3.1.70 \(\int \frac {(3-x+2 x^2)^{3/2}}{(2+3 x+5 x^2)^2} \, dx\) [70]

Optimal. Leaf size=232 \[ \frac {4}{155} (4-5 x) \sqrt {3-x+2 x^2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {2}{25} \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )+\frac {\sqrt {\frac {11}{31} \left (3169333+2265350 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (3169333+2265350 \sqrt {2}\right )}} \left (3514+2963 \sqrt {2}+\left (9440+6477 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1550}-\frac {\sqrt {\frac {11}{31} \left (-3169333+2265350 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (-3169333+2265350 \sqrt {2}\right )}} \left (3514-2963 \sqrt {2}+\left (9440-6477 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1550} \]

[Out]

1/31*(3+10*x)*(2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)-2/25*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)+4/155*(4-5*x)*(2*x^2
-x+3)^(1/2)-1/48050*arctanh(1/62*(3514+x*(9440-6477*2^(1/2))-2963*2^(1/2))*682^(1/2)/(-3169333+2265350*2^(1/2)
)^(1/2)/(2*x^2-x+3)^(1/2))*(-1080742553+772484350*2^(1/2))^(1/2)+1/48050*arctan(1/62*(3514+2963*2^(1/2)+x*(944
0+6477*2^(1/2)))*682^(1/2)/(3169333+2265350*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(1080742553+772484350*2^(1/2))^(
1/2)

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Rubi [A]
time = 0.37, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {985, 1080, 1090, 633, 221, 1049, 1043, 212, 210} \begin {gather*} \frac {\sqrt {\frac {11}{31} \left (3169333+2265350 \sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {11}{62 \left (3169333+2265350 \sqrt {2}\right )}} \left (\left (9440+6477 \sqrt {2}\right ) x+2963 \sqrt {2}+3514\right )}{\sqrt {2 x^2-x+3}}\right )}{1550}+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{31 \left (5 x^2+3 x+2\right )}+\frac {4}{155} (4-5 x) \sqrt {2 x^2-x+3}-\frac {\sqrt {\frac {11}{31} \left (2265350 \sqrt {2}-3169333\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (2265350 \sqrt {2}-3169333\right )}} \left (\left (9440-6477 \sqrt {2}\right ) x-2963 \sqrt {2}+3514\right )}{\sqrt {2 x^2-x+3}}\right )}{1550}-\frac {2}{25} \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(4 - 5*x)*Sqrt[3 - x + 2*x^2])/155 + ((3 + 10*x)*(3 - x + 2*x^2)^(3/2))/(31*(2 + 3*x + 5*x^2)) - (2*Sqrt[2]
*ArcSinh[(1 - 4*x)/Sqrt[23]])/25 + (Sqrt[(11*(3169333 + 2265350*Sqrt[2]))/31]*ArcTan[(Sqrt[11/(62*(3169333 + 2
265350*Sqrt[2]))]*(3514 + 2963*Sqrt[2] + (9440 + 6477*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/1550 - (Sqrt[(11*(-31
69333 + 2265350*Sqrt[2]))/31]*ArcTanh[(Sqrt[11/(62*(-3169333 + 2265350*Sqrt[2]))]*(3514 - 2963*Sqrt[2] + (9440
- 6477*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/1550

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 221

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

Rule 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 985

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

Rule 1080

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[(B*c*f*(2*p + 2*q + 3) + C*(b*f*p - c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*(
a + b*x + c*x^2)^p*((d + e*x + f*x^2)^(q + 1)/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3))), x] - Dist[1/(2*c*f^2*(p
+ q + 1)*(2*p + 2*q + 3)), Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[p*(b*d - a*e)*(C*(c*e - b*f)
*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*
(B*e - 2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (
p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q +
3))))*x + (p*(c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 -
4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x] /; F
reeQ[{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] && GtQ[p, 0] && NeQ[p +
q + 1, 0] && NeQ[2*p + 2*q + 3, 0] &&  !IGtQ[p, 0] &&  !IGtQ[q, 0]

Rule 1090

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x
_)^2]), x_Symbol] :> Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + (B*c - b*C)*x)
/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; 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 {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx &=\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{31} \int \frac {\sqrt {3-x+2 x^2} \left (-\frac {69}{2}+13 x+40 x^2\right )}{2+3 x+5 x^2} \, dx\\ &=\frac {4}{155} (4-5 x) \sqrt {3-x+2 x^2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {13070-5750 x+2480 x^2}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{3100}\\ &=\frac {4}{155} (4-5 x) \sqrt {3-x+2 x^2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {60390-36190 x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{15500}+\frac {4}{25} \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx\\ &=\frac {4}{155} (4-5 x) \sqrt {3-x+2 x^2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{25} \left (2 \sqrt {\frac {2}{23}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )+\frac {\int \frac {-1210 \left (878-549 \sqrt {2}\right )+1210 \left (220-329 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{341000 \sqrt {2}}-\frac {\int \frac {-1210 \left (878+549 \sqrt {2}\right )+1210 \left (220+329 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{341000 \sqrt {2}}\\ &=\frac {4}{155} (4-5 x) \sqrt {3-x+2 x^2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {2}{25} \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )+\frac {1}{155} \left (1331 \left (4530700-3169333 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-90774200 \left (3169333-2265350 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {-1210 \left (3514-2963 \sqrt {2}\right )-1210 \left (9440-6477 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )+\frac {1}{155} \left (1331 \left (4530700+3169333 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-90774200 \left (3169333+2265350 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {-1210 \left (3514+2963 \sqrt {2}\right )-1210 \left (9440+6477 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )\\ &=\frac {4}{155} (4-5 x) \sqrt {3-x+2 x^2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {2}{25} \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )+\frac {\sqrt {\frac {11}{31} \left (3169333+2265350 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (3169333+2265350 \sqrt {2}\right )}} \left (3514+2963 \sqrt {2}+\left (9440+6477 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1550}-\frac {\sqrt {\frac {11}{31} \left (-3169333+2265350 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{62 \left (-3169333+2265350 \sqrt {2}\right )}} \left (3514-2963 \sqrt {2}+\left (9440-6477 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1550}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.52, size = 416, normalized size = 1.79 \begin {gather*} \frac {50 \left (\frac {55 (7+13 x) \sqrt {3-x+2 x^2}}{2+3 x+5 x^2}-62 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )\right )+682 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {999 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+310 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+100 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+11 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-72888 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+8230 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+2025 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{38750} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(50*((55*(7 + 13*x)*Sqrt[3 - x + 2*x^2])/(2 + 3*x + 5*x^2) - 62*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]]
) + 682*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (999*Log[-(Sqrt[2]*x) + Sqrt[3 - x
+ 2*x^2] - #1] + 310*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 + 100*Log[-(Sqrt[2]*x) + Sqrt[3
- x + 2*x^2] - #1]*#1^2)/(-13*Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ] + 11*RootSum[-56 - 26*Sqrt[2]*#1
+ 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (-72888*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 8230*Sqrt[2]*L
og[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 + 2025*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1^2)/(-13*
Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ])/38750

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(28184\) vs. \(2(175)=350\).
time = 0.98, size = 28185, normalized size = 121.49

method	result	size
trager	\(\text {Expression too large to display}\)	\(603\)
risch	\(\frac {11 \left (7+13 x \right ) \sqrt {2 x^{2}-x +3}}{155 \left (5 x^{2}+3 x +2\right )}+\frac {2 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{25}+\frac {\sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}\, \sqrt {2}\, \left (126130 \sqrt {2}\, \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}+178601 \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}[phone] \arctanh \left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right ) \sqrt {2}-248376216 \arctanh \left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right )\right )}{1489550 \sqrt {\frac {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}{\left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right )^{2}}}\, \left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right ) \left (8+3 \sqrt {2}\right ) \sqrt {-8866+6820 \sqrt {2}}}\)	\(730\)
default	\(\text {Expression too large to display}\)	\(28185\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2150 vs. \(2 (175) = 350\).
time = 2.79, size = 2150, normalized size = 9.27 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/90746855745853600*(10421084*1987037073032^(1/4)*sqrt(45307)*sqrt(62)*sqrt(2)*(5*x^2 + 3*x + 2)*sqrt(3169333*
sqrt(2) + 4530700)*arctan(1/172758074198807633719789*(64607782*sqrt(45307)*(2*1987037073032^(3/4)*sqrt(62)*(24
33118*x^7 - 9616349*x^6 + 20077988*x^5 - 32895253*x^4 + 16664280*x^3 - 8289000*x^2 - sqrt(2)*(1842432*x^7 - 69
16062*x^6 + 14611071*x^5 - 22920229*x^4 + 11367152*x^3 - 5107176*x^2 - 12897792*x + 8726400) - 17452800*x + 12
897792) + 1404517*1987037073032^(1/4)*sqrt(62)*(373384*x^7 - 5757834*x^6 + 30631880*x^5 - 70476664*x^4 + 91370
880*x^3 - 59457600*x^2 - sqrt(2)*(276977*x^7 - 4232733*x^6 + 22218448*x^5 - 50249260*x^4 + 64668384*x^3 - 3947
9328*x^2 - 46697472*x + 32016384) - 64032768*x + 46697472))*sqrt(2*x^2 - x + 3)*sqrt(3169333*sqrt(2) + 4530700
) + 490410017080486186043272*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) - sqrt(45307/2711)*(sqrt(45307)*(2*1987037073032^(3/
4)*sqrt(62)*(8480726*x^7 - 12210811*x^6 + 39548601*x^5 - 16962480*x^4 + 21434760*x^3 + 14432256*x^2 - sqrt(2)*
(6779042*x^7 - 9704193*x^6 + 31062363*x^5 - 11094928*x^4 + 12114072*x^3 + 16301952*x^2 - 16301952*x) - 1443225
6*x) + 1404517*1987037073032^(1/4)*sqrt(62)*(1312966*x^7 - 16987736*x^6 + 65572040*x^5 - 85530240*x^4 + 112374
720*x^3 + 57314304*x^2 - sqrt(2)*(1011501*x^7 - 13081364*x^6 + 50391260*x^5 - 64806336*x^4 + 81634464*x^3 + 56
070144*x^2 - 56070144*x) - 57314304*x))*sqrt(2*x^2 - x + 3)*sqrt(3169333*sqrt(2) + 4530700) + 7590571938849196
*sqrt(31)*sqrt(2)*(123408*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 - 10368
00*x) + 3276288*x) + 345025997220418*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781
920*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(-(1987037073032^(1/4)*sqrt(45307)*sqrt(62)*sqrt(31)*sqrt(2*x^2 -
x + 3)*(sqrt(2)*(1867*x + 1425) - 3292*x - 442)*sqrt(3169333*sqrt(2) + 4530700) - 11567627293306*x^2 - 103872
57161336*sqrt(2)*(2*x^2 - x + 3) + 35647177985494*x - 47214805278800)/x^2) + 5572841103187343023219*sqrt(31)*(
2828123*x^8 - 9696916*x^7 + 53385560*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) + 2230
64064*x - 94887936))/(2585191*x^8 - 4661200*x^7 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34
615296*x^2 - 24772608*x + 18579456)) + 10421084*1987037073032^(1/4)*sqrt(45307)*sqrt(62)*sqrt(2)*(5*x^2 + 3*x
+ 2)*sqrt(3169333*sqrt(2) + 4530700)*arctan(1/172758074198807633719789*(64607782*sqrt(45307)*(2*1987037073032^
(3/4)*sqrt(62)*(2433118*x^7 - 9616349*x^6 + 20077988*x^5 - 32895253*x^4 + 16664280*x^3 - 8289000*x^2 - sqrt(2)
*(1842432*x^7 - 6916062*x^6 + 14611071*x^5 - 22920229*x^4 + 11367152*x^3 - 5107176*x^2 - 12897792*x + 8726400)
- 17452800*x + 12897792) + 1404517*1987037073032^(1/4)*sqrt(62)*(373384*x^7 - 5757834*x^6 + 30631880*x^5 - 70
476664*x^4 + 91370880*x^3 - 59457600*x^2 - sqrt(2)*(276977*x^7 - 4232733*x^6 + 22218448*x^5 - 50249260*x^4 + 6
4668384*x^3 - 39479328*x^2 - 46697472*x + 32016384) - 64032768*x + 46697472))*sqrt(2*x^2 - x + 3)*sqrt(3169333
*sqrt(2) + 4530700) - 490410017080486186043272*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 + 13207
10*x^4 - 752088*x^3 + 396144*x^2 + 546048*x - 539136) + 1154304*x - 456192) - sqrt(45307/2711)*(sqrt(45307)*(2
*1987037073032^(3/4)*sqrt(62)*(8480726*x^7 - 12210811*x^6 + 39548601*x^5 - 16962480*x^4 + 21434760*x^3 + 14432
256*x^2 - sqrt(2)*(6779042*x^7 - 9704193*x^6 + 31062363*x^5 - 11094928*x^4 + 12114072*x^3 + 16301952*x^2 - 163
01952*x) - 14432256*x) + 1404517*1987037073032^(1/4)*sqrt(62)*(1312966*x^7 - 16987736*x^6 + 65572040*x^5 - 855
30240*x^4 + 112374720*x^3 + 57314304*x^2 - sqrt(2)*(1011501*x^7 - 13081364*x^6 + 50391260*x^5 - 64806336*x^4 +
81634464*x^3 + 56070144*x^2 - 56070144*x) - 57314304*x))*sqrt(2*x^2 - x + 3)*sqrt(3169333*sqrt(2) + 4530700)
- 7590571938849196*sqrt(31)*sqrt(2)*(123408*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 + 1
209600*x^2 - 1036800*x) + 3276288*x) - 345025997220418*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 908
66808*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((1987037073032^(1/4)*sqrt(45307)*sqrt(62)*sqrt
(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(1867*x + 142...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{174900625,[8]%%%}+%%%{%%{[-419761500,0]:[1,0,-2]%%},[7]%
%%}+%%%{-68

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

[Out]

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

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