3.14.61 \(\int \frac {-8+8 x+6 x^2+7 x^3+x^4+(-8-10 x+16 x^2+2 x^3) \log (\frac {4+9 x+x^2}{15 x+5 e^5 x})}{-4 x^2-5 x^3+8 x^4+x^5} \, dx\)

Optimal. Leaf size=33 \[ \log (-1+x)-\frac {2 \log \left (\frac {x+\frac {1}{5} (2+x)^2}{\left (3+e^5\right ) x}\right )}{x} \]

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Rubi [B]  time = 1.01, antiderivative size = 333, normalized size of antiderivative = 10.09, number of steps used = 34, number of rules used = 9, integrand size = 78, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6741, 6742, 705, 31, 632, 893, 800, 1628, 2525} \begin {gather*} -\frac {2 \log \left (\frac {x^2+9 x+4}{5 \left (3+e^5\right ) x}\right )}{x}+\log (1-x)+\frac {\left (2795+347 \sqrt {65}\right ) \log \left (2 x-\sqrt {65}+9\right )}{1820}+\frac {1}{455} \left (325+41 \sqrt {65}\right ) \log \left (2 x-\sqrt {65}+9\right )-\frac {3}{910} \left (65+11 \sqrt {65}\right ) \log \left (2 x-\sqrt {65}+9\right )-\frac {1}{4} \left (9+\sqrt {65}\right ) \log \left (2 x-\sqrt {65}+9\right )-\frac {1}{260} \left (65-17 \sqrt {65}\right ) \log \left (2 x-\sqrt {65}+9\right )+\frac {\left (845-109 \sqrt {65}\right ) \log \left (2 x-\sqrt {65}+9\right )}{1820}+\frac {\left (845+109 \sqrt {65}\right ) \log \left (2 x+\sqrt {65}+9\right )}{1820}-\frac {1}{260} \left (65+17 \sqrt {65}\right ) \log \left (2 x+\sqrt {65}+9\right )-\frac {1}{4} \left (9-\sqrt {65}\right ) \log \left (2 x+\sqrt {65}+9\right )-\frac {3}{910} \left (65-11 \sqrt {65}\right ) \log \left (2 x+\sqrt {65}+9\right )+\frac {1}{455} \left (325-41 \sqrt {65}\right ) \log \left (2 x+\sqrt {65}+9\right )+\frac {\left (2795-347 \sqrt {65}\right ) \log \left (2 x+\sqrt {65}+9\right )}{1820} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8 + 8*x + 6*x^2 + 7*x^3 + x^4 + (-8 - 10*x + 16*x^2 + 2*x^3)*Log[(4 + 9*x + x^2)/(15*x + 5*E^5*x)])/(-4*
x^2 - 5*x^3 + 8*x^4 + x^5),x]

[Out]

Log[1 - x] + ((845 - 109*Sqrt[65])*Log[9 - Sqrt[65] + 2*x])/1820 - ((65 - 17*Sqrt[65])*Log[9 - Sqrt[65] + 2*x]
)/260 - ((9 + Sqrt[65])*Log[9 - Sqrt[65] + 2*x])/4 - (3*(65 + 11*Sqrt[65])*Log[9 - Sqrt[65] + 2*x])/910 + ((32
5 + 41*Sqrt[65])*Log[9 - Sqrt[65] + 2*x])/455 + ((2795 + 347*Sqrt[65])*Log[9 - Sqrt[65] + 2*x])/1820 + ((2795
- 347*Sqrt[65])*Log[9 + Sqrt[65] + 2*x])/1820 + ((325 - 41*Sqrt[65])*Log[9 + Sqrt[65] + 2*x])/455 - (3*(65 - 1
1*Sqrt[65])*Log[9 + Sqrt[65] + 2*x])/910 - ((9 - Sqrt[65])*Log[9 + Sqrt[65] + 2*x])/4 - ((65 + 17*Sqrt[65])*Lo
g[9 + Sqrt[65] + 2*x])/260 + ((845 + 109*Sqrt[65])*Log[9 + Sqrt[65] + 2*x])/1820 - (2*Log[(4 + 9*x + x^2)/(5*(
3 + E^5)*x)])/x

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 705

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

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 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 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, 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 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8-8 x-6 x^2-7 x^3-x^4-\left (-8-10 x+16 x^2+2 x^3\right ) \log \left (\frac {4+9 x+x^2}{15 x+5 e^5 x}\right )}{x^2 \left (4+5 x-8 x^2-x^3\right )} \, dx\\ &=\int \left (\frac {6}{(-1+x) \left (4+9 x+x^2\right )}-\frac {8}{(-1+x) x^2 \left (4+9 x+x^2\right )}+\frac {8}{(-1+x) x \left (4+9 x+x^2\right )}+\frac {7 x}{(-1+x) \left (4+9 x+x^2\right )}+\frac {x^2}{(-1+x) \left (4+9 x+x^2\right )}+\frac {2 \log \left (\frac {4+9 x+x^2}{5 \left (3+e^5\right ) x}\right )}{x^2}\right ) \, dx\\ &=2 \int \frac {\log \left (\frac {4+9 x+x^2}{5 \left (3+e^5\right ) x}\right )}{x^2} \, dx+6 \int \frac {1}{(-1+x) \left (4+9 x+x^2\right )} \, dx+7 \int \frac {x}{(-1+x) \left (4+9 x+x^2\right )} \, dx-8 \int \frac {1}{(-1+x) x^2 \left (4+9 x+x^2\right )} \, dx+8 \int \frac {1}{(-1+x) x \left (4+9 x+x^2\right )} \, dx+\int \frac {x^2}{(-1+x) \left (4+9 x+x^2\right )} \, dx\\ &=-\frac {2 \log \left (\frac {4+9 x+x^2}{5 \left (3+e^5\right ) x}\right )}{x}+\frac {3}{7} \int \frac {1}{-1+x} \, dx+\frac {3}{7} \int \frac {-10-x}{4+9 x+x^2} \, dx+2 \int \frac {-4+x^2}{x^2 \left (4+9 x+x^2\right )} \, dx+7 \int \left (\frac {1}{14 (-1+x)}+\frac {4-x}{14 \left (4+9 x+x^2\right )}\right ) \, dx-8 \int \left (\frac {1}{14 (-1+x)}-\frac {1}{4 x^2}+\frac {5}{16 x}+\frac {-367-43 x}{112 \left (4+9 x+x^2\right )}\right ) \, dx+8 \int \left (\frac {1}{14 (-1+x)}-\frac {1}{4 x}+\frac {43+5 x}{28 \left (4+9 x+x^2\right )}\right ) \, dx+\int \left (\frac {1}{14 (-1+x)}+\frac {4+13 x}{14 \left (4+9 x+x^2\right )}\right ) \, dx\\ &=-\frac {2}{x}+\log (1-x)-\frac {9 \log (x)}{2}-\frac {2 \log \left (\frac {4+9 x+x^2}{5 \left (3+e^5\right ) x}\right )}{x}-\frac {1}{14} \int \frac {-367-43 x}{4+9 x+x^2} \, dx+\frac {1}{14} \int \frac {4+13 x}{4+9 x+x^2} \, dx+\frac {2}{7} \int \frac {43+5 x}{4+9 x+x^2} \, dx+\frac {1}{2} \int \frac {4-x}{4+9 x+x^2} \, dx+2 \int \left (-\frac {1}{x^2}+\frac {9}{4 x}+\frac {-73-9 x}{4 \left (4+9 x+x^2\right )}\right ) \, dx-\frac {1}{910} \left (3 \left (65-11 \sqrt {65}\right )\right ) \int \frac {1}{\frac {9}{2}+\frac {\sqrt {65}}{2}+x} \, dx-\frac {1}{910} \left (3 \left (65+11 \sqrt {65}\right )\right ) \int \frac {1}{\frac {9}{2}-\frac {\sqrt {65}}{2}+x} \, dx\\ &=\log (1-x)-\frac {3}{910} \left (65+11 \sqrt {65}\right ) \log \left (9-\sqrt {65}+2 x\right )-\frac {3}{910} \left (65-11 \sqrt {65}\right ) \log \left (9+\sqrt {65}+2 x\right )-\frac {2 \log \left (\frac {4+9 x+x^2}{5 \left (3+e^5\right ) x}\right )}{x}+\frac {1}{2} \int \frac {-73-9 x}{4+9 x+x^2} \, dx+\frac {\left (845-109 \sqrt {65}\right ) \int \frac {1}{\frac {9}{2}-\frac {\sqrt {65}}{2}+x} \, dx}{1820}+\frac {1}{455} \left (325-41 \sqrt {65}\right ) \int \frac {1}{\frac {9}{2}+\frac {\sqrt {65}}{2}+x} \, dx+\frac {1}{260} \left (-65+17 \sqrt {65}\right ) \int \frac {1}{\frac {9}{2}-\frac {\sqrt {65}}{2}+x} \, dx-\frac {1}{260} \left (65+17 \sqrt {65}\right ) \int \frac {1}{\frac {9}{2}+\frac {\sqrt {65}}{2}+x} \, dx+\frac {1}{455} \left (325+41 \sqrt {65}\right ) \int \frac {1}{\frac {9}{2}-\frac {\sqrt {65}}{2}+x} \, dx+\frac {\left (845+109 \sqrt {65}\right ) \int \frac {1}{\frac {9}{2}+\frac {\sqrt {65}}{2}+x} \, dx}{1820}-\frac {\left (-2795+347 \sqrt {65}\right ) \int \frac {1}{\frac {9}{2}+\frac {\sqrt {65}}{2}+x} \, dx}{1820}+\frac {\left (2795+347 \sqrt {65}\right ) \int \frac {1}{\frac {9}{2}-\frac {\sqrt {65}}{2}+x} \, dx}{1820}\\ &=\log (1-x)+\frac {\left (845-109 \sqrt {65}\right ) \log \left (9-\sqrt {65}+2 x\right )}{1820}-\frac {1}{260} \left (65-17 \sqrt {65}\right ) \log \left (9-\sqrt {65}+2 x\right )-\frac {3}{910} \left (65+11 \sqrt {65}\right ) \log \left (9-\sqrt {65}+2 x\right )+\frac {1}{455} \left (325+41 \sqrt {65}\right ) \log \left (9-\sqrt {65}+2 x\right )+\frac {\left (2795+347 \sqrt {65}\right ) \log \left (9-\sqrt {65}+2 x\right )}{1820}+\frac {\left (2795-347 \sqrt {65}\right ) \log \left (9+\sqrt {65}+2 x\right )}{1820}+\frac {1}{455} \left (325-41 \sqrt {65}\right ) \log \left (9+\sqrt {65}+2 x\right )-\frac {3}{910} \left (65-11 \sqrt {65}\right ) \log \left (9+\sqrt {65}+2 x\right )-\frac {1}{260} \left (65+17 \sqrt {65}\right ) \log \left (9+\sqrt {65}+2 x\right )+\frac {\left (845+109 \sqrt {65}\right ) \log \left (9+\sqrt {65}+2 x\right )}{1820}-\frac {2 \log \left (\frac {4+9 x+x^2}{5 \left (3+e^5\right ) x}\right )}{x}+\frac {1}{4} \left (-9-\sqrt {65}\right ) \int \frac {1}{\frac {9}{2}-\frac {\sqrt {65}}{2}+x} \, dx+\frac {1}{4} \left (-9+\sqrt {65}\right ) \int \frac {1}{\frac {9}{2}+\frac {\sqrt {65}}{2}+x} \, dx\\ &=\log (1-x)+\frac {\left (845-109 \sqrt {65}\right ) \log \left (9-\sqrt {65}+2 x\right )}{1820}-\frac {1}{260} \left (65-17 \sqrt {65}\right ) \log \left (9-\sqrt {65}+2 x\right )-\frac {1}{4} \left (9+\sqrt {65}\right ) \log \left (9-\sqrt {65}+2 x\right )-\frac {3}{910} \left (65+11 \sqrt {65}\right ) \log \left (9-\sqrt {65}+2 x\right )+\frac {1}{455} \left (325+41 \sqrt {65}\right ) \log \left (9-\sqrt {65}+2 x\right )+\frac {\left (2795+347 \sqrt {65}\right ) \log \left (9-\sqrt {65}+2 x\right )}{1820}+\frac {\left (2795-347 \sqrt {65}\right ) \log \left (9+\sqrt {65}+2 x\right )}{1820}+\frac {1}{455} \left (325-41 \sqrt {65}\right ) \log \left (9+\sqrt {65}+2 x\right )-\frac {3}{910} \left (65-11 \sqrt {65}\right ) \log \left (9+\sqrt {65}+2 x\right )-\frac {1}{4} \left (9-\sqrt {65}\right ) \log \left (9+\sqrt {65}+2 x\right )-\frac {1}{260} \left (65+17 \sqrt {65}\right ) \log \left (9+\sqrt {65}+2 x\right )+\frac {\left (845+109 \sqrt {65}\right ) \log \left (9+\sqrt {65}+2 x\right )}{1820}-\frac {2 \log \left (\frac {4+9 x+x^2}{5 \left (3+e^5\right ) x}\right )}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 35, normalized size = 1.06 \begin {gather*} \log (1-x)-\frac {2 \log \left (\frac {4+9 x+x^2}{5 \left (3+e^5\right ) x}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 + 8*x + 6*x^2 + 7*x^3 + x^4 + (-8 - 10*x + 16*x^2 + 2*x^3)*Log[(4 + 9*x + x^2)/(15*x + 5*E^5*x)]
)/(-4*x^2 - 5*x^3 + 8*x^4 + x^5),x]

[Out]

Log[1 - x] - (2*Log[(4 + 9*x + x^2)/(5*(3 + E^5)*x)])/x

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fricas [A]  time = 0.65, size = 34, normalized size = 1.03 \begin {gather*} \frac {x \log \left (x - 1\right ) - 2 \, \log \left (\frac {x^{2} + 9 \, x + 4}{5 \, {\left (x e^{5} + 3 \, x\right )}}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+16*x^2-10*x-8)*log((x^2+9*x+4)/(5*x*exp(5)+15*x))+x^4+7*x^3+6*x^2+8*x-8)/(x^5+8*x^4-5*x^3-4*
x^2),x, algorithm="fricas")

[Out]

(x*log(x - 1) - 2*log(1/5*(x^2 + 9*x + 4)/(x*e^5 + 3*x)))/x

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giac [A]  time = 0.69, size = 34, normalized size = 1.03 \begin {gather*} \frac {x \log \left (x - 1\right ) - 2 \, \log \left (\frac {x^{2} + 9 \, x + 4}{5 \, {\left (x e^{5} + 3 \, x\right )}}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+16*x^2-10*x-8)*log((x^2+9*x+4)/(5*x*exp(5)+15*x))+x^4+7*x^3+6*x^2+8*x-8)/(x^5+8*x^4-5*x^3-4*
x^2),x, algorithm="giac")

[Out]

(x*log(x - 1) - 2*log(1/5*(x^2 + 9*x + 4)/(x*e^5 + 3*x)))/x

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maple [A]  time = 0.14, size = 32, normalized size = 0.97

method	result	size
norman	\(-\frac {2 \ln \left (\frac {x^{2}+9 x +4}{5 x \,{\mathrm e}^{5}+15 x}\right )}{x}+\ln \left (x -1\right )\)	\(32\)
risch	\(-\frac {2 \ln \left (\frac {x^{2}+9 x +4}{5 x \,{\mathrm e}^{5}+15 x}\right )}{x}+\ln \left (x -1\right )\)	\(32\)
default	\(\ln \left (x -1\right )-\frac {2 \ln \left (\frac {x^{2}+9 x +4}{x}\right )}{x}+\frac {2 \ln \left (3+{\mathrm e}^{5}\right )}{x}+\frac {2 \ln \relax (5)}{x}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^3+16*x^2-10*x-8)*ln((x^2+9*x+4)/(5*x*exp(5)+15*x))+x^4+7*x^3+6*x^2+8*x-8)/(x^5+8*x^4-5*x^3-4*x^2),x,
method=_RETURNVERBOSE)

[Out]

-2*ln((x^2+9*x+4)/(5*x*exp(5)+15*x))/x+ln(x-1)

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maxima [B]  time = 0.71, size = 67, normalized size = 2.03 \begin {gather*} -\frac {{\left (9 \, x + 8\right )} \log \left (x^{2} + 9 \, x + 4\right ) - 2 \, {\left (9 \, x + 4\right )} \log \relax (x) - 8 \, \log \relax (5) - 8 \, \log \left (e^{5} + 3\right ) - 8}{4 \, x} - \frac {2}{x} + \frac {9}{4} \, \log \left (x^{2} + 9 \, x + 4\right ) + \log \left (x - 1\right ) - \frac {9}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+16*x^2-10*x-8)*log((x^2+9*x+4)/(5*x*exp(5)+15*x))+x^4+7*x^3+6*x^2+8*x-8)/(x^5+8*x^4-5*x^3-4*
x^2),x, algorithm="maxima")

[Out]

-1/4*((9*x + 8)*log(x^2 + 9*x + 4) - 2*(9*x + 4)*log(x) - 8*log(5) - 8*log(e^5 + 3) - 8)/x - 2/x + 9/4*log(x^2
+ 9*x + 4) + log(x - 1) - 9/2*log(x)

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mupad [B]  time = 1.21, size = 31, normalized size = 0.94 \begin {gather*} \ln \left (x-1\right )-\frac {2\,\ln \left (\frac {x^2+9\,x+4}{15\,x+5\,x\,{\mathrm {e}}^5}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x - log((9*x + x^2 + 4)/(15*x + 5*x*exp(5)))*(10*x - 16*x^2 - 2*x^3 + 8) + 6*x^2 + 7*x^3 + x^4 - 8)/(4
*x^2 + 5*x^3 - 8*x^4 - x^5),x)

[Out]

log(x - 1) - (2*log((9*x + x^2 + 4)/(15*x + 5*x*exp(5))))/x

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sympy [A]  time = 0.21, size = 27, normalized size = 0.82 \begin {gather*} \log {\left (x - 1 \right )} - \frac {2 \log {\left (\frac {x^{2} + 9 x + 4}{15 x + 5 x e^{5}} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3+16*x**2-10*x-8)*ln((x**2+9*x+4)/(5*x*exp(5)+15*x))+x**4+7*x**3+6*x**2+8*x-8)/(x**5+8*x**4-5
*x**3-4*x**2),x)

[Out]

log(x - 1) - 2*log((x**2 + 9*x + 4)/(15*x + 5*x*exp(5)))/x

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