\(\int \frac {(3-x+2 x^2)^{5/2}}{2+3 x+5 x^2} \, dx\) [110]

Optimal result

Integrand size = 27, antiderivative size = 236 \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx=-\frac {226249 \sqrt {3-x+2 x^2}}{80000}+\frac {4981 x \sqrt {3-x+2 x^2}}{4000}-\frac {1}{600} (103-60 x) \left (3-x+2 x^2\right )^{3/2}-\frac {7216203 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{800000 \sqrt {2}}-\frac {121 \sqrt {\frac {11}{31} \left (-15457+25000 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (-15457+25000 \sqrt {2}\right )}} \left (196-443 \sqrt {2}-\left (690+247 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{3125}+\frac {121 \sqrt {\frac {11}{31} \left (15457+25000 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (15457+25000 \sqrt {2}\right )}} \left (196+443 \sqrt {2}-\left (690-247 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{3125} \] Output:

-226249/80000*(2*x^2-x+3)^(1/2)+4981/4000*x*(2*x^2-x+3)^(1/2)-1/600*(103-6 
0*x)*(2*x^2-x+3)^(3/2)-7216203/1600000*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1 
/2)-121/96875*(-5270837+8525000*2^(1/2))^(1/2)*arctan(11^(1/2)/(-958334+15 
50000*2^(1/2))^(1/2)*(196-443*2^(1/2)-(690+247*2^(1/2))*x)/(2*x^2-x+3)^(1/ 
2))+121/96875*(5270837+8525000*2^(1/2))^(1/2)*arctanh(11^(1/2)/(958334+155 
0000*2^(1/2))^(1/2)*(196+443*2^(1/2)-(690-247*2^(1/2))*x)/(2*x^2-x+3)^(1/2 
))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.85 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.01 \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx=\frac {20 \sqrt {3-x+2 x^2} \left (-802347+412060 x-106400 x^2+48000 x^3\right )-21648609 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )-2044416 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {368 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+22 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-119 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{4800000} \] Input:

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

Output:

(20*Sqrt[3 - x + 2*x^2]*(-802347 + 412060*x - 106400*x^2 + 48000*x^3) - 21 
648609*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]] - 2044416*RootSum[-5 
6 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (368*Log[-(Sqrt[ 
2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 22*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - 
 x + 2*x^2] - #1]*#1 - 119*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) & ])/4800000
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {1308, 27, 2138, 27, 2143, 27, 1090, 222, 1368, 27, 1362, 217, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (2 x^2-x+3\right )^{5/2}}{5 x^2+3 x+2} \, dx\)

\(\Big \downarrow \) 1308

\(\displaystyle -\frac {1}{300} \int -\frac {3 \sqrt {2 x^2-x+3} \left (4981 x^2-2045 x+3154\right )}{4 \left (5 x^2+3 x+2\right )}dx-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{400} \int \frac {\sqrt {2 x^2-x+3} \left (4981 x^2-2045 x+3154\right )}{5 x^2+3 x+2}dx-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\)

\(\Big \downarrow \) 2138

\(\displaystyle \frac {1}{400} \left (-\frac {1}{100} \int -\frac {7216203 x^2-3779795 x+2136862}{4 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx-\frac {1}{200} \sqrt {2 x^2-x+3} (226249-99620 x)\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{400} \left (\frac {1}{400} \int \frac {7216203 x^2-3779795 x+2136862}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\)

\(\Big \downarrow \) 2143

\(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx+\frac {1}{5} \int -\frac {340736 (119 x+11)}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {340736}{5} \int \frac {119 x+11}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203 \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)}{5 \sqrt {46}}-\frac {340736}{5} \int \frac {119 x+11}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}-\frac {340736}{5} \int \frac {119 x+11}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\)

\(\Big \downarrow \) 1368

\(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}-\frac {340736}{5} \left (\frac {\int \frac {11 \left (\left (130+119 \sqrt {2}\right ) x+11 \sqrt {2}+108\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int \frac {11 \left (\left (130-119 \sqrt {2}\right ) x-11 \sqrt {2}+108\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}-\frac {340736}{5} \left (\frac {\int \frac {\left (130+119 \sqrt {2}\right ) x+11 \sqrt {2}+108}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {\left (130-119 \sqrt {2}\right ) x-11 \sqrt {2}+108}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\)

\(\Big \downarrow \) 1362

\(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}-\frac {340736}{5} \left (\sqrt {2} \left (15457+25000 \sqrt {2}\right ) \int \frac {1}{62 \left (15457+25000 \sqrt {2}\right )-\frac {11 \left (-\left (\left (690-247 \sqrt {2}\right ) x\right )+443 \sqrt {2}+196\right )^2}{2 x^2-x+3}}d\left (-\frac {-\left (\left (690-247 \sqrt {2}\right ) x\right )+443 \sqrt {2}+196}{\sqrt {2 x^2-x+3}}\right )-\sqrt {2} \left (15457-25000 \sqrt {2}\right ) \int \frac {1}{62 \left (15457-25000 \sqrt {2}\right )-\frac {11 \left (-\left (\left (690+247 \sqrt {2}\right ) x\right )-443 \sqrt {2}+196\right )^2}{2 x^2-x+3}}d\left (-\frac {-\left (\left (690+247 \sqrt {2}\right ) x\right )-443 \sqrt {2}+196}{\sqrt {2 x^2-x+3}}\right )\right )\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}-\frac {340736}{5} \left (\sqrt {2} \left (15457+25000 \sqrt {2}\right ) \int \frac {1}{62 \left (15457+25000 \sqrt {2}\right )-\frac {11 \left (-\left (\left (690-247 \sqrt {2}\right ) x\right )+443 \sqrt {2}+196\right )^2}{2 x^2-x+3}}d\left (-\frac {-\left (\left (690-247 \sqrt {2}\right ) x\right )+443 \sqrt {2}+196}{\sqrt {2 x^2-x+3}}\right )-\frac {\left (15457-25000 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {\frac {11}{62 \left (25000 \sqrt {2}-15457\right )}} \left (-\left (\left (690+247 \sqrt {2}\right ) x\right )-443 \sqrt {2}+196\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {341 \left (25000 \sqrt {2}-15457\right )}}\right )\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{400} \left (\frac {1}{400} \left (\frac {7216203 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}-\frac {340736}{5} \left (-\frac {\left (15457-25000 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {\frac {11}{62 \left (25000 \sqrt {2}-15457\right )}} \left (-\left (\left (690+247 \sqrt {2}\right ) x\right )-443 \sqrt {2}+196\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {341 \left (25000 \sqrt {2}-15457\right )}}-\sqrt {\frac {1}{341} \left (15457+25000 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (15457+25000 \sqrt {2}\right )}} \left (-\left (\left (690-247 \sqrt {2}\right ) x\right )+443 \sqrt {2}+196\right )}{\sqrt {2 x^2-x+3}}\right )\right )\right )-\frac {1}{200} (226249-99620 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{600} (103-60 x) \left (2 x^2-x+3\right )^{3/2}\)

Input:

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

Output:

-1/600*((103 - 60*x)*(3 - x + 2*x^2)^(3/2)) + (-1/200*((226249 - 99620*x)* 
Sqrt[3 - x + 2*x^2]) + ((7216203*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(5*Sqrt[2]) 
 - (340736*(-(((15457 - 25000*Sqrt[2])*ArcTan[(Sqrt[11/(62*(-15457 + 25000 
*Sqrt[2]))]*(196 - 443*Sqrt[2] - (690 + 247*Sqrt[2])*x))/Sqrt[3 - x + 2*x^ 
2]])/Sqrt[341*(-15457 + 25000*Sqrt[2])]) - Sqrt[(15457 + 25000*Sqrt[2])/34 
1]*ArcTanh[(Sqrt[11/(62*(15457 + 25000*Sqrt[2]))]*(196 + 443*Sqrt[2] - (69 
0 - 247*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]]))/5)/400)/400
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 222

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 1090

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* 
(c/(b^2 - 4*a*c)))^p)   Subst[Int[Simp[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 1308

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

rule 1362

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + 
(e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[-2*g*(g*b - 2*a*h)   Subst[I 
nt[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, Simp[ 
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] && Ne 
Q[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 1368

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + 
(e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d 
 - a*e)*(c*e - b*f), 2]}, Simp[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)*Sqr 
t[d + e*x + f*x^2]), x], x] - Simp[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] && N 
eQ[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 2138

Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. 
)*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] 
, C = Coeff[Px, x, 2]}, 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] - Simp[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*Si 
mp[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]] /; FreeQ[{a, b, c, d, 
e, f, q}, x] && PolyQ[Px, x, 2] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2* 
p + 2*q + 3, 0] &&  !IGtQ[p, 0] &&  !IGtQ[q, 0]
 

rule 2143

Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_ 
.)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C 
 = Coeff[Px, x, 2]}, Simp[C/c   Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[ 
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}, x] && PolyQ[Px, x, 2]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.76 (sec) , antiderivative size = 508, normalized size of antiderivative = 2.15

method	result	size
trager	\(\text {Expression too large to display}\)	\(508\)
risch	\(\frac {\left (48000 x^{3}-106400 x^{2}+412060 x -802347\right ) \sqrt {2 x^{2}-x +3}}{240000}+\frac {7216203 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{1600000}+\frac {121 \sqrt {\frac {8 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}\, \sqrt {2}\, \left (6955 \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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (-1+\sqrt {2}+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (-1+\sqrt {2}+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}+10111 \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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (-1+\sqrt {2}+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (-1+\sqrt {2}+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (-1+\sqrt {2}+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] \,\operatorname {arctanh}\left (\frac {31 \sqrt {\frac {8 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right ) \sqrt {2}+993674 \,\operatorname {arctanh}\left (\frac {31 \sqrt {\frac {8 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right )\right )}{3003125 \sqrt {\frac {\frac {8 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}{\left (1+\frac {-1+\sqrt {2}+x}{\sqrt {2}+1-x}\right )^{2}}}\, \left (1+\frac {-1+\sqrt {2}+x}{\sqrt {2}+1-x}\right ) \left (8+3 \sqrt {2}\right ) \sqrt {-8866+6820 \sqrt {2}}}\)	\(728\)
default	\(\text {Expression too large to display}\)	\(4860\)

Input:

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

Output:

(1/5*x^3-133/300*x^2+20603/12000*x-267449/80000)*(2*x^2-x+3)^(1/2)-7216203 
/1600000*RootOf(_Z^2-2)*ln(-4*RootOf(_Z^2-2)*x+RootOf(_Z^2-2)+4*(2*x^2-x+3 
)^(1/2))+1/80000*RootOf(24025*_Z^4-1264358596886528*_Z^2+87031555002938163 
200000000)*ln((27172875625*x*RootOf(24025*_Z^4-1264358596886528*_Z^2+87031 
555002938163200000000)^5-2369193093325432422400*RootOf(24025*_Z^4-12643585 
96886528*_Z^2+87031555002938163200000000)^3*x+4594429410426634240000*RootO 
f(24025*_Z^4-1264358596886528*_Z^2+87031555002938163200000000)^3+713801017 
384745500672000000*RootOf(24025*_Z^4-1264358596886528*_Z^2+870315550029381 
63200000000)^2*(2*x^2-x+3)^(1/2)-7230675965381243295156924317696*RootOf(24 
025*_Z^4-1264358596886528*_Z^2+87031555002938163200000000)*x[phone] 
0850814746112819200*RootOf(24025*_Z^4-1264358596886528*_Z^2+87031555002938 
163200000000)-16823935022206520125808353466122240000*(2*x^2-x+3)^(1/2))/(7 
75*x*RootOf(24025*_Z^4-1264358596886528*_Z^2+87031555002938163200000000)^2 
[phone]*x[phone]))+1/12400000*RootOf(_Z^2+24025*RootOf(2402 
5*_Z^4-1264358596886528*_Z^2+87031555002938163200000000)^2-126435859688652 
8)*ln((43476601*RootOf(_Z^2+24025*RootOf(24025*_Z^4-1264358596886528*_Z^2+ 
87031555002938163200000000)^2-1264358596886528)*RootOf(24025*_Z^4-12643585 
96886528*_Z^2+87031555002938163200000000)^4*x-785358833218781184*RootOf(24 
025*_Z^4-1264358596886528*_Z^2+87031555002938163200000000)^2*RootOf(_Z^2+2 
4025*RootOf(24025*_Z^4-1264358596886528*_Z^2+87031555002938163200000000...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (172) = 344\).

Time = 0.12 (sec) , antiderivative size = 565, normalized size of antiderivative = 2.39 \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-121/6250*sqrt(11/31)*sqrt(25000*sqrt(2) - 15457)*arctan(-1/177041*(248*sq 
rt(11/31)*(115*x^3 - 1149*x^2 - sqrt(2)*(993*x^3 - 1628*x^2 + 1128*x + 129 
6) - 2768*x - 264)*sqrt(2*x^2 - x + 3)*sqrt(25000*sqrt(2) - 15457) + (171* 
x^4 + 1212*x^3 - 1640*x^2 - 176*sqrt(2)*(6*x^4 + 5*x^3 + 5*x^2 + 12*x) - 3 
936*x)*sqrt(25000*sqrt(2) + 15457)*sqrt(25000*sqrt(2) - 15457))/(343*x^4 - 
 400*x^3 + 1136*x^2 + 384*x - 576)) + 121/6250*sqrt(11/31)*sqrt(25000*sqrt 
(2) - 15457)*arctan(1/177041*(248*sqrt(11/31)*(115*x^3 - 1149*x^2 - sqrt(2 
)*(993*x^3 - 1628*x^2 + 1128*x + 1296) - 2768*x - 264)*sqrt(2*x^2 - x + 3) 
*sqrt(25000*sqrt(2) - 15457) - (171*x^4 + 1212*x^3 - 1640*x^2 - 176*sqrt(2 
)*(6*x^4 + 5*x^3 + 5*x^2 + 12*x) - 3936*x)*sqrt(25000*sqrt(2) + 15457)*sqr 
t(25000*sqrt(2) - 15457))/(343*x^4 - 400*x^3 + 1136*x^2 + 384*x - 576)) + 
121/12500*sqrt(11/31)*sqrt(25000*sqrt(2) + 15457)*log(121*(2*sqrt(11/31)*s 
qrt(2*x^2 - x + 3)*(sqrt(2)*(47*x + 1972) - 2019*x + 1925)*sqrt(25000*sqrt 
(2) + 15457) + 279839*x^2 + 251284*sqrt(2)*(2*x^2 - x + 3) - 862361*x + 11 
42200)/x^2) - 121/12500*sqrt(11/31)*sqrt(25000*sqrt(2) + 15457)*log(-121*( 
2*sqrt(11/31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(47*x + 1972) - 2019*x + 1925)* 
sqrt(25000*sqrt(2) + 15457) - 279839*x^2 - 251284*sqrt(2)*(2*x^2 - x + 3) 
+ 862361*x - 1142200)/x^2) + 1/240000*(48000*x^3 - 106400*x^2 + 412060*x - 
 802347)*sqrt(2*x^2 - x + 3) + 7216203/3200000*sqrt(2)*log(-4*sqrt(2)*sqrt 
(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)
 

Sympy [F]

\[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx=\int \frac {\left (2 x^{2} - x + 3\right )^{\frac {5}{2}}}{5 x^{2} + 3 x + 2}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx=\int { \frac {{\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{5 \, x^{2} + 3 \, x + 2} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Francis algorithm failure for[-1.0, 
infinity,infinity,infinity,infinity]proot error [1.0,infinity,infinity,inf 
inity,inf
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx=\int \frac {{\left (2\,x^2-x+3\right )}^{5/2}}{5\,x^2+3\,x+2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{2+3 x+5 x^2} \, dx=\frac {\sqrt {2 x^{2}-x +3}\, x^{3}}{5}-\frac {133 \sqrt {2 x^{2}-x +3}\, x^{2}}{300}+\frac {20603 \sqrt {2 x^{2}-x +3}\, x}{12000}-\frac {329345961 \sqrt {2 x^{2}-x +3}}{50000000}+\frac {464238163 \sqrt {2}\, \mathrm {log}\left (-2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}-4 x +1\right )}{40000000}+\frac {1108723 \sqrt {2}\, \mathrm {log}\left (2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}-4 x +1\right )}{156250}-\frac {3097237 \left (\int \frac {\sqrt {2 x^{2}-x +3}}{10 x^{4}+x^{3}+16 x^{2}+7 x +6}d x \right )}{390625}+\frac {2534224 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{3}}{10 x^{4}+x^{3}+16 x^{2}+7 x +6}d x \right )}{78125}+\frac {4434892 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{2}}{10 x^{4}+x^{3}+16 x^{2}+7 x +6}d x \right )}{390625}-\frac {3326169 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x}{10 x^{4}+x^{3}+16 x^{2}+7 x +6}d x \right )}{78125} \] Input:

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

Output:

(120000000*sqrt(2*x**2 - x + 3)*x**3 - 266000000*sqrt(2*x**2 - x + 3)*x**2 
 + 1030150000*sqrt(2*x**2 - x + 3)*x - 3952151532*sqrt(2*x**2 - x + 3) + 6 
963572445*sqrt(2)*log( - 2*sqrt(2*x**2 - x + 3)*sqrt(2) - 4*x + 1) + 42574 
96320*sqrt(2)*log(2*sqrt(2*x**2 - x + 3)*sqrt(2) - 4*x + 1) - 4757356032*i 
nt(sqrt(2*x**2 - x + 3)/(10*x**4 + x**3 + 16*x**2 + 7*x + 6),x) + 19462840 
320*int((sqrt(2*x**2 - x + 3)*x**3)/(10*x**4 + x**3 + 16*x**2 + 7*x + 6),x 
) + 6811994112*int((sqrt(2*x**2 - x + 3)*x**2)/(10*x**4 + x**3 + 16*x**2 + 
 7*x + 6),x) - 25544977920*int((sqrt(2*x**2 - x + 3)*x)/(10*x**4 + x**3 + 
16*x**2 + 7*x + 6),x))/600000000