x^2)]

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fricas [A]  time = 0.70, size = 17, normalized size = 1.00 \begin {gather*} x \log \left (9216 \, x^{3} - 18432 \, x^{2} + 9215 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9216*x^2-18432*x+9215)*log(9216*x^3-18432*x^2+9215*x)+27648*x^2-36864*x+9215)/(9216*x^2-18432*x+92
15),x, algorithm="fricas")

[Out]

x*log(9216*x^3 - 18432*x^2 + 9215*x)

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giac [A]  time = 0.23, size = 17, normalized size = 1.00 \begin {gather*} x \log \left (9216 \, x^{3} - 18432 \, x^{2} + 9215 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9216*x^2-18432*x+9215)*log(9216*x^3-18432*x^2+9215*x)+27648*x^2-36864*x+9215)/(9216*x^2-18432*x+92
15),x, algorithm="giac")

[Out]

x*log(9216*x^3 - 18432*x^2 + 9215*x)

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maple [A]  time = 0.42, size = 18, normalized size = 1.06

method	result	size
default	\(\ln \left (9216 x^{3}-18432 x^{2}+9215 x \right ) x\)	\(18\)
norman	\(\ln \left (9216 x^{3}-18432 x^{2}+9215 x \right ) x\)	\(18\)
risch	\(\ln \left (9216 x^{3}-18432 x^{2}+9215 x \right ) x\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((9216*x^2-18432*x+9215)*ln(9216*x^3-18432*x^2+9215*x)+27648*x^2-36864*x+9215)/(9216*x^2-18432*x+9215),x,m
ethod=_RETURNVERBOSE)

[Out]

ln(9216*x^3-18432*x^2+9215*x)*x

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maxima [B]  time = 0.51, size = 47, normalized size = 2.76 \begin {gather*} \frac {1}{96} \, {\left (96 \, x - 95\right )} \log \left (96 \, x - 95\right ) + \frac {1}{96} \, {\left (96 \, x - 97\right )} \log \left (96 \, x - 97\right ) + x \log \relax (x) + \frac {95}{96} \, \log \left (96 \, x - 95\right ) + \frac {97}{96} \, \log \left (96 \, x - 97\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9216*x^2-18432*x+9215)*log(9216*x^3-18432*x^2+9215*x)+27648*x^2-36864*x+9215)/(9216*x^2-18432*x+92
15),x, algorithm="maxima")

[Out]

1/96*(96*x - 95)*log(96*x - 95) + 1/96*(96*x - 97)*log(96*x - 97) + x*log(x) + 95/96*log(96*x - 95) + 97/96*lo
g(96*x - 97)

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mupad [B]  time = 1.24, size = 15, normalized size = 0.88 \begin {gather*} x\,\ln \left (x\,\left (9216\,x^2-18432\,x+9215\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((27648*x^2 - 36864*x + log(9215*x - 18432*x^2 + 9216*x^3)*(9216*x^2 - 18432*x + 9215) + 9215)/(9216*x^2 -
18432*x + 9215),x)

[Out]

x*log(x*(9216*x^2 - 18432*x + 9215))

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sympy [A]  time = 0.15, size = 15, normalized size = 0.88 \begin {gather*} x \log {\left (9216 x^{3} - 18432 x^{2} + 9215 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9216*x**2-18432*x+9215)*ln(9216*x**3-18432*x**2+9215*x)+27648*x**2-36864*x+9215)/(9216*x**2-18432*
x+9215),x)

[Out]

x*log(9216*x**3 - 18432*x**2 + 9215*x)

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