3.76.30 \(\int \frac {(-120 x^3+1075 x^4-2500 x^5-41 x^6+200 x^7-4 x^9) \log (3)}{25-250 x+625 x^2+10 x^3-50 x^4+x^6} \, dx\)
Optimal. Leaf size=24 \[ x^3 \left (-x+\frac {1}{25-\frac {5}{x}-x^2}\right ) \log (3) \]
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Rubi [C] time = 28.16, antiderivative size = 2412, normalized size of antiderivative =
100.50, number of steps used = 19, number of rules used = 10, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170,
Rules used = {12, 2074, 2101, 2079, 822, 800, 634, 618, 206, 628}
result too large to display
Antiderivative was successfully verified.
[In]
Int[((-120*x^3 + 1075*x^4 - 2500*x^5 - 41*x^6 + 200*x^7 - 4*x^9)*Log[3])/(25 - 250*x + 625*x^2 + 10*x^3 - 50*x
^4 + x^6),x]
[Out]
-(x*Log[3]) - x^4*Log[3] - (150*(9 - I*Sqrt[7419])^(1/3)*Log[3])/(30^(1/3)*(10*15^(1/3) + 2^(1/3)*(9 - I*Sqrt[
7419])^(2/3)) + 6*(9 - I*Sqrt[7419])^(1/3)*x) + (45*2^(2/3)*(9 - I*Sqrt[7419])^(4/3)*(123650 - ((5*3^(1/6)*(24
73*Sqrt[3] + (15*I)*Sqrt[2473])*(10*(45 - (5*I)*Sqrt[7419]))^(1/3) - 2^(2/3)*15^(1/3)*(7419 + (1259*I)*Sqrt[74
19]))*x)/(9 - I*Sqrt[7419])^(2/3))*Log[3])/((10*3^(1/6)*5^(2/3)*(2473*Sqrt[3] + (9*I)*Sqrt[2473]) - 30^(1/3)*(
9 - I*Sqrt[7419])^(2/3)*(2473 + (3*I)*Sqrt[7419]))*((5*15^(2/3))/((9 - I*Sqrt[7419])/2)^(1/3) + ((15*(9 - I*Sq
rt[7419]))/2)^(1/3) + 3*x)*(150 - (750*15^(1/3))/((9 - I*Sqrt[7419])/2)^(2/3) - 2^(1/3)*(15*(9 - I*Sqrt[7419])
)^(2/3) + 6*((5*15^(2/3))/((9 - I*Sqrt[7419])/2)^(1/3) + ((15*(9 - I*Sqrt[7419]))/2)^(1/3))*x - 18*x^2)) - (12
50*Log[3])/(3*(5 - 25*x + x^3)) + (29676000*3^(1/6)*10^(1/3)*(10*5^(1/3)*((747743*I)*Sqrt[3] - 7419*Sqrt[2473]
) - (731111*I)*2^(1/3)*3^(1/6)*(9 - I*Sqrt[7419])^(2/3) + 3071*2^(1/3)*Sqrt[2473]*(27 - (3*I)*Sqrt[7419])^(2/3
))*ArcTanh[(30^(1/3)*((10*I)*(15*(9 - I*Sqrt[7419]))^(1/3) + 2^(1/3)*(9*I + Sqrt[7419])) - (12*I)*(9 - I*Sqrt[
7419])^(2/3)*x)/(6*Sqrt[-(2^(1/3)*3^(1/6)*5^(2/3)*(1223*Sqrt[3] + (9*I)*Sqrt[2473])) + 125*15^(1/3)*(18 - (2*I
)*Sqrt[7419])^(2/3) + (50*I)*(9 - I*Sqrt[7419])^(1/3)*(9*I + Sqrt[7419])])]*Log[3])/((250*2^(2/3)*15^(1/3) - 1
00*(9 - I*Sqrt[7419])^(2/3) + 3^(1/6)*(3*Sqrt[3] - I*Sqrt[2473])*(10*(45 - (5*I)*Sqrt[7419]))^(1/3))*(250*2^(2
/3)*15^(1/3) + 50*(9 - I*Sqrt[7419])^(2/3) + 3^(1/6)*(3*Sqrt[3] - I*Sqrt[2473])*(10*(45 - (5*I)*Sqrt[7419]))^(
1/3))^3*Sqrt[-(2^(1/3)*3^(1/6)*5^(2/3)*(1223*Sqrt[3] + (9*I)*Sqrt[2473])) + 125*15^(1/3)*(18 - (2*I)*Sqrt[7419
])^(2/3) + (50*I)*(9 - I*Sqrt[7419])^(1/3)*(9*I + Sqrt[7419])]) - (10*3^(1/6)*10^(1/3)*(5*5^(1/3)*((1241*I)*Sq
rt[3] - 3*Sqrt[2473]) - (616*I)*2^(1/3)*3^(1/6)*(9 - I*Sqrt[7419])^(2/3) + 2^(1/3)*Sqrt[2473]*(27 - (3*I)*Sqrt
[7419])^(2/3))*ArcTanh[(30^(1/3)*((10*I)*(15*(9 - I*Sqrt[7419]))^(1/3) + 2^(1/3)*(9*I + Sqrt[7419])) - (12*I)*
(9 - I*Sqrt[7419])^(2/3)*x)/(6*Sqrt[-(2^(1/3)*3^(1/6)*5^(2/3)*(1223*Sqrt[3] + (9*I)*Sqrt[2473])) + 125*15^(1/3
)*(18 - (2*I)*Sqrt[7419])^(2/3) + (50*I)*(9 - I*Sqrt[7419])^(1/3)*(9*I + Sqrt[7419])])]*Log[3])/((250*2^(2/3)*
15^(1/3) + 50*(9 - I*Sqrt[7419])^(2/3) + 3^(1/6)*(3*Sqrt[3] - I*Sqrt[2473])*(10*(45 - (5*I)*Sqrt[7419]))^(1/3)
)*Sqrt[-(2^(1/3)*3^(1/6)*5^(2/3)*(1223*Sqrt[3] + (9*I)*Sqrt[2473])) + 125*15^(1/3)*(18 - (2*I)*Sqrt[7419])^(2/
3) + (50*I)*(9 - I*Sqrt[7419])^(1/3)*(9*I + Sqrt[7419])]) - (5*(9 - I*Sqrt[7419])^(1/3)*(50*2^(1/3)*15^(2/3) +
12*(9 - I*Sqrt[7419])^(1/3) + 5*15^(1/3)*(18 - (2*I)*Sqrt[7419])^(2/3))*Log[3]*Log[30^(1/3)*(10*15^(1/3) + 2^
(1/3)*(9 - I*Sqrt[7419])^(2/3)) + 6*(9 - I*Sqrt[7419])^(1/3)*x])/(3*(250*2^(2/3)*15^(1/3) + 50*(9 - I*Sqrt[741
9])^(2/3) + 3^(1/6)*(3*Sqrt[3] - I*Sqrt[2473])*(10*(45 - (5*I)*Sqrt[7419]))^(1/3))) + (5*((7338*I - 18*Sqrt[74
19])*(9 - I*Sqrt[7419])^(2/3) + 5*3^(1/6)*((6115*I)*Sqrt[3] - 45*Sqrt[2473])*(10*(45 - (5*I)*Sqrt[7419]))^(1/3
) + 10*2^(2/3)*15^(1/3)*(8316*I + 299*Sqrt[7419]))*Log[3]*Log[30^(1/3)*(10*15^(1/3) + 2^(1/3)*(9 - I*Sqrt[7419
])^(2/3)) + 6*(9 - I*Sqrt[7419])^(1/3)*x])/(3*(125*2^(2/3)*15^(1/3)*(1223*I - 3*Sqrt[7419]) + (30575*I - 75*Sq
rt[7419])*(9 - I*Sqrt[7419])^(2/3) + 3^(1/6)*((5544*I)*Sqrt[3] + 598*Sqrt[2473])*(10*(45 - (5*I)*Sqrt[7419]))^
(1/3))) + (5*(9 - I*Sqrt[7419])^(2/3)*(12 + (50*15^(2/3))/((9 - I*Sqrt[7419])/2)^(1/3) + 5*2^(2/3)*(15*(9 - I*
Sqrt[7419]))^(1/3))*Log[3]*Log[I*(250*2^(2/3)*15^(1/3) - 50*(9 - I*Sqrt[7419])^(2/3) + 3^(1/6)*(3*Sqrt[3] - I*
Sqrt[2473])*(10*(45 - (5*I)*Sqrt[7419]))^(1/3)) - 2^(2/3)*15^(1/3)*(9*I + Sqrt[7419] + (5*I)*2^(2/3)*(15*(9 -
I*Sqrt[7419]))^(1/3))*x + (6*I)*(9 - I*Sqrt[7419])^(2/3)*x^2])/(6*(250*2^(2/3)*15^(1/3) + 50*(9 - I*Sqrt[7419]
)^(2/3) + 3^(1/6)*(3*Sqrt[3] - I*Sqrt[2473])*(10*(45 - (5*I)*Sqrt[7419]))^(1/3))) - (5*(6*(9 - I*Sqrt[7419])^(
2/3)*(1223 + (3*I)*Sqrt[7419]) + 25*3^(1/6)*(1223*Sqrt[3] + (9*I)*Sqrt[2473])*(10*(45 - (5*I)*Sqrt[7419]))^(1/
3) + 10*2^(2/3)*15^(1/3)*(8316 - (299*I)*Sqrt[7419]))*Log[3]*Log[I*(250*2^(2/3)*15^(1/3) - 50*(9 - I*Sqrt[7419
])^(2/3) + 3^(1/6)*(3*Sqrt[3] - I*Sqrt[2473])*(10*(45 - (5*I)*Sqrt[7419]))^(1/3)) - 2^(2/3)*15^(1/3)*(9*I + Sq
rt[7419] + (5*I)*2^(2/3)*(15*(9 - I*Sqrt[7419]))^(1/3))*x + (6*I)*(9 - I*Sqrt[7419])^(2/3)*x^2])/(6*(125*2^(2/
3)*15^(1/3)*(1223 + (3*I)*Sqrt[7419]) + 25*(9 - I*Sqrt[7419])^(2/3)*(1223 + (3*I)*Sqrt[7419]) + 3^(1/6)*(5544*
Sqrt[3] - (598*I)*Sqrt[2473])*(10*(45 - (5*I)*Sqrt[7419]))^(1/3)))
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 618
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
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 634
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), 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 822
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])
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]
Rule 2079
Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + S
qrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[(e + f*x)^m*Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/
3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r
/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0
]
Rule 2101
Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Qn^(p + 1)
)/(n*(p + 1)*Coeff[Qn, x, n]), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[P
m, x, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && NeQ[p,
-1]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\log (3) \int \frac {-120 x^3+1075 x^4-2500 x^5-41 x^6+200 x^7-4 x^9}{25-250 x+625 x^2+10 x^3-50 x^4+x^6} \, dx\\ &=\log (3) \int \left (-1-4 x^3+\frac {25 \left (3-25 x+50 x^2\right )}{\left (5-25 x+x^3\right )^2}+\frac {5 (-2+5 x)}{5-25 x+x^3}\right ) \, dx\\ &=-x \log (3)-x^4 \log (3)+(5 \log (3)) \int \frac {-2+5 x}{5-25 x+x^3} \, dx+(25 \log (3)) \int \frac {3-25 x+50 x^2}{\left (5-25 x+x^3\right )^2} \, dx\\ &=-x \log (3)-x^4 \log (3)-\frac {1250 \log (3)}{3 \left (5-25 x+x^3\right )}+(5 \log (3)) \int \frac {-2+5 x}{\left (\frac {1}{3} \left (\frac {5\ 15^{2/3}}{\sqrt [3]{\frac {1}{2} \left (9-i \sqrt {7419}\right )}}+\sqrt [3]{\frac {15}{2} \left (9-i \sqrt {7419}\right )}\right )+x\right ) \left (\frac {1}{18} \left (-150+\frac {750 \sqrt [3]{15}}{\left (\frac {1}{2} \left (9-i \sqrt {7419}\right )\right )^{2/3}}+\sqrt [3]{2} \left (15 \left (9-i \sqrt {7419}\right )\right )^{2/3}\right )-\frac {1}{3} \left (\frac {5\ 15^{2/3}}{\sqrt [3]{\frac {1}{2} \left (9-i \sqrt {7419}\right )}}+\sqrt [3]{\frac {15}{2} \left (9-i \sqrt {7419}\right )}\right ) x+x^2\right )} \, dx+\frac {1}{3} (25 \log (3)) \int \frac {1259-75 x}{\left (5-25 x+x^3\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.04 \begin {gather*} -\frac {x^4 \left (6-25 x+x^3\right ) \log (3)}{5-25 x+x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((-120*x^3 + 1075*x^4 - 2500*x^5 - 41*x^6 + 200*x^7 - 4*x^9)*Log[3])/(25 - 250*x + 625*x^2 + 10*x^3
- 50*x^4 + x^6),x]
[Out]
-((x^4*(6 - 25*x + x^3)*Log[3])/(5 - 25*x + x^3))
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fricas [A] time = 0.52, size = 28, normalized size = 1.17 \begin {gather*} -\frac {{\left (x^{7} - 25 \, x^{5} + 6 \, x^{4}\right )} \log \relax (3)}{x^{3} - 25 \, x + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-4*x^9+200*x^7-41*x^6-2500*x^5+1075*x^4-120*x^3)*log(3)/(x^6-50*x^4+10*x^3+625*x^2-250*x+25),x, alg
orithm="fricas")
[Out]
-(x^7 - 25*x^5 + 6*x^4)*log(3)/(x^3 - 25*x + 5)
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giac [A] time = 0.14, size = 30, normalized size = 1.25 \begin {gather*} -{\left (x^{4} + x + \frac {5 \, {\left (5 \, x^{2} - x\right )}}{x^{3} - 25 \, x + 5}\right )} \log \relax (3) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-4*x^9+200*x^7-41*x^6-2500*x^5+1075*x^4-120*x^3)*log(3)/(x^6-50*x^4+10*x^3+625*x^2-250*x+25),x, alg
orithm="giac")
[Out]
-(x^4 + x + 5*(5*x^2 - x)/(x^3 - 25*x + 5))*log(3)
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maple [A] time = 0.05, size = 26, normalized size = 1.08
|
|
|
| method |
result |
size |
|
|
|
| gosper |
\(-\frac {x^{4} \left (x^{3}-25 x +6\right ) \ln \relax (3)}{x^{3}-25 x +5}\) |
\(26\) |
| default |
\(\ln \relax (3) \left (-x^{4}-x +\frac {-25 x^{2}+5 x}{x^{3}-25 x +5}\right )\) |
\(32\) |
| norman |
\(\frac {-6 x^{4} \ln \relax (3)+25 x^{5} \ln \relax (3)-\ln \relax (3) x^{7}}{x^{3}-25 x +5}\) |
\(34\) |
| risch |
\(-x^{4} \ln \relax (3)-x \ln \relax (3)+\frac {\ln \relax (3) \left (-25 x^{2}+5 x \right )}{x^{3}-25 x +5}\) |
\(36\) |
|
|
|
| |
| |
| |
| |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-4*x^9+200*x^7-41*x^6-2500*x^5+1075*x^4-120*x^3)*ln(3)/(x^6-50*x^4+10*x^3+625*x^2-250*x+25),x,method=_RET
URNVERBOSE)
[Out]
-x^4*(x^3-25*x+6)*ln(3)/(x^3-25*x+5)
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maxima [A] time = 0.37, size = 30, normalized size = 1.25 \begin {gather*} -{\left (x^{4} + x + \frac {5 \, {\left (5 \, x^{2} - x\right )}}{x^{3} - 25 \, x + 5}\right )} \log \relax (3) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-4*x^9+200*x^7-41*x^6-2500*x^5+1075*x^4-120*x^3)*log(3)/(x^6-50*x^4+10*x^3+625*x^2-250*x+25),x, alg
orithm="maxima")
[Out]
-(x^4 + x + 5*(5*x^2 - x)/(x^3 - 25*x + 5))*log(3)
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mupad [B] time = 0.12, size = 25, normalized size = 1.04 \begin {gather*} -\frac {x^4\,\ln \relax (3)\,\left (x^3-25\,x+6\right )}{x^3-25\,x+5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(log(3)*(120*x^3 - 1075*x^4 + 2500*x^5 + 41*x^6 - 200*x^7 + 4*x^9))/(625*x^2 - 250*x + 10*x^3 - 50*x^4 +
x^6 + 25),x)
[Out]
-(x^4*log(3)*(x^3 - 25*x + 6))/(x^3 - 25*x + 5)
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sympy [B] time = 0.28, size = 36, normalized size = 1.50 \begin {gather*} - x^{4} \log {\relax (3 )} - x \log {\relax (3 )} - \frac {25 x^{2} \log {\relax (3 )} - 5 x \log {\relax (3 )}}{x^{3} - 25 x + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-4*x**9+200*x**7-41*x**6-2500*x**5+1075*x**4-120*x**3)*ln(3)/(x**6-50*x**4+10*x**3+625*x**2-250*x+2
5),x)
[Out]
-x**4*log(3) - x*log(3) - (25*x**2*log(3) - 5*x*log(3))/(x**3 - 25*x + 5)
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