3.43 \(\int \frac{1}{(3-x+2 x^2) (2+3 x+5 x^2)^3} \, dx\)

Optimal. Leaf size=115 \[ \frac{65 x+4}{1364 \left (5 x^2+3 x+2\right )^2}+\frac{21605 x+7923}{465124 \left (5 x^2+3 x+2\right )}-\frac{\log \left (2 x^2-x+3\right )}{21296}+\frac{\log \left (5 x^2+3 x+2\right )}{21296}-\frac{45 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{10648 \sqrt{23}}+\frac{847793 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{10232728 \sqrt{31}} \]

[Out]

(4 + 65*x)/(1364*(2 + 3*x + 5*x^2)^2) + (7923 + 21605*x)/(465124*(2 + 3*x + 5*x^2)) - (45*ArcTan[(1 - 4*x)/Sqr
t[23]])/(10648*Sqrt[23]) + (847793*ArcTan[(3 + 10*x)/Sqrt[31]])/(10232728*Sqrt[31]) - Log[3 - x + 2*x^2]/21296
+ Log[2 + 3*x + 5*x^2]/21296

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Rubi [A]  time = 0.124178, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {974, 1060, 1072, 634, 618, 204, 628} \[ \frac{65 x+4}{1364 \left (5 x^2+3 x+2\right )^2}+\frac{21605 x+7923}{465124 \left (5 x^2+3 x+2\right )}-\frac{\log \left (2 x^2-x+3\right )}{21296}+\frac{\log \left (5 x^2+3 x+2\right )}{21296}-\frac{45 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{10648 \sqrt{23}}+\frac{847793 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{10232728 \sqrt{31}} \]

Antiderivative was successfully verified.

[In]

Int[1/((3 - x + 2*x^2)*(2 + 3*x + 5*x^2)^3),x]

[Out]

(4 + 65*x)/(1364*(2 + 3*x + 5*x^2)^2) + (7923 + 21605*x)/(465124*(2 + 3*x + 5*x^2)) - (45*ArcTan[(1 - 4*x)/Sqr
t[23]])/(10648*Sqrt[23]) + (847793*ArcTan[(3 + 10*x)/Sqrt[31]])/(10232728*Sqrt[31]) - Log[3 - x + 2*x^2]/21296
+ Log[2 + 3*x + 5*x^2]/21296

Rule 974

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[((2*a
*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p +
1)*(d + e*x + f*x^2)^(q + 1))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] - Dist[1/
((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*
x^2)^q*Simp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(
p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^
2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f*(p + 1) - c*e*(2*p +
q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e,
f, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e
- b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 1060

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e - b*(c
*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b
*c*d - 2*a*c*e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)
*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a
+ b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f)
)*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C
*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c
*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(
c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*
e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b,
c, d, e, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 -
(b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 1072

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)
), x_Symbol] :> With[{q = c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b*e*f + a^2*f^2}, Dist[1/q, In
t[(A*c^2*d - a*c*C*d - A*b*c*e + a*B*c*e + A*b^2*f - a*b*B*f - a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d - A*c*e +
a*C*e + A*b*f - a*B*f)*x)/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[(c*C*d^2 - B*c*d*e + A*c*e^2 + b*B*d*f - A
*c*d*f - a*C*d*f - A*b*e*f + a*A*f^2 - f*(B*c*d - b*C*d - A*c*e + a*C*e + A*b*f - a*B*f)*x)/(d + e*x + f*x^2),
x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{1}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^3} \, dx &=\frac{4+65 x}{1364 \left (2+3 x+5 x^2\right )^2}-\frac{\int \frac{-5753+3509 x-4290 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2} \, dx}{15004}\\ &=\frac{4+65 x}{1364 \left (2+3 x+5 x^2\right )^2}+\frac{7923+21605 x}{465124 \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-14522420+3833038 x-10456820 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )} \, dx}{112560008}\\ &=\frac{4+65 x}{1364 \left (2+3 x+5 x^2\right )^2}+\frac{7923+21605 x}{465124 \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-58838186+5116364 x}{3-x+2 x^2} \, dx}{27239521936}-\frac{\int \frac{-1132249756-12790910 x}{2+3 x+5 x^2} \, dx}{27239521936}\\ &=\frac{4+65 x}{1364 \left (2+3 x+5 x^2\right )^2}+\frac{7923+21605 x}{465124 \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-1+4 x}{3-x+2 x^2} \, dx}{21296}+\frac{\int \frac{3+10 x}{2+3 x+5 x^2} \, dx}{21296}+\frac{45 \int \frac{1}{3-x+2 x^2} \, dx}{21296}+\frac{847793 \int \frac{1}{2+3 x+5 x^2} \, dx}{20465456}\\ &=\frac{4+65 x}{1364 \left (2+3 x+5 x^2\right )^2}+\frac{7923+21605 x}{465124 \left (2+3 x+5 x^2\right )}-\frac{\log \left (3-x+2 x^2\right )}{21296}+\frac{\log \left (2+3 x+5 x^2\right )}{21296}-\frac{45 \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,-1+4 x\right )}{10648}-\frac{847793 \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )}{10232728}\\ &=\frac{4+65 x}{1364 \left (2+3 x+5 x^2\right )^2}+\frac{7923+21605 x}{465124 \left (2+3 x+5 x^2\right )}-\frac{45 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{10648 \sqrt{23}}+\frac{847793 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{10232728 \sqrt{31}}-\frac{\log \left (3-x+2 x^2\right )}{21296}+\frac{\log \left (2+3 x+5 x^2\right )}{21296}\\ \end{align*}

Mathematica [A]  time = 0.143564, size = 104, normalized size = 0.9 \[ \frac{31 \left (\frac{44 \left (108025 x^3+104430 x^2+89144 x+17210\right )}{\left (5 x^2+3 x+2\right )^2}-961 \log \left (2 x^2-x+3\right )+961 \log \left (5 x^2+3 x+2\right )\right )+1695586 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{634429136}+\frac{45 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{10648 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((3 - x + 2*x^2)*(2 + 3*x + 5*x^2)^3),x]

[Out]

(45*ArcTan[(-1 + 4*x)/Sqrt[23]])/(10648*Sqrt[23]) + (1695586*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt[31]] + 31*((44*(1
7210 + 89144*x + 104430*x^2 + 108025*x^3))/(2 + 3*x + 5*x^2)^2 - 961*Log[3 - x + 2*x^2] + 961*Log[2 + 3*x + 5*
x^2]))/634429136

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Maple [A]  time = 0.049, size = 89, normalized size = 0.8 \begin{align*}{\frac{25}{10648\, \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{2}} \left ({\frac{95062\,{x}^{3}}{961}}+{\frac{459492\,{x}^{2}}{4805}}+{\frac{1961168\,x}{24025}}+{\frac{75724}{4805}} \right ) }+{\frac{\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{21296}}+{\frac{847793\,\sqrt{31}}{317214568}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) }-{\frac{\ln \left ( 2\,{x}^{2}-x+3 \right ) }{21296}}+{\frac{45\,\sqrt{23}}{244904}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{23}}{23}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x^2-x+3)/(5*x^2+3*x+2)^3,x)

[Out]

25/10648*(95062/961*x^3+459492/4805*x^2+1961168/24025*x+75724/4805)/(5*x^2+3*x+2)^2+1/21296*ln(5*x^2+3*x+2)+84
7793/317214568*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)-1/21296*ln(2*x^2-x+3)+45/244904*23^(1/2)*arctan(1/23*(-
1+4*x)*23^(1/2))

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Maxima [A]  time = 1.43887, size = 132, normalized size = 1.15 \begin{align*} \frac{847793}{317214568} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{45}{244904} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{108025 \, x^{3} + 104430 \, x^{2} + 89144 \, x + 17210}{465124 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} + \frac{1}{21296} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac{1}{21296} \, \log \left (2 \, x^{2} - x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^2-x+3)/(5*x^2+3*x+2)^3,x, algorithm="maxima")

[Out]

847793/317214568*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 45/244904*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)
) + 1/465124*(108025*x^3 + 104430*x^2 + 89144*x + 17210)/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) + 1/21296*log(5
*x^2 + 3*x + 2) - 1/21296*log(2*x^2 - x + 3)

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Fricas [A]  time = 1.0267, size = 578, normalized size = 5.03 \begin{align*} \frac{3388960300 \, x^{3} + 38998478 \, \sqrt{31}{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + 2681190 \, \sqrt{23}{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + 3276177960 \, x^{2} + 685193 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 685193 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \log \left (2 \, x^{2} - x + 3\right ) + 2796625568 \, x + 539912120}{14591870128 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^2-x+3)/(5*x^2+3*x+2)^3,x, algorithm="fricas")

[Out]

1/14591870128*(3388960300*x^3 + 38998478*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*arctan(1/31*sqrt(31)*(
10*x + 3)) + 2681190*sqrt(23)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*arctan(1/23*sqrt(23)*(4*x - 1)) + 32761779
60*x^2 + 685193*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*log(5*x^2 + 3*x + 2) - 685193*(25*x^4 + 30*x^3 + 29*x^2
+ 12*x + 4)*log(2*x^2 - x + 3) + 2796625568*x + 539912120)/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)

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Sympy [A]  time = 0.399983, size = 119, normalized size = 1.03 \begin{align*} \frac{108025 x^{3} + 104430 x^{2} + 89144 x + 17210}{11628100 x^{4} + 13953720 x^{3} + 13488596 x^{2} + 5581488 x + 1860496} - \frac{\log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{21296} + \frac{\log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{21296} + \frac{45 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{244904} + \frac{847793 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{317214568} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x**2-x+3)/(5*x**2+3*x+2)**3,x)

[Out]

(108025*x**3 + 104430*x**2 + 89144*x + 17210)/(11628100*x**4 + 13953720*x**3 + 13488596*x**2 + 5581488*x + 186
0496) - log(x**2 - x/2 + 3/2)/21296 + log(x**2 + 3*x/5 + 2/5)/21296 + 45*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(
23)/23)/244904 + 847793*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/317214568

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Giac [A]  time = 1.25983, size = 119, normalized size = 1.03 \begin{align*} \frac{847793}{317214568} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{45}{244904} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{108025 \, x^{3} + 104430 \, x^{2} + 89144 \, x + 17210}{465124 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} + \frac{1}{21296} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac{1}{21296} \, \log \left (2 \, x^{2} - x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^2-x+3)/(5*x^2+3*x+2)^3,x, algorithm="giac")

[Out]

847793/317214568*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 45/244904*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)
) + 1/465124*(108025*x^3 + 104430*x^2 + 89144*x + 17210)/(5*x^2 + 3*x + 2)^2 + 1/21296*log(5*x^2 + 3*x + 2) -
1/21296*log(2*x^2 - x + 3)