\(\int (1+4 x-7 x^2)^3 (2+5 x+x^2) \sqrt {3+2 x+5 x^2} \, dx\) [2]

Optimal result

Integrand size = 35, antiderivative size = 208 \[ \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=-\frac {77159983 (1+5 x) \sqrt {3+2 x+5 x^2}}{31250000}-\frac {1968340667 \left (3+2 x+5 x^2\right )^{3/2}}{131250000}+\frac {1045360143 x \left (3+2 x+5 x^2\right )^{3/2}}{43750000}+\frac {98060877 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{4375000}-\frac {90960857 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{1575000}-\frac {888751 x^4 \left (3+2 x+5 x^2\right )^{3/2}}{105000}+\frac {190939 x^5 \left (3+2 x+5 x^2\right )^{3/2}}{3000}-\frac {50519 x^6 \left (3+2 x+5 x^2\right )^{3/2}}{2250}-\frac {343}{50} x^7 \left (3+2 x+5 x^2\right )^{3/2}-\frac {540119881 \text {arcsinh}\left (\frac {1+5 x}{\sqrt {14}}\right )}{15625000 \sqrt {5}} \] Output:

-77159983/31250000*(1+5*x)*(5*x^2+2*x+3)^(1/2)-1968340667/131250000*(5*x^2 
+2*x+3)^(3/2)[phone]/43750000*x*(5*x^2+2*x+3)^(3/2)[phone]/4375000*x 
^2*(5*x^2+2*x+3)^(3/2)-90960857/1575000*x^3*(5*x^2+2*x+3)^(3/2)-888751/105 
000*x^4*(5*x^2+2*x+3)^(3/2)+190939/3000*x^5*(5*x^2+2*x+3)^(3/2)-50519/2250 
*x^6*(5*x^2+2*x+3)^(3/2)-343/50*x^7*(5*x^2+2*x+3)^(3/2)-540119881/78125000 
*arcsinh(1/14*(1+5*x)*14^(1/2))*5^(1/2)
 

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.48 \[ \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=\frac {\sqrt {3+2 x+5 x^2} \left (-93436408944+57768004650 x+78839046795 x^2-17642392275 x^3-56757413000 x^4-225922362500 x^5+34674656250 x^6+497593468750 x^7-248031875000 x^8-67528125000 x^9\right )}{1968750000}+\frac {540119881 \log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )}{15625000 \sqrt {5}} \] Input:

Integrate[(1 + 4*x - 7*x^2)^3*(2 + 5*x + x^2)*Sqrt[3 + 2*x + 5*x^2],x]
 

Output:

(Sqrt[3 + 2*x + 5*x^2]*(-93436408944 + 57768004650*x + 78839046795*x^2 - 1 
7642392275*x^3 - 56757413000*x^4 - 225922362500*x^5 + 34674656250*x^6 + 49 
7593468750*x^7 - 248031875000*x^8 - 67528125000*x^9))/1968750000 + (540119 
881*Log[-1 - 5*x + Sqrt[5]*Sqrt[3 + 2*x + 5*x^2]])/(15625000*Sqrt[5])
 

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.20, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {2192, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 25, 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.

\(\displaystyle \int \left (-7 x^2+4 x+1\right )^3 \left (x^2+5 x+2\right ) \sqrt {5 x^2+2 x+3} \, dx\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{50} \int \sqrt {5 x^2+2 x+3} \left (-50519 x^7+110453 x^6+6350 x^5-43550 x^4-3050 x^3+5750 x^2+1450 x+100\right )dx-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{50} \left (\frac {1}{45} \int 6 \sqrt {5 x^2+2 x+3} \left (954695 x^6+199182 x^5-326625 x^4-22875 x^3+43125 x^2+10875 x+750\right )dx-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{50} \left (\frac {2}{15} \int \sqrt {5 x^2+2 x+3} \left (954695 x^6+199182 x^5-326625 x^4-22875 x^3+43125 x^2+10875 x+750\right )dx-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{50} \left (\frac {2}{15} \left (\frac {1}{40} \int 5 \sqrt {5 x^2+2 x+3} \left (-888751 x^5-5477085 x^4-183000 x^3+345000 x^2+87000 x+6000\right )dx+\frac {190939}{8} \left (5 x^2+2 x+3\right )^{3/2} x^5\right )-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{50} \left (\frac {2}{15} \left (\frac {1}{8} \int \sqrt {5 x^2+2 x+3} \left (-888751 x^5-5477085 x^4-183000 x^3+345000 x^2+87000 x+6000\right )dx+\frac {190939}{8} \left (5 x^2+2 x+3\right )^{3/2} x^5\right )-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{50} \left (\frac {2}{15} \left (\frac {1}{8} \left (\frac {1}{35} \int 2 \sqrt {5 x^2+2 x+3} \left (-90960857 x^4+2130006 x^3+6037500 x^2+1522500 x+105000\right )dx-\frac {888751}{35} x^4 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {190939}{8} \left (5 x^2+2 x+3\right )^{3/2} x^5\right )-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{50} \left (\frac {2}{15} \left (\frac {1}{8} \left (\frac {2}{35} \int \sqrt {5 x^2+2 x+3} \left (-90960857 x^4+2130006 x^3+6037500 x^2+1522500 x+105000\right )dx-\frac {888751}{35} x^4 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {190939}{8} \left (5 x^2+2 x+3\right )^{3/2} x^5\right )-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{50} \left (\frac {2}{15} \left (\frac {1}{8} \left (\frac {2}{35} \left (\frac {1}{30} \int 9 \sqrt {5 x^2+2 x+3} \left (98060877 x^3+111085857 x^2+5075000 x+350000\right )dx-\frac {90960857}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {888751}{35} x^4 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {190939}{8} \left (5 x^2+2 x+3\right )^{3/2} x^5\right )-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{50} \left (\frac {2}{15} \left (\frac {1}{8} \left (\frac {2}{35} \left (\frac {3}{10} \int \sqrt {5 x^2+2 x+3} \left (98060877 x^3+111085857 x^2+5075000 x+350000\right )dx-\frac {90960857}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {888751}{35} x^4 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {190939}{8} \left (5 x^2+2 x+3\right )^{3/2} x^5\right )-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{50} \left (\frac {2}{15} \left (\frac {1}{8} \left (\frac {2}{35} \left (\frac {3}{10} \left (\frac {1}{25} \int 2 \sqrt {5 x^2+2 x+3} \left (1045360143 x^2-230745131 x+4375000\right )dx+\frac {98060877}{25} \left (5 x^2+2 x+3\right )^{3/2} x^2\right )-\frac {90960857}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {888751}{35} x^4 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {190939}{8} \left (5 x^2+2 x+3\right )^{3/2} x^5\right )-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{50} \left (\frac {2}{15} \left (\frac {1}{8} \left (\frac {2}{35} \left (\frac {3}{10} \left (\frac {2}{25} \int \sqrt {5 x^2+2 x+3} \left (1045360143 x^2-230745131 x+4375000\right )dx+\frac {98060877}{25} \left (5 x^2+2 x+3\right )^{3/2} x^2\right )-\frac {90960857}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {888751}{35} x^4 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {190939}{8} \left (5 x^2+2 x+3\right )^{3/2} x^5\right )-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{50} \left (\frac {2}{15} \left (\frac {1}{8} \left (\frac {2}{35} \left (\frac {3}{10} \left (\frac {2}{25} \left (\frac {1}{20} \int -\left ((9841703335 x+3048580429) \sqrt {5 x^2+2 x+3}\right )dx+\frac {1045360143}{20} x \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {98060877}{25} \left (5 x^2+2 x+3\right )^{3/2} x^2\right )-\frac {90960857}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {888751}{35} x^4 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {190939}{8} \left (5 x^2+2 x+3\right )^{3/2} x^5\right )-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{50} \left (\frac {2}{15} \left (\frac {1}{8} \left (\frac {2}{35} \left (\frac {3}{10} \left (\frac {2}{25} \left (\frac {1045360143}{20} x \left (5 x^2+2 x+3\right )^{3/2}-\frac {1}{20} \int (9841703335 x+3048580429) \sqrt {5 x^2+2 x+3}dx\right )+\frac {98060877}{25} \left (5 x^2+2 x+3\right )^{3/2} x^2\right )-\frac {90960857}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {888751}{35} x^4 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {190939}{8} \left (5 x^2+2 x+3\right )^{3/2} x^5\right )-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{50} \left (\frac {2}{15} \left (\frac {1}{8} \left (\frac {2}{35} \left (\frac {3}{10} \left (\frac {2}{25} \left (\frac {1}{20} \left (-1080239762 \int \sqrt {5 x^2+2 x+3}dx-\frac {1968340667}{3} \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {1045360143}{20} x \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {98060877}{25} \left (5 x^2+2 x+3\right )^{3/2} x^2\right )-\frac {90960857}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {888751}{35} x^4 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {190939}{8} \left (5 x^2+2 x+3\right )^{3/2} x^5\right )-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{50} \left (\frac {2}{15} \left (\frac {1}{8} \left (\frac {2}{35} \left (\frac {3}{10} \left (\frac {2}{25} \left (\frac {1}{20} \left (-1080239762 \left (\frac {7}{5} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx+\frac {1}{10} \sqrt {5 x^2+2 x+3} (5 x+1)\right )-\frac {1968340667}{3} \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {1045360143}{20} x \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {98060877}{25} \left (5 x^2+2 x+3\right )^{3/2} x^2\right )-\frac {90960857}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {888751}{35} x^4 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {190939}{8} \left (5 x^2+2 x+3\right )^{3/2} x^5\right )-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{50} \left (\frac {2}{15} \left (\frac {1}{8} \left (\frac {2}{35} \left (\frac {3}{10} \left (\frac {2}{25} \left (\frac {1}{20} \left (-1080239762 \left (\frac {1}{10} \sqrt {\frac {7}{10}} \int \frac {1}{\sqrt {\frac {1}{56} (10 x+2)^2+1}}d(10 x+2)+\frac {1}{10} \sqrt {5 x^2+2 x+3} (5 x+1)\right )-\frac {1968340667}{3} \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {1045360143}{20} x \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {98060877}{25} \left (5 x^2+2 x+3\right )^{3/2} x^2\right )-\frac {90960857}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {888751}{35} x^4 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {190939}{8} \left (5 x^2+2 x+3\right )^{3/2} x^5\right )-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{50} \left (\frac {2}{15} \left (\frac {1}{8} \left (\frac {2}{35} \left (\frac {3}{10} \left (\frac {2}{25} \left (\frac {1}{20} \left (-1080239762 \left (\frac {7 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{5 \sqrt {5}}+\frac {1}{10} \sqrt {5 x^2+2 x+3} (5 x+1)\right )-\frac {1968340667}{3} \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {1045360143}{20} x \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {98060877}{25} \left (5 x^2+2 x+3\right )^{3/2} x^2\right )-\frac {90960857}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {888751}{35} x^4 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {190939}{8} \left (5 x^2+2 x+3\right )^{3/2} x^5\right )-\frac {50519}{45} x^6 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {343}{50} x^7 \left (5 x^2+2 x+3\right )^{3/2}\)

Input:

Int[(1 + 4*x - 7*x^2)^3*(2 + 5*x + x^2)*Sqrt[3 + 2*x + 5*x^2],x]
 

Output:

(-343*x^7*(3 + 2*x + 5*x^2)^(3/2))/50 + ((-50519*x^6*(3 + 2*x + 5*x^2)^(3/ 
2))/45 + (2*((190939*x^5*(3 + 2*x + 5*x^2)^(3/2))/8 + ((-888751*x^4*(3 + 2 
*x + 5*x^2)^(3/2))/35 + (2*((-90960857*x^3*(3 + 2*x + 5*x^2)^(3/2))/30 + ( 
3*((98060877*x^2*(3 + 2*x + 5*x^2)^(3/2))/25 + (2*((1045360143*x*(3 + 2*x 
+ 5*x^2)^(3/2))/20 + ((-1968340667*(3 + 2*x + 5*x^2)^(3/2))/3 - 1080239762 
*(((1 + 5*x)*Sqrt[3 + 2*x + 5*x^2])/10 + (7*ArcSinh[(2 + 10*x)/(2*Sqrt[14] 
)])/(5*Sqrt[5])))/20))/25))/10))/35)/8))/15)/50
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 = 0.68 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.36

method	result	size
risch	\(-\frac {\left (67528125000 x^{9}[phone] x^{8}-497593468750 x^{7}-34674656250 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}-78839046795 x^{2}-57768004650 x [phone]\right ) \sqrt {5 x^{2}+2 x +3}}{1968750000}-\frac {540119881 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{78125000}\)	\(75\)
trager	\(\left (-\frac {343}{10} x^{9}-\frac {56693}{450} x^{8}+\frac {2274713}{9000} x^{7}+\frac {369863}{21000} x^{6}-\frac {18073789}{157500} x^{5}-\frac {56757413}{1968750} x^{4}-\frac {235231897}{26250000} x^{3}+\frac {5255936453}{131250000} x^{2}+\frac {385120031}{13125000} x -\frac {648863951}{13671875}\right ) \sqrt {5 x^{2}+2 x +3}+\frac {540119881 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )+5 \sqrt {5 x^{2}+2 x +3}\right )}{78125000}\)	\(101\)
default	\(-\frac {77159983 \left (10 x +2\right ) \sqrt {5 x^{2}+2 x +3}}{62500000}-\frac {540119881 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{78125000}-\frac {1968340667 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{131250000}+\frac {1045360143 x \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{43750000}+\frac {98060877 x^{2} \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{4375000}-\frac {90960857 x^{3} \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{1575000}-\frac {888751 x^{4} \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{105000}+\frac {190939 x^{5} \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{3000}-\frac {50519 x^{6} \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{2250}-\frac {343 x^{7} \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{50}\)	\(166\)

Input:

int((-7*x^2+4*x+1)^3*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x,method=_RETURNVERBO 
SE)
 

Output:

-1/1968750000*(67528125000*x^9+248031875000*x^8-497593468750*x^7-346746562 
50*x^6+225922362500*x^5+56757413000*x^4+17642392275*x^3-78839046795*x^2-57 
768004650*x[phone])*(5*x^2+2*x+3)^(1/2)-540119881/78125000*5^(1/2)*ar 
csinh(5/14*14^(1/2)*(x+1/5))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.47 \[ \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=-\frac {1}{1968750000} \, {\left (67528125000 \, x^{9} + 248031875000 \, x^{8} - 497593468750 \, x^{7} - 34674656250 \, x^{6} + 225922362500 \, x^{5} + 56757413000 \, x^{4} + 17642392275 \, x^{3} - 78839046795 \, x^{2} - 57768004650 \, x + 93436408944\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {540119881}{156250000} \, \sqrt {5} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \] Input:

integrate((-7*x^2+4*x+1)^3*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x, algorithm="f 
ricas")
 

Output:

-1/1968750000*(67528125000*x^9 + 248031875000*x^8 - 497593468750*x^7 - 346 
74656250*x^6 + 225922362500*x^5 + 56757413000*x^4 + 17642392275*x^3 - 7883 
9046795*x^2 - 57768004650*x + 93436408944)*sqrt(5*x^2 + 2*x + 3) + 5401198 
81/156250000*sqrt(5)*log(sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 
- 10*x - 8)
 

Sympy [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.48 \[ \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=\sqrt {5 x^{2} + 2 x + 3} \left (- \frac {343 x^{9}}{10} - \frac {56693 x^{8}}{450} + \frac {2274713 x^{7}}{9000} + \frac {369863 x^{6}}{21000} - \frac {18073789 x^{5}}{157500} - \frac {56757413 x^{4}}{1968750} - \frac {235231897 x^{3}}{26250000} + \frac {5255936453 x^{2}}{131250000} + \frac {385120031 x}{13125000} - \frac {648863951}{13671875}\right ) - \frac {540119881 \sqrt {5} \operatorname {asinh}{\left (\frac {5 \sqrt {14} \left (x + \frac {1}{5}\right )}{14} \right )}}{78125000} \] Input:

integrate((-7*x**2+4*x+1)**3*(x**2+5*x+2)*(5*x**2+2*x+3)**(1/2),x)
 

Output:

sqrt(5*x**2 + 2*x + 3)*(-343*x**9/10 - 56693*x**8/450 + 2274713*x**7/9000 
+ 369863*x**6/21000 - 18073789*x**5/157500 - 56757413*x**4/1968750 - 23523 
1897*x**3/26250000 + 5255936453*x**2/131250000 + 385120031*x/13125000 - 64 
8863951/13671875) - 540119881*sqrt(5)*asinh(5*sqrt(14)*(x + 1/5)/14)/78125 
000
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.85 \[ \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=-\frac {343}{50} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{7} - \frac {50519}{2250} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{6} + \frac {190939}{3000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{5} - \frac {888751}{105000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {90960857}{1575000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{3} + \frac {98060877}{4375000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {1045360143}{43750000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x - \frac {1968340667}{131250000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} - \frac {77159983}{6250000} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x - \frac {540119881}{78125000} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {77159983}{31250000} \, \sqrt {5 \, x^{2} + 2 \, x + 3} \] Input:

integrate((-7*x^2+4*x+1)^3*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x, algorithm="m 
axima")
 

Output:

-343/50*(5*x^2 + 2*x + 3)^(3/2)*x^7 - 50519/2250*(5*x^2 + 2*x + 3)^(3/2)*x 
^6 + 190939/3000*(5*x^2 + 2*x + 3)^(3/2)*x^5 - 888751/105000*(5*x^2 + 2*x 
+ 3)^(3/2)*x^4 - 90960857/1575000*(5*x^2 + 2*x + 3)^(3/2)*x^3 + 98060877/4 
375000*(5*x^2 + 2*x + 3)^(3/2)*x^2 + 1045360143/43750000*(5*x^2 + 2*x + 3) 
^(3/2)*x - 1968340667/131250000*(5*x^2 + 2*x + 3)^(3/2) - 77159983/6250000 
*sqrt(5*x^2 + 2*x + 3)*x - 540119881/78125000*sqrt(5)*arcsinh(1/14*sqrt(14 
)*(5*x + 1)) - 77159983/31250000*sqrt(5*x^2 + 2*x + 3)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.44 \[ \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=-\frac {1}{1968750000} \, {\left (5 \, {\left ({\left (5 \, {\left (10 \, {\left (25 \, {\left (5 \, {\left (49 \, {\left (140 \, {\left (315 \, x + 1157\right )} x - 324959\right )} x - 1109589\right )} x + 36147578\right )} x + 227029652\right )} x + 705695691\right )} x - 15767809359\right )} x - 11553600930\right )} x + 93436408944\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {540119881}{78125000} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \] Input:

integrate((-7*x^2+4*x+1)^3*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x, algorithm="g 
iac")
 

Output:

-1/1968750000*(5*((5*(10*(25*(5*(49*(140*(315*x + 1157)*x - 324959)*x - 11 
09589)*x + 36147578)*x + 227029652)*x + 705695691)*x - 15767809359)*x - 11 
553600930)*x + 93436408944)*sqrt(5*x^2 + 2*x + 3) + 540119881/78125000*sqr 
t(5)*log(-sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)) - 1)
 

Mupad [B] (verification not implemented)

Time = 20.51 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.06 \[ \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=\frac {98060877\,x^2\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{4375000}-\frac {90960857\,x^3\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{1575000}-\frac {888751\,x^4\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{105000}+\frac {190939\,x^5\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{3000}-\frac {50519\,x^6\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{2250}-\frac {343\,x^7\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{50}-\frac {3048580429\,\sqrt {5}\,\ln \left (\sqrt {5\,x^2+2\,x+3}+\frac {\sqrt {5}\,\left (5\,x+1\right )}{5}\right )}{156250000}-\frac {3048580429\,\left (\frac {x}{2}+\frac {1}{10}\right )\,\sqrt {5\,x^2+2\,x+3}}{43750000}-\frac {1968340667\,\sqrt {5\,x^2+2\,x+3}\,\left (200\,x^2+20\,x+108\right )}{5250000000}+\frac {1045360143\,x\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{43750000}+\frac {1968340667\,\sqrt {5}\,\ln \left (2\,\sqrt {5\,x^2+2\,x+3}+\frac {\sqrt {5}\,\left (10\,x+2\right )}{5}\right )}{156250000} \] Input:

int((5*x + x^2 + 2)*(2*x + 5*x^2 + 3)^(1/2)*(4*x - 7*x^2 + 1)^3,x)
                                                                                    
                                                                                    
 

Output:

(98060877*x^2*(2*x + 5*x^2 + 3)^(3/2))/4375000 - (90960857*x^3*(2*x + 5*x^ 
2 + 3)^(3/2))/1575000 - (888751*x^4*(2*x + 5*x^2 + 3)^(3/2))/105000 + (190 
939*x^5*(2*x + 5*x^2 + 3)^(3/2))/3000 - (50519*x^6*(2*x + 5*x^2 + 3)^(3/2) 
)/2250 - (343*x^7*(2*x + 5*x^2 + 3)^(3/2))/50 - (3048580429*5^(1/2)*log((2 
*x + 5*x^2 + 3)^(1/2) + (5^(1/2)*(5*x + 1))/5))/156250000 - (3048580429*(x 
/2 + 1/10)*(2*x + 5*x^2 + 3)^(1/2))/43750000 - (1968340667*(2*x + 5*x^2 + 
3)^(1/2)*(20*x + 200*x^2 + 108))/5250000000 + (1045360143*x*(2*x + 5*x^2 + 
 3)^(3/2))/43750000 + (1968340667*5^(1/2)*log(2*(2*x + 5*x^2 + 3)^(1/2) + 
(5^(1/2)*(10*x + 2))/5))/156250000
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.89 \[ \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=-\frac {343 \sqrt {5 x^{2}+2 x +3}\, x^{9}}{10}-\frac {56693 \sqrt {5 x^{2}+2 x +3}\, x^{8}}{450}+\frac {2274713 \sqrt {5 x^{2}+2 x +3}\, x^{7}}{9000}+\frac {369863 \sqrt {5 x^{2}+2 x +3}\, x^{6}}{21000}-\frac {18073789 \sqrt {5 x^{2}+2 x +3}\, x^{5}}{157500}-\frac {56757413 \sqrt {5 x^{2}+2 x +3}\, x^{4}}{1968750}-\frac {235231897 \sqrt {5 x^{2}+2 x +3}\, x^{3}}{26250000}+\frac {5255936453 \sqrt {5 x^{2}+2 x +3}\, x^{2}}{131250000}+\frac {385120031 \sqrt {5 x^{2}+2 x +3}\, x}{13125000}-\frac {648863951 \sqrt {5 x^{2}+2 x +3}}{13671875}-\frac {540119881 \sqrt {5}\, \mathrm {log}\left (\frac {\sqrt {5 x^{2}+2 x +3}\, \sqrt {5}+5 x +1}{\sqrt {14}}\right )}{78125000} \] Input:

int((-7*x^2+4*x+1)^3*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x)
 

Output:

( - 337640625000*sqrt(5*x**2 + 2*x + 3)*x**9 - 1240159375000*sqrt(5*x**2 + 
 2*x + 3)*x**8 + 2487967343750*sqrt(5*x**2 + 2*x + 3)*x**7 + 173373281250* 
sqrt(5*x**2 + 2*x + 3)*x**6 - 1129611812500*sqrt(5*x**2 + 2*x + 3)*x**5 - 
283787065000*sqrt(5*x**2 + 2*x + 3)*x**4 - 88211961375*sqrt(5*x**2 + 2*x + 
 3)*x**3 + 394195233975*sqrt(5*x**2 + 2*x + 3)*x**2 + 288840023250*sqrt(5* 
x**2 + 2*x + 3)*x - 467182044720*sqrt(5*x**2 + 2*x + 3) - 68055105006*sqrt 
(5)*log((sqrt(5*x**2 + 2*x + 3)*sqrt(5) + 5*x + 1)/sqrt(14)))/9843750000