\(\int \sqrt {3-x+2 x^2} (2+3 x+5 x^2)^4 \, dx\) [92]

Optimal result

Integrand size = 27, antiderivative size = 208 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^4 \, dx=-\frac {359471503 (1-4 x) \sqrt {3-x+2 x^2}}{67108864}+\frac {27185733541 \left (3-x+2 x^2\right )^{3/2}}{440401920}+\frac {804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac {83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac {8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac {4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac {233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}-\frac {8267844569 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{134217728 \sqrt {2}} \] Output:

-359471503/67108864*(1-4*x)*(2*x^2-x+3)^(1/2)[phone]/440401920*(2*x^2 
-x+3)^(3/2)[phone]/36700160*x*(2*x^2-x+3)^(3/2)-83948353/2293760*x^2*(2 
*x^2-x+3)^(3/2)+8325631/1032192*x^3*(2*x^2-x+3)^(3/2)+4796405/43008*x^4*(2 
*x^2-x+3)^(3/2)+233225/1536*x^5*(2*x^2-x+3)^(3/2)+14125/144*x^6*(2*x^2-x+3 
)^(3/2)+125/4*x^7*(2*x^2-x+3)^(3/2)-8267844569/268435456*arcsinh(1/23*(1-4 
*x)*23^(1/2))*2^(1/2)
 

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.46 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^4 \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (3801512106459+537752185764 x-174418077792 x^2+2211683657856 x^3+5354741991424 x^4+7612808028160 x^5+7725962035200 x^6+6327795712000 x^7+3486515200000 x^8+1321205760000 x^9\right )-2604371039235 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{84557168640} \] Input:

Integrate[Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)^4,x]
 

Output:

(4*Sqrt[3 - x + 2*x^2]*(3801512106459 + 537752185764*x - 174418077792*x^2 
+ 2211683657856*x^3 + 5354741991424*x^4 + 7612808028160*x^5 + 772596203520 
0*x^6 + 6327795712000*x^7 + 3486515200000*x^8 + 1321205760000*x^9) - 26043 
71039235*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/84557168640
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.19, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 1160, 1087, 1090, 222}

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.

\(\displaystyle \int \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^4 \, dx\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{20} \int \frac {5}{2} \sqrt {2 x^2-x+3} \left (14125 x^7+13550 x^6+18720 x^5+14088 x^4+7488 x^3+3008 x^2+768 x+128\right )dx+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \int \sqrt {2 x^2-x+3} \left (14125 x^7+13550 x^6+18720 x^5+14088 x^4+7488 x^3+3008 x^2+768 x+128\right )dx+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{8} \left (\frac {1}{18} \int \frac {3}{2} \sqrt {2 x^2-x+3} \left (233225 x^6+55140 x^5+169056 x^4+89856 x^3+36096 x^2+9216 x+1536\right )dx+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {1}{12} \int \sqrt {2 x^2-x+3} \left (233225 x^6+55140 x^5+169056 x^4+89856 x^3+36096 x^2+9216 x+1536\right )dx+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{16} \int \frac {1}{2} \sqrt {2 x^2-x+3} \left (4796405 x^5-1586958 x^4+2875392 x^3+1155072 x^2+294912 x+49152\right )dx+\frac {233225}{16} \left (2 x^2-x+3\right )^{3/2} x^5\right )+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \int \sqrt {2 x^2-x+3} \left (4796405 x^5-1586958 x^4+2875392 x^3+1155072 x^2+294912 x+49152\right )dx+\frac {233225}{16} \left (2 x^2-x+3\right )^{3/2} x^5\right )+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {1}{14} \int \frac {1}{2} \sqrt {2 x^2-x+3} \left (8325631 x^4-34602744 x^3+32342016 x^2+8257536 x+1376256\right )dx+\frac {4796405}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {233225}{16} \left (2 x^2-x+3\right )^{3/2} x^5\right )+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {1}{28} \int \sqrt {2 x^2-x+3} \left (8325631 x^4-34602744 x^3+32342016 x^2+8257536 x+1376256\right )dx+\frac {4796405}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {233225}{16} \left (2 x^2-x+3\right )^{3/2} x^5\right )+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{12} \int \frac {9}{2} \sqrt {2 x^2-x+3} \left (-83948353 x^3+69594114 x^2+22020096 x+3670016\right )dx+\frac {8325631}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {4796405}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {233225}{16} \left (2 x^2-x+3\right )^{3/2} x^5\right )+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {1}{28} \left (\frac {3}{8} \int \sqrt {2 x^2-x+3} \left (-83948353 x^3+69594114 x^2+22020096 x+3670016\right )dx+\frac {8325631}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {4796405}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {233225}{16} \left (2 x^2-x+3\right )^{3/2} x^5\right )+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {1}{28} \left (\frac {3}{8} \left (\frac {1}{10} \int \frac {1}{2} \sqrt {2 x^2-x+3} \left (804243809 x^2+1447782156 x+73400320\right )dx-\frac {83948353}{10} x^2 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {8325631}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {4796405}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {233225}{16} \left (2 x^2-x+3\right )^{3/2} x^5\right )+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {1}{28} \left (\frac {3}{8} \left (\frac {1}{20} \int \sqrt {2 x^2-x+3} \left (804243809 x^2+1447782156 x+73400320\right )dx-\frac {83948353}{10} x^2 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {8325631}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {4796405}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {233225}{16} \left (2 x^2-x+3\right )^{3/2} x^5\right )+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {1}{28} \left (\frac {3}{8} \left (\frac {1}{20} \left (\frac {1}{8} \int -\frac {1}{2} (3651057734-27185733541 x) \sqrt {2 x^2-x+3}dx+\frac {804243809}{8} x \left (2 x^2-x+3\right )^{3/2}\right )-\frac {83948353}{10} x^2 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {8325631}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {4796405}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {233225}{16} \left (2 x^2-x+3\right )^{3/2} x^5\right )+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {1}{28} \left (\frac {3}{8} \left (\frac {1}{20} \left (\frac {804243809}{8} x \left (2 x^2-x+3\right )^{3/2}-\frac {1}{16} \int (3651057734-27185733541 x) \sqrt {2 x^2-x+3}dx\right )-\frac {83948353}{10} x^2 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {8325631}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {4796405}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {233225}{16} \left (2 x^2-x+3\right )^{3/2} x^5\right )+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {1}{28} \left (\frac {3}{8} \left (\frac {1}{20} \left (\frac {1}{16} \left (\frac {12581502605}{4} \int \sqrt {2 x^2-x+3}dx+\frac {27185733541}{6} \left (2 x^2-x+3\right )^{3/2}\right )+\frac {804243809}{8} x \left (2 x^2-x+3\right )^{3/2}\right )-\frac {83948353}{10} x^2 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {8325631}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {4796405}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {233225}{16} \left (2 x^2-x+3\right )^{3/2} x^5\right )+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {1}{28} \left (\frac {3}{8} \left (\frac {1}{20} \left (\frac {1}{16} \left (\frac {12581502605}{4} \left (\frac {23}{16} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )+\frac {27185733541}{6} \left (2 x^2-x+3\right )^{3/2}\right )+\frac {804243809}{8} x \left (2 x^2-x+3\right )^{3/2}\right )-\frac {83948353}{10} x^2 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {8325631}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {4796405}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {233225}{16} \left (2 x^2-x+3\right )^{3/2} x^5\right )+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {1}{28} \left (\frac {3}{8} \left (\frac {1}{20} \left (\frac {1}{16} \left (\frac {12581502605}{4} \left (\frac {1}{16} \sqrt {\frac {23}{2}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )+\frac {27185733541}{6} \left (2 x^2-x+3\right )^{3/2}\right )+\frac {804243809}{8} x \left (2 x^2-x+3\right )^{3/2}\right )-\frac {83948353}{10} x^2 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {8325631}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {4796405}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {233225}{16} \left (2 x^2-x+3\right )^{3/2} x^5\right )+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {1}{28} \left (\frac {3}{8} \left (\frac {1}{20} \left (\frac {1}{16} \left (\frac {12581502605}{4} \left (\frac {23 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{16 \sqrt {2}}-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )+\frac {27185733541}{6} \left (2 x^2-x+3\right )^{3/2}\right )+\frac {804243809}{8} x \left (2 x^2-x+3\right )^{3/2}\right )-\frac {83948353}{10} x^2 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {8325631}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {4796405}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {233225}{16} \left (2 x^2-x+3\right )^{3/2} x^5\right )+\frac {14125}{18} \left (2 x^2-x+3\right )^{3/2} x^6\right )+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7\)

Input:

Int[Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)^4,x]
 

Output:

(125*x^7*(3 - x + 2*x^2)^(3/2))/4 + ((14125*x^6*(3 - x + 2*x^2)^(3/2))/18 
+ ((233225*x^5*(3 - x + 2*x^2)^(3/2))/16 + ((4796405*x^4*(3 - x + 2*x^2)^( 
3/2))/14 + ((8325631*x^3*(3 - x + 2*x^2)^(3/2))/12 + (3*((-83948353*x^2*(3 
 - x + 2*x^2)^(3/2))/10 + ((804243809*x*(3 - x + 2*x^2)^(3/2))/8 + ((27185 
733541*(3 - x + 2*x^2)^(3/2))/6 + (12581502605*(-1/8*((1 - 4*x)*Sqrt[3 - x 
 + 2*x^2]) + (23*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(16*Sqrt[2])))/4)/16)/20))/ 
8)/28)/32)/12)/8
 

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 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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 2.42 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.36

method	result	size
risch	\(\frac {\left (1321205760000 x^{9}[phone] x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}-174418077792 x^{2}[phone] x [phone]\right ) \sqrt {2 x^{2}-x +3}}{21139292160}+\frac {8267844569 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{268435456}\)	\(75\)
trager	\(\left (\frac {125}{2} x^{9}+\frac {11875}{72} x^{8}+\frac {689675}{2304} x^{7}+\frac {7859255}{21504} x^{6}+\frac {185859571}{516096} x^{5}+\frac {373517159}{1474560} x^{4}+\frac {5759592859}{55050240} x^{3}-\frac {259550711}{31457280} x^{2}+\frac {44812682147}{1761607680} x +\frac {422390234051}{2348810240}\right ) \sqrt {2 x^{2}-x +3}-\frac {8267844569 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{268435456}\)	\(99\)
default	\(\frac {359471503 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{67108864}+\frac {8267844569 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{268435456}+\frac {27185733541 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{440401920}+\frac {804243809 x \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{36700160}-\frac {83948353 x^{2} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{2293760}+\frac {8325631 x^{3} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{1032192}+\frac {4796405 x^{4} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{43008}+\frac {233225 x^{5} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{1536}+\frac {14125 x^{6} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{144}+\frac {125 x^{7} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{4}\)	\(166\)

Input:

int((2*x^2-x+3)^(1/2)*(5*x^2+3*x+2)^4,x,method=_RETURNVERBOSE)
 

Output:

1/21139292160*(1321205760000*x^9+3486515200000*x^8+6327795712000*x^7+77259 
62035200*x^6+7612808028160*x^5+5354741991424*x^4+2211683657856*x^3-1744180 
77792*x^2+537752185764*x[phone])*(2*x^2-x+3)^(1/2)[phone]/26843 
5456*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.47 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^4 \, dx=\frac {1}{21139292160} \, {\left (1321205760000 \, x^{9} + 3486515200000 \, x^{8} + 6327795712000 \, x^{7} + 7725962035200 \, x^{6} + 7612808028160 \, x^{5} + 5354741991424 \, x^{4} + 2211683657856 \, x^{3} - 174418077792 \, x^{2} + 537752185764 \, x + 3801512106459\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {8267844569}{536870912} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \] Input:

integrate((2*x^2-x+3)^(1/2)*(5*x^2+3*x+2)^4,x, algorithm="fricas")
 

Output:

1/21139292160*(1321205760000*x^9 + 3486515200000*x^8 + 6327795712000*x^7 + 
 7725962035200*x^6 + 7612808028160*x^5 + 5354741991424*x^4 + 2211683657856 
*x^3 - 174418077792*x^2 + 537752185764*x + 3801512106459)*sqrt(2*x^2 - x + 
 3) + 8267844569/536870912*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x 
 - 1) - 32*x^2 + 16*x - 25)
 

Sympy [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.47 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^4 \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {125 x^{9}}{2} + \frac {11875 x^{8}}{72} + \frac {689675 x^{7}}{2304} + \frac {7859255 x^{6}}{21504} + \frac {185859571 x^{5}}{516096} + \frac {373517159 x^{4}}{1474560} + \frac {5759592859 x^{3}}{55050240} - \frac {259550711 x^{2}}{31457280} + \frac {44812682147 x}{1761607680} + \frac {422390234051}{2348810240}\right ) + \frac {8267844569 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{268435456} \] Input:

integrate((2*x**2-x+3)**(1/2)*(5*x**2+3*x+2)**4,x)
 

Output:

sqrt(2*x**2 - x + 3)*(125*x**9/2 + 11875*x**8/72 + 689675*x**7/2304 + 7859 
255*x**6/21504 + 185859571*x**5/516096 + 373517159*x**4/1474560 + 57595928 
59*x**3/55050240 - 259550711*x**2/31457280 + 44812682147*x/1761607680 + 42 
2390234051/2348810240) + 8267844569*sqrt(2)*asinh(4*sqrt(23)*(x - 1/4)/23) 
/268435456
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.85 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^4 \, dx=\frac {125}{4} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{7} + \frac {14125}{144} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{6} + \frac {233225}{1536} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{5} + \frac {4796405}{43008} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} + \frac {8325631}{1032192} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {83948353}{2293760} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {804243809}{36700160} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {27185733541}{440401920} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {359471503}{16777216} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {8267844569}{268435456} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {359471503}{67108864} \, \sqrt {2 \, x^{2} - x + 3} \] Input:

integrate((2*x^2-x+3)^(1/2)*(5*x^2+3*x+2)^4,x, algorithm="maxima")
 

Output:

125/4*(2*x^2 - x + 3)^(3/2)*x^7 + 14125/144*(2*x^2 - x + 3)^(3/2)*x^6 + 23 
3225/1536*(2*x^2 - x + 3)^(3/2)*x^5 + 4796405/43008*(2*x^2 - x + 3)^(3/2)* 
x^4 + 8325631/1032192*(2*x^2 - x + 3)^(3/2)*x^3 - 83948353/2293760*(2*x^2 
- x + 3)^(3/2)*x^2 + 804243809/36700160*(2*x^2 - x + 3)^(3/2)*x + 27185733 
541/440401920*(2*x^2 - x + 3)^(3/2) + 359471503/16777216*sqrt(2*x^2 - x + 
3)*x + 8267844569/268435456*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 359 
471503/67108864*sqrt(2*x^2 - x + 3)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.45 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^4 \, dx=\frac {1}{21139292160} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (40 \, {\left (140 \, {\left (160 \, {\left (36 \, x + 95\right )} x + 27587\right )} x + 4715553\right )} x + 185859571\right )} x + 2614620113\right )} x + 17278778577\right )} x - 5450564931\right )} x + 134438046441\right )} x + 3801512106459\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {8267844569}{268435456} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \] Input:

integrate((2*x^2-x+3)^(1/2)*(5*x^2+3*x+2)^4,x, algorithm="giac")
 

Output:

1/21139292160*(4*(8*(4*(16*(20*(40*(140*(160*(36*x + 95)*x + 27587)*x + 47 
15553)*x + 185859571)*x + 2614620113)*x + 17278778577)*x - 5450564931)*x + 
 134438046441)*x + 3801512106459)*sqrt(2*x^2 - x + 3) - 8267844569/2684354 
56*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)
 

Mupad [B] (verification not implemented)

Time = 18.13 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.06 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^4 \, dx=\frac {8325631\,x^3\,{\left (2\,x^2-x+3\right )}^{3/2}}{1032192}-\frac {83948353\,x^2\,{\left (2\,x^2-x+3\right )}^{3/2}}{2293760}+\frac {4796405\,x^4\,{\left (2\,x^2-x+3\right )}^{3/2}}{43008}+\frac {233225\,x^5\,{\left (2\,x^2-x+3\right )}^{3/2}}{1536}+\frac {14125\,x^6\,{\left (2\,x^2-x+3\right )}^{3/2}}{144}+\frac {125\,x^7\,{\left (2\,x^2-x+3\right )}^{3/2}}{4}-\frac {41987163941\,\sqrt {2}\,\ln \left (\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (2\,x-\frac {1}{2}\right )}{2}\right )}{1174405120}-\frac {1825528867\,\left (\frac {x}{2}-\frac {1}{8}\right )\,\sqrt {2\,x^2-x+3}}{36700160}+\frac {27185733541\,\sqrt {2\,x^2-x+3}\,\left (32\,x^2-4\,x+45\right )}{7046430720}+\frac {804243809\,x\,{\left (2\,x^2-x+3\right )}^{3/2}}{36700160}+\frac {625271871443\,\sqrt {2}\,\ln \left (2\,\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (4\,x-1\right )}{2}\right )}{9395240960} \] Input:

int((2*x^2 - x + 3)^(1/2)*(3*x + 5*x^2 + 2)^4,x)
 

Output:

(8325631*x^3*(2*x^2 - x + 3)^(3/2))/1032192 - (83948353*x^2*(2*x^2 - x + 3 
)^(3/2))/2293760 + (4796405*x^4*(2*x^2 - x + 3)^(3/2))/43008 + (233225*x^5 
*(2*x^2 - x + 3)^(3/2))/1536 + (14125*x^6*(2*x^2 - x + 3)^(3/2))/144 + (12 
5*x^7*(2*x^2 - x + 3)^(3/2))/4 - (41987163941*2^(1/2)*log((2*x^2 - x + 3)^ 
(1/2) + (2^(1/2)*(2*x - 1/2))/2))/1174405120 - (1825528867*(x/2 - 1/8)*(2* 
x^2 - x + 3)^(1/2))/36700160 + (27185733541*(2*x^2 - x + 3)^(1/2)*(32*x^2 
- 4*x + 45))/7046430720 + (804243809*x*(2*x^2 - x + 3)^(3/2))/36700160 + ( 
625271871443*2^(1/2)*log(2*(2*x^2 - x + 3)^(1/2) + (2^(1/2)*(4*x - 1))/2)) 
/9395240960
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.89 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^4 \, dx=\frac {125 \sqrt {2 x^{2}-x +3}\, x^{9}}{2}+\frac {11875 \sqrt {2 x^{2}-x +3}\, x^{8}}{72}+\frac {689675 \sqrt {2 x^{2}-x +3}\, x^{7}}{2304}+\frac {7859255 \sqrt {2 x^{2}-x +3}\, x^{6}}{21504}+\frac {185859571 \sqrt {2 x^{2}-x +3}\, x^{5}}{516096}+\frac {373517159 \sqrt {2 x^{2}-x +3}\, x^{4}}{1474560}+\frac {5759592859 \sqrt {2 x^{2}-x +3}\, x^{3}}{55050240}-\frac {259550711 \sqrt {2 x^{2}-x +3}\, x^{2}}{31457280}+\frac {44812682147 \sqrt {2 x^{2}-x +3}\, x}{1761607680}+\frac {422390234051 \sqrt {2 x^{2}-x +3}}{2348810240}+\frac {8267844569 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right )}{268435456} \] Input:

int((2*x^2-x+3)^(1/2)*(5*x^2+3*x+2)^4,x)
 

Output:

(5284823040000*sqrt(2*x**2 - x + 3)*x**9 + 13946060800000*sqrt(2*x**2 - x 
+ 3)*x**8 + 25311182848000*sqrt(2*x**2 - x + 3)*x**7 + 30903848140800*sqrt 
(2*x**2 - x + 3)*x**6 + 30451232112640*sqrt(2*x**2 - x + 3)*x**5 + 2141896 
7965696*sqrt(2*x**2 - x + 3)*x**4 + 8846734631424*sqrt(2*x**2 - x + 3)*x** 
3 - 697672311168*sqrt(2*x**2 - x + 3)*x**2 + 2151008743056*sqrt(2*x**2 - x 
 + 3)*x + 15206048425836*sqrt(2*x**2 - x + 3) + 2604371039235*sqrt(2)*log( 
(2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23)))/84557168640