Integrand size = 33, antiderivative size = 147 \[ \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx=-\frac {128779 (1+5 x) \sqrt {3+2 x+5 x^2}}{250000}-\frac {18397 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{150000}+\frac {149509 \left (3+2 x+5 x^2\right )^{5/2}}{262500}+\frac {2809 x \left (3+2 x+5 x^2\right )^{5/2}}{5250}-\frac {1163 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{1400}-\frac {7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}-\frac {901453 \text {arcsinh}\left (\frac {1+5 x}{\sqrt {14}}\right )}{125000 \sqrt {5}} \]
-18397/150000*(1+5*x)*(5*x^2+2*x+3)^(3/2)+149509/262500*(5*x^2+2*x+3)^(5/2 )+2809/5250*x*(5*x^2+2*x+3)^(5/2)-1163/1400*x^2*(5*x^2+2*x+3)^(5/2)-7/40*x ^3*(5*x^2+2*x+3)^(5/2)-901453/625000*arcsinh(1/14*(1+5*x)*14^(1/2))*5^(1/2 )-128779/250000*(1+5*x)*(5*x^2+2*x+3)^(1/2)
Time = 0.51 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.61 \[ \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx=\frac {\sqrt {3+2 x+5 x^2} \left (22275576+36695150 x+86464445 x^2+78608475 x^3-28373000 x^4-48237500 x^5-127406250 x^6-22968750 x^7\right )}{5250000}+\frac {901453 \log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )}{125000 \sqrt {5}} \]
(Sqrt[3 + 2*x + 5*x^2]*(22275576 + 36695150*x + 86464445*x^2 + 78608475*x^ 3 - 28373000*x^4 - 48237500*x^5 - 127406250*x^6 - 22968750*x^7))/5250000 + (901453*Log[-1 - 5*x + Sqrt[5]*Sqrt[3 + 2*x + 5*x^2]])/(125000*Sqrt[5])
Time = 0.43 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {2192, 2192, 27, 2192, 27, 1160, 1087, 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.
| \(\displaystyle \int \left (-7 x^2+4 x+1\right ) \left (x^2+5 x+2\right ) \left (5 x^2+2 x+3\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{40} \int \left (5 x^2+2 x+3\right )^{3/2} \left (-1163 x^3+343 x^2+520 x+80\right )dx-\frac {7}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{40} \left (\frac {1}{35} \int 2 \left (5 x^2+2 x+3\right )^{3/2} \left (11236 x^2+12589 x+1400\right )dx-\frac {1163}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {7}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{40} \left (\frac {2}{35} \int \left (5 x^2+2 x+3\right )^{3/2} \left (11236 x^2+12589 x+1400\right )dx-\frac {1163}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {7}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{40} \left (\frac {2}{35} \left (\frac {1}{30} \int 2 (149509 x+4146) \left (5 x^2+2 x+3\right )^{3/2}dx+\frac {5618}{15} x \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {1163}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {7}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{40} \left (\frac {2}{35} \left (\frac {1}{15} \int (149509 x+4146) \left (5 x^2+2 x+3\right )^{3/2}dx+\frac {5618}{15} x \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {1163}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {7}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{40} \left (\frac {2}{35} \left (\frac {1}{15} \left (\frac {149509}{25} \left (5 x^2+2 x+3\right )^{5/2}-\frac {128779}{5} \int \left (5 x^2+2 x+3\right )^{3/2}dx\right )+\frac {5618}{15} x \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {1163}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {7}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{40} \left (\frac {2}{35} \left (\frac {1}{15} \left (\frac {149509}{25} \left (5 x^2+2 x+3\right )^{5/2}-\frac {128779}{5} \left (\frac {21}{10} \int \sqrt {5 x^2+2 x+3}dx+\frac {1}{20} (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}\right )\right )+\frac {5618}{15} x \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {1163}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {7}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{40} \left (\frac {2}{35} \left (\frac {1}{15} \left (\frac {149509}{25} \left (5 x^2+2 x+3\right )^{5/2}-\frac {128779}{5} \left (\frac {21}{10} \left (\frac {7}{5} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx+\frac {1}{10} \sqrt {5 x^2+2 x+3} (5 x+1)\right )+\frac {1}{20} (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}\right )\right )+\frac {5618}{15} x \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {1163}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {7}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{40} \left (\frac {2}{35} \left (\frac {1}{15} \left (\frac {149509}{25} \left (5 x^2+2 x+3\right )^{5/2}-\frac {128779}{5} \left (\frac {21}{10} \left (\frac {1}{10} \sqrt {\frac {7}{10}} \int \frac {1}{\sqrt {\frac {1}{56} (10 x+2)^2+1}}d(10 x+2)+\frac {1}{10} \sqrt {5 x^2+2 x+3} (5 x+1)\right )+\frac {1}{20} (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}\right )\right )+\frac {5618}{15} x \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {1163}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {7}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{40} \left (\frac {2}{35} \left (\frac {1}{15} \left (\frac {149509}{25} \left (5 x^2+2 x+3\right )^{5/2}-\frac {128779}{5} \left (\frac {21}{10} \left (\frac {7 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{5 \sqrt {5}}+\frac {1}{10} \sqrt {5 x^2+2 x+3} (5 x+1)\right )+\frac {1}{20} (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}\right )\right )+\frac {5618}{15} x \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {1163}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {7}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\) |
(-7*x^3*(3 + 2*x + 5*x^2)^(5/2))/40 + ((-1163*x^2*(3 + 2*x + 5*x^2)^(5/2)) /35 + (2*((5618*x*(3 + 2*x + 5*x^2)^(5/2))/15 + ((149509*(3 + 2*x + 5*x^2) ^(5/2))/25 - (128779*(((1 + 5*x)*(3 + 2*x + 5*x^2)^(3/2))/20 + (21*(((1 + 5*x)*Sqrt[3 + 2*x + 5*x^2])/10 + (7*ArcSinh[(2 + 10*x)/(2*Sqrt[14])])/(5*S qrt[5])))/10))/5)/15))/35)/40
3.4.82.3.1 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[(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])
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 = 0.53 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.44
| method | result | size |
| risch | \(-\frac {\left (22968750 x^{7}+127406250 x^{6}+48237500 x^{5}+28373000 x^{4}-78608475 x^{3}-86464445 x^{2}-36695150 x -22275576\right ) \sqrt {5 x^{2}+2 x +3}}{5250000}-\frac {901453 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{625000}\) | \(65\) |
| trager | \(\left (-\frac {35}{8} x^{7}-\frac {1359}{56} x^{6}-\frac {3859}{420} x^{5}-\frac {28373}{5250} x^{4}+\frac {1048113}{70000} x^{3}+\frac {17292889}{1050000} x^{2}+\frac {733903}{105000} x +\frac {928149}{218750}\right ) \sqrt {5 x^{2}+2 x +3}-\frac {901453 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )+5 \sqrt {5 x^{2}+2 x +3}\right )}{625000}\) | \(89\) |
| default | \(-\frac {18397 \left (10 x +2\right ) \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{300000}-\frac {128779 \left (10 x +2\right ) \sqrt {5 x^{2}+2 x +3}}{500000}-\frac {901453 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{625000}+\frac {149509 \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{262500}-\frac {7 x^{3} \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{40}-\frac {1163 x^{2} \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{1400}+\frac {2809 x \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{5250}\) | \(117\) |
-1/5250000*(22968750*x^7+127406250*x^6+48237500*x^5+28373000*x^4-78608475* x^3-86464445*x^2-36695150*x-22275576)*(5*x^2+2*x+3)^(1/2)-901453/625000*5^ (1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))
Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.59 \[ \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx=-\frac {1}{5250000} \, {\left (22968750 \, x^{7} + 127406250 \, x^{6} + 48237500 \, x^{5} + 28373000 \, x^{4} - 78608475 \, x^{3} - 86464445 \, x^{2} - 36695150 \, x - 22275576\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {901453}{1250000} \, \sqrt {5} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \]
-1/5250000*(22968750*x^7 + 127406250*x^6 + 48237500*x^5 + 28373000*x^4 - 7 8608475*x^3 - 86464445*x^2 - 36695150*x - 22275576)*sqrt(5*x^2 + 2*x + 3) + 901453/1250000*sqrt(5)*log(sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25* x^2 - 10*x - 8)
Time = 0.46 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.58 \[ \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx=\sqrt {5 x^{2} + 2 x + 3} \left (- \frac {35 x^{7}}{8} - \frac {1359 x^{6}}{56} - \frac {3859 x^{5}}{420} - \frac {28373 x^{4}}{5250} + \frac {1048113 x^{3}}{70000} + \frac {17292889 x^{2}}{1050000} + \frac {733903 x}{105000} + \frac {928149}{218750}\right ) - \frac {901453 \sqrt {5} \operatorname {asinh}{\left (\frac {5 \sqrt {14} \left (x + \frac {1}{5}\right )}{14} \right )}}{625000} \]
sqrt(5*x**2 + 2*x + 3)*(-35*x**7/8 - 1359*x**6/56 - 3859*x**5/420 - 28373* x**4/5250 + 1048113*x**3/70000 + 17292889*x**2/1050000 + 733903*x/105000 + 928149/218750) - 901453*sqrt(5)*asinh(5*sqrt(14)*(x + 1/5)/14)/625000
Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.94 \[ \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx=-\frac {7}{40} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x^{3} - \frac {1163}{1400} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x^{2} + \frac {2809}{5250} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x + \frac {149509}{262500} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} - \frac {18397}{30000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x - \frac {18397}{150000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} - \frac {128779}{50000} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x - \frac {901453}{625000} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {128779}{250000} \, \sqrt {5 \, x^{2} + 2 \, x + 3} \]
-7/40*(5*x^2 + 2*x + 3)^(5/2)*x^3 - 1163/1400*(5*x^2 + 2*x + 3)^(5/2)*x^2 + 2809/5250*(5*x^2 + 2*x + 3)^(5/2)*x + 149509/262500*(5*x^2 + 2*x + 3)^(5 /2) - 18397/30000*(5*x^2 + 2*x + 3)^(3/2)*x - 18397/150000*(5*x^2 + 2*x + 3)^(3/2) - 128779/50000*sqrt(5*x^2 + 2*x + 3)*x - 901453/625000*sqrt(5)*ar csinh(1/14*sqrt(14)*(5*x + 1)) - 128779/250000*sqrt(5*x^2 + 2*x + 3)
Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.56 \[ \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx=-\frac {1}{5250000} \, {\left (5 \, {\left ({\left (5 \, {\left (10 \, {\left (25 \, {\left (15 \, {\left (245 \, x + 1359\right )} x + 7718\right )} x + 113492\right )} x - 3144339\right )} x - 17292889\right )} x - 7339030\right )} x - 22275576\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {901453}{625000} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \]
-1/5250000*(5*((5*(10*(25*(15*(245*x + 1359)*x + 7718)*x + 113492)*x - 314 4339)*x - 17292889)*x - 7339030)*x - 22275576)*sqrt(5*x^2 + 2*x + 3) + 901 453/625000*sqrt(5)*log(-sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)) - 1)
Timed out. \[ \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx=\int \left (x^2+5\,x+2\right )\,{\left (5\,x^2+2\,x+3\right )}^{3/2}\,\left (-7\,x^2+4\,x+1\right ) \,d x \]