3.70.40 \(\int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx\)
Optimal. Leaf size=29 \[ x-\frac {1}{x-\frac {6 (5-4 x)}{x (1+2 x)}}-\log (x) \]
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Rubi [B] time = 7.46, antiderivative size = 1356, normalized size of antiderivative = 46.76,
number of steps used = 20, number of rules used = 10, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.149, Rules used
= {2074, 2101, 2081, 2079, 822, 800, 634, 618, 204, 628}
result too large to display
Warning: Unable to verify antiderivative.
[In]
Int[(-900 + 2370*x - 1836*x^2 + 541*x^3 - 165*x^4 + 97*x^5 + 4*x^7)/(900*x - 1440*x^2 + 516*x^3 - 72*x^4 + 97*
x^5 + 4*x^6 + 4*x^7),x]
[Out]
x - 16/(30 - 24*x - x^2 - 2*x^3) - (6*(1835 + 18*Sqrt[19418])^(1/3))/(143 - (1835 + 18*Sqrt[19418])^(2/3) + (1
835 + 18*Sqrt[19418])^(1/3)*(1 + 6*x)) + (108*(2*(349524 + 1835*Sqrt[19418]))^(1/3)*(19418 - ((9709 + 619*Sqrt
[19418] - (1835 + 18*Sqrt[19418])^(1/3)*(9709 + 47*Sqrt[19418]))*(1 + 6*x))/(1835 + 18*Sqrt[19418])^(2/3)))/(9
709^(2/3)*(143 + (1835 + 18*Sqrt[19418])^(2/3))*(1 + 143/(1835 + 18*Sqrt[19418])^(1/3) - (1835 + 18*Sqrt[19418
])^(1/3) + 6*x)*(143 + 20449/(1835 + 18*Sqrt[19418])^(2/3) + (1835 + 18*Sqrt[19418])^(2/3) - ((143 - (1835 + 1
8*Sqrt[19418])^(2/3))*(1 + 6*x))/(1835 + 18*Sqrt[19418])^(1/3) + (1 + 6*x)^2)) + (6*(3394489 + 23736*Sqrt[1941
8] + (5593 - 12*Sqrt[19418])*(1835 + 18*Sqrt[19418])^(2/3))*Sqrt[3/(9658657 + 66060*Sqrt[19418] + 20449*(1835
+ 18*Sqrt[19418])^(2/3) + 286*(1835 + 18*Sqrt[19418])^(4/3))]*ArcTan[(1835 + 18*Sqrt[19418] - 143*(1835 + 18*S
qrt[19418])^(1/3) + 2*(1835 + 18*Sqrt[19418])^(2/3)*(1 + 6*x))/Sqrt[3*(9658657 + 66060*Sqrt[19418] + 20449*(18
35 + 18*Sqrt[19418])^(2/3) + 286*(1835 + 18*Sqrt[19418])^(4/3))]])/(20449 - 143*(1835 + 18*Sqrt[19418])^(2/3)
+ (1835 + 18*Sqrt[19418])^(4/3)) - (150994368*Sqrt[3/(9658657 + 66060*Sqrt[19418] + 20449*(1835 + 18*Sqrt[1941
8])^(2/3) + 286*(1835 + 18*Sqrt[19418])^(4/3))]*(63233632048153 + 453497825892*Sqrt[19418] + (1835 + 18*Sqrt[1
9418])^(2/3)*(38627831641 + 253569696*Sqrt[19418]))*ArcTan[(1835 + 18*Sqrt[19418] - 143*(1835 + 18*Sqrt[19418]
)^(1/3) + 2*(1835 + 18*Sqrt[19418])^(2/3)*(1 + 6*x))/Sqrt[3*(9658657 + 66060*Sqrt[19418] + 20449*(1835 + 18*Sq
rt[19418])^(2/3) + 286*(1835 + 18*Sqrt[19418])^(4/3))]])/((143 + (1835 + 18*Sqrt[19418])^(2/3))^2*(20449 - 143
*(1835 + 18*Sqrt[19418])^(2/3) + (1835 + 18*Sqrt[19418])^(4/3))^3) - Log[x] - ((1835 + 18*Sqrt[19418] - 143*(1
835 + 18*Sqrt[19418])^(1/3) + 2*(1835 + 18*Sqrt[19418])^(2/3))*Log[3*(1238 + Sqrt[19418] + (94 + Sqrt[19418])*
(1835 + 18*Sqrt[19418])^(1/3) + 8*(1835 + 18*Sqrt[19418])^(2/3)) + 1835*x + 18*Sqrt[19418]*x - 143*(1835 + 18*
Sqrt[19418])^(1/3)*x + 2*(1835 + 18*Sqrt[19418])^(2/3)*x + 6*(1835 + 18*Sqrt[19418])^(2/3)*x^2])/(20449 - 143*
(1835 + 18*Sqrt[19418])^(2/3) + (1835 + 18*Sqrt[19418])^(4/3)) + (25165728*(40813191035 + 295075926*Sqrt[19418
] - 143*(1835 + 18*Sqrt[19418])^(1/3)*(9658657 + 66060*Sqrt[19418]) + 2*(1835 + 18*Sqrt[19418])^(2/3)*(9658657
+ 66060*Sqrt[19418]))*Log[3*(1238 + Sqrt[19418] + (94 + Sqrt[19418])*(1835 + 18*Sqrt[19418])^(1/3) + 8*(1835
+ 18*Sqrt[19418])^(2/3)) + 1835*x + 18*Sqrt[19418]*x - 143*(1835 + 18*Sqrt[19418])^(1/3)*x + 2*(1835 + 18*Sqrt
[19418])^(2/3)*x + 6*(1835 + 18*Sqrt[19418])^(2/3)*x^2])/((143 + (1835 + 18*Sqrt[19418])^(2/3))^2*(20449 - 143
*(1835 + 18*Sqrt[19418])^(2/3) + (1835 + 18*Sqrt[19418])^(4/3))^3) + (50331456*(1835 + 18*Sqrt[19418])^(7/3)*(
143 - 2*(1835 + 18*Sqrt[19418])^(1/3) - (1835 + 18*Sqrt[19418])^(2/3))*Log[143 - (1835 + 18*Sqrt[19418])^(2/3)
+ (1835 + 18*Sqrt[19418])^(1/3)*(1 + 6*x)])/((143 + (1835 + 18*Sqrt[19418])^(2/3))^2*(20449 - 143*(1835 + 18*
Sqrt[19418])^(2/3) + (1835 + 18*Sqrt[19418])^(4/3))^3) - (2*(1835 + 18*Sqrt[19418])^(1/3)*(11 - (1835 + 18*Sqr
t[19418])^(1/3))*(13 + (1835 + 18*Sqrt[19418])^(1/3))*Log[143 - (1835 + 18*Sqrt[19418])^(2/3) + (1835 + 18*Sqr
t[19418])^(1/3)*(1 + 6*x)])/(20449 - 143*(1835 + 18*Sqrt[19418])^(2/3) + (1835 + 18*Sqrt[19418])^(4/3))
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[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 2081
Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C
oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2
)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x
] && PolyQ[P3, x, 3]
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} &=\int \left (1-\frac {1}{x}-\frac {12 \left (-5-13 x+8 x^2\right )}{\left (-30+24 x+x^2+2 x^3\right )^2}+\frac {1+2 x}{-30+24 x+x^2+2 x^3}\right ) \, dx\\ &=x-\log (x)-12 \int \frac {-5-13 x+8 x^2}{\left (-30+24 x+x^2+2 x^3\right )^2} \, dx+\int \frac {1+2 x}{-30+24 x+x^2+2 x^3} \, dx\\ &=x-\frac {16}{30-24 x-x^2-2 x^3}-\log (x)-2 \int \frac {-222-94 x}{\left (-30+24 x+x^2+2 x^3\right )^2} \, dx+\operatorname {Subst}\left (\int \frac {\frac {2}{3}+2 x}{-\frac {1835}{54}+\frac {143 x}{6}+2 x^3} \, dx,x,\frac {1}{6}+x\right )\\ &=x-\frac {16}{30-24 x-x^2-2 x^3}-\log (x)-2 \operatorname {Subst}\left (\int \frac {-\frac {619}{3}-94 x}{\left (-\frac {1835}{54}+\frac {143 x}{6}+2 x^3\right )^2} \, dx,x,\frac {1}{6}+x\right )+4 \operatorname {Subst}\left (\int \frac {\frac {2}{3}+2 x}{\left (\frac {143-\left (1835+18 \sqrt {19418}\right )^{2/3}}{3 \sqrt [3]{1835+18 \sqrt {19418}}}+2 x\right ) \left (\frac {1}{9} \left (143+\frac {20449}{\left (1835+18 \sqrt {19418}\right )^{2/3}}+\left (1835+18 \sqrt {19418}\right )^{2/3}\right )-\frac {2 \left (143-\left (1835+18 \sqrt {19418}\right )^{2/3}\right ) x}{3 \sqrt [3]{1835+18 \sqrt {19418}}}+4 x^2\right )} \, dx,x,\frac {1}{6}+x\right )\\ &=x-\frac {16}{30-24 x-x^2-2 x^3}-\log (x)+4 \operatorname {Subst}\left (\int \left (\frac {3 \left (1835+18 \sqrt {19418}\right )^{2/3} \left (-143+2 \sqrt [3]{1835+18 \sqrt {19418}}+\left (1835+18 \sqrt {19418}\right )^{2/3}\right )}{\left (20449-143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\left (1835+18 \sqrt {19418}\right )^{4/3}\right ) \left (143-\left (1835+18 \sqrt {19418}\right )^{2/3}+6 \sqrt [3]{1835+18 \sqrt {19418}} x\right )}+\frac {3 \left (1835+18 \sqrt {19418}\right )^{2/3} \left (13109-72 \sqrt {19418}+143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\sqrt [3]{1835+18 \sqrt {19418}} \left (2407+18 \sqrt {19418}\right )-6 \left (1835+18 \sqrt {19418}-143 \sqrt [3]{1835+18 \sqrt {19418}}+2 \left (1835+18 \sqrt {19418}\right )^{2/3}\right ) x\right )}{\left (20449-143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\left (1835+18 \sqrt {19418}\right )^{4/3}\right ) \left (20449+143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\left (1835+18 \sqrt {19418}\right )^{4/3}+6 \left (1835+18 \sqrt {19418}-143 \sqrt [3]{1835+18 \sqrt {19418}}\right ) x+36 \left (1835+18 \sqrt {19418}\right )^{2/3} x^2\right )}\right ) \, dx,x,\frac {1}{6}+x\right )-32 \operatorname {Subst}\left (\int \frac {-\frac {619}{3}-94 x}{\left (\frac {143-\left (1835+18 \sqrt {19418}\right )^{2/3}}{3 \sqrt [3]{1835+18 \sqrt {19418}}}+2 x\right )^2 \left (\frac {1}{9} \left (143+\frac {20449}{\left (1835+18 \sqrt {19418}\right )^{2/3}}+\left (1835+18 \sqrt {19418}\right )^{2/3}\right )-\frac {2 \left (143-\left (1835+18 \sqrt {19418}\right )^{2/3}\right ) x}{3 \sqrt [3]{1835+18 \sqrt {19418}}}+4 x^2\right )^2} \, dx,x,\frac {1}{6}+x\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 31, normalized size = 1.07 \begin {gather*} x+\frac {-x-2 x^2}{-30+24 x+x^2+2 x^3}-\log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-900 + 2370*x - 1836*x^2 + 541*x^3 - 165*x^4 + 97*x^5 + 4*x^7)/(900*x - 1440*x^2 + 516*x^3 - 72*x^4
+ 97*x^5 + 4*x^6 + 4*x^7),x]
[Out]
x + (-x - 2*x^2)/(-30 + 24*x + x^2 + 2*x^3) - Log[x]
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fricas [A] time = 0.84, size = 50, normalized size = 1.72 \begin {gather*} \frac {2 \, x^{4} + x^{3} + 22 \, x^{2} - {\left (2 \, x^{3} + x^{2} + 24 \, x - 30\right )} \log \relax (x) - 31 \, x}{2 \, x^{3} + x^{2} + 24 \, x - 30} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((4*x^7+97*x^5-165*x^4+541*x^3-1836*x^2+2370*x-900)/(4*x^7+4*x^6+97*x^5-72*x^4+516*x^3-1440*x^2+900*x
),x, algorithm="fricas")
[Out]
(2*x^4 + x^3 + 22*x^2 - (2*x^3 + x^2 + 24*x - 30)*log(x) - 31*x)/(2*x^3 + x^2 + 24*x - 30)
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giac [A] time = 0.27, size = 31, normalized size = 1.07 \begin {gather*} x - \frac {2 \, x^{2} + x}{2 \, x^{3} + x^{2} + 24 \, x - 30} - \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((4*x^7+97*x^5-165*x^4+541*x^3-1836*x^2+2370*x-900)/(4*x^7+4*x^6+97*x^5-72*x^4+516*x^3-1440*x^2+900*x
),x, algorithm="giac")
[Out]
x - (2*x^2 + x)/(2*x^3 + x^2 + 24*x - 30) - log(abs(x))
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maple [A] time = 0.04, size = 32, normalized size = 1.10
|
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| method |
result |
size |
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| default |
\(x -\ln \relax (x )+\frac {-x^{2}-\frac {1}{2} x}{x^{3}+\frac {1}{2} x^{2}+12 x -15}\) |
\(32\) |
| risch |
\(x -\ln \relax (x )+\frac {-x^{2}-\frac {1}{2} x}{x^{3}+\frac {1}{2} x^{2}+12 x -15}\) |
\(32\) |
| norman |
\(\frac {2 x^{4}-43 x^{3}-559 x +660}{2 x^{3}+x^{2}+24 x -30}-\ln \relax (x )\) |
\(37\) |
|
|
|
| |
| |
| |
| |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((4*x^7+97*x^5-165*x^4+541*x^3-1836*x^2+2370*x-900)/(4*x^7+4*x^6+97*x^5-72*x^4+516*x^3-1440*x^2+900*x),x,me
thod=_RETURNVERBOSE)
[Out]
x-ln(x)+(-x^2-1/2*x)/(x^3+1/2*x^2+12*x-15)
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maxima [A] time = 0.38, size = 30, normalized size = 1.03 \begin {gather*} x - \frac {2 \, x^{2} + x}{2 \, x^{3} + x^{2} + 24 \, x - 30} - \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((4*x^7+97*x^5-165*x^4+541*x^3-1836*x^2+2370*x-900)/(4*x^7+4*x^6+97*x^5-72*x^4+516*x^3-1440*x^2+900*x
),x, algorithm="maxima")
[Out]
x - (2*x^2 + x)/(2*x^3 + x^2 + 24*x - 30) - log(x)
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mupad [B] time = 0.07, size = 30, normalized size = 1.03 \begin {gather*} x-\ln \relax (x)-\frac {x^2+\frac {x}{2}}{x^3+\frac {x^2}{2}+12\,x-15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2370*x - 1836*x^2 + 541*x^3 - 165*x^4 + 97*x^5 + 4*x^7 - 900)/(900*x - 1440*x^2 + 516*x^3 - 72*x^4 + 97*x
^5 + 4*x^6 + 4*x^7),x)
[Out]
x - log(x) - (x/2 + x^2)/(12*x + x^2/2 + x^3 - 15)
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sympy [A] time = 0.14, size = 26, normalized size = 0.90 \begin {gather*} x + \frac {- 2 x^{2} - x}{2 x^{3} + x^{2} + 24 x - 30} - \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((4*x**7+97*x**5-165*x**4+541*x**3-1836*x**2+2370*x-900)/(4*x**7+4*x**6+97*x**5-72*x**4+516*x**3-1440
*x**2+900*x),x)
[Out]
x + (-2*x**2 - x)/(2*x**3 + x**2 + 24*x - 30) - log(x)
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