Integrand size = 25, antiderivative size = 105 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right ) \, dx=-\frac {4117 (1-4 x) \sqrt {3-x+2 x^2}}{8192}-\frac {179 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {107}{240} \left (3-x+2 x^2\right )^{5/2}+\frac {5}{12} x \left (3-x+2 x^2\right )^{5/2}-\frac {94691 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{16384 \sqrt {2}} \]
-179/1536*(1-4*x)*(2*x^2-x+3)^(3/2)+107/240*(2*x^2-x+3)^(5/2)+5/12*x*(2*x^ 2-x+3)^(5/2)-94691/32768*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-4117/8192* (1-4*x)*(2*x^2-x+3)^(1/2)
Time = 0.36 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.71 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (388341+565276 x+319072 x^2+561024 x^3+14336 x^4+204800 x^5\right )-1420365 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{491520} \]
(4*Sqrt[3 - x + 2*x^2]*(388341 + 565276*x + 319072*x^2 + 561024*x^3 + 1433 6*x^4 + 204800*x^5) - 1420365*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2 ]])/491520
Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {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.
\(\displaystyle \int \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right ) \, dx\) |
\(\Big \downarrow \) 2192 |
\(\displaystyle \frac {1}{12} \int \frac {1}{2} (107 x+18) \left (2 x^2-x+3\right )^{3/2}dx+\frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{24} \int (107 x+18) \left (2 x^2-x+3\right )^{3/2}dx+\frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {1}{24} \left (\frac {179}{4} \int \left (2 x^2-x+3\right )^{3/2}dx+\frac {107}{10} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {1}{24} \left (\frac {179}{4} \left (\frac {69}{32} \int \sqrt {2 x^2-x+3}dx-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {107}{10} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {1}{24} \left (\frac {179}{4} \left (\frac {69}{32} \left (\frac {23}{16} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {107}{10} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {1}{24} \left (\frac {179}{4} \left (\frac {69}{32} \left (\frac {1}{16} \sqrt {\frac {23}{2}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {107}{10} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{24} \left (\frac {179}{4} \left (\frac {69}{32} \left (\frac {23 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{16 \sqrt {2}}-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {107}{10} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}\) |
(5*x*(3 - x + 2*x^2)^(5/2))/12 + ((107*(3 - x + 2*x^2)^(5/2))/10 + (179*(- 1/16*((1 - 4*x)*(3 - x + 2*x^2)^(3/2)) + (69*(-1/8*((1 - 4*x)*Sqrt[3 - x + 2*x^2]) + (23*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(16*Sqrt[2])))/32))/4)/24
3.1.68.3.1 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 = 0.69 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {\left (204800 x^{5}+14336 x^{4}+561024 x^{3}+319072 x^{2}+565276 x +388341\right ) \sqrt {2 x^{2}-x +3}}{122880}+\frac {94691 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{32768}\) | \(55\) |
trager | \(\left (\frac {5}{3} x^{5}+\frac {7}{60} x^{4}+\frac {1461}{320} x^{3}+\frac {9971}{3840} x^{2}+\frac {141319}{30720} x +\frac {129447}{40960}\right ) \sqrt {2 x^{2}-x +3}+\frac {94691 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{32768}\) | \(81\) |
default | \(\frac {179 \left (-1+4 x \right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{1536}+\frac {4117 \left (-1+4 x \right ) \sqrt {2 x^{2}-x +3}}{8192}+\frac {94691 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{32768}+\frac {107 \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{240}+\frac {5 x \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{12}\) | \(83\) |
1/122880*(204800*x^5+14336*x^4+561024*x^3+319072*x^2+565276*x+388341)*(2*x ^2-x+3)^(1/2)+94691/32768*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.74 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {1}{122880} \, {\left (204800 \, x^{5} + 14336 \, x^{4} + 561024 \, x^{3} + 319072 \, x^{2} + 565276 \, x + 388341\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {94691}{65536} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
1/122880*(204800*x^5 + 14336*x^4 + 561024*x^3 + 319072*x^2 + 565276*x + 38 8341)*sqrt(2*x^2 - x + 3) + 94691/65536*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)
Time = 0.37 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.67 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right ) \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {5 x^{5}}{3} + \frac {7 x^{4}}{60} + \frac {1461 x^{3}}{320} + \frac {9971 x^{2}}{3840} + \frac {141319 x}{30720} + \frac {129447}{40960}\right ) + \frac {94691 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{32768} \]
sqrt(2*x**2 - x + 3)*(5*x**5/3 + 7*x**4/60 + 1461*x**3/320 + 9971*x**2/384 0 + 141319*x/30720 + 129447/40960) + 94691*sqrt(2)*asinh(4*sqrt(23)*(x - 1 /4)/23)/32768
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {5}{12} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {107}{240} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {179}{384} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {179}{1536} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {4117}{2048} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {94691}{32768} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {4117}{8192} \, \sqrt {2 \, x^{2} - x + 3} \]
5/12*(2*x^2 - x + 3)^(5/2)*x + 107/240*(2*x^2 - x + 3)^(5/2) + 179/384*(2* x^2 - x + 3)^(3/2)*x - 179/1536*(2*x^2 - x + 3)^(3/2) + 4117/2048*sqrt(2*x ^2 - x + 3)*x + 94691/32768*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 411 7/8192*sqrt(2*x^2 - x + 3)
Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {1}{122880} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x + 7\right )} x + 4383\right )} x + 9971\right )} x + 141319\right )} x + 388341\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {94691}{32768} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]
1/122880*(4*(8*(4*(16*(100*x + 7)*x + 4383)*x + 9971)*x + 141319)*x + 3883 41)*sqrt(2*x^2 - x + 3) - 94691/32768*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)
Timed out. \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right ) \, dx=\int {\left (2\,x^2-x+3\right )}^{3/2}\,\left (5\,x^2+3\,x+2\right ) \,d x \]