Integrand size = 35, antiderivative size = 231 \[ \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx=-\frac {479652579 (1+5 x) \sqrt {3+2 x+5 x^2}}{312500000}-\frac {22840599 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{62500000}-\frac {6133820867 \left (3+2 x+5 x^2\right )^{5/2}}{1203125000}+\frac {837379699 x \left (3+2 x+5 x^2\right )^{5/2}}{72187500}+\frac {2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac {190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac {796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac {1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac {61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac {343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}-\frac {3357568053 \text {arcsinh}\left (\frac {1+5 x}{\sqrt {14}}\right )}{156250000 \sqrt {5}} \]
-22840599/62500000*(1+5*x)*(5*x^2+2*x+3)^(3/2)-6133820867/1203125000*(5*x^ 2+2*x+3)^(5/2)+837379699/72187500*x*(5*x^2+2*x+3)^(5/2)+2173004363/1732500 00*x^2*(5*x^2+2*x+3)^(5/2)-190236913/4950000*x^3*(5*x^2+2*x+3)^(5/2)-79655 9/123750*x^4*(5*x^2+2*x+3)^(5/2)+1031177/20625*x^5*(5*x^2+2*x+3)^(5/2)-611 03/3300*x^6*(5*x^2+2*x+3)^(5/2)-343/60*x^7*(5*x^2+2*x+3)^(5/2)-3357568053/ 781250000*arcsinh(1/14*(1+5*x)*14^(1/2))*5^(1/2)-479652579/312500000*(1+5* x)*(5*x^2+2*x+3)^(1/2)
Time = 0.86 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.47 \[ \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx=\frac {\sqrt {3+2 x+5 x^2} \left (-10506617068392+6352777129950 x+15865844408685 x^2+19041688239675 x^3+2573089891000 x^4-85130334087500 x^5-52106830406250 x^6+72918247281250 x^7+30505457500000 x^8+148393743750000 x^9-125007421875000 x^{10}-30950390625000 x^{11}\right )}{216562500000}+\frac {3357568053 \log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )}{156250000 \sqrt {5}} \]
(Sqrt[3 + 2*x + 5*x^2]*(-10506617068392 + 6352777129950*x + 15865844408685 *x^2 + 19041688239675*x^3 + 2573089891000*x^4 - 85130334087500*x^5 - 52106 830406250*x^6 + 72918247281250*x^7 + 30505457500000*x^8 + 148393743750000* x^9 - 125007421875000*x^10 - 30950390625000*x^11))/216562500000 + (3357568 053*Log[-1 - 5*x + Sqrt[5]*Sqrt[3 + 2*x + 5*x^2]])/(156250000*Sqrt[5])
Time = 0.89 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.21, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.457, Rules used = {2192, 2192, 27, 2192, 27, 2192, 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.
| \(\displaystyle \int \left (-7 x^2+4 x+1\right )^3 \left (x^2+5 x+2\right ) \left (5 x^2+2 x+3\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{60} \int \left (5 x^2+2 x+3\right )^{3/2} \left (-61103 x^7+131103 x^6+7620 x^5-52260 x^4-3660 x^3+6900 x^2+1740 x+120\right )dx-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{60} \left (\frac {1}{55} \int 2 \left (5 x^2+2 x+3\right )^{3/2} \left (4124708 x^6+759477 x^5-1437150 x^4-100650 x^3+189750 x^2+47850 x+3300\right )dx-\frac {61103}{55} x^6 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{60} \left (\frac {2}{55} \int \left (5 x^2+2 x+3\right )^{3/2} \left (4124708 x^6+759477 x^5-1437150 x^4-100650 x^3+189750 x^2+47850 x+3300\right )dx-\frac {61103}{55} x^6 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{60} \left (\frac {2}{55} \left (\frac {1}{50} \int 30 \left (5 x^2+2 x+3\right )^{3/2} \left (-796559 x^5-4457604 x^4-167750 x^3+316250 x^2+79750 x+5500\right )dx+\frac {2062354}{25} \left (5 x^2+2 x+3\right )^{5/2} x^5\right )-\frac {61103}{55} x^6 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{60} \left (\frac {2}{55} \left (\frac {3}{5} \int \left (5 x^2+2 x+3\right )^{3/2} \left (-796559 x^5-4457604 x^4-167750 x^3+316250 x^2+79750 x+5500\right )dx+\frac {2062354}{25} \left (5 x^2+2 x+3\right )^{5/2} x^5\right )-\frac {61103}{55} x^6 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{60} \left (\frac {2}{55} \left (\frac {3}{5} \left (\frac {1}{45} \int \left (5 x^2+2 x+3\right )^{3/2} \left (-190236913 x^4+2009958 x^3+14231250 x^2+3588750 x+247500\right )dx-\frac {796559}{45} x^4 \left (5 x^2+2 x+3\right )^{5/2}\right )+\frac {2062354}{25} \left (5 x^2+2 x+3\right )^{5/2} x^5\right )-\frac {61103}{55} x^6 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{60} \left (\frac {2}{55} \left (\frac {3}{5} \left (\frac {1}{45} \left (\frac {1}{40} \int \left (5 x^2+2 x+3\right )^{3/2} \left (2173004363 x^3+2281382217 x^2+143550000 x+9900000\right )dx-\frac {190236913}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {796559}{45} x^4 \left (5 x^2+2 x+3\right )^{5/2}\right )+\frac {2062354}{25} \left (5 x^2+2 x+3\right )^{5/2} x^5\right )-\frac {61103}{55} x^6 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{60} \left (\frac {2}{55} \left (\frac {3}{5} \left (\frac {1}{45} \left (\frac {1}{40} \left (\frac {1}{35} \int 6 \left (5 x^2+2 x+3\right )^{3/2} \left (10048556388 x^2-1335629363 x+57750000\right )dx+\frac {2173004363}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {190236913}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {796559}{45} x^4 \left (5 x^2+2 x+3\right )^{5/2}\right )+\frac {2062354}{25} \left (5 x^2+2 x+3\right )^{5/2} x^5\right )-\frac {61103}{55} x^6 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{60} \left (\frac {2}{55} \left (\frac {3}{5} \left (\frac {1}{45} \left (\frac {1}{40} \left (\frac {6}{35} \int \left (5 x^2+2 x+3\right )^{3/2} \left (10048556388 x^2-1335629363 x+57750000\right )dx+\frac {2173004363}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {190236913}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {796559}{45} x^4 \left (5 x^2+2 x+3\right )^{5/2}\right )+\frac {2062354}{25} \left (5 x^2+2 x+3\right )^{5/2} x^5\right )-\frac {61103}{55} x^6 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{60} \left (\frac {2}{55} \left (\frac {3}{5} \left (\frac {1}{45} \left (\frac {1}{40} \left (\frac {6}{35} \left (\frac {1}{30} \int -18 (6133820867 x+1578509398) \left (5 x^2+2 x+3\right )^{3/2}dx+\frac {1674759398}{5} x \left (5 x^2+2 x+3\right )^{5/2}\right )+\frac {2173004363}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {190236913}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {796559}{45} x^4 \left (5 x^2+2 x+3\right )^{5/2}\right )+\frac {2062354}{25} \left (5 x^2+2 x+3\right )^{5/2} x^5\right )-\frac {61103}{55} x^6 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{60} \left (\frac {2}{55} \left (\frac {3}{5} \left (\frac {1}{45} \left (\frac {1}{40} \left (\frac {6}{35} \left (\frac {1674759398}{5} x \left (5 x^2+2 x+3\right )^{5/2}-\frac {3}{5} \int (6133820867 x+1578509398) \left (5 x^2+2 x+3\right )^{3/2}dx\right )+\frac {2173004363}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {190236913}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {796559}{45} x^4 \left (5 x^2+2 x+3\right )^{5/2}\right )+\frac {2062354}{25} \left (5 x^2+2 x+3\right )^{5/2} x^5\right )-\frac {61103}{55} x^6 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{60} \left (\frac {2}{55} \left (\frac {3}{5} \left (\frac {1}{45} \left (\frac {1}{40} \left (\frac {6}{35} \left (\frac {1674759398}{5} x \left (5 x^2+2 x+3\right )^{5/2}-\frac {3}{5} \left (\frac {1758726123}{5} \int \left (5 x^2+2 x+3\right )^{3/2}dx+\frac {6133820867}{25} \left (5 x^2+2 x+3\right )^{5/2}\right )\right )+\frac {2173004363}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {190236913}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {796559}{45} x^4 \left (5 x^2+2 x+3\right )^{5/2}\right )+\frac {2062354}{25} \left (5 x^2+2 x+3\right )^{5/2} x^5\right )-\frac {61103}{55} x^6 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{60} \left (\frac {2}{55} \left (\frac {3}{5} \left (\frac {1}{45} \left (\frac {1}{40} \left (\frac {6}{35} \left (\frac {1674759398}{5} x \left (5 x^2+2 x+3\right )^{5/2}-\frac {3}{5} \left (\frac {1758726123}{5} \left (\frac {21}{10} \int \sqrt {5 x^2+2 x+3}dx+\frac {1}{20} (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {6133820867}{25} \left (5 x^2+2 x+3\right )^{5/2}\right )\right )+\frac {2173004363}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {190236913}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {796559}{45} x^4 \left (5 x^2+2 x+3\right )^{5/2}\right )+\frac {2062354}{25} \left (5 x^2+2 x+3\right )^{5/2} x^5\right )-\frac {61103}{55} x^6 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{60} \left (\frac {2}{55} \left (\frac {3}{5} \left (\frac {1}{45} \left (\frac {1}{40} \left (\frac {6}{35} \left (\frac {1674759398}{5} x \left (5 x^2+2 x+3\right )^{5/2}-\frac {3}{5} \left (\frac {1758726123}{5} \left (\frac {21}{10} \left (\frac {7}{5} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx+\frac {1}{10} \sqrt {5 x^2+2 x+3} (5 x+1)\right )+\frac {1}{20} (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {6133820867}{25} \left (5 x^2+2 x+3\right )^{5/2}\right )\right )+\frac {2173004363}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {190236913}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {796559}{45} x^4 \left (5 x^2+2 x+3\right )^{5/2}\right )+\frac {2062354}{25} \left (5 x^2+2 x+3\right )^{5/2} x^5\right )-\frac {61103}{55} x^6 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{60} \left (\frac {2}{55} \left (\frac {3}{5} \left (\frac {1}{45} \left (\frac {1}{40} \left (\frac {6}{35} \left (\frac {1674759398}{5} x \left (5 x^2+2 x+3\right )^{5/2}-\frac {3}{5} \left (\frac {1758726123}{5} \left (\frac {21}{10} \left (\frac {1}{10} \sqrt {\frac {7}{10}} \int \frac {1}{\sqrt {\frac {1}{56} (10 x+2)^2+1}}d(10 x+2)+\frac {1}{10} \sqrt {5 x^2+2 x+3} (5 x+1)\right )+\frac {1}{20} (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {6133820867}{25} \left (5 x^2+2 x+3\right )^{5/2}\right )\right )+\frac {2173004363}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {190236913}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {796559}{45} x^4 \left (5 x^2+2 x+3\right )^{5/2}\right )+\frac {2062354}{25} \left (5 x^2+2 x+3\right )^{5/2} x^5\right )-\frac {61103}{55} x^6 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{60} \left (\frac {2}{55} \left (\frac {3}{5} \left (\frac {1}{45} \left (\frac {1}{40} \left (\frac {6}{35} \left (\frac {1674759398}{5} x \left (5 x^2+2 x+3\right )^{5/2}-\frac {3}{5} \left (\frac {1758726123}{5} \left (\frac {21}{10} \left (\frac {7 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{5 \sqrt {5}}+\frac {1}{10} \sqrt {5 x^2+2 x+3} (5 x+1)\right )+\frac {1}{20} (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {6133820867}{25} \left (5 x^2+2 x+3\right )^{5/2}\right )\right )+\frac {2173004363}{35} x^2 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {190236913}{40} x^3 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {796559}{45} x^4 \left (5 x^2+2 x+3\right )^{5/2}\right )+\frac {2062354}{25} \left (5 x^2+2 x+3\right )^{5/2} x^5\right )-\frac {61103}{55} x^6 \left (5 x^2+2 x+3\right )^{5/2}\right )-\frac {343}{60} x^7 \left (5 x^2+2 x+3\right )^{5/2}\) |
(-343*x^7*(3 + 2*x + 5*x^2)^(5/2))/60 + ((-61103*x^6*(3 + 2*x + 5*x^2)^(5/ 2))/55 + (2*((2062354*x^5*(3 + 2*x + 5*x^2)^(5/2))/25 + (3*((-796559*x^4*( 3 + 2*x + 5*x^2)^(5/2))/45 + ((-190236913*x^3*(3 + 2*x + 5*x^2)^(5/2))/40 + ((2173004363*x^2*(3 + 2*x + 5*x^2)^(5/2))/35 + (6*((1674759398*x*(3 + 2* x + 5*x^2)^(5/2))/5 - (3*((6133820867*(3 + 2*x + 5*x^2)^(5/2))/25 + (17587 26123*(((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*Sqrt[5])))/10))/ 5))/5))/35)/40)/45))/5))/55)/60
3.4.80.3.1 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 = 0.24 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.37
| method | result | size |
| risch | \(-\frac {\left (30950390625000 x^{11}+125007421875000 x^{10}-148393743750000 x^{9}-30505457500000 x^{8}-72918247281250 x^{7}+52106830406250 x^{6}+85130334087500 x^{5}-2573089891000 x^{4}-19041688239675 x^{3}-15865844408685 x^{2}-6352777129950 x +10506617068392\right ) \sqrt {5 x^{2}+2 x +3}}{216562500000}-\frac {3357568053 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{781250000}\) | \(85\) |
| trager | \(\left (-\frac {1715}{12} x^{11}-\frac {76195}{132} x^{10}+\frac {376873}{550} x^{9}+\frac {1743169}{12375} x^{8}+\frac {333340559}{990000} x^{7}-\frac {555806191}{2310000} x^{6}-\frac {6810426727}{17325000} x^{5}+\frac {2573089891}{216562500} x^{4}+\frac {253889176529}{2887500000} x^{3}+\frac {352574320193}{4812500000} x^{2}+\frac {14117282511}{481250000} x -\frac {145925237061}{3007812500}\right ) \sqrt {5 x^{2}+2 x +3}-\frac {3357568053 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )+5 \sqrt {5 x^{2}+2 x +3}\right )}{781250000}\) | \(109\) |
| default | \(-\frac {22840599 \left (10 x +2\right ) \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{125000000}-\frac {479652579 \left (10 x +2\right ) \sqrt {5 x^{2}+2 x +3}}{625000000}-\frac {3357568053 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{781250000}-\frac {6133820867 \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{1203125000}+\frac {2173004363 x^{2} \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{173250000}+\frac {837379699 x \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{72187500}-\frac {343 x^{7} \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{60}-\frac {61103 x^{6} \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{3300}+\frac {1031177 x^{5} \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{20625}-\frac {796559 x^{4} \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{123750}-\frac {190236913 x^{3} \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{4950000}\) | \(185\) |
-1/216562500000*(30950390625000*x^11+125007421875000*x^10-148393743750000* x^9-30505457500000*x^8-72918247281250*x^7+52106830406250*x^6+8513033408750 0*x^5-2573089891000*x^4-19041688239675*x^3-15865844408685*x^2-635277712995 0*x+10506617068392)*(5*x^2+2*x+3)^(1/2)-3357568053/781250000*5^(1/2)*arcsi nh(5/14*14^(1/2)*(x+1/5))
Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.46 \[ \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx=-\frac {1}{216562500000} \, {\left (30950390625000 \, x^{11} + 125007421875000 \, x^{10} - 148393743750000 \, x^{9} - 30505457500000 \, x^{8} - 72918247281250 \, x^{7} + 52106830406250 \, x^{6} + 85130334087500 \, x^{5} - 2573089891000 \, x^{4} - 19041688239675 \, x^{3} - 15865844408685 \, x^{2} - 6352777129950 \, x + 10506617068392\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {3357568053}{1562500000} \, \sqrt {5} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \]
-1/216562500000*(30950390625000*x^11 + 125007421875000*x^10 - 148393743750 000*x^9 - 30505457500000*x^8 - 72918247281250*x^7 + 52106830406250*x^6 + 8 5130334087500*x^5 - 2573089891000*x^4 - 19041688239675*x^3 - 1586584440868 5*x^2 - 6352777129950*x + 10506617068392)*sqrt(5*x^2 + 2*x + 3) + 33575680 53/1562500000*sqrt(5)*log(sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8)
Time = 0.63 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.48 \[ \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx=\sqrt {5 x^{2} + 2 x + 3} \left (- \frac {1715 x^{11}}{12} - \frac {76195 x^{10}}{132} + \frac {376873 x^{9}}{550} + \frac {1743169 x^{8}}{12375} + \frac {333340559 x^{7}}{990000} - \frac {555806191 x^{6}}{2310000} - \frac {6810426727 x^{5}}{17325000} + \frac {2573089891 x^{4}}{216562500} + \frac {253889176529 x^{3}}{2887500000} + \frac {352574320193 x^{2}}{4812500000} + \frac {14117282511 x}{481250000} - \frac {145925237061}{3007812500}\right ) - \frac {3357568053 \sqrt {5} \operatorname {asinh}{\left (\frac {5 \sqrt {14} \left (x + \frac {1}{5}\right )}{14} \right )}}{781250000} \]
sqrt(5*x**2 + 2*x + 3)*(-1715*x**11/12 - 76195*x**10/132 + 376873*x**9/550 + 1743169*x**8/12375 + 333340559*x**7/990000 - 555806191*x**6/2310000 - 6 810426727*x**5/17325000 + 2573089891*x**4/216562500 + 253889176529*x**3/28 87500000 + 352574320193*x**2/4812500000 + 14117282511*x/481250000 - 145925 237061/3007812500) - 3357568053*sqrt(5)*asinh(5*sqrt(14)*(x + 1/5)/14)/781 250000
Time = 0.28 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.89 \[ \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx=-\frac {343}{60} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x^{7} - \frac {61103}{3300} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x^{6} + \frac {1031177}{20625} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x^{5} - \frac {796559}{123750} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x^{4} - \frac {190236913}{4950000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x^{3} + \frac {2173004363}{173250000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x^{2} + \frac {837379699}{72187500} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x - \frac {6133820867}{1203125000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} - \frac {22840599}{12500000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x - \frac {22840599}{62500000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} - \frac {479652579}{62500000} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x - \frac {3357568053}{781250000} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {479652579}{312500000} \, \sqrt {5 \, x^{2} + 2 \, x + 3} \]
-343/60*(5*x^2 + 2*x + 3)^(5/2)*x^7 - 61103/3300*(5*x^2 + 2*x + 3)^(5/2)*x ^6 + 1031177/20625*(5*x^2 + 2*x + 3)^(5/2)*x^5 - 796559/123750*(5*x^2 + 2* x + 3)^(5/2)*x^4 - 190236913/4950000*(5*x^2 + 2*x + 3)^(5/2)*x^3 + 2173004 363/173250000*(5*x^2 + 2*x + 3)^(5/2)*x^2 + 837379699/72187500*(5*x^2 + 2* x + 3)^(5/2)*x - 6133820867/1203125000*(5*x^2 + 2*x + 3)^(5/2) - 22840599/ 12500000*(5*x^2 + 2*x + 3)^(3/2)*x - 22840599/62500000*(5*x^2 + 2*x + 3)^( 3/2) - 479652579/62500000*sqrt(5*x^2 + 2*x + 3)*x - 3357568053/781250000*s qrt(5)*arcsinh(1/14*sqrt(14)*(5*x + 1)) - 479652579/312500000*sqrt(5*x^2 + 2*x + 3)
Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.44 \[ \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx=-\frac {1}{216562500000} \, {\left (5 \, {\left ({\left (5 \, {\left (10 \, {\left (25 \, {\left (5 \, {\left (7 \, {\left (20 \, {\left (105 \, {\left (875 \, {\left (77 \, x + 311\right )} x - 323034\right )} x - 6972676\right )} x - 333340559\right )} x + 1667418573\right )} x + 13620853454\right )} x - 10292359564\right )} x - 761667529587\right )} x - 3173168881737\right )} x - 1270555425990\right )} x + 10506617068392\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {3357568053}{781250000} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \]
-1/216562500000*(5*((5*(10*(25*(5*(7*(20*(105*(875*(77*x + 311)*x - 323034 )*x - 6972676)*x - 333340559)*x + 1667418573)*x + 13620853454)*x - 1029235 9564)*x - 761667529587)*x - 3173168881737)*x - 1270555425990)*x + 10506617 068392)*sqrt(5*x^2 + 2*x + 3) + 3357568053/781250000*sqrt(5)*log(-sqrt(5)* (sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)) - 1)
Timed out. \[ \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx=\int \left (x^2+5\,x+2\right )\,{\left (5\,x^2+2\,x+3\right )}^{3/2}\,{\left (-7\,x^2+4\,x+1\right )}^3 \,d x \]