3.1.84 \(\int \frac {1}{\sqrt {3-x+2 x^2} (2+3 x+5 x^2)^2} \, dx\) [84]

Optimal. Leaf size=188 \[ \frac {(4+65 x) \sqrt {3-x+2 x^2}}{682 \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {1}{682} \left (2343727+1678700 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (2343727+1678700 \sqrt {2}\right )}} \left (2119+1816 \sqrt {2}+\left (5751+3935 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1364}-\frac {\sqrt {\frac {1}{682} \left (-2343727+1678700 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (-2343727+1678700 \sqrt {2}\right )}} \left (2119-1816 \sqrt {2}+\left (5751-3935 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1364} \]

[Out]

1/682*(4+65*x)*(2*x^2-x+3)^(1/2)/(5*x^2+3*x+2)-1/930248*arctanh(1/31*(2119+x*(5751-3935*2^(1/2))-1816*2^(1/2))
*341^(1/2)/(-2343727+1678700*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(-1598421814+1144873400*2^(1/2))^(1/2)+1/930248
*arctan(1/31*(2119+1816*2^(1/2)+x*(5751+3935*2^(1/2)))*341^(1/2)/(2343727+1678700*2^(1/2))^(1/2)/(2*x^2-x+3)^(
1/2))*(1598421814+1144873400*2^(1/2))^(1/2)

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Rubi [A]
time = 0.28, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {988, 1049, 1043, 212, 210} \begin {gather*} \frac {\sqrt {\frac {1}{682} \left (2343727+1678700 \sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {11}{31 \left (2343727+1678700 \sqrt {2}\right )}} \left (\left (5751+3935 \sqrt {2}\right ) x+1816 \sqrt {2}+2119\right )}{\sqrt {2 x^2-x+3}}\right )}{1364}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{682 \left (5 x^2+3 x+2\right )}-\frac {\sqrt {\frac {1}{682} \left (1678700 \sqrt {2}-2343727\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (1678700 \sqrt {2}-2343727\right )}} \left (\left (5751-3935 \sqrt {2}\right ) x-1816 \sqrt {2}+2119\right )}{\sqrt {2 x^2-x+3}}\right )}{1364} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4 + 65*x)*Sqrt[3 - x + 2*x^2])/(682*(2 + 3*x + 5*x^2)) + (Sqrt[(2343727 + 1678700*Sqrt[2])/682]*ArcTan[(Sqrt
[11/(31*(2343727 + 1678700*Sqrt[2]))]*(2119 + 1816*Sqrt[2] + (5751 + 3935*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/1
364 - (Sqrt[(-2343727 + 1678700*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-2343727 + 1678700*Sqrt[2]))]*(2119 - 1816
*Sqrt[2] + (5751 - 3935*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/1364

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 988

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 1043

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> Dist[-2*g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, S
imp[g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[
b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(
c*e - b*f), 0]

Rule 1049

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Dist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*
d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - D
ist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x
+ c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e
^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx &=\frac {(4+65 x) \sqrt {3-x+2 x^2}}{682 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-1826+\frac {2255 x}{2}}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{7502}\\ &=\frac {(4+65 x) \sqrt {3-x+2 x^2}}{682 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {\frac {121}{2} \left (537-332 \sqrt {2}\right )-\frac {121}{2} \left (127-205 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{165044 \sqrt {2}}+\frac {\int \frac {\frac {121}{2} \left (537+332 \sqrt {2}\right )-\frac {121}{2} \left (127+205 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{165044 \sqrt {2}}\\ &=\frac {(4+65 x) \sqrt {3-x+2 x^2}}{682 \left (2+3 x+5 x^2\right )}-\frac {1}{496} \left (11 \left (3357400-2343727 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {453871}{4} \left (2343727-1678700 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {\frac {121}{2} \left (2119-1816 \sqrt {2}\right )+\frac {121}{2} \left (5751-3935 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )-\frac {1}{496} \left (11 \left (3357400+2343727 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {453871}{4} \left (2343727+1678700 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {\frac {121}{2} \left (2119+1816 \sqrt {2}\right )+\frac {121}{2} \left (5751+3935 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )\\ &=\frac {(4+65 x) \sqrt {3-x+2 x^2}}{682 \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {1}{682} \left (2343727+1678700 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (2343727+1678700 \sqrt {2}\right )}} \left (2119+1816 \sqrt {2}+\left (5751+3935 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1364}-\frac {\sqrt {\frac {1}{682} \left (-2343727+1678700 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (-2343727+1678700 \sqrt {2}\right )}} \left (2119-1816 \sqrt {2}+\left (5751-3935 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1364}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.36, size = 230, normalized size = 1.22 \begin {gather*} \frac {(4+65 x) \sqrt {3-x+2 x^2}}{682 \left (2+3 x+5 x^2\right )}+\frac {\text {RootSum}\left [-10580-2024 \sqrt {2} \text {$\#$1}+68 \text {$\#$1}^2+44 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-9430 \sqrt {2} \log \left (\sqrt {2} (-1+4 x)-4 \sqrt {3-x+2 x^2}+\text {$\#$1}\right )+4492 \log \left (\sqrt {2} (-1+4 x)-4 \sqrt {3-x+2 x^2}+\text {$\#$1}\right ) \text {$\#$1}+205 \sqrt {2} \log \left (\sqrt {2} (-1+4 x)-4 \sqrt {3-x+2 x^2}+\text {$\#$1}\right ) \text {$\#$1}^2}{-506 \sqrt {2}+34 \text {$\#$1}+33 \sqrt {2} \text {$\#$1}^2-5 \text {$\#$1}^3}\&\right ]}{682 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4 + 65*x)*Sqrt[3 - x + 2*x^2])/(682*(2 + 3*x + 5*x^2)) + RootSum[-10580 - 2024*Sqrt[2]*#1 + 68*#1^2 + 44*Sqr
t[2]*#1^3 - 5*#1^4 & , (-9430*Sqrt[2]*Log[Sqrt[2]*(-1 + 4*x) - 4*Sqrt[3 - x + 2*x^2] + #1] + 4492*Log[Sqrt[2]*
(-1 + 4*x) - 4*Sqrt[3 - x + 2*x^2] + #1]*#1 + 205*Sqrt[2]*Log[Sqrt[2]*(-1 + 4*x) - 4*Sqrt[3 - x + 2*x^2] + #1]
*#1^2)/(-506*Sqrt[2] + 34*#1 + 33*Sqrt[2]*#1^2 - 5*#1^3) & ]/(682*Sqrt[2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(5224\) vs. \(2(140)=280\).
time = 0.63, size = 5225, normalized size = 27.79 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2102 vs. \(2 (140) = 280\).
time = 3.57, size = 2102, normalized size = 11.18 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/263043507934399808*(8422204*563606738^(1/4)*sqrt(33574)*sqrt(341)*sqrt(2)*(5*x^2 + 3*x + 2)*sqrt(2343727*sqr
t(2) + 3357400)*arctan(1/7101900221517254683789*(47876524*sqrt(33574)*(22*563606738^(3/4)*sqrt(341)*(2950932*x
^7 - 11691762*x^6 + 24397746*x^5 - 40053004*x^4 + 20309552*x^3 - 10145376*x^2 - sqrt(2)*(2248634*x^7 - 8421787
*x^6 + 17801494*x^5 - 27869789*x^4 + 13808040*x^3 - 6172200*x^2 - 15724800*x + 10596096) - 21192192*x + 157248
00) + 520397*563606738^(1/4)*sqrt(341)*(226651*x^7 - 3496629*x^6 + 18614024*x^5 - 42860780*x^4 + 55586592*x^3
- 36274464*x^2 - sqrt(2)*(168871*x^7 - 2579646*x^6 + 13533020*x^5 - 30582616*x^4 + 39345120*x^3 - 23947200*x^2
- 28449792*x + 19450368) - 38900736*x + 28449792))*sqrt(2*x^2 - x + 3)*sqrt(2343727*sqrt(2) + 3357400) + 2016
0232886887690715272*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1385256*x^5 + 1549144*x^4 - 642048
*x^3 - 98496*x^2 - sqrt(2)*(8746*x^8 - 102335*x^7 + 396104*x^6 - 783113*x^5 + 1320710*x^4 - 752088*x^3 + 39614
4*x^2 + 546048*x - 539136) + 1154304*x - 456192) - 2*sqrt(33574/2191)*(sqrt(33574)*(22*563606738^(3/4)*sqrt(34
1)*(10257392*x^7 - 14773368*x^6 + 47877288*x^5 - 20710528*x^4 + 26321472*x^3 + 17079552*x^2 - sqrt(2)*(8292238
*x^7 - 11867543*x^6 + 37968813*x^5 - 13449840*x^4 + 14570280*x^3 + 20176128*x^2 - 20176128*x) - 17079552*x) +
520397*563606738^(1/4)*sqrt(341)*(795513*x^7 - 10292932*x^6 + 39734380*x^5 - 51864768*x^4 + 68281632*x^3 + 342
55872*x^2 - 8*sqrt(2)*(77213*x^7 - 998548*x^6 + 3846220*x^5 - 4943520*x^4 + 6215760*x^3 + 4318272*x^2 - 431827
2*x) - 34255872*x))*sqrt(2*x^2 - x + 3)*sqrt(2343727*sqrt(2) + 3357400) + 421088065768678*sqrt(31)*sqrt(2)*(12
3408*x^8 - 914152*x^7 + 1578888*x^6 - 3293072*x^5 + 396480*x^4 + 798336*x^3 - 3822336*x^2 - sqrt(2)*(15550*x^8
- 118051*x^7 + 244047*x^6 - 707374*x^5 + 1053960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) +
19140366625849*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219328*x^3
- 168956928*x^2 - 15488*sqrt(2)*(4*x^8 - 76*x^7 + 517*x^6 - 1536*x^5 + 2385*x^4 - 3618*x^3 + 2268*x^2 - 1944*
x) + 144820224*x))*sqrt(-(563606738^(1/4)*sqrt(33574)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(1123*x
+ 898) - 2021*x - 225)*sqrt(2343727*sqrt(2) + 3357400) - 1731948347213*x^2 - 1555218924028*sqrt(2)*(2*x^2 - x
+ 3) + 5337228580187*x - 7069176927400)/x^2) + 229093555532814667219*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 533
85560*x^6 - 142835344*x^5 + 254146592*x^4 - 249300096*x^3 + 37981440*x^2 - 7744*sqrt(2)*(1348*x^8 - 2692*x^7 +
9789*x^6 - 10070*x^5 + 15569*x^4 - 5568*x^3 + 1080*x^2 + 4320*x - 5184) + 223064064*x - 94887936))/(2585191*x
^8 - 4661200*x^7 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 18579
456)) + 8422204*563606738^(1/4)*sqrt(33574)*sqrt(341)*sqrt(2)*(5*x^2 + 3*x + 2)*sqrt(2343727*sqrt(2) + 3357400
)*arctan(1/7101900221517254683789*(47876524*sqrt(33574)*(22*563606738^(3/4)*sqrt(341)*(2950932*x^7 - 11691762*
x^6 + 24397746*x^5 - 40053004*x^4 + 20309552*x^3 - 10145376*x^2 - sqrt(2)*(2248634*x^7 - 8421787*x^6 + 1780149
4*x^5 - 27869789*x^4 + 13808040*x^3 - 6172200*x^2 - 15724800*x + 10596096) - 21192192*x + 15724800) + 520397*5
63606738^(1/4)*sqrt(341)*(226651*x^7 - 3496629*x^6 + 18614024*x^5 - 42860780*x^4 + 55586592*x^3 - 36274464*x^2
- sqrt(2)*(168871*x^7 - 2579646*x^6 + 13533020*x^5 - 30582616*x^4 + 39345120*x^3 - 23947200*x^2 - 28449792*x
+ 19450368) - 38900736*x + 28449792))*sqrt(2*x^2 - x + 3)*sqrt(2343727*sqrt(2) + 3357400) - 201602328868876907
15272*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1385256*x^5 + 1549144*x^4 - 642048*x^3 - 98496*x
^2 - sqrt(2)*(8746*x^8 - 102335*x^7 + 396104*x^6 - 783113*x^5 + 1320710*x^4 - 752088*x^3 + 396144*x^2 + 546048
*x - 539136) + 1154304*x - 456192) - 2*sqrt(33574/2191)*(sqrt(33574)*(22*563606738^(3/4)*sqrt(341)*(10257392*x
^7 - 14773368*x^6 + 47877288*x^5 - 20710528*x^4 + 26321472*x^3 + 17079552*x^2 - sqrt(2)*(8292238*x^7 - 1186754
3*x^6 + 37968813*x^5 - 13449840*x^4 + 14570280*x^3 + 20176128*x^2 - 20176128*x) - 17079552*x) + 520397*5636067
38^(1/4)*sqrt(341)*(795513*x^7 - 10292932*x^6 + 39734380*x^5 - 51864768*x^4 + 68281632*x^3 + 34255872*x^2 - 8*
sqrt(2)*(77213*x^7 - 998548*x^6 + 3846220*x^5 - 4943520*x^4 + 6215760*x^3 + 4318272*x^2 - 4318272*x) - 3425587
2*x))*sqrt(2*x^2 - x + 3)*sqrt(2343727*sqrt(2) + 3357400) - 421088065768678*sqrt(31)*sqrt(2)*(123408*x^8 - 914
152*x^7 + 1578888*x^6 - 3293072*x^5 + 396480*x^4 + 798336*x^3 - 3822336*x^2 - sqrt(2)*(15550*x^8 - 118051*x^7
+ 244047*x^6 - 707374*x^5 + 1053960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) - 19140366625849
*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219328*x^3 - 168956928*x
^2 - 15488*sqrt(2)*(4*x^8 - 76*x^7 + 517*x^6 - 1536*x^5 + 2385*x^4 - 3618*x^3 + 2268*x^2 - 1944*x) + 144820224
*x))*sqrt((563606738^(1/4)*sqrt(33574)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(1123*x + 898) - 2021*x
- 225)*sqrt(2343727*sqrt(2) + 3357400) + 17319...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Francis algorithm failure for[-1.0,infinity,infinity,infinity,infinity]proot error [1.0,infinity,infinity,i
nfinity,inf

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {2\,x^2-x+3}\,{\left (5\,x^2+3\,x+2\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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