3.49.76 \(\int \frac {-39-83 x+24 x^2+43 x^3-4 x^5+(18 x+3 x^2-9 x^3-3 x^4) \log (-3+x+x^2)}{-15+5 x+5 x^2} \, dx\)

Optimal. Leaf size=28 \[ \frac {1}{5} (3+x) \left (10+x+x^2 \left (5-x-\log \left (-3+x+x^2\right )\right )\right ) \]

________________________________________________________________________________________

Rubi [B]  time = 0.33, antiderivative size = 201, normalized size of antiderivative = 7.18, number of steps used = 21, number of rules used = 7, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.117, Rules used = {6728, 1657, 632, 31, 2528, 2525, 800} \begin {gather*} -\frac {x^4}{5}+\frac {2 x^3}{5}+\frac {16 x^2}{5}-\frac {3}{5} x^2 \log \left (x^2+x-3\right )-\frac {1}{5} x^3 \log \left (x^2+x-3\right )+\frac {13 x}{5}-\frac {1}{10} \left (11+\sqrt {13}\right ) \log \left (2 x-\sqrt {13}+1\right )+\frac {3}{10} \left (7-\sqrt {13}\right ) \log \left (2 x-\sqrt {13}+1\right )-\frac {1}{5} \left (5-2 \sqrt {13}\right ) \log \left (2 x-\sqrt {13}+1\right )-\frac {1}{5} \left (5+2 \sqrt {13}\right ) \log \left (2 x+\sqrt {13}+1\right )+\frac {3}{10} \left (7+\sqrt {13}\right ) \log \left (2 x+\sqrt {13}+1\right )-\frac {1}{10} \left (11-\sqrt {13}\right ) \log \left (2 x+\sqrt {13}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-39 - 83*x + 24*x^2 + 43*x^3 - 4*x^5 + (18*x + 3*x^2 - 9*x^3 - 3*x^4)*Log[-3 + x + x^2])/(-15 + 5*x + 5*x
^2),x]

[Out]

(13*x)/5 + (16*x^2)/5 + (2*x^3)/5 - x^4/5 - ((5 - 2*Sqrt[13])*Log[1 - Sqrt[13] + 2*x])/5 + (3*(7 - Sqrt[13])*L
og[1 - Sqrt[13] + 2*x])/10 - ((11 + Sqrt[13])*Log[1 - Sqrt[13] + 2*x])/10 - ((11 - Sqrt[13])*Log[1 + Sqrt[13]
+ 2*x])/10 + (3*(7 + Sqrt[13])*Log[1 + Sqrt[13] + 2*x])/10 - ((5 + 2*Sqrt[13])*Log[1 + Sqrt[13] + 2*x])/5 - (3
*x^2*Log[-3 + x + x^2])/5 - (x^3*Log[-3 + x + x^2])/5

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 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

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 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, 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 {-39-83 x+24 x^2+43 x^3-4 x^5}{5 \left (-3+x+x^2\right )}-\frac {3}{5} x (2+x) \log \left (-3+x+x^2\right )\right ) \, dx\\ &=\frac {1}{5} \int \frac {-39-83 x+24 x^2+43 x^3-4 x^5}{-3+x+x^2} \, dx-\frac {3}{5} \int x (2+x) \log \left (-3+x+x^2\right ) \, dx\\ &=\frac {1}{5} \int \left (9+27 x+4 x^2-4 x^3-\frac {12+11 x}{-3+x+x^2}\right ) \, dx-\frac {3}{5} \int \left (2 x \log \left (-3+x+x^2\right )+x^2 \log \left (-3+x+x^2\right )\right ) \, dx\\ &=\frac {9 x}{5}+\frac {27 x^2}{10}+\frac {4 x^3}{15}-\frac {x^4}{5}-\frac {1}{5} \int \frac {12+11 x}{-3+x+x^2} \, dx-\frac {3}{5} \int x^2 \log \left (-3+x+x^2\right ) \, dx-\frac {6}{5} \int x \log \left (-3+x+x^2\right ) \, dx\\ &=\frac {9 x}{5}+\frac {27 x^2}{10}+\frac {4 x^3}{15}-\frac {x^4}{5}-\frac {3}{5} x^2 \log \left (-3+x+x^2\right )-\frac {1}{5} x^3 \log \left (-3+x+x^2\right )+\frac {1}{5} \int \frac {x^3 (1+2 x)}{-3+x+x^2} \, dx+\frac {3}{5} \int \frac {x^2 (1+2 x)}{-3+x+x^2} \, dx-\frac {1}{10} \left (11-\sqrt {13}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {13}}{2}+x} \, dx-\frac {1}{10} \left (11+\sqrt {13}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {13}}{2}+x} \, dx\\ &=\frac {9 x}{5}+\frac {27 x^2}{10}+\frac {4 x^3}{15}-\frac {x^4}{5}-\frac {1}{10} \left (11+\sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )-\frac {1}{10} \left (11-\sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )-\frac {3}{5} x^2 \log \left (-3+x+x^2\right )-\frac {1}{5} x^3 \log \left (-3+x+x^2\right )+\frac {1}{5} \int \left (7-x+2 x^2+\frac {21-10 x}{-3+x+x^2}\right ) \, dx+\frac {3}{5} \int \left (-1+2 x-\frac {3-7 x}{-3+x+x^2}\right ) \, dx\\ &=\frac {13 x}{5}+\frac {16 x^2}{5}+\frac {2 x^3}{5}-\frac {x^4}{5}-\frac {1}{10} \left (11+\sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )-\frac {1}{10} \left (11-\sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )-\frac {3}{5} x^2 \log \left (-3+x+x^2\right )-\frac {1}{5} x^3 \log \left (-3+x+x^2\right )+\frac {1}{5} \int \frac {21-10 x}{-3+x+x^2} \, dx-\frac {3}{5} \int \frac {3-7 x}{-3+x+x^2} \, dx\\ &=\frac {13 x}{5}+\frac {16 x^2}{5}+\frac {2 x^3}{5}-\frac {x^4}{5}-\frac {1}{10} \left (11+\sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )-\frac {1}{10} \left (11-\sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )-\frac {3}{5} x^2 \log \left (-3+x+x^2\right )-\frac {1}{5} x^3 \log \left (-3+x+x^2\right )+\frac {1}{5} \left (-5-2 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {13}}{2}+x} \, dx+\frac {1}{10} \left (3 \left (7-\sqrt {13}\right )\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {13}}{2}+x} \, dx+\frac {1}{10} \left (3 \left (7+\sqrt {13}\right )\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {13}}{2}+x} \, dx+\frac {1}{5} \left (-5+2 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {13}}{2}+x} \, dx\\ &=\frac {13 x}{5}+\frac {16 x^2}{5}+\frac {2 x^3}{5}-\frac {x^4}{5}-\frac {1}{5} \left (5-2 \sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )+\frac {3}{10} \left (7-\sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )-\frac {1}{10} \left (11+\sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )-\frac {1}{10} \left (11-\sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )+\frac {3}{10} \left (7+\sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )-\frac {1}{5} \left (5+2 \sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )-\frac {3}{5} x^2 \log \left (-3+x+x^2\right )-\frac {1}{5} x^3 \log \left (-3+x+x^2\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 47, normalized size = 1.68 \begin {gather*} \frac {1}{5} \left (13 x+16 x^2+2 x^3-x^4-3 x^2 \log \left (-3+x+x^2\right )-x^3 \log \left (-3+x+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-39 - 83*x + 24*x^2 + 43*x^3 - 4*x^5 + (18*x + 3*x^2 - 9*x^3 - 3*x^4)*Log[-3 + x + x^2])/(-15 + 5*x
+ 5*x^2),x]

[Out]

(13*x + 16*x^2 + 2*x^3 - x^4 - 3*x^2*Log[-3 + x + x^2] - x^3*Log[-3 + x + x^2])/5

________________________________________________________________________________________

fricas [A]  time = 1.38, size = 37, normalized size = 1.32 \begin {gather*} -\frac {1}{5} \, x^{4} + \frac {2}{5} \, x^{3} + \frac {16}{5} \, x^{2} - \frac {1}{5} \, {\left (x^{3} + 3 \, x^{2}\right )} \log \left (x^{2} + x - 3\right ) + \frac {13}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^4-9*x^3+3*x^2+18*x)*log(x^2+x-3)-4*x^5+43*x^3+24*x^2-83*x-39)/(5*x^2+5*x-15),x, algorithm="fr
icas")

[Out]

-1/5*x^4 + 2/5*x^3 + 16/5*x^2 - 1/5*(x^3 + 3*x^2)*log(x^2 + x - 3) + 13/5*x

________________________________________________________________________________________

giac [A]  time = 0.17, size = 37, normalized size = 1.32 \begin {gather*} -\frac {1}{5} \, x^{4} + \frac {2}{5} \, x^{3} + \frac {16}{5} \, x^{2} - \frac {1}{5} \, {\left (x^{3} + 3 \, x^{2}\right )} \log \left (x^{2} + x - 3\right ) + \frac {13}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^4-9*x^3+3*x^2+18*x)*log(x^2+x-3)-4*x^5+43*x^3+24*x^2-83*x-39)/(5*x^2+5*x-15),x, algorithm="gi
ac")

[Out]

-1/5*x^4 + 2/5*x^3 + 16/5*x^2 - 1/5*(x^3 + 3*x^2)*log(x^2 + x - 3) + 13/5*x

________________________________________________________________________________________

maple [A]  time = 0.23, size = 39, normalized size = 1.39

method	result	size
risch	\(\left (-\frac {1}{5} x^{3}-\frac {3}{5} x^{2}\right ) \ln \left (x^{2}+x -3\right )-\frac {x^{4}}{5}+\frac {2 x^{3}}{5}+\frac {16 x^{2}}{5}+\frac {13 x}{5}\)	\(39\)
default	\(-\frac {x^{4}}{5}+\frac {2 x^{3}}{5}+\frac {16 x^{2}}{5}+\frac {13 x}{5}-\frac {\ln \left (x^{2}+x -3\right ) x^{3}}{5}-\frac {3 \ln \left (x^{2}+x -3\right ) x^{2}}{5}\)	\(44\)
norman	\(-\frac {x^{4}}{5}+\frac {2 x^{3}}{5}+\frac {16 x^{2}}{5}+\frac {13 x}{5}-\frac {\ln \left (x^{2}+x -3\right ) x^{3}}{5}-\frac {3 \ln \left (x^{2}+x -3\right ) x^{2}}{5}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x^4-9*x^3+3*x^2+18*x)*ln(x^2+x-3)-4*x^5+43*x^3+24*x^2-83*x-39)/(5*x^2+5*x-15),x,method=_RETURNVERBOSE
)

[Out]

(-1/5*x^3-3/5*x^2)*ln(x^2+x-3)-1/5*x^4+2/5*x^3+16/5*x^2+13/5*x

________________________________________________________________________________________

maxima [B]  time = 0.49, size = 49, normalized size = 1.75 \begin {gather*} -\frac {1}{5} \, x^{4} + \frac {2}{5} \, x^{3} + \frac {16}{5} \, x^{2} - \frac {1}{10} \, {\left (2 \, x^{3} + 6 \, x^{2} - 11\right )} \log \left (x^{2} + x - 3\right ) + \frac {13}{5} \, x - \frac {11}{10} \, \log \left (x^{2} + x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^4-9*x^3+3*x^2+18*x)*log(x^2+x-3)-4*x^5+43*x^3+24*x^2-83*x-39)/(5*x^2+5*x-15),x, algorithm="ma
xima")

[Out]

-1/5*x^4 + 2/5*x^3 + 16/5*x^2 - 1/10*(2*x^3 + 6*x^2 - 11)*log(x^2 + x - 3) + 13/5*x - 11/10*log(x^2 + x - 3)

________________________________________________________________________________________

mupad [B]  time = 0.25, size = 41, normalized size = 1.46 \begin {gather*} \frac {13\,x}{5}-x^3\,\left (\frac {\ln \left (x^2+x-3\right )}{5}-\frac {2}{5}\right )-x^2\,\left (\frac {3\,\ln \left (x^2+x-3\right )}{5}-\frac {16}{5}\right )-\frac {x^4}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(83*x - log(x + x^2 - 3)*(18*x + 3*x^2 - 9*x^3 - 3*x^4) - 24*x^2 - 43*x^3 + 4*x^5 + 39)/(5*x + 5*x^2 - 15
),x)

[Out]

(13*x)/5 - x^3*(log(x + x^2 - 3)/5 - 2/5) - x^2*((3*log(x + x^2 - 3))/5 - 16/5) - x^4/5

________________________________________________________________________________________

sympy [A]  time = 0.24, size = 44, normalized size = 1.57 \begin {gather*} - \frac {x^{4}}{5} + \frac {2 x^{3}}{5} + \frac {16 x^{2}}{5} + \frac {13 x}{5} + \left (- \frac {x^{3}}{5} - \frac {3 x^{2}}{5}\right ) \log {\left (x^{2} + x - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x**4-9*x**3+3*x**2+18*x)*ln(x**2+x-3)-4*x**5+43*x**3+24*x**2-83*x-39)/(5*x**2+5*x-15),x)

[Out]

-x**4/5 + 2*x**3/5 + 16*x**2/5 + 13*x/5 + (-x**3/5 - 3*x**2/5)*log(x**2 + x - 3)

________________________________________________________________________________________