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

Optimal. Leaf size=181 \[ \frac {5 (302-35 x)}{64009 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}+\frac {15 (7140435 x+2618306)}{14886061058 \left (5 x^2+3 x+2\right )}-\frac {5 (77020 x+223707)}{87308276 \left (5 x^2+3 x+2\right )^2}+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}+\frac {405 \log \left (2 x^2-x+3\right )}{1288408}-\frac {405 \log \left (5 x^2+3 x+2\right )}{1288408}-\frac {880575 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{340783916 \sqrt {23}}+\frac {2768835 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{619080044 \sqrt {31}} \]

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Rubi [A]  time = 0.20, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {974, 1060, 1072, 634, 618, 204, 628} \begin {gather*} \frac {5 (302-35 x)}{64009 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}+\frac {15 (7140435 x+2618306)}{14886061058 \left (5 x^2+3 x+2\right )}-\frac {5 (77020 x+223707)}{87308276 \left (5 x^2+3 x+2\right )^2}+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}+\frac {405 \log \left (2 x^2-x+3\right )}{1288408}-\frac {405 \log \left (5 x^2+3 x+2\right )}{1288408}-\frac {880575 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{340783916 \sqrt {23}}+\frac {2768835 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{619080044 \sqrt {31}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(223707 + 77020*x))/(87308276*(2 + 3*x + 5*x^2)^2) + (13 - 6*x)/(1012*(3 - x + 2*x^2)^2*(2 + 3*x + 5*x^2)^
2) + (5*(302 - 35*x))/(64009*(3 - x + 2*x^2)*(2 + 3*x + 5*x^2)^2) + (15*(2618306 + 7140435*x))/(14886061058*(2
+ 3*x + 5*x^2)) - (880575*ArcTan[(1 - 4*x)/Sqrt[23]])/(340783916*Sqrt[23]) + (2768835*ArcTan[(3 + 10*x)/Sqrt[
31]])/(619080044*Sqrt[31]) + (405*Log[3 - x + 2*x^2])/1288408 - (405*Log[2 + 3*x + 5*x^2])/1288408

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 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]

Rubi steps

\begin {align*} \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^3} \, dx &=\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}-\frac {\int \frac {-4510-4400 x+2310 x^2}{\left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^3} \, dx}{11132}\\ &=\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}+\frac {5 (302-35 x)}{64009 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}-\frac {\int \frac {-16501980-41902300 x+4235000 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^3} \, dx}{61960712}\\ &=-\frac {5 (223707+77020 x)}{87308276 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}+\frac {5 (302-35 x)}{64009 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}-\frac {\int \frac {-28908042240+73138343520 x+24603268800 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2} \, dx}{929658522848}\\ &=-\frac {5 (223707+77020 x)}{87308276 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}+\frac {5 (302-35 x)}{64009 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {15 (2618306+7140435 x)}{14886061058 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-40694764915200+36795056089440 x-100361384481600 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )} \, dx}{6974298238405696}\\ &=-\frac {5 (223707+77020 x)}{87308276 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}+\frac {5 (302-35 x)}{64009 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {15 (2618306+7140435 x)}{14886061058 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-1650046422874080-2122156864428480 x}{3-x+2 x^2} \, dx}{1687780173694178432}-\frac {\int \frac {-2182680087910080+5305392161071200 x}{2+3 x+5 x^2} \, dx}{1687780173694178432}\\ &=-\frac {5 (223707+77020 x)}{87308276 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}+\frac {5 (302-35 x)}{64009 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {15 (2618306+7140435 x)}{14886061058 \left (2+3 x+5 x^2\right )}+\frac {405 \int \frac {-1+4 x}{3-x+2 x^2} \, dx}{1288408}-\frac {405 \int \frac {3+10 x}{2+3 x+5 x^2} \, dx}{1288408}+\frac {880575 \int \frac {1}{3-x+2 x^2} \, dx}{681567832}+\frac {2768835 \int \frac {1}{2+3 x+5 x^2} \, dx}{1238160088}\\ &=-\frac {5 (223707+77020 x)}{87308276 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}+\frac {5 (302-35 x)}{64009 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {15 (2618306+7140435 x)}{14886061058 \left (2+3 x+5 x^2\right )}+\frac {405 \log \left (3-x+2 x^2\right )}{1288408}-\frac {405 \log \left (2+3 x+5 x^2\right )}{1288408}-\frac {880575 \operatorname {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )}{340783916}-\frac {2768835 \operatorname {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )}{619080044}\\ &=-\frac {5 (223707+77020 x)}{87308276 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}+\frac {5 (302-35 x)}{64009 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {15 (2618306+7140435 x)}{14886061058 \left (2+3 x+5 x^2\right )}-\frac {880575 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{340783916 \sqrt {23}}+\frac {2768835 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{619080044 \sqrt {31}}+\frac {405 \log \left (3-x+2 x^2\right )}{1288408}-\frac {405 \log \left (2+3 x+5 x^2\right )}{1288408}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 151, normalized size = 0.83 \begin {gather*} \frac {405 \log \left (2 x^2-x+3\right )}{1288408}-\frac {405 \log \left (5 x^2+3 x+2\right )}{1288408}+\frac {6850 x^3-9275 x^2+11154 x-4342}{345092 \left (10 x^4+x^3+16 x^2+7 x+6\right )^2}+\frac {5 \left (42842610 x^3-5711469 x^2+51156233 x+14085977\right )}{14886061058 \left (10 x^4+x^3+16 x^2+7 x+6\right )}+\frac {880575 \tan ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{340783916 \sqrt {23}}+\frac {2768835 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{619080044 \sqrt {31}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4342 + 11154*x - 9275*x^2 + 6850*x^3)/(345092*(6 + 7*x + 16*x^2 + x^3 + 10*x^4)^2) + (5*(14085977 + 51156233
*x - 5711469*x^2 + 42842610*x^3))/(14886061058*(6 + 7*x + 16*x^2 + x^3 + 10*x^4)) + (880575*ArcTan[(-1 + 4*x)/
Sqrt[23]])/(340783916*Sqrt[23]) + (2768835*ArcTan[(3 + 10*x)/Sqrt[31]])/(619080044*Sqrt[31]) + (405*Log[3 - x
+ 2*x^2])/1288408 - (405*Log[2 + 3*x + 5*x^2])/1288408

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 297, normalized size = 1.64 \begin {gather*} \frac {67202918046000 \, x^{7} - 2238718468800 \, x^{6} + 186872434930060 \, x^{5} + 62827256425340 \, x^{4} + 173919793526820 \, x^{3} + 67376830890 \, \sqrt {31} {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 52466419650 \, \sqrt {23} {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 73595926401690 \, x^{2} - 146799174285 \, {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 146799174285 \, {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )} \log \left (2 \, x^{2} - x + 3\right ) + 78707350628632 \, x + 7381223830244}{467005507511576 \, {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/467005507511576*(67202918046000*x^7 - 2238718468800*x^6 + 186872434930060*x^5 + 62827256425340*x^4 + 1739197
93526820*x^3 + 67376830890*sqrt(31)*(100*x^8 + 20*x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x^3 + 241*x^2 + 84*x
+ 36)*arctan(1/31*sqrt(31)*(10*x + 3)) + 52466419650*sqrt(23)*(100*x^8 + 20*x^7 + 321*x^6 + 172*x^5 + 390*x^4
+ 236*x^3 + 241*x^2 + 84*x + 36)*arctan(1/23*sqrt(23)*(4*x - 1)) + 73595926401690*x^2 - 146799174285*(100*x^8
+ 20*x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x^3 + 241*x^2 + 84*x + 36)*log(5*x^2 + 3*x + 2) + 146799174285*(
100*x^8 + 20*x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x^3 + 241*x^2 + 84*x + 36)*log(2*x^2 - x + 3) + 787073506
28632*x + 7381223830244)/(100*x^8 + 20*x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x^3 + 241*x^2 + 84*x + 36)

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giac [A]  time = 0.22, size = 116, normalized size = 0.64 \begin {gather*} \frac {2768835}{19191481364} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {880575}{7838030068} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {4284261000 \, x^{7} - 142720800 \, x^{6} + 11913326210 \, x^{5} + 4005307690 \, x^{4} + 11087580870 \, x^{3} + 4691822415 \, x^{2} + 5017681412 \, x + 470561254}{29772122116 \, {\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )}^{2}} - \frac {405}{1288408} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {405}{1288408} \, \log \left (2 \, x^{2} - x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2768835/19191481364*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 880575/7838030068*sqrt(23)*arctan(1/23*sqrt(23
)*(4*x - 1)) + 1/29772122116*(4284261000*x^7 - 142720800*x^6 + 11913326210*x^5 + 4005307690*x^4 + 11087580870*
x^3 + 4691822415*x^2 + 5017681412*x + 470561254)/(10*x^4 + x^3 + 16*x^2 + 7*x + 6)^2 - 405/1288408*log(5*x^2 +
3*x + 2) + 405/1288408*log(2*x^2 - x + 3)

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maple [A]  time = 0.01, size = 118, normalized size = 0.65 \begin {gather*} \frac {2768835 \sqrt {31}\, \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right )}{19191481364}+\frac {880575 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{7838030068}+\frac {405 \ln \left (2 x^{2}-x +3\right )}{1288408}-\frac {405 \ln \left (5 x^{2}+3 x +2\right )}{1288408}-\frac {25 \left (-\frac {3013197}{961} x^{3}-\frac {14516062}{4805} x^{2}-\frac {51193868}{24025} x -\frac {5423968}{24025}\right )}{2576816 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {\frac {302907}{529} x^{3}-\frac {368291}{529} x^{2}+\frac {2501587}{2116} x -\frac {665819}{1058}}{644204 \left (2 x^{2}-x +3\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/2576816*(-3013197/961*x^3-14516062/4805*x^2-51193868/24025*x-5423968/24025)/(5*x^2+3*x+2)^2-405/1288408*ln
(5*x^2+3*x+2)+2768835/19191481364*31^(1/2)*arctan(1/31*(10*x+3)*31^(1/2))+1/644204*(302907/529*x^3-368291/529*
x^2+2501587/2116*x-665819/1058)/(2*x^2-x+3)^2+405/1288408*ln(2*x^2-x+3)+880575/7838030068*23^(1/2)*arctan(1/23
*(4*x-1)*23^(1/2))

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maxima [A]  time = 0.97, size = 138, normalized size = 0.76 \begin {gather*} \frac {2768835}{19191481364} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {880575}{7838030068} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {4284261000 \, x^{7} - 142720800 \, x^{6} + 11913326210 \, x^{5} + 4005307690 \, x^{4} + 11087580870 \, x^{3} + 4691822415 \, x^{2} + 5017681412 \, x + 470561254}{29772122116 \, {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )}} - \frac {405}{1288408} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {405}{1288408} \, \log \left (2 \, x^{2} - x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2768835/19191481364*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 880575/7838030068*sqrt(23)*arctan(1/23*sqrt(23
)*(4*x - 1)) + 1/29772122116*(4284261000*x^7 - 142720800*x^6 + 11913326210*x^5 + 4005307690*x^4 + 11087580870*
x^3 + 4691822415*x^2 + 5017681412*x + 470561254)/(100*x^8 + 20*x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x^3 + 2
41*x^2 + 84*x + 36) - 405/1288408*log(5*x^2 + 3*x + 2) + 405/1288408*log(2*x^2 - x + 3)

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mupad [B]  time = 3.59, size = 155, normalized size = 0.86 \begin {gather*} \frac {\frac {21421305\,x^7}{14886061058}-\frac {356802\,x^6}{7443030529}+\frac {1191332621\,x^5}{297721221160}+\frac {400530769\,x^4}{297721221160}+\frac {1108758087\,x^3}{297721221160}+\frac {938364483\,x^2}{595442442320}+\frac {1254420353\,x}{744303052900}+\frac {235280627}{1488606105800}}{x^8+\frac {x^7}{5}+\frac {321\,x^6}{100}+\frac {43\,x^5}{25}+\frac {39\,x^4}{10}+\frac {59\,x^3}{25}+\frac {241\,x^2}{100}+\frac {21\,x}{25}+\frac {9}{25}}+\ln \left (x-\frac {1}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (\frac {405}{1288408}+\frac {\sqrt {23}\,880575{}\mathrm {i}}{15676060136}\right )-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {405}{1288408}+\frac {\sqrt {31}\,2768835{}\mathrm {i}}{38382962728}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {405}{1288408}+\frac {\sqrt {31}\,2768835{}\mathrm {i}}{38382962728}\right )-\ln \left (x-\frac {1}{4}-\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (-\frac {405}{1288408}+\frac {\sqrt {23}\,880575{}\mathrm {i}}{15676060136}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*880575i)/15676060136 + 405/1288408) - log(x - (23^(1/2)*1i)/4 - 1/4)
*((23^(1/2)*880575i)/15676060136 - 405/1288408) - log(x - (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*2768835i)/383829
62728 + 405/1288408) + log(x + (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*2768835i)/38382962728 - 405/1288408) + ((12
54420353*x)/744303052900 + (938364483*x^2)/595442442320 + (1108758087*x^3)/297721221160 + (400530769*x^4)/2977
21221160 + (1191332621*x^5)/297721221160 - (356802*x^6)/7443030529 + (21421305*x^7)/14886061058 + 235280627/14
88606105800)/((21*x)/25 + (241*x^2)/100 + (59*x^3)/25 + (39*x^4)/10 + (43*x^5)/25 + (321*x^6)/100 + x^7/5 + x^
8 + 9/25)

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sympy [A]  time = 0.47, size = 163, normalized size = 0.90 \begin {gather*} \frac {4284261000 x^{7} - 142720800 x^{6} + 11913326210 x^{5} + 4005307690 x^{4} + 11087580870 x^{3} + 4691822415 x^{2} + 5017681412 x + 470561254}{2977212211600 x^{8} + 595442442320 x^{7} + 9556851199236 x^{6} + 5120805003952 x^{5} + 11611127625240 x^{4} + 7026220819376 x^{3} + 7175081429956 x^{2} + 2500858257744 x + 1071796396176} + \frac {405 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{1288408} - \frac {405 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{1288408} + \frac {880575 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{7838030068} + \frac {2768835 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{19191481364} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4284261000*x**7 - 142720800*x**6 + 11913326210*x**5 + 4005307690*x**4 + 11087580870*x**3 + 4691822415*x**2 +
5017681412*x + 470561254)/(2977212211600*x**8 + 595442442320*x**7 + 9556851199236*x**6 + 5120805003952*x**5 +
11611127625240*x**4 + 7026220819376*x**3 + 7175081429956*x**2 + 2500858257744*x + 1071796396176) + 405*log(x**
2 - x/2 + 3/2)/1288408 - 405*log(x**2 + 3*x/5 + 2/5)/1288408 + 880575*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)
/23)/7838030068 + 2768835*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/19191481364

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