Integrand size = 25, antiderivative size = 128 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right ) \, dx=-\frac {732665 (1-4 x) \sqrt {3-x+2 x^2}}{524288}-\frac {31855 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{98304}-\frac {277 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{3072}+\frac {141}{448} \left (3-x+2 x^2\right )^{7/2}+\frac {5}{16} x \left (3-x+2 x^2\right )^{7/2}-\frac {16851295 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1048576 \sqrt {2}} \]
-31855/98304*(1-4*x)*(2*x^2-x+3)^(3/2)-277/3072*(1-4*x)*(2*x^2-x+3)^(5/2)+ 141/448*(2*x^2-x+3)^(7/2)+5/16*x*(2*x^2-x+3)^(7/2)-16851295/2097152*arcsin h(1/23*(1-4*x)*23^(1/2))*2^(1/2)-732665/524288*(1-4*x)*(2*x^2-x+3)^(1/2)
Time = 0.55 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.66 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (58536675+148957444 x+67272352 x^2+172684416 x^3-1619968 x^4+118808576 x^5-13565952 x^6+27525120 x^7\right )-353877195 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{44040192} \]
(4*Sqrt[3 - x + 2*x^2]*(58536675 + 148957444*x + 67272352*x^2 + 172684416* x^3 - 1619968*x^4 + 118808576*x^5 - 13565952*x^6 + 27525120*x^7) - 3538771 95*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/44040192
Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2192, 27, 1160, 1087, 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.
\(\displaystyle \int \left (2 x^2-x+3\right )^{5/2} \left (5 x^2+3 x+2\right ) \, dx\) |
\(\Big \downarrow \) 2192 |
\(\displaystyle \frac {1}{16} \int \frac {1}{2} (141 x+34) \left (2 x^2-x+3\right )^{5/2}dx+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{32} \int (141 x+34) \left (2 x^2-x+3\right )^{5/2}dx+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {1}{32} \left (\frac {277}{4} \int \left (2 x^2-x+3\right )^{5/2}dx+\frac {141}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {1}{32} \left (\frac {277}{4} \left (\frac {115}{48} \int \left (2 x^2-x+3\right )^{3/2}dx-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {141}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {1}{32} \left (\frac {277}{4} \left (\frac {115}{48} \left (\frac {69}{32} \int \sqrt {2 x^2-x+3}dx-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {141}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {1}{32} \left (\frac {277}{4} \left (\frac {115}{48} \left (\frac {69}{32} \left (\frac {23}{16} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {141}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {1}{32} \left (\frac {277}{4} \left (\frac {115}{48} \left (\frac {69}{32} \left (\frac {1}{16} \sqrt {\frac {23}{2}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {141}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{32} \left (\frac {277}{4} \left (\frac {115}{48} \left (\frac {69}{32} \left (\frac {23 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{16 \sqrt {2}}-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {141}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\) |
(5*x*(3 - x + 2*x^2)^(7/2))/16 + ((141*(3 - x + 2*x^2)^(7/2))/14 + (277*(- 1/24*((1 - 4*x)*(3 - x + 2*x^2)^(5/2)) + (115*(-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))/48))/4)/32
3.1.75.3.1 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 = 0.70 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.51
method | result | size |
risch | \(\frac {\left (27525120 x^{7}-13565952 x^{6}+118808576 x^{5}-1619968 x^{4}+172684416 x^{3}+67272352 x^{2}+148957444 x +58536675\right ) \sqrt {2 x^{2}-x +3}}{11010048}+\frac {16851295 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{2097152}\) | \(65\) |
trager | \(\left (\frac {5}{2} x^{7}-\frac {69}{56} x^{6}+\frac {14503}{1344} x^{5}-\frac {113}{768} x^{4}+\frac {449699}{28672} x^{3}+\frac {300323}{49152} x^{2}+\frac {37239361}{2752512} x +\frac {19512225}{3670016}\right ) \sqrt {2 x^{2}-x +3}+\frac {16851295 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{2097152}\) | \(91\) |
default | \(\frac {277 \left (-1+4 x \right ) \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{3072}+\frac {31855 \left (-1+4 x \right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{98304}+\frac {732665 \left (-1+4 x \right ) \sqrt {2 x^{2}-x +3}}{524288}+\frac {16851295 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{2097152}+\frac {141 \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{448}+\frac {5 x \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{16}\) | \(102\) |
1/11010048*(27525120*x^7-13565952*x^6+118808576*x^5-1619968*x^4+172684416* x^3+67272352*x^2+148957444*x+58536675)*(2*x^2-x+3)^(1/2)+16851295/2097152* 2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.69 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {1}{11010048} \, {\left (27525120 \, x^{7} - 13565952 \, x^{6} + 118808576 \, x^{5} - 1619968 \, x^{4} + 172684416 \, x^{3} + 67272352 \, x^{2} + 148957444 \, x + 58536675\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {16851295}{4194304} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
1/11010048*(27525120*x^7 - 13565952*x^6 + 118808576*x^5 - 1619968*x^4 + 17 2684416*x^3 + 67272352*x^2 + 148957444*x + 58536675)*sqrt(2*x^2 - x + 3) + 16851295/4194304*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 3 2*x^2 + 16*x - 25)
Time = 0.48 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right ) \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {5 x^{7}}{2} - \frac {69 x^{6}}{56} + \frac {14503 x^{5}}{1344} - \frac {113 x^{4}}{768} + \frac {449699 x^{3}}{28672} + \frac {300323 x^{2}}{49152} + \frac {37239361 x}{2752512} + \frac {19512225}{3670016}\right ) + \frac {16851295 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{2097152} \]
sqrt(2*x**2 - x + 3)*(5*x**7/2 - 69*x**6/56 + 14503*x**5/1344 - 113*x**4/7 68 + 449699*x**3/28672 + 300323*x**2/49152 + 37239361*x/2752512 + 19512225 /3670016) + 16851295*sqrt(2)*asinh(4*sqrt(23)*(x - 1/4)/23)/2097152
Time = 0.27 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.04 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {5}{16} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x + \frac {141}{448} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {277}{768} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {277}{3072} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {31855}{24576} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {31855}{98304} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {732665}{131072} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {16851295}{2097152} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {732665}{524288} \, \sqrt {2 \, x^{2} - x + 3} \]
5/16*(2*x^2 - x + 3)^(7/2)*x + 141/448*(2*x^2 - x + 3)^(7/2) + 277/768*(2* x^2 - x + 3)^(5/2)*x - 277/3072*(2*x^2 - x + 3)^(5/2) + 31855/24576*(2*x^2 - x + 3)^(3/2)*x - 31855/98304*(2*x^2 - x + 3)^(3/2) + 732665/131072*sqrt (2*x^2 - x + 3)*x + 16851295/2097152*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 732665/524288*sqrt(2*x^2 - x + 3)
Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {1}{11010048} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (24 \, {\left (140 \, x - 69\right )} x + 14503\right )} x - 791\right )} x + 1349097\right )} x + 2102261\right )} x + 37239361\right )} x + 58536675\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {16851295}{2097152} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]
1/11010048*(4*(8*(4*(16*(4*(24*(140*x - 69)*x + 14503)*x - 791)*x + 134909 7)*x + 2102261)*x + 37239361)*x + 58536675)*sqrt(2*x^2 - x + 3) - 16851295 /2097152*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)
Timed out. \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right ) \, dx=\int {\left (2\,x^2-x+3\right )}^{5/2}\,\left (5\,x^2+3\,x+2\right ) \,d x \]