Integrand size = 33, antiderivative size = 68 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {89+219 x}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac {1465+2604 x}{2116 \sqrt {3-x+2 x^2}}-\frac {5 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4 \sqrt {2}} \] Output:
1/276*(89+219*x)/(2*x^2-x+3)^(3/2)-1/2116*(1465+2604*x)/(2*x^2-x+3)^(1/2)- 5/8*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)
Time = 0.65 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.96 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {-5569-7002 x-489 x^2-7812 x^3}{3174 \left (3-x+2 x^2\right )^{3/2}}-\frac {5 \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{4 \sqrt {2}} \] Input:
Integrate[(2 + x + 3*x^2 - x^3 + 5*x^4)/(3 - x + 2*x^2)^(5/2),x]
Output:
(-5569 - 7002*x - 489*x^2 - 7812*x^3)/(3174*(3 - x + 2*x^2)^(3/2)) - (5*Lo g[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/(4*Sqrt[2])
Time = 0.43 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2191, 27, 2191, 27, 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.
\(\displaystyle \int \frac {5 x^4-x^3+3 x^2+x+2}{\left (2 x^2-x+3\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {2}{69} \int -\frac {3 \left (-460 x^2-138 x+53\right )}{16 \left (2 x^2-x+3\right )^{3/2}}dx+\frac {219 x+89}{276 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {219 x+89}{276 \left (2 x^2-x+3\right )^{3/2}}-\frac {1}{184} \int \frac {-460 x^2-138 x+53}{\left (2 x^2-x+3\right )^{3/2}}dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {1}{184} \left (-\frac {2}{23} \int -\frac {2645}{\sqrt {2 x^2-x+3}}dx-\frac {2 (2604 x+1465)}{23 \sqrt {2 x^2-x+3}}\right )+\frac {219 x+89}{276 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{184} \left (230 \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {2 (2604 x+1465)}{23 \sqrt {2 x^2-x+3}}\right )+\frac {219 x+89}{276 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {1}{184} \left (5 \sqrt {46} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {2 (2604 x+1465)}{23 \sqrt {2 x^2-x+3}}\right )+\frac {219 x+89}{276 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{184} \left (115 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )-\frac {2 (2604 x+1465)}{23 \sqrt {2 x^2-x+3}}\right )+\frac {219 x+89}{276 \left (2 x^2-x+3\right )^{3/2}}\) |
Input:
Int[(2 + x + 3*x^2 - x^3 + 5*x^4)/(3 - x + 2*x^2)^(5/2),x]
Output:
(89 + 219*x)/(276*(3 - x + 2*x^2)^(3/2)) + ((-2*(1465 + 2604*x))/(23*Sqrt[ 3 - x + 2*x^2]) + 115*Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23]])/184
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[(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]
Time = 0.12 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.66
method | result | size |
risch | \(-\frac {7812 x^{3}+489 x^{2}+7002 x +5569}{3174 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {5 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8}\) | \(45\) |
trager | \(-\frac {7812 x^{3}+489 x^{2}+7002 x +5569}{3174 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{8}\) | \(72\) |
default | \(-\frac {271}{768 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {\frac {2423 x}{4416}-\frac {2423}{17664}}{\left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {\frac {692 x}{1587}-\frac {173}{1587}}{\sqrt {2 x^{2}-x +3}}-\frac {47 x}{64 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {x^{2}}{8 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {5 x^{3}}{6 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {5 x}{4 \sqrt {2 x^{2}-x +3}}-\frac {5}{16 \sqrt {2 x^{2}-x +3}}+\frac {5 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8}\) | \(146\) |
Input:
int((5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/3174*(7812*x^3+489*x^2+7002*x+5569)/(2*x^2-x+3)^(3/2)+5/8*2^(1/2)*arcsi nh(4/23*23^(1/2)*(x-1/4))
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (55) = 110\).
Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.65 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {7935 \, \sqrt {2} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) - 8 \, {\left (7812 \, x^{3} + 489 \, x^{2} + 7002 \, x + 5569\right )} \sqrt {2 \, x^{2} - x + 3}}{25392 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \] Input:
integrate((5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(5/2),x, algorithm="fricas")
Output:
1/25392*(7935*sqrt(2)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*log(-4*sqrt(2)*sq rt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) - 8*(7812*x^3 + 489*x^2 + 7002*x + 5569)*sqrt(2*x^2 - x + 3))/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)
\[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\int \frac {5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\left (2 x^{2} - x + 3\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((5*x**4-x**3+3*x**2+x+2)/(2*x**2-x+3)**(5/2),x)
Output:
Integral((5*x**4 - x**3 + 3*x**2 + x + 2)/(2*x**2 - x + 3)**(5/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (55) = 110\).
Time = 0.11 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.72 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {5}{6348} \, x {\left (\frac {284 \, x}{\sqrt {2 \, x^{2} - x + 3}} - \frac {3174 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {71}{\sqrt {2 \, x^{2} - x + 3}} + \frac {805 \, x}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {3243}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} + \frac {5}{8} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {355}{3174} \, \sqrt {2 \, x^{2} - x + 3} - \frac {58 \, x}{1587 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {x^{2}}{2 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1897}{6348 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {95 \, x}{276 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {41}{276 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:
integrate((5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(5/2),x, algorithm="maxima")
Output:
5/6348*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)) + 5/8*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 355/3174*sqrt(2*x^2 - x + 3) - 58/1587*x/sqrt(2*x^2 - x + 3) + 1/2*x^2/(2*x^2 - x + 3)^(3/2) - 1897/6348/sqrt(2*x^2 - x + 3) - 95/276*x/(2*x^2 - x + 3)^(3/2) + 41/276/ (2*x^2 - x + 3)^(3/2)
Time = 0.20 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx=-\frac {5}{8} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac {3 \, {\left ({\left (2604 \, x + 163\right )} x + 2334\right )} x + 5569}{3174 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:
integrate((5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(5/2),x, algorithm="giac")
Output:
-5/8*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) - 1/317 4*(3*((2604*x + 163)*x + 2334)*x + 5569)/(2*x^2 - x + 3)^(3/2)
Timed out. \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\int \frac {5\,x^4-x^3+3\,x^2+x+2}{{\left (2\,x^2-x+3\right )}^{5/2}} \,d x \] Input:
int((x + 3*x^2 - x^3 + 5*x^4 + 2)/(2*x^2 - x + 3)^(5/2),x)
Output:
int((x + 3*x^2 - x^3 + 5*x^4 + 2)/(2*x^2 - x + 3)^(5/2), x)
Time = 0.17 (sec) , antiderivative size = 273, normalized size of antiderivative = 4.01 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {-31248 \sqrt {2 x^{2}-x +3}\, x^{3}-1956 \sqrt {2 x^{2}-x +3}\, x^{2}-28008 \sqrt {2 x^{2}-x +3}\, x -22276 \sqrt {2 x^{2}-x +3}+31740 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right ) x^{4}-31740 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right ) x^{3}+103155 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right ) x^{2}-47610 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right ) x +71415 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right )+1072 \sqrt {2}\, x^{4}-1072 \sqrt {2}\, x^{3}+3484 \sqrt {2}\, x^{2}-1608 \sqrt {2}\, x +2412 \sqrt {2}}{50784 x^{4}-50784 x^{3}+165048 x^{2}-76176 x +114264} \] Input:
int((5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(5/2),x)
Output:
( - 31248*sqrt(2*x**2 - x + 3)*x**3 - 1956*sqrt(2*x**2 - x + 3)*x**2 - 280 08*sqrt(2*x**2 - x + 3)*x - 22276*sqrt(2*x**2 - x + 3) + 31740*sqrt(2)*log ((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23))*x**4 - 31740*sqrt(2) *log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23))*x**3 + 103155*sq rt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23))*x**2 - 4761 0*sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23))*x + 714 15*sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23)) + 1072 *sqrt(2)*x**4 - 1072*sqrt(2)*x**3 + 3484*sqrt(2)*x**2 - 1608*sqrt(2)*x + 2 412*sqrt(2))/(12696*(4*x**4 - 4*x**3 + 13*x**2 - 6*x + 9))