Integrand size = 27, antiderivative size = 68 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {121 (19-7 x)}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac {11 (7351+2336 x)}{6348 \sqrt {3-x+2 x^2}}-\frac {25 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4 \sqrt {2}} \] Output:
121/276*(19-7*x)/(2*x^2-x+3)^(3/2)-11/6348*(7351+2336*x)/(2*x^2-x+3)^(1/2) -25/8*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)
Time = 0.90 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.96 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^{5/2}} \, dx=-\frac {11 \left (8623+714 x+6183 x^2+2336 x^3\right )}{3174 \left (3-x+2 x^2\right )^{3/2}}-\frac {25 \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{4 \sqrt {2}} \] Input:
Integrate[(2 + 3*x + 5*x^2)^2/(3 - x + 2*x^2)^(5/2),x]
Output:
(-11*(8623 + 714*x + 6183*x^2 + 2336*x^3))/(3174*(3 - x + 2*x^2)^(3/2)) - (25*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/(4*Sqrt[2])
Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, 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.
\(\displaystyle \int \frac {\left (5 x^2+3 x+2\right )^2}{\left (2 x^2-x+3\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {2}{69} \int \frac {6900 x^2+11730 x+131}{16 \left (2 x^2-x+3\right )^{3/2}}dx+\frac {121 (19-7 x)}{276 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{552} \int \frac {6900 x^2+11730 x+131}{\left (2 x^2-x+3\right )^{3/2}}dx+\frac {121 (19-7 x)}{276 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {1}{552} \left (\frac {2}{23} \int \frac {39675}{\sqrt {2 x^2-x+3}}dx-\frac {22 (2336 x+7351)}{23 \sqrt {2 x^2-x+3}}\right )+\frac {121 (19-7 x)}{276 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{552} \left (3450 \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {22 (2336 x+7351)}{23 \sqrt {2 x^2-x+3}}\right )+\frac {121 (19-7 x)}{276 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {1}{552} \left (75 \sqrt {46} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {22 (2336 x+7351)}{23 \sqrt {2 x^2-x+3}}\right )+\frac {121 (19-7 x)}{276 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{552} \left (1725 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )-\frac {22 (2336 x+7351)}{23 \sqrt {2 x^2-x+3}}\right )+\frac {121 (19-7 x)}{276 \left (2 x^2-x+3\right )^{3/2}}\) |
Input:
Int[(2 + 3*x + 5*x^2)^2/(3 - x + 2*x^2)^(5/2),x]
Output:
(121*(19 - 7*x))/(276*(3 - x + 2*x^2)^(3/2)) + ((-22*(7351 + 2336*x))/(23* Sqrt[3 - x + 2*x^2]) + 1725*Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23]])/552
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 = 2.50 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.66
method | result | size |
risch | \(-\frac {11 \left (2336 x^{3}+6183 x^{2}+714 x +8623\right )}{3174 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {25 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8}\) | \(45\) |
trager | \(-\frac {11 \left (2336 x^{3}+6183 x^{2}+714 x +8623\right )}{3174 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {25 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{8}\) | \(72\) |
default | \(\frac {\frac {8493 x}{1472}-\frac {8493}{5888}}{\left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {\frac {2267 x}{529}-\frac {2267}{2116}}{\sqrt {2 x^{2}-x +3}}-\frac {15775}{768 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {319 x}{64 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {145 x^{2}}{8 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {25 x^{3}}{6 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {25 x}{4 \sqrt {2 x^{2}-x +3}}-\frac {25}{16 \sqrt {2 x^{2}-x +3}}+\frac {25 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8}\) | \(146\) |
Input:
int((5*x^2+3*x+2)^2/(2*x^2-x+3)^(5/2),x,method=_RETURNVERBOSE)
Output:
-11/3174*(2336*x^3+6183*x^2+714*x+8623)/(2*x^2-x+3)^(3/2)+25/8*2^(1/2)*arc sinh(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 = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.65 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {39675 \, \sqrt {2} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) - 88 \, {\left (2336 \, x^{3} + 6183 \, x^{2} + 714 \, x + 8623\right )} \sqrt {2 \, x^{2} - x + 3}}{25392 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \] Input:
integrate((5*x^2+3*x+2)^2/(2*x^2-x+3)^(5/2),x, algorithm="fricas")
Output:
1/25392*(39675*sqrt(2)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*log(-4*sqrt(2)*s qrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) - 88*(2336*x^3 + 6183*x ^2 + 714*x + 8623)*sqrt(2*x^2 - x + 3))/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)
\[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\int \frac {\left (5 x^{2} + 3 x + 2\right )^{2}}{\left (2 x^{2} - x + 3\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((5*x**2+3*x+2)**2/(2*x**2-x+3)**(5/2),x)
Output:
Integral((5*x**2 + 3*x + 2)**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 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {25}{6348} \, x {\left (\frac {284 \, x}{\sqrt {2 \, x^{2} - x + 3}} - \frac {3174 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {71}{\sqrt {2 \, x^{2} - x + 3}} + \frac {805 \, x}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {3243}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} + \frac {25}{8} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {1775}{3174} \, \sqrt {2 \, x^{2} - x + 3} + \frac {1017 \, x}{529 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {15 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {6413}{3174 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {x}{138 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {2593}{138 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:
integrate((5*x^2+3*x+2)^2/(2*x^2-x+3)^(5/2),x, algorithm="maxima")
Output:
25/6348*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)) + 25/8*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 1775/3174*sqrt(2* x^2 - x + 3) + 1017/529*x/sqrt(2*x^2 - x + 3) - 15*x^2/(2*x^2 - x + 3)^(3/ 2) - 6413/3174/sqrt(2*x^2 - x + 3) - 1/138*x/(2*x^2 - x + 3)^(3/2) - 2593/ 138/(2*x^2 - x + 3)^(3/2)
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^{5/2}} \, dx=-\frac {25}{8} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac {11 \, {\left ({\left ({\left (2336 \, x + 6183\right )} x + 714\right )} x + 8623\right )}}{3174 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:
integrate((5*x^2+3*x+2)^2/(2*x^2-x+3)^(5/2),x, algorithm="giac")
Output:
-25/8*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) - 11/3 174*(((2336*x + 6183)*x + 714)*x + 8623)/(2*x^2 - x + 3)^(3/2)
Timed out. \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (5\,x^2+3\,x+2\right )}^2}{{\left (2\,x^2-x+3\right )}^{5/2}} \,d x \] Input:
int((3*x + 5*x^2 + 2)^2/(2*x^2 - x + 3)^(5/2),x)
Output:
int((3*x + 5*x^2 + 2)^2/(2*x^2 - x + 3)^(5/2), x)
Time = 0.20 (sec) , antiderivative size = 273, normalized size of antiderivative = 4.01 \[ \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^{5/2}} \, dx=\frac {-102784 \sqrt {2 x^{2}-x +3}\, x^{3}-272052 \sqrt {2 x^{2}-x +3}\, x^{2}-31416 \sqrt {2 x^{2}-x +3}\, x -379412 \sqrt {2 x^{2}-x +3}+158700 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right ) x^{4}-158700 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right ) x^{3}+515775 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right ) x^{2}-238050 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right ) x +357075 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right )+70400 \sqrt {2}\, x^{4}-70400 \sqrt {2}\, x^{3}+228800 \sqrt {2}\, x^{2}-105600 \sqrt {2}\, x +158400 \sqrt {2}}{50784 x^{4}-50784 x^{3}+165048 x^{2}-76176 x +114264} \] Input:
int((5*x^2+3*x+2)^2/(2*x^2-x+3)^(5/2),x)
Output:
( - 102784*sqrt(2*x**2 - x + 3)*x**3 - 272052*sqrt(2*x**2 - x + 3)*x**2 - 31416*sqrt(2*x**2 - x + 3)*x - 379412*sqrt(2*x**2 - x + 3) + 158700*sqrt(2 )*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23))*x**4 - 158700*s qrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23))*x**3 + 515 775*sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23))*x**2 - 238050*sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23))* x + 357075*sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23) ) + 70400*sqrt(2)*x**4 - 70400*sqrt(2)*x**3 + 228800*sqrt(2)*x**2 - 105600 *sqrt(2)*x + 158400*sqrt(2))/(12696*(4*x**4 - 4*x**3 + 13*x**2 - 6*x + 9))