Integrand size = 32, antiderivative size = 86 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{5/2}} \, dx=\frac {2 (12839-3871 x)}{5589 \left (2-x+3 x^2\right )^{3/2}}-\frac {28 (35809+42240 x)}{128547 \sqrt {2-x+3 x^2}}+\frac {32}{27} \sqrt {2-x+3 x^2}-\frac {296 \text {arcsinh}\left (\frac {1-6 x}{\sqrt {23}}\right )}{27 \sqrt {3}} \] Output:
2/5589*(12839-3871*x)/(3*x^2-x+2)^(3/2)-28/128547*(35809+42240*x)/(3*x^2-x +2)^(1/2)+32/27*(3*x^2-x+2)^(1/2)-296/81*arcsinh(1/23*(1-6*x)*23^(1/2))*3^ (1/2)
Time = 0.93 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.81 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{5/2}} \, dx=\frac {2 \left (-44739-119459 x+8630 x^2-247904 x^3+76176 x^4\right )}{14283 \left (2-x+3 x^2\right )^{3/2}}-\frac {296 \log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )}{27 \sqrt {3}} \] Input:
Integrate[((1 + 2*x)^3*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(5/2),x]
Output:
(2*(-44739 - 119459*x + 8630*x^2 - 247904*x^3 + 76176*x^4))/(14283*(2 - x + 3*x^2)^(3/2)) - (296*Log[1 - 6*x + 2*Sqrt[6 - 3*x + 9*x^2]])/(27*Sqrt[3] )
Time = 0.55 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2191, 27, 2191, 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.
| \(\displaystyle \int \frac {(2 x+1)^3 \left (4 x^2+3 x+1\right )}{\left (3 x^2-x+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {2}{69} \int -\frac {-29808 x^3-77004 x^2-69138 x+4361}{81 \left (3 x^2-x+2\right )^{3/2}}dx+\frac {2 (12839-3871 x)}{5589 \left (3 x^2-x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (12839-3871 x)}{5589 \left (3 x^2-x+2\right )^{3/2}}-\frac {2 \int \frac {-29808 x^3-77004 x^2-69138 x+4361}{\left (3 x^2-x+2\right )^{3/2}}dx}{5589}\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {2 (12839-3871 x)}{5589 \left (3 x^2-x+2\right )^{3/2}}-\frac {2 \left (\frac {2}{23} \int -\frac {9522 (12 x+35)}{\sqrt {3 x^2-x+2}}dx+\frac {14 (42240 x+35809)}{23 \sqrt {3 x^2-x+2}}\right )}{5589}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (12839-3871 x)}{5589 \left (3 x^2-x+2\right )^{3/2}}-\frac {2 \left (\frac {14 (42240 x+35809)}{23 \sqrt {3 x^2-x+2}}-828 \int \frac {12 x+35}{\sqrt {3 x^2-x+2}}dx\right )}{5589}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {2 (12839-3871 x)}{5589 \left (3 x^2-x+2\right )^{3/2}}-\frac {2 \left (\frac {14 (42240 x+35809)}{23 \sqrt {3 x^2-x+2}}-828 \left (37 \int \frac {1}{\sqrt {3 x^2-x+2}}dx+4 \sqrt {3 x^2-x+2}\right )\right )}{5589}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {2 (12839-3871 x)}{5589 \left (3 x^2-x+2\right )^{3/2}}-\frac {2 \left (\frac {14 (42240 x+35809)}{23 \sqrt {3 x^2-x+2}}-828 \left (\frac {37 \int \frac {1}{\sqrt {\frac {1}{23} (6 x-1)^2+1}}d(6 x-1)}{\sqrt {69}}+4 \sqrt {3 x^2-x+2}\right )\right )}{5589}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {2 (12839-3871 x)}{5589 \left (3 x^2-x+2\right )^{3/2}}-\frac {2 \left (\frac {14 (42240 x+35809)}{23 \sqrt {3 x^2-x+2}}-828 \left (\frac {37 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{\sqrt {3}}+4 \sqrt {3 x^2-x+2}\right )\right )}{5589}\) |
Input:
Int[((1 + 2*x)^3*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(5/2),x]
Output:
(2*(12839 - 3871*x))/(5589*(2 - x + 3*x^2)^(3/2)) - (2*((14*(35809 + 42240 *x))/(23*Sqrt[2 - x + 3*x^2]) - 828*(4*Sqrt[2 - x + 3*x^2] + (37*ArcSinh[( -1 + 6*x)/Sqrt[23]])/Sqrt[3])))/5589
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]
Time = 0.14 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(\frac {\frac {32}{3} x^{4}-\frac {495808}{14283} x^{3}+\frac {17260}{14283} x^{2}-\frac {238918}{14283} x -\frac {3314}{529}}{\left (3 x^{2}-x +2\right )^{\frac {3}{2}}}+\frac {296 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{81}\) | \(50\) |
| trager | \(\frac {\frac {32}{3} x^{4}-\frac {495808}{14283} x^{3}+\frac {17260}{14283} x^{2}-\frac {238918}{14283} x -\frac {3314}{529}}{\left (3 x^{2}-x +2\right )^{\frac {3}{2}}}-\frac {296 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}-x +2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{81}\) | \(75\) |
| default | \(\frac {\frac {13763 x}{5589}-\frac {13763}{33534}}{\left (3 x^{2}-x +2\right )^{\frac {3}{2}}}+\frac {\frac {130528 x}{42849}-\frac {65264}{128547}}{\sqrt {3 x^{2}-x +2}}-\frac {1727}{1458 \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}-\frac {461 x}{81 \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}+\frac {8 x^{2}}{27 \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}-\frac {296 x^{3}}{27 \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}-\frac {296 x}{27 \sqrt {3 x^{2}-x +2}}-\frac {148}{81 \sqrt {3 x^{2}-x +2}}+\frac {296 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{81}+\frac {32 x^{4}}{3 \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}\) | \(163\) |
Input:
int((1+2*x)^3*(4*x^2+3*x+1)/(3*x^2-x+2)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/14283*(76176*x^4-247904*x^3+8630*x^2-119459*x-44739)/(3*x^2-x+2)^(3/2)+2 96/81*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))
Time = 0.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.36 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (39146 \, \sqrt {3} {\left (9 \, x^{4} - 6 \, x^{3} + 13 \, x^{2} - 4 \, x + 4\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 3 \, {\left (76176 \, x^{4} - 247904 \, x^{3} + 8630 \, x^{2} - 119459 \, x - 44739\right )} \sqrt {3 \, x^{2} - x + 2}\right )}}{42849 \, {\left (9 \, x^{4} - 6 \, x^{3} + 13 \, x^{2} - 4 \, x + 4\right )}} \] Input:
integrate((1+2*x)^3*(4*x^2+3*x+1)/(3*x^2-x+2)^(5/2),x, algorithm="fricas")
Output:
2/42849*(39146*sqrt(3)*(9*x^4 - 6*x^3 + 13*x^2 - 4*x + 4)*log(-4*sqrt(3)*s qrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2 + 24*x - 25) + 3*(76176*x^4 - 247904 *x^3 + 8630*x^2 - 119459*x - 44739)*sqrt(3*x^2 - x + 2))/(9*x^4 - 6*x^3 + 13*x^2 - 4*x + 4)
\[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{5/2}} \, dx=\int \frac {\left (2 x + 1\right )^{3} \cdot \left (4 x^{2} + 3 x + 1\right )}{\left (3 x^{2} - x + 2\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((1+2*x)**3*(4*x**2+3*x+1)/(3*x**2-x+2)**(5/2),x)
Output:
Integral((2*x + 1)**3*(4*x**2 + 3*x + 1)/(3*x**2 - x + 2)**(5/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (69) = 138\).
Time = 0.11 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.35 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{5/2}} \, dx=\frac {32 \, x^{4}}{3 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} + \frac {296}{42849} \, x {\left (\frac {426 \, x}{\sqrt {3 \, x^{2} - x + 2}} - \frac {4761 \, x^{2}}{{\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} - \frac {71}{\sqrt {3 \, x^{2} - x + 2}} + \frac {805 \, x}{{\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} - \frac {2162}{{\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}}\right )} + \frac {296}{81} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (6 \, x - 1\right )}\right ) - \frac {42032}{42849} \, \sqrt {3 \, x^{2} - x + 2} - \frac {47072 \, x}{42849 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {52 \, x^{2}}{9 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} - \frac {23104}{14283 \, \sqrt {3 \, x^{2} - x + 2}} - \frac {7742 \, x}{1863 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} + \frac {1666}{1863 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} \] Input:
integrate((1+2*x)^3*(4*x^2+3*x+1)/(3*x^2-x+2)^(5/2),x, algorithm="maxima")
Output:
32/3*x^4/(3*x^2 - x + 2)^(3/2) + 296/42849*x*(426*x/sqrt(3*x^2 - x + 2) - 4761*x^2/(3*x^2 - x + 2)^(3/2) - 71/sqrt(3*x^2 - x + 2) + 805*x/(3*x^2 - x + 2)^(3/2) - 2162/(3*x^2 - x + 2)^(3/2)) + 296/81*sqrt(3)*arcsinh(1/23*sq rt(23)*(6*x - 1)) - 42032/42849*sqrt(3*x^2 - x + 2) - 47072/42849*x/sqrt(3 *x^2 - x + 2) + 52/9*x^2/(3*x^2 - x + 2)^(3/2) - 23104/14283/sqrt(3*x^2 - x + 2) - 7742/1863*x/(3*x^2 - x + 2)^(3/2) + 1666/1863/(3*x^2 - x + 2)^(3/ 2)
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.78 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{5/2}} \, dx=-\frac {296}{81} \, \sqrt {3} \log \left (-2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} + 1\right ) + \frac {2 \, {\left ({\left (2 \, {\left (8 \, {\left (4761 \, x - 15494\right )} x + 4315\right )} x - 119459\right )} x - 44739\right )}}{14283 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} \] Input:
integrate((1+2*x)^3*(4*x^2+3*x+1)/(3*x^2-x+2)^(5/2),x, algorithm="giac")
Output:
-296/81*sqrt(3)*log(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) + 1) + 2/ 14283*((2*(8*(4761*x - 15494)*x + 4315)*x - 119459)*x - 44739)/(3*x^2 - x + 2)^(3/2)
Timed out. \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{5/2}} \, dx=\int \frac {{\left (2\,x+1\right )}^3\,\left (4\,x^2+3\,x+1\right )}{{\left (3\,x^2-x+2\right )}^{5/2}} \,d x \] Input:
int(((2*x + 1)^3*(3*x + 4*x^2 + 1))/(3*x^2 - x + 2)^(5/2),x)
Output:
int(((2*x + 1)^3*(3*x + 4*x^2 + 1))/(3*x^2 - x + 2)^(5/2), x)
Time = 0.22 (sec) , antiderivative size = 290, normalized size of antiderivative = 3.37 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{5/2}} \, dx=\frac {457056 \sqrt {3 x^{2}-x +2}\, x^{4}-1487424 \sqrt {3 x^{2}-x +2}\, x^{3}+51780 \sqrt {3 x^{2}-x +2}\, x^{2}-716754 \sqrt {3 x^{2}-x +2}\, x -268434 \sqrt {3 x^{2}-x +2}+1409256 \sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}-x +2}\, \sqrt {3}+6 x -1}{\sqrt {23}}\right ) x^{4}-939504 \sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}-x +2}\, \sqrt {3}+6 x -1}{\sqrt {23}}\right ) x^{3}+2035592 \sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}-x +2}\, \sqrt {3}+6 x -1}{\sqrt {23}}\right ) x^{2}-626336 \sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}-x +2}\, \sqrt {3}+6 x -1}{\sqrt {23}}\right ) x +626336 \sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}-x +2}\, \sqrt {3}+6 x -1}{\sqrt {23}}\right )+518112 \sqrt {3}\, x^{4}-345408 \sqrt {3}\, x^{3}+748384 \sqrt {3}\, x^{2}-230272 \sqrt {3}\, x +230272 \sqrt {3}}{385641 x^{4}-257094 x^{3}+557037 x^{2}-171396 x +171396} \] Input:
int((1+2*x)^3*(4*x^2+3*x+1)/(3*x^2-x+2)^(5/2),x)
Output:
(2*(228528*sqrt(3*x**2 - x + 2)*x**4 - 743712*sqrt(3*x**2 - x + 2)*x**3 + 25890*sqrt(3*x**2 - x + 2)*x**2 - 358377*sqrt(3*x**2 - x + 2)*x - 134217*s qrt(3*x**2 - x + 2) + 704628*sqrt(3)*log((2*sqrt(3*x**2 - x + 2)*sqrt(3) + 6*x - 1)/sqrt(23))*x**4 - 469752*sqrt(3)*log((2*sqrt(3*x**2 - x + 2)*sqrt (3) + 6*x - 1)/sqrt(23))*x**3 + 1017796*sqrt(3)*log((2*sqrt(3*x**2 - x + 2 )*sqrt(3) + 6*x - 1)/sqrt(23))*x**2 - 313168*sqrt(3)*log((2*sqrt(3*x**2 - x + 2)*sqrt(3) + 6*x - 1)/sqrt(23))*x + 313168*sqrt(3)*log((2*sqrt(3*x**2 - x + 2)*sqrt(3) + 6*x - 1)/sqrt(23)) + 259056*sqrt(3)*x**4 - 172704*sqrt( 3)*x**3 + 374192*sqrt(3)*x**2 - 115136*sqrt(3)*x + 115136*sqrt(3)))/(42849 *(9*x**4 - 6*x**3 + 13*x**2 - 4*x + 4))