Integrand size = 27, antiderivative size = 105 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{5/2}} \, dx=-\frac {1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac {121 (10679-6744 x)}{8464 \sqrt {3-x+2 x^2}}+\frac {3175}{64} \sqrt {3-x+2 x^2}+\frac {125}{16} x \sqrt {3-x+2 x^2}-\frac {7495 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{128 \sqrt {2}} \]
-1331/1104*(17-45*x)/(2*x^2-x+3)^(3/2)-7495/256*arcsinh(1/23*(1-4*x)*23^(1 /2))*2^(1/2)+121/8464*(10679-6744*x)/(2*x^2-x+3)^(1/2)+3175/64*(2*x^2-x+3) ^(1/2)+125/16*x*(2*x^2-x+3)^(1/2)
Time = 0.77 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.71 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {89784565-62463282 x+101546529 x^2-29423976 x^3+16980900 x^4+3174000 x^5}{101568 \left (3-x+2 x^2\right )^{3/2}}-\frac {7495 \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{128 \sqrt {2}} \]
(89784565 - 62463282*x + 101546529*x^2 - 29423976*x^3 + 16980900*x^4 + 317 4000*x^5)/(101568*(3 - x + 2*x^2)^(3/2)) - (7495*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/(128*Sqrt[2])
Time = 0.36 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2191, 27, 2191, 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.
| \(\displaystyle \int \frac {\left (5 x^2+3 x+2\right )^3}{\left (2 x^2-x+3\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {2}{69} \int -\frac {3 \left (-46000 x^4-105800 x^3-88780 x^2+38134 x+30425\right )}{64 \left (2 x^2-x+3\right )^{3/2}}dx-\frac {1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{736} \int \frac {-46000 x^4-105800 x^3-88780 x^2+38134 x+30425}{\left (2 x^2-x+3\right )^{3/2}}dx-\frac {1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {1}{736} \left (\frac {242 (10679-6744 x)}{23 \sqrt {2 x^2-x+3}}-\frac {2}{23} \int -\frac {2645 \left (100 x^2+280 x+183\right )}{\sqrt {2 x^2-x+3}}dx\right )-\frac {1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{736} \left (230 \int \frac {100 x^2+280 x+183}{\sqrt {2 x^2-x+3}}dx+\frac {242 (10679-6744 x)}{23 \sqrt {2 x^2-x+3}}\right )-\frac {1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{736} \left (230 \left (\frac {1}{4} \int \frac {2 (635 x+216)}{\sqrt {2 x^2-x+3}}dx+25 \sqrt {2 x^2-x+3} x\right )+\frac {242 (10679-6744 x)}{23 \sqrt {2 x^2-x+3}}\right )-\frac {1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{736} \left (230 \left (\frac {1}{2} \int \frac {635 x+216}{\sqrt {2 x^2-x+3}}dx+25 \sqrt {2 x^2-x+3} x\right )+\frac {242 (10679-6744 x)}{23 \sqrt {2 x^2-x+3}}\right )-\frac {1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{736} \left (230 \left (\frac {1}{2} \left (\frac {1499}{4} \int \frac {1}{\sqrt {2 x^2-x+3}}dx+\frac {635}{2} \sqrt {2 x^2-x+3}\right )+25 \sqrt {2 x^2-x+3} x\right )+\frac {242 (10679-6744 x)}{23 \sqrt {2 x^2-x+3}}\right )-\frac {1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{736} \left (230 \left (\frac {1}{2} \left (\frac {1499 \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)}{4 \sqrt {46}}+\frac {635}{2} \sqrt {2 x^2-x+3}\right )+25 \sqrt {2 x^2-x+3} x\right )+\frac {242 (10679-6744 x)}{23 \sqrt {2 x^2-x+3}}\right )-\frac {1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{736} \left (230 \left (\frac {1}{2} \left (\frac {1499 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{4 \sqrt {2}}+\frac {635}{2} \sqrt {2 x^2-x+3}\right )+25 \sqrt {2 x^2-x+3} x\right )+\frac {242 (10679-6744 x)}{23 \sqrt {2 x^2-x+3}}\right )-\frac {1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}\) |
(-1331*(17 - 45*x))/(1104*(3 - x + 2*x^2)^(3/2)) + ((242*(10679 - 6744*x)) /(23*Sqrt[3 - x + 2*x^2]) + 230*(25*x*Sqrt[3 - x + 2*x^2] + ((635*Sqrt[3 - x + 2*x^2])/2 + (1499*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(4*Sqrt[2]))/2))/736
3.1.94.3.1 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 = 0.80 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(\frac {3174000 x^{5}+16980900 x^{4}-29423976 x^{3}+101546529 x^{2}-62463282 x +89784565}{101568 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {7495 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{256}\) | \(55\) |
| trager | \(\frac {3174000 x^{5}+16980900 x^{4}-29423976 x^{3}+101546529 x^{2}-62463282 x +89784565}{101568 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {7495 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{256}\) | \(82\) |
| default | \(-\frac {14081711 \left (-1+4 x \right )}{565248 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {3391139 \left (-1+4 x \right )}{203136 \sqrt {2 x^{2}-x +3}}+\frac {20961031}{24576 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {125 x^{5}}{4 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {2675 x^{4}}{16 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {7495 x^{3}}{192 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {222809 x^{2}}{256 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {281177 x}{2048 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {7495 x}{128 \sqrt {2 x^{2}-x +3}}-\frac {7495}{512 \sqrt {2 x^{2}-x +3}}+\frac {7495 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{256}\) | \(180\) |
1/101568*(3174000*x^5+16980900*x^4-29423976*x^3+101546529*x^2-62463282*x+8 9784565)/(2*x^2-x+3)^(3/2)+7495/256*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.16 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {11894565 \, \sqrt {2} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 8 \, {\left (3174000 \, x^{5} + 16980900 \, x^{4} - 29423976 \, x^{3} + 101546529 \, x^{2} - 62463282 \, x + 89784565\right )} \sqrt {2 \, x^{2} - x + 3}}{812544 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \]
1/812544*(11894565*sqrt(2)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*log(-4*sqrt( 2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 8*(3174000*x^5 + 16980900*x^4 - 29423976*x^3 + 101546529*x^2 - 62463282*x + 89784565)*sqrt( 2*x^2 - x + 3))/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)
\[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\int \frac {\left (5 x^{2} + 3 x + 2\right )^{3}}{\left (2 x^{2} - x + 3\right )^{\frac {5}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (84) = 168\).
Time = 0.27 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.09 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {125 \, x^{5}}{4 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2675 \, x^{4}}{16 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {7495}{203136} \, x {\left (\frac {284 \, x}{\sqrt {2 \, x^{2} - x + 3}} - \frac {3174 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {71}{\sqrt {2 \, x^{2} - x + 3}} + \frac {805 \, x}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {3243}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} + \frac {7495}{256} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {532145}{101568} \, \sqrt {2 \, x^{2} - x + 3} - \frac {4515389 \, x}{50784 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {7197 \, x^{2}}{8 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {396211}{50784 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {269783 \, x}{1104 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1002137}{1104 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
125/4*x^5/(2*x^2 - x + 3)^(3/2) + 2675/16*x^4/(2*x^2 - x + 3)^(3/2) + 7495 /203136*x*(284*x/sqrt(2*x^2 - x + 3) - 3174*x^2/(2*x^2 - x + 3)^(3/2) - 71 /sqrt(2*x^2 - x + 3) + 805*x/(2*x^2 - x + 3)^(3/2) - 3243/(2*x^2 - x + 3)^ (3/2)) + 7495/256*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 532145/101568 *sqrt(2*x^2 - x + 3) - 4515389/50784*x/sqrt(2*x^2 - x + 3) + 7197/8*x^2/(2 *x^2 - x + 3)^(3/2) + 396211/50784/sqrt(2*x^2 - x + 3) - 269783/1104*x/(2* x^2 - x + 3)^(3/2) + 1002137/1104/(2*x^2 - x + 3)^(3/2)
Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.69 \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{5/2}} \, dx=-\frac {7495}{256} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac {3 \, {\left ({\left (4 \, {\left (13225 \, {\left (20 \, x + 107\right )} x - 2451998\right )} x + 33848843\right )} x - 20821094\right )} x + 89784565}{101568 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
-7495/256*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 1/101568*(3*((4*(13225*(20*x + 107)*x - 2451998)*x + 33848843)*x - 2082109 4)*x + 89784565)/(2*x^2 - x + 3)^(3/2)
Timed out. \[ \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (5\,x^2+3\,x+2\right )}^3}{{\left (2\,x^2-x+3\right )}^{5/2}} \,d x \]