Integrand size = 35, antiderivative size = 143 \[ \int \frac {\left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx=-\frac {22053 \sqrt {3+2 x+5 x^2}}{31250}+\frac {36073 x \sqrt {3+2 x+5 x^2}}{1875}-\frac {207427 x^2 \sqrt {3+2 x+5 x^2}}{37500}-\frac {33259 x^3 \sqrt {3+2 x+5 x^2}}{2500}+\frac {5131}{750} x^4 \sqrt {3+2 x+5 x^2}+\frac {49}{30} x^5 \sqrt {3+2 x+5 x^2}-\frac {1719097 \text {arcsinh}\left (\frac {1+5 x}{\sqrt {14}}\right )}{31250 \sqrt {5}} \]
-1719097/156250*arcsinh(1/14*(1+5*x)*14^(1/2))*5^(1/2)-22053/31250*(5*x^2+ 2*x+3)^(1/2)+36073/1875*x*(5*x^2+2*x+3)^(1/2)-207427/37500*x^2*(5*x^2+2*x+ 3)^(1/2)-33259/2500*x^3*(5*x^2+2*x+3)^(1/2)+5131/750*x^4*(5*x^2+2*x+3)^(1/ 2)+49/30*x^5*(5*x^2+2*x+3)^(1/2)
Time = 0.43 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.55 \[ \int \frac {\left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {\sqrt {3+2 x+5 x^2} \left (-132318+3607300 x-1037135 x^2-2494425 x^3+1282750 x^4+306250 x^5\right )}{187500}+\frac {1719097 \log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )}{31250 \sqrt {5}} \]
(Sqrt[3 + 2*x + 5*x^2]*(-132318 + 3607300*x - 1037135*x^2 - 2494425*x^3 + 1282750*x^4 + 306250*x^5))/187500 + (1719097*Log[-1 - 5*x + Sqrt[5]*Sqrt[3 + 2*x + 5*x^2]])/(31250*Sqrt[5])
Time = 0.54 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.17, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {2192, 2192, 27, 2192, 2192, 27, 2192, 27, 1160, 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.
| \(\displaystyle \int \frac {\left (-7 x^2+4 x+1\right )^2 \left (x^2+5 x+2\right )}{\sqrt {5 x^2+2 x+3}} \, dx\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{30} \int \frac {5131 x^5-6135 x^4-2820 x^3+1350 x^2+630 x+60}{\sqrt {5 x^2+2 x+3}}dx+\frac {49}{30} \sqrt {5 x^2+2 x+3} x^5\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{30} \left (\frac {1}{25} \int \frac {6 \left (-33259 x^4-22012 x^3+5625 x^2+2625 x+250\right )}{\sqrt {5 x^2+2 x+3}}dx+\frac {5131}{25} \sqrt {5 x^2+2 x+3} x^4\right )+\frac {49}{30} \sqrt {5 x^2+2 x+3} x^5\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {6}{25} \int \frac {-33259 x^4-22012 x^3+5625 x^2+2625 x+250}{\sqrt {5 x^2+2 x+3}}dx+\frac {5131}{25} \sqrt {5 x^2+2 x+3} x^4\right )+\frac {49}{30} \sqrt {5 x^2+2 x+3} x^5\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{30} \left (\frac {6}{25} \left (\frac {1}{20} \int \frac {-207427 x^3+411831 x^2+52500 x+5000}{\sqrt {5 x^2+2 x+3}}dx-\frac {33259}{20} x^3 \sqrt {5 x^2+2 x+3}\right )+\frac {5131}{25} \sqrt {5 x^2+2 x+3} x^4\right )+\frac {49}{30} \sqrt {5 x^2+2 x+3} x^5\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{30} \left (\frac {6}{25} \left (\frac {1}{20} \left (\frac {1}{15} \int \frac {2 \left (3607300 x^2+1016031 x+37500\right )}{\sqrt {5 x^2+2 x+3}}dx-\frac {207427}{15} x^2 \sqrt {5 x^2+2 x+3}\right )-\frac {33259}{20} x^3 \sqrt {5 x^2+2 x+3}\right )+\frac {5131}{25} \sqrt {5 x^2+2 x+3} x^4\right )+\frac {49}{30} \sqrt {5 x^2+2 x+3} x^5\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {6}{25} \left (\frac {1}{20} \left (\frac {2}{15} \int \frac {3607300 x^2+1016031 x+37500}{\sqrt {5 x^2+2 x+3}}dx-\frac {207427}{15} x^2 \sqrt {5 x^2+2 x+3}\right )-\frac {33259}{20} x^3 \sqrt {5 x^2+2 x+3}\right )+\frac {5131}{25} \sqrt {5 x^2+2 x+3} x^4\right )+\frac {49}{30} \sqrt {5 x^2+2 x+3} x^5\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{30} \left (\frac {6}{25} \left (\frac {1}{20} \left (\frac {2}{15} \left (\frac {1}{10} \int -\frac {30 (22053 x+348230)}{\sqrt {5 x^2+2 x+3}}dx+360730 \sqrt {5 x^2+2 x+3} x\right )-\frac {207427}{15} x^2 \sqrt {5 x^2+2 x+3}\right )-\frac {33259}{20} x^3 \sqrt {5 x^2+2 x+3}\right )+\frac {5131}{25} \sqrt {5 x^2+2 x+3} x^4\right )+\frac {49}{30} \sqrt {5 x^2+2 x+3} x^5\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \left (\frac {6}{25} \left (\frac {1}{20} \left (\frac {2}{15} \left (360730 x \sqrt {5 x^2+2 x+3}-3 \int \frac {22053 x+348230}{\sqrt {5 x^2+2 x+3}}dx\right )-\frac {207427}{15} x^2 \sqrt {5 x^2+2 x+3}\right )-\frac {33259}{20} x^3 \sqrt {5 x^2+2 x+3}\right )+\frac {5131}{25} \sqrt {5 x^2+2 x+3} x^4\right )+\frac {49}{30} \sqrt {5 x^2+2 x+3} x^5\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{30} \left (\frac {6}{25} \left (\frac {1}{20} \left (\frac {2}{15} \left (360730 x \sqrt {5 x^2+2 x+3}-3 \left (\frac {1719097}{5} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx+\frac {22053}{5} \sqrt {5 x^2+2 x+3}\right )\right )-\frac {207427}{15} x^2 \sqrt {5 x^2+2 x+3}\right )-\frac {33259}{20} x^3 \sqrt {5 x^2+2 x+3}\right )+\frac {5131}{25} \sqrt {5 x^2+2 x+3} x^4\right )+\frac {49}{30} \sqrt {5 x^2+2 x+3} x^5\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{30} \left (\frac {6}{25} \left (\frac {1}{20} \left (\frac {2}{15} \left (360730 x \sqrt {5 x^2+2 x+3}-3 \left (\frac {1719097 \int \frac {1}{\sqrt {\frac {1}{56} (10 x+2)^2+1}}d(10 x+2)}{10 \sqrt {70}}+\frac {22053}{5} \sqrt {5 x^2+2 x+3}\right )\right )-\frac {207427}{15} x^2 \sqrt {5 x^2+2 x+3}\right )-\frac {33259}{20} x^3 \sqrt {5 x^2+2 x+3}\right )+\frac {5131}{25} \sqrt {5 x^2+2 x+3} x^4\right )+\frac {49}{30} \sqrt {5 x^2+2 x+3} x^5\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{30} \left (\frac {6}{25} \left (\frac {1}{20} \left (\frac {2}{15} \left (360730 x \sqrt {5 x^2+2 x+3}-3 \left (\frac {1719097 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{5 \sqrt {5}}+\frac {22053}{5} \sqrt {5 x^2+2 x+3}\right )\right )-\frac {207427}{15} x^2 \sqrt {5 x^2+2 x+3}\right )-\frac {33259}{20} x^3 \sqrt {5 x^2+2 x+3}\right )+\frac {5131}{25} \sqrt {5 x^2+2 x+3} x^4\right )+\frac {49}{30} \sqrt {5 x^2+2 x+3} x^5\) |
(49*x^5*Sqrt[3 + 2*x + 5*x^2])/30 + ((5131*x^4*Sqrt[3 + 2*x + 5*x^2])/25 + (6*((-33259*x^3*Sqrt[3 + 2*x + 5*x^2])/20 + ((-207427*x^2*Sqrt[3 + 2*x + 5*x^2])/15 + (2*(360730*x*Sqrt[3 + 2*x + 5*x^2] - 3*((22053*Sqrt[3 + 2*x + 5*x^2])/5 + (1719097*ArcSinh[(2 + 10*x)/(2*Sqrt[14])])/(5*Sqrt[5]))))/15) /20))/25)/30
3.4.87.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
Time = 0.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.38
| method | result | size |
| risch | \(\frac {\left (306250 x^{5}+1282750 x^{4}-2494425 x^{3}-1037135 x^{2}+3607300 x -132318\right ) \sqrt {5 x^{2}+2 x +3}}{187500}-\frac {1719097 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{156250}\) | \(55\) |
| trager | \(\left (\frac {49}{30} x^{5}+\frac {5131}{750} x^{4}-\frac {33259}{2500} x^{3}-\frac {207427}{37500} x^{2}+\frac {36073}{1875} x -\frac {22053}{31250}\right ) \sqrt {5 x^{2}+2 x +3}+\frac {1719097 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +5 \sqrt {5 x^{2}+2 x +3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )\right )}{156250}\) | \(81\) |
| default | \(-\frac {1719097 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{156250}-\frac {22053 \sqrt {5 x^{2}+2 x +3}}{31250}+\frac {49 x^{5} \sqrt {5 x^{2}+2 x +3}}{30}+\frac {5131 x^{4} \sqrt {5 x^{2}+2 x +3}}{750}-\frac {33259 x^{3} \sqrt {5 x^{2}+2 x +3}}{2500}-\frac {207427 x^{2} \sqrt {5 x^{2}+2 x +3}}{37500}+\frac {36073 x \sqrt {5 x^{2}+2 x +3}}{1875}\) | \(113\) |
1/187500*(306250*x^5+1282750*x^4-2494425*x^3-1037135*x^2+3607300*x-132318) *(5*x^2+2*x+3)^(1/2)-1719097/156250*5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))
Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.54 \[ \int \frac {\left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {1}{187500} \, {\left (306250 \, x^{5} + 1282750 \, x^{4} - 2494425 \, x^{3} - 1037135 \, x^{2} + 3607300 \, x - 132318\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {1719097}{312500} \, \sqrt {5} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \]
1/187500*(306250*x^5 + 1282750*x^4 - 2494425*x^3 - 1037135*x^2 + 3607300*x - 132318)*sqrt(5*x^2 + 2*x + 3) + 1719097/312500*sqrt(5)*log(sqrt(5)*sqrt (5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8)
Time = 0.45 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.50 \[ \int \frac {\left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx=\sqrt {5 x^{2} + 2 x + 3} \cdot \left (\frac {49 x^{5}}{30} + \frac {5131 x^{4}}{750} - \frac {33259 x^{3}}{2500} - \frac {207427 x^{2}}{37500} + \frac {36073 x}{1875} - \frac {22053}{31250}\right ) - \frac {1719097 \sqrt {5} \operatorname {asinh}{\left (\frac {5 \sqrt {14} \left (x + \frac {1}{5}\right )}{14} \right )}}{156250} \]
sqrt(5*x**2 + 2*x + 3)*(49*x**5/30 + 5131*x**4/750 - 33259*x**3/2500 - 207 427*x**2/37500 + 36073*x/1875 - 22053/31250) - 1719097*sqrt(5)*asinh(5*sqr t(14)*(x + 1/5)/14)/156250
Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.80 \[ \int \frac {\left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {49}{30} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x^{5} + \frac {5131}{750} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x^{4} - \frac {33259}{2500} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x^{3} - \frac {207427}{37500} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x^{2} + \frac {36073}{1875} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x - \frac {1719097}{156250} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {22053}{31250} \, \sqrt {5 \, x^{2} + 2 \, x + 3} \]
49/30*sqrt(5*x^2 + 2*x + 3)*x^5 + 5131/750*sqrt(5*x^2 + 2*x + 3)*x^4 - 332 59/2500*sqrt(5*x^2 + 2*x + 3)*x^3 - 207427/37500*sqrt(5*x^2 + 2*x + 3)*x^2 + 36073/1875*sqrt(5*x^2 + 2*x + 3)*x - 1719097/156250*sqrt(5)*arcsinh(1/1 4*sqrt(14)*(5*x + 1)) - 22053/31250*sqrt(5*x^2 + 2*x + 3)
Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.50 \[ \int \frac {\left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {1}{187500} \, {\left (5 \, {\left ({\left (5 \, {\left (70 \, {\left (175 \, x + 733\right )} x - 99777\right )} x - 207427\right )} x + 721460\right )} x - 132318\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {1719097}{156250} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \]
1/187500*(5*((5*(70*(175*x + 733)*x - 99777)*x - 207427)*x + 721460)*x - 1 32318)*sqrt(5*x^2 + 2*x + 3) + 1719097/156250*sqrt(5)*log(-sqrt(5)*(sqrt(5 )*x - sqrt(5*x^2 + 2*x + 3)) - 1)
Timed out. \[ \int \frac {\left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx=\int \frac {\left (x^2+5\,x+2\right )\,{\left (-7\,x^2+4\,x+1\right )}^2}{\sqrt {5\,x^2+2\,x+3}} \,d x \]