Integrand size = 27, antiderivative size = 166 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{3/2}} \, dx=-\frac {14641 (101+79 x)}{1472 \sqrt {3-x+2 x^2}}-\frac {31009685 \sqrt {3-x+2 x^2}}{65536}-\frac {8992487 x \sqrt {3-x+2 x^2}}{16384}-\frac {111315 x^2 \sqrt {3-x+2 x^2}}{2048}+\frac {79425}{512} x^3 \sqrt {3-x+2 x^2}+\frac {10075}{96} x^4 \sqrt {3-x+2 x^2}+\frac {625}{24} x^5 \sqrt {3-x+2 x^2}-\frac {310445587 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{131072 \sqrt {2}} \]
-310445587/262144*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-14641/1472*(101+7 9*x)/(2*x^2-x+3)^(1/2)-31009685/65536*(2*x^2-x+3)^(1/2)-8992487/16384*x*(2 *x^2-x+3)^(1/2)-111315/2048*x^2*(2*x^2-x+3)^(1/2)+79425/512*x^3*(2*x^2-x+3 )^(1/2)+10075/96*x^4*(2*x^2-x+3)^(1/2)+625/24*x^5*(2*x^2-x+3)^(1/2)
Time = 0.89 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.51 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {\frac {4 \left (-10961697147-8859305979 x-2534760678 x^2-2613624504 x^3+230669760 x^4+1281670400 x^5+831385600 x^6+235520000 x^7\right )}{\sqrt {3-x+2 x^2}}-21420745503 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{18087936} \]
((4*(-10961697147 - 8859305979*x - 2534760678*x^2 - 2613624504*x^3 + 23066 9760*x^4 + 1281670400*x^5 + 831385600*x^6 + 235520000*x^7))/Sqrt[3 - x + 2 *x^2] - 21420745503*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/180879 36
Time = 0.62 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.16, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {2191, 27, 2192, 27, 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.
| \(\displaystyle \int \frac {\left (5 x^2+3 x+2\right )^4}{\left (2 x^2-x+3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {2}{23} \int \frac {23 \left (40000 x^6+116000 x^5+148400 x^4+49960 x^3-84916 x^2-57494 x+122691\right )}{256 \sqrt {2 x^2-x+3}}dx-\frac {14641 (79 x+101)}{1472 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{128} \int \frac {40000 x^6+116000 x^5+148400 x^4+49960 x^3-84916 x^2-57494 x+122691}{\sqrt {2 x^2-x+3}}dx-\frac {14641 (79 x+101)}{1472 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{128} \left (\frac {1}{12} \int \frac {4 \left (403000 x^5+295200 x^4+149880 x^3-254748 x^2-172482 x+368073\right )}{\sqrt {2 x^2-x+3}}dx+\frac {10000}{3} \sqrt {2 x^2-x+3} x^5\right )-\frac {14641 (79 x+101)}{1472 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{128} \left (\frac {1}{3} \int \frac {403000 x^5+295200 x^4+149880 x^3-254748 x^2-172482 x+368073}{\sqrt {2 x^2-x+3}}dx+\frac {10000}{3} \sqrt {2 x^2-x+3} x^5\right )-\frac {14641 (79 x+101)}{1472 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{128} \left (\frac {1}{3} \left (\frac {1}{10} \int \frac {30 \left (158850 x^4-111240 x^3-84916 x^2-57494 x+122691\right )}{\sqrt {2 x^2-x+3}}dx+40300 \sqrt {2 x^2-x+3} x^4\right )+\frac {10000}{3} \sqrt {2 x^2-x+3} x^5\right )-\frac {14641 (79 x+101)}{1472 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{128} \left (\frac {1}{3} \left (3 \int \frac {158850 x^4-111240 x^3-84916 x^2-57494 x+122691}{\sqrt {2 x^2-x+3}}dx+40300 \sqrt {2 x^2-x+3} x^4\right )+\frac {10000}{3} \sqrt {2 x^2-x+3} x^5\right )-\frac {14641 (79 x+101)}{1472 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{128} \left (\frac {1}{3} \left (3 \left (\frac {1}{8} \int \frac {-333945 x^3-2108978 x^2-459952 x+981528}{\sqrt {2 x^2-x+3}}dx+\frac {79425}{4} \sqrt {2 x^2-x+3} x^3\right )+40300 \sqrt {2 x^2-x+3} x^4\right )+\frac {10000}{3} \sqrt {2 x^2-x+3} x^5\right )-\frac {14641 (79 x+101)}{1472 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{128} \left (\frac {1}{3} \left (3 \left (\frac {1}{8} \left (\frac {1}{6} \int \frac {3 \left (-8992487 x^2-504028 x+3926112\right )}{2 \sqrt {2 x^2-x+3}}dx-\frac {111315}{2} x^2 \sqrt {2 x^2-x+3}\right )+\frac {79425}{4} \sqrt {2 x^2-x+3} x^3\right )+40300 \sqrt {2 x^2-x+3} x^4\right )+\frac {10000}{3} \sqrt {2 x^2-x+3} x^5\right )-\frac {14641 (79 x+101)}{1472 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{128} \left (\frac {1}{3} \left (3 \left (\frac {1}{8} \left (\frac {1}{4} \int \frac {-8992487 x^2-504028 x+3926112}{\sqrt {2 x^2-x+3}}dx-\frac {111315}{2} x^2 \sqrt {2 x^2-x+3}\right )+\frac {79425}{4} \sqrt {2 x^2-x+3} x^3\right )+40300 \sqrt {2 x^2-x+3} x^4\right )+\frac {10000}{3} \sqrt {2 x^2-x+3} x^5\right )-\frac {14641 (79 x+101)}{1472 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{128} \left (\frac {1}{3} \left (3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{4} \int \frac {85363818-31009685 x}{2 \sqrt {2 x^2-x+3}}dx-\frac {8992487}{4} x \sqrt {2 x^2-x+3}\right )-\frac {111315}{2} x^2 \sqrt {2 x^2-x+3}\right )+\frac {79425}{4} \sqrt {2 x^2-x+3} x^3\right )+40300 \sqrt {2 x^2-x+3} x^4\right )+\frac {10000}{3} \sqrt {2 x^2-x+3} x^5\right )-\frac {14641 (79 x+101)}{1472 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{128} \left (\frac {1}{3} \left (3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{8} \int \frac {85363818-31009685 x}{\sqrt {2 x^2-x+3}}dx-\frac {8992487}{4} x \sqrt {2 x^2-x+3}\right )-\frac {111315}{2} x^2 \sqrt {2 x^2-x+3}\right )+\frac {79425}{4} \sqrt {2 x^2-x+3} x^3\right )+40300 \sqrt {2 x^2-x+3} x^4\right )+\frac {10000}{3} \sqrt {2 x^2-x+3} x^5\right )-\frac {14641 (79 x+101)}{1472 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{128} \left (\frac {1}{3} \left (3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {310445587}{4} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {31009685}{2} \sqrt {2 x^2-x+3}\right )-\frac {8992487}{4} x \sqrt {2 x^2-x+3}\right )-\frac {111315}{2} x^2 \sqrt {2 x^2-x+3}\right )+\frac {79425}{4} \sqrt {2 x^2-x+3} x^3\right )+40300 \sqrt {2 x^2-x+3} x^4\right )+\frac {10000}{3} \sqrt {2 x^2-x+3} x^5\right )-\frac {14641 (79 x+101)}{1472 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{128} \left (\frac {1}{3} \left (3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {310445587 \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)}{4 \sqrt {46}}-\frac {31009685}{2} \sqrt {2 x^2-x+3}\right )-\frac {8992487}{4} x \sqrt {2 x^2-x+3}\right )-\frac {111315}{2} x^2 \sqrt {2 x^2-x+3}\right )+\frac {79425}{4} \sqrt {2 x^2-x+3} x^3\right )+40300 \sqrt {2 x^2-x+3} x^4\right )+\frac {10000}{3} \sqrt {2 x^2-x+3} x^5\right )-\frac {14641 (79 x+101)}{1472 \sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{128} \left (\frac {1}{3} \left (3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {310445587 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{4 \sqrt {2}}-\frac {31009685}{2} \sqrt {2 x^2-x+3}\right )-\frac {8992487}{4} x \sqrt {2 x^2-x+3}\right )-\frac {111315}{2} x^2 \sqrt {2 x^2-x+3}\right )+\frac {79425}{4} \sqrt {2 x^2-x+3} x^3\right )+40300 \sqrt {2 x^2-x+3} x^4\right )+\frac {10000}{3} \sqrt {2 x^2-x+3} x^5\right )-\frac {14641 (79 x+101)}{1472 \sqrt {2 x^2-x+3}}\) |
(-14641*(101 + 79*x))/(1472*Sqrt[3 - x + 2*x^2]) + ((10000*x^5*Sqrt[3 - x + 2*x^2])/3 + (40300*x^4*Sqrt[3 - x + 2*x^2] + 3*((79425*x^3*Sqrt[3 - x + 2*x^2])/4 + ((-111315*x^2*Sqrt[3 - x + 2*x^2])/2 + ((-8992487*x*Sqrt[3 - x + 2*x^2])/4 + ((-31009685*Sqrt[3 - x + 2*x^2])/2 + (310445587*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(4*Sqrt[2]))/8)/4)/8))/3)/128
3.1.86.3.1 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 = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[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 = 0.76 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.39
| method | result | size |
| risch | \(\frac {235520000 x^{7}+831385600 x^{6}+1281670400 x^{5}+230669760 x^{4}-2613624504 x^{3}-2534760678 x^{2}-8859305979 x -10961697147}{4521984 \sqrt {2 x^{2}-x +3}}+\frac {310445587 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{262144}\) | \(65\) |
| trager | \(\frac {235520000 x^{7}+831385600 x^{6}+1281670400 x^{5}+230669760 x^{4}-2613624504 x^{3}-2534760678 x^{2}-8859305979 x -10961697147}{4521984 \sqrt {2 x^{2}-x +3}}+\frac {310445587 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{262144}\) | \(92\) |
| default | \(\frac {-\frac {1234044515}{12058624}+\frac {1234044515 x}{3014656}}{\sqrt {2 x^{2}-x +3}}-\frac {1217267299}{524288 \sqrt {2 x^{2}-x +3}}+\frac {625 x^{7}}{12 \sqrt {2 x^{2}-x +3}}+\frac {8825 x^{6}}{48 \sqrt {2 x^{2}-x +3}}+\frac {217675 x^{5}}{768 \sqrt {2 x^{2}-x +3}}+\frac {52235 x^{4}}{1024 \sqrt {2 x^{2}-x +3}}-\frac {4734827 x^{3}}{8192 \sqrt {2 x^{2}-x +3}}-\frac {18367831 x^{2}}{32768 \sqrt {2 x^{2}-x +3}}-\frac {310445587 x}{131072 \sqrt {2 x^{2}-x +3}}+\frac {310445587 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{262144}\) | \(166\) |
1/4521984*(235520000*x^7+831385600*x^6+1281670400*x^5+230669760*x^4-261362 4504*x^3-2534760678*x^2-8859305979*x-10961697147)/(2*x^2-x+3)^(1/2)+310445 587/262144*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.67 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {21420745503 \, \sqrt {2} {\left (2 \, x^{2} - x + 3\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 8 \, {\left (235520000 \, x^{7} + 831385600 \, x^{6} + 1281670400 \, x^{5} + 230669760 \, x^{4} - 2613624504 \, x^{3} - 2534760678 \, x^{2} - 8859305979 \, x - 10961697147\right )} \sqrt {2 \, x^{2} - x + 3}}{36175872 \, {\left (2 \, x^{2} - x + 3\right )}} \]
1/36175872*(21420745503*sqrt(2)*(2*x^2 - x + 3)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 8*(235520000*x^7 + 831385600*x^ 6 + 1281670400*x^5 + 230669760*x^4 - 2613624504*x^3 - 2534760678*x^2 - 885 9305979*x - 10961697147)*sqrt(2*x^2 - x + 3))/(2*x^2 - x + 3)
\[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{3/2}} \, dx=\int \frac {\left (5 x^{2} + 3 x + 2\right )^{4}}{\left (2 x^{2} - x + 3\right )^{\frac {3}{2}}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.89 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{3/2}} \, dx=\frac {625 \, x^{7}}{12 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {8825 \, x^{6}}{48 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {217675 \, x^{5}}{768 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {52235 \, x^{4}}{1024 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {4734827 \, x^{3}}{8192 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {18367831 \, x^{2}}{32768 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {310445587}{262144} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {2953101993 \, x}{1507328 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {3653899049}{1507328 \, \sqrt {2 \, x^{2} - x + 3}} \]
625/12*x^7/sqrt(2*x^2 - x + 3) + 8825/48*x^6/sqrt(2*x^2 - x + 3) + 217675/ 768*x^5/sqrt(2*x^2 - x + 3) + 52235/1024*x^4/sqrt(2*x^2 - x + 3) - 4734827 /8192*x^3/sqrt(2*x^2 - x + 3) - 18367831/32768*x^2/sqrt(2*x^2 - x + 3) + 3 10445587/262144*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 2953101993/1507 328*x/sqrt(2*x^2 - x + 3) - 3653899049/1507328/sqrt(2*x^2 - x + 3)
Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.49 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{3/2}} \, dx=-\frac {310445587}{262144} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac {{\left (46 \, {\left (4 \, {\left (40 \, {\left (20 \, {\left (16 \, {\left (100 \, x + 353\right )} x + 8707\right )} x + 31341\right )} x - 14204481\right )} x - 55103493\right )} x - 8859305979\right )} x - 10961697147}{4521984 \, \sqrt {2 \, x^{2} - x + 3}} \]
-310445587/262144*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 1/4521984*((46*(4*(40*(20*(16*(100*x + 353)*x + 8707)*x + 31341)*x - 14204481)*x - 55103493)*x - 8859305979)*x - 10961697147)/sqrt(2*x^2 - x + 3)
Timed out. \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (5\,x^2+3\,x+2\right )}^4}{{\left (2\,x^2-x+3\right )}^{3/2}} \,d x \]