Integrand size = 32, antiderivative size = 89 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \sqrt {2-x+3 x^2}} \, dx=-\frac {\sqrt {2-x+3 x^2}}{26 (1+2 x)^2}+\frac {7 \sqrt {2-x+3 x^2}}{169 (1+2 x)}-\frac {581 \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right )}{676 \sqrt {13}} \] Output:
-1/26*(3*x^2-x+2)^(1/2)/(1+2*x)^2+7*(3*x^2-x+2)^(1/2)/(169+338*x)-581/8788 *13^(1/2)*arctanh(1/26*(9-8*x)*13^(1/2)/(3*x^2-x+2)^(1/2))
Time = 0.51 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.87 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \sqrt {2-x+3 x^2}} \, dx=\frac {\frac {13 (1+28 x) \sqrt {2-x+3 x^2}}{(1+2 x)^2}+581 \sqrt {13} \text {arctanh}\left (\frac {\sqrt {3}+2 \sqrt {3} x-2 \sqrt {2-x+3 x^2}}{\sqrt {13}}\right )}{4394} \] Input:
Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^3*Sqrt[2 - x + 3*x^2]),x]
Output:
((13*(1 + 28*x)*Sqrt[2 - x + 3*x^2])/(1 + 2*x)^2 + 581*Sqrt[13]*ArcTanh[(S qrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 - x + 3*x^2])/Sqrt[13]])/4394
Time = 0.47 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2181, 27, 1228, 1154, 219}
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.
\(\displaystyle \int \frac {4 x^2+3 x+1}{(2 x+1)^3 \sqrt {3 x^2-x+2}} \, dx\) |
\(\Big \downarrow \) 2181 |
\(\displaystyle -\frac {1}{26} \int -\frac {7 (14 x+5)}{2 (2 x+1)^2 \sqrt {3 x^2-x+2}}dx-\frac {\sqrt {3 x^2-x+2}}{26 (2 x+1)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {7}{52} \int \frac {14 x+5}{(2 x+1)^2 \sqrt {3 x^2-x+2}}dx-\frac {\sqrt {3 x^2-x+2}}{26 (2 x+1)^2}\) |
\(\Big \downarrow \) 1228 |
\(\displaystyle \frac {7}{52} \left (\frac {83}{13} \int \frac {1}{(2 x+1) \sqrt {3 x^2-x+2}}dx+\frac {4 \sqrt {3 x^2-x+2}}{13 (2 x+1)}\right )-\frac {\sqrt {3 x^2-x+2}}{26 (2 x+1)^2}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {7}{52} \left (\frac {4 \sqrt {3 x^2-x+2}}{13 (2 x+1)}-\frac {166}{13} \int \frac {1}{52-\frac {(9-8 x)^2}{3 x^2-x+2}}d\frac {9-8 x}{\sqrt {3 x^2-x+2}}\right )-\frac {\sqrt {3 x^2-x+2}}{26 (2 x+1)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {7}{52} \left (\frac {4 \sqrt {3 x^2-x+2}}{13 (2 x+1)}-\frac {83 \text {arctanh}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{13 \sqrt {13}}\right )-\frac {\sqrt {3 x^2-x+2}}{26 (2 x+1)^2}\) |
Input:
Int[(1 + 3*x + 4*x^2)/((1 + 2*x)^3*Sqrt[2 - x + 3*x^2]),x]
Output:
-1/26*Sqrt[2 - x + 3*x^2]/(1 + 2*x)^2 + (7*((4*Sqrt[2 - x + 3*x^2])/(13*(1 + 2*x)) - (83*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/(13*Sq rt[13])))/52
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) ^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R *(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
Time = 0.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76
method | result | size |
risch | \(\frac {84 x^{3}-25 x^{2}+55 x +2}{338 \left (1+2 x \right )^{2} \sqrt {3 x^{2}-x +2}}-\frac {581 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-16 x}}\right )}{8788}\) | \(68\) |
default | \(-\frac {581 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-16 x}}\right )}{8788}-\frac {\sqrt {3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-4 x}}{104 \left (\frac {1}{2}+x \right )^{2}}+\frac {7 \sqrt {3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-4 x}}{338 \left (\frac {1}{2}+x \right )}\) | \(74\) |
trager | \(\frac {\left (28 x +1\right ) \sqrt {3 x^{2}-x +2}}{338 \left (1+2 x \right )^{2}}-\frac {581 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right ) x +26 \sqrt {3 x^{2}-x +2}+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-13\right )}{1+2 x}\right )}{8788}\) | \(77\) |
Input:
int((4*x^2+3*x+1)/(1+2*x)^3/(3*x^2-x+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/338*(84*x^3-25*x^2+55*x+2)/(1+2*x)^2/(3*x^2-x+2)^(1/2)-581/8788*13^(1/2) *arctanh(2/13*(9/2-4*x)*13^(1/2)/(12*(1/2+x)^2+5-16*x)^(1/2))
Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.08 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \sqrt {2-x+3 x^2}} \, dx=\frac {581 \, \sqrt {13} {\left (4 \, x^{2} + 4 \, x + 1\right )} \log \left (-\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 52 \, \sqrt {3 \, x^{2} - x + 2} {\left (28 \, x + 1\right )}}{17576 \, {\left (4 \, x^{2} + 4 \, x + 1\right )}} \] Input:
integrate((4*x^2+3*x+1)/(1+2*x)^3/(3*x^2-x+2)^(1/2),x, algorithm="fricas")
Output:
1/17576*(581*sqrt(13)*(4*x^2 + 4*x + 1)*log(-(4*sqrt(13)*sqrt(3*x^2 - x + 2)*(8*x - 9) + 220*x^2 - 196*x + 185)/(4*x^2 + 4*x + 1)) + 52*sqrt(3*x^2 - x + 2)*(28*x + 1))/(4*x^2 + 4*x + 1)
\[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \sqrt {2-x+3 x^2}} \, dx=\int \frac {4 x^{2} + 3 x + 1}{\left (2 x + 1\right )^{3} \sqrt {3 x^{2} - x + 2}}\, dx \] Input:
integrate((4*x**2+3*x+1)/(1+2*x)**3/(3*x**2-x+2)**(1/2),x)
Output:
Integral((4*x**2 + 3*x + 1)/((2*x + 1)**3*sqrt(3*x**2 - x + 2)), x)
Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.92 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \sqrt {2-x+3 x^2}} \, dx=\frac {581}{8788} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) - \frac {\sqrt {3 \, x^{2} - x + 2}}{26 \, {\left (4 \, x^{2} + 4 \, x + 1\right )}} + \frac {7 \, \sqrt {3 \, x^{2} - x + 2}}{169 \, {\left (2 \, x + 1\right )}} \] Input:
integrate((4*x^2+3*x+1)/(1+2*x)^3/(3*x^2-x+2)^(1/2),x, algorithm="maxima")
Output:
581/8788*sqrt(13)*arcsinh(8/23*sqrt(23)*x/abs(2*x + 1) - 9/23*sqrt(23)/abs (2*x + 1)) - 1/26*sqrt(3*x^2 - x + 2)/(4*x^2 + 4*x + 1) + 7/169*sqrt(3*x^2 - x + 2)/(2*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (71) = 142\).
Time = 0.23 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.29 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \sqrt {2-x+3 x^2}} \, dx=\frac {581}{8788} \, \sqrt {13} \log \left (-\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {13} - 2 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} - x + 2} \right |}}{2 \, {\left (2 \, \sqrt {3} x - \sqrt {13} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} - x + 2}\right )}}\right ) + \frac {190 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{3} - 53 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{2} - 489 \, \sqrt {3} x + 289 \, \sqrt {3} + 489 \, \sqrt {3 \, x^{2} - x + 2}}{338 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{2} + 2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} - 5\right )}^{2}} \] Input:
integrate((4*x^2+3*x+1)/(1+2*x)^3/(3*x^2-x+2)^(1/2),x, algorithm="giac")
Output:
581/8788*sqrt(13)*log(-1/2*abs(-4*sqrt(3)*x - 2*sqrt(13) - 2*sqrt(3) + 4*s qrt(3*x^2 - x + 2))/(2*sqrt(3)*x - sqrt(13) + sqrt(3) - 2*sqrt(3*x^2 - x + 2))) + 1/338*(190*(sqrt(3)*x - sqrt(3*x^2 - x + 2))^3 - 53*sqrt(3)*(sqrt( 3)*x - sqrt(3*x^2 - x + 2))^2 - 489*sqrt(3)*x + 289*sqrt(3) + 489*sqrt(3*x ^2 - x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 - x + 2))^2 + 2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) - 5)^2
Timed out. \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \sqrt {2-x+3 x^2}} \, dx=\int \frac {4\,x^2+3\,x+1}{{\left (2\,x+1\right )}^3\,\sqrt {3\,x^2-x+2}} \,d x \] Input:
int((3*x + 4*x^2 + 1)/((2*x + 1)^3*(3*x^2 - x + 2)^(1/2)),x)
Output:
int((3*x + 4*x^2 + 1)/((2*x + 1)^3*(3*x^2 - x + 2)^(1/2)), x)
Time = 0.22 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.73 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \sqrt {2-x+3 x^2}} \, dx=\frac {728 \sqrt {3 x^{2}-x +2}\, x +26 \sqrt {3 x^{2}-x +2}+2324 \sqrt {13}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {13}+8 x -9\right ) x^{2}+2324 \sqrt {13}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {13}+8 x -9\right ) x +581 \sqrt {13}\, \mathrm {log}\left (2 \sqrt {3 x^{2}-x +2}\, \sqrt {13}+8 x -9\right )-2324 \sqrt {13}\, \mathrm {log}\left (2 x +1\right ) x^{2}-2324 \sqrt {13}\, \mathrm {log}\left (2 x +1\right ) x -581 \sqrt {13}\, \mathrm {log}\left (2 x +1\right )}{35152 x^{2}+35152 x +8788} \] Input:
int((4*x^2+3*x+1)/(1+2*x)^3/(3*x^2-x+2)^(1/2),x)
Output:
(728*sqrt(3*x**2 - x + 2)*x + 26*sqrt(3*x**2 - x + 2) + 2324*sqrt(13)*log( 2*sqrt(3*x**2 - x + 2)*sqrt(13) + 8*x - 9)*x**2 + 2324*sqrt(13)*log(2*sqrt (3*x**2 - x + 2)*sqrt(13) + 8*x - 9)*x + 581*sqrt(13)*log(2*sqrt(3*x**2 - x + 2)*sqrt(13) + 8*x - 9) - 2324*sqrt(13)*log(2*x + 1)*x**2 - 2324*sqrt(1 3)*log(2*x + 1)*x - 581*sqrt(13)*log(2*x + 1))/(8788*(4*x**2 + 4*x + 1))