Integrand size = 27, antiderivative size = 147 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2 \, dx=\frac {558739 (1-4 x) \sqrt {3-x+2 x^2}}{1048576}+\frac {24293 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{196608}+\frac {73861 \left (3-x+2 x^2\right )^{5/2}}{215040}+\frac {24499 x \left (3-x+2 x^2\right )^{5/2}}{10752}+\frac {1235}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac {12850997 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2097152 \sqrt {2}} \] Output:
558739/1048576*(1-4*x)*(2*x^2-x+3)^(1/2)+24293/196608*(1-4*x)*(2*x^2-x+3)^ (3/2)+73861/215040*(2*x^2-x+3)^(5/2)+24499/10752*x*(2*x^2-x+3)^(5/2)+1235/ 448*x^2*(2*x^2-x+3)^(5/2)+25/16*x^3*(2*x^2-x+3)^(5/2)+12850997/4194304*arc sinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)
Time = 0.91 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.58 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2 \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (439831323+1619403428 x+1799647136 x^2+2728413312 x^3+2061273088 x^4+2025840640 x^5+525926400 x^6+688128000 x^7\right )+1349354685 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{440401920} \] Input:
Integrate[(3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^2,x]
Output:
(4*Sqrt[3 - x + 2*x^2]*(439831323 + 1619403428*x + 1799647136*x^2 + 272841 3312*x^3 + 2061273088*x^4 + 2025840640*x^5 + 525926400*x^6 + 688128000*x^7 ) + 1349354685*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/440401920
Time = 0.41 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.17, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2192, 27, 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.
| \(\displaystyle \int \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2 \, dx\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{16} \int \frac {1}{2} \left (2 x^2-x+3\right )^{3/2} \left (1235 x^3+478 x^2+384 x+128\right )dx+\frac {25}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \int \left (2 x^2-x+3\right )^{3/2} \left (1235 x^3+478 x^2+384 x+128\right )dx+\frac {25}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{14} \int \frac {1}{2} \left (2 x^2-x+3\right )^{3/2} \left (24499 x^2-4068 x+3584\right )dx+\frac {1235}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {25}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \int \left (2 x^2-x+3\right )^{3/2} \left (24499 x^2-4068 x+3584\right )dx+\frac {1235}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {25}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{12} \int -\frac {1}{2} (60978-73861 x) \left (2 x^2-x+3\right )^{3/2}dx+\frac {24499}{12} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {1235}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {25}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {24499}{12} x \left (2 x^2-x+3\right )^{5/2}-\frac {1}{24} \int (60978-73861 x) \left (2 x^2-x+3\right )^{3/2}dx\right )+\frac {1235}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {25}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {73861}{10} \left (2 x^2-x+3\right )^{5/2}-\frac {170051}{4} \int \left (2 x^2-x+3\right )^{3/2}dx\right )+\frac {24499}{12} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {1235}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {25}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {73861}{10} \left (2 x^2-x+3\right )^{5/2}-\frac {170051}{4} \left (\frac {69}{32} \int \sqrt {2 x^2-x+3}dx-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )\right )+\frac {24499}{12} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {1235}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {25}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {73861}{10} \left (2 x^2-x+3\right )^{5/2}-\frac {170051}{4} \left (\frac {69}{32} \left (\frac {23}{16} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )\right )+\frac {24499}{12} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {1235}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {25}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {73861}{10} \left (2 x^2-x+3\right )^{5/2}-\frac {170051}{4} \left (\frac {69}{32} \left (\frac {1}{16} \sqrt {\frac {23}{2}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )\right )+\frac {24499}{12} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {1235}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {25}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {73861}{10} \left (2 x^2-x+3\right )^{5/2}-\frac {170051}{4} \left (\frac {69}{32} \left (\frac {23 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{16 \sqrt {2}}-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )\right )+\frac {24499}{12} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {1235}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {25}{16} \left (2 x^2-x+3\right )^{5/2} x^3\) |
Input:
Int[(3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^2,x]
Output:
(25*x^3*(3 - x + 2*x^2)^(5/2))/16 + ((1235*x^2*(3 - x + 2*x^2)^(5/2))/14 + ((24499*x*(3 - x + 2*x^2)^(5/2))/12 + ((73861*(3 - x + 2*x^2)^(5/2))/10 - (170051*(-1/16*((1 - 4*x)*(3 - x + 2*x^2)^(3/2)) + (69*(-1/8*((1 - 4*x)*S qrt[3 - x + 2*x^2]) + (23*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(16*Sqrt[2])))/32) )/4)/24)/28)/32
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 = 3.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.44
| method | result | size |
| risch | \(\frac {\left (688128000 x^{7}+525926400 x^{6}+2025840640 x^{5}+2061273088 x^{4}+2728413312 x^{3}+1799647136 x^{2}+1619403428 x +439831323\right ) \sqrt {2 x^{2}-x +3}}{110100480}-\frac {12850997 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4194304}\) | \(65\) |
| trager | \(\left (\frac {25}{4} x^{7}+\frac {535}{112} x^{6}+\frac {49459}{2688} x^{5}+\frac {143783}{7680} x^{4}+\frac {7105243}{286720} x^{3}+\frac {8034139}{491520} x^{2}+\frac {404850857}{27525120} x +\frac {146610441}{36700160}\right ) \sqrt {2 x^{2}-x +3}+\frac {12850997 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{4194304}\) | \(89\) |
| default | \(-\frac {24293 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{196608}-\frac {558739 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{1048576}-\frac {12850997 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4194304}+\frac {73861 \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{215040}+\frac {24499 x \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{10752}+\frac {1235 x^{2} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{448}+\frac {25 x^{3} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{16}\) | \(117\) |
Input:
int((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
Output:
1/110100480*(688128000*x^7+525926400*x^6+2025840640*x^5+2061273088*x^4+272 8413312*x^3+1799647136*x^2+1619403428*x+439831323)*(2*x^2-x+3)^(1/2)-12850 997/4194304*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.60 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2 \, dx=\frac {1}{110100480} \, {\left (688128000 \, x^{7} + 525926400 \, x^{6} + 2025840640 \, x^{5} + 2061273088 \, x^{4} + 2728413312 \, x^{3} + 1799647136 \, x^{2} + 1619403428 \, x + 439831323\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {12850997}{8388608} \, \sqrt {2} \log \left (4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \] Input:
integrate((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^2,x, algorithm="fricas")
Output:
1/110100480*(688128000*x^7 + 525926400*x^6 + 2025840640*x^5 + 2061273088*x ^4 + 2728413312*x^3 + 1799647136*x^2 + 1619403428*x + 439831323)*sqrt(2*x^ 2 - x + 3) + 12850997/8388608*sqrt(2)*log(4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4 *x - 1) - 32*x^2 + 16*x - 25)
Time = 0.46 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.56 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2 \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {25 x^{7}}{4} + \frac {535 x^{6}}{112} + \frac {49459 x^{5}}{2688} + \frac {143783 x^{4}}{7680} + \frac {7105243 x^{3}}{286720} + \frac {8034139 x^{2}}{491520} + \frac {404850857 x}{27525120} + \frac {146610441}{36700160}\right ) - \frac {12850997 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{4194304} \] Input:
integrate((2*x**2-x+3)**(3/2)*(5*x**2+3*x+2)**2,x)
Output:
sqrt(2*x**2 - x + 3)*(25*x**7/4 + 535*x**6/112 + 49459*x**5/2688 + 143783* x**4/7680 + 7105243*x**3/286720 + 8034139*x**2/491520 + 404850857*x/275251 20 + 146610441/36700160) - 12850997*sqrt(2)*asinh(4*sqrt(23)*(x - 1/4)/23) /4194304
Time = 0.11 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.94 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2 \, dx=\frac {25}{16} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{3} + \frac {1235}{448} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} + \frac {24499}{10752} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {73861}{215040} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} - \frac {24293}{49152} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {24293}{196608} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {558739}{262144} \, \sqrt {2 \, x^{2} - x + 3} x - \frac {12850997}{4194304} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {558739}{1048576} \, \sqrt {2 \, x^{2} - x + 3} \] Input:
integrate((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^2,x, algorithm="maxima")
Output:
25/16*(2*x^2 - x + 3)^(5/2)*x^3 + 1235/448*(2*x^2 - x + 3)^(5/2)*x^2 + 244 99/10752*(2*x^2 - x + 3)^(5/2)*x + 73861/215040*(2*x^2 - x + 3)^(5/2) - 24 293/49152*(2*x^2 - x + 3)^(3/2)*x + 24293/196608*(2*x^2 - x + 3)^(3/2) - 5 58739/262144*sqrt(2*x^2 - x + 3)*x - 12850997/4194304*sqrt(2)*arcsinh(1/23 *sqrt(23)*(4*x - 1)) + 558739/1048576*sqrt(2*x^2 - x + 3)
Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.56 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2 \, dx=\frac {1}{110100480} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (120 \, {\left (140 \, x + 107\right )} x + 49459\right )} x + 1006481\right )} x + 21315729\right )} x + 56238973\right )} x + 404850857\right )} x + 439831323\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {12850997}{4194304} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \] Input:
integrate((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^2,x, algorithm="giac")
Output:
1/110100480*(4*(8*(4*(16*(20*(120*(140*x + 107)*x + 49459)*x + 1006481)*x + 21315729)*x + 56238973)*x + 404850857)*x + 439831323)*sqrt(2*x^2 - x + 3 ) + 12850997/4194304*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)
Timed out. \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2 \, dx=\int {\left (2\,x^2-x+3\right )}^{3/2}\,{\left (5\,x^2+3\,x+2\right )}^2 \,d x \] Input:
int((2*x^2 - x + 3)^(3/2)*(3*x + 5*x^2 + 2)^2,x)
Output:
int((2*x^2 - x + 3)^(3/2)*(3*x + 5*x^2 + 2)^2, x)
Time = 0.23 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.05 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2 \, dx=\frac {25 \sqrt {2 x^{2}-x +3}\, x^{7}}{4}+\frac {535 \sqrt {2 x^{2}-x +3}\, x^{6}}{112}+\frac {49459 \sqrt {2 x^{2}-x +3}\, x^{5}}{2688}+\frac {143783 \sqrt {2 x^{2}-x +3}\, x^{4}}{7680}+\frac {7105243 \sqrt {2 x^{2}-x +3}\, x^{3}}{286720}+\frac {8034139 \sqrt {2 x^{2}-x +3}\, x^{2}}{491520}+\frac {404850857 \sqrt {2 x^{2}-x +3}\, x}{27525120}+\frac {146610441 \sqrt {2 x^{2}-x +3}}{36700160}-\frac {12850997 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right )}{4194304} \] Input:
int((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^2,x)
Output:
(2752512000*sqrt(2*x**2 - x + 3)*x**7 + 2103705600*sqrt(2*x**2 - x + 3)*x* *6 + 8103362560*sqrt(2*x**2 - x + 3)*x**5 + 8245092352*sqrt(2*x**2 - x + 3 )*x**4 + 10913653248*sqrt(2*x**2 - x + 3)*x**3 + 7198588544*sqrt(2*x**2 - x + 3)*x**2 + 6477613712*sqrt(2*x**2 - x + 3)*x + 1759325292*sqrt(2*x**2 - x + 3) - 1349354685*sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1 )/sqrt(23)))/440401920