Integrand size = 14, antiderivative size = 56 \[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=-\frac {2 x^2 (2+x)}{3 \sqrt {1+x+x^2}}+\frac {1}{3} (5+2 x) \sqrt {1+x+x^2}-\frac {3}{2} \text {arcsinh}\left (\frac {1+2 x}{\sqrt {3}}\right ) \] Output:
-3/2*arcsinh(1/3*(1+2*x)*3^(1/2))-2/3*x^2*(2+x)/(x^2+x+1)^(1/2)+1/3*(5+2*x )*(x^2+x+1)^(1/2)
Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.84 \[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=\frac {5+7 x+3 x^2}{3 \sqrt {1+x+x^2}}+\frac {3}{2} \log \left (-1-2 x+2 \sqrt {1+x+x^2}\right ) \] Input:
Integrate[x^3/(1 + x + x^2)^(3/2),x]
Output:
(5 + 7*x + 3*x^2)/(3*Sqrt[1 + x + x^2]) + (3*Log[-1 - 2*x + 2*Sqrt[1 + x + x^2]])/2
Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1164, 27, 1225, 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.
\(\displaystyle \int \frac {x^3}{\left (x^2+x+1\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1164 |
\(\displaystyle \frac {2}{3} \int \frac {2 x (x+2)}{\sqrt {x^2+x+1}}dx-\frac {2 x^2 (x+2)}{3 \sqrt {x^2+x+1}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4}{3} \int \frac {x (x+2)}{\sqrt {x^2+x+1}}dx-\frac {2 x^2 (x+2)}{3 \sqrt {x^2+x+1}}\) |
\(\Big \downarrow \) 1225 |
\(\displaystyle \frac {4}{3} \left (\frac {1}{4} (2 x+5) \sqrt {x^2+x+1}-\frac {9}{8} \int \frac {1}{\sqrt {x^2+x+1}}dx\right )-\frac {2 x^2 (x+2)}{3 \sqrt {x^2+x+1}}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {4}{3} \left (\frac {1}{4} (2 x+5) \sqrt {x^2+x+1}-\frac {3}{8} \sqrt {3} \int \frac {1}{\sqrt {\frac {1}{3} (2 x+1)^2+1}}d(2 x+1)\right )-\frac {2 x^2 (x+2)}{3 \sqrt {x^2+x+1}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {4}{3} \left (\frac {1}{4} (2 x+5) \sqrt {x^2+x+1}-\frac {9}{8} \text {arcsinh}\left (\frac {2 x+1}{\sqrt {3}}\right )\right )-\frac {2 x^2 (x+2)}{3 \sqrt {x^2+x+1}}\) |
Input:
Int[x^3/(1 + x + x^2)^(3/2),x]
Output:
(-2*x^2*(2 + x))/(3*Sqrt[1 + x + x^2]) + (4*(((5 + 2*x)*Sqrt[1 + x + x^2]) /4 - (9*ArcSinh[(1 + 2*x)/Sqrt[3]])/8))/3
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_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*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[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int QuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.59
method | result | size |
risch | \(\frac {3 x^{2}+7 x +5}{3 \sqrt {x^{2}+x +1}}-\frac {3 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{2}\) | \(33\) |
trager | \(\frac {3 x^{2}+7 x +5}{3 \sqrt {x^{2}+x +1}}-\frac {3 \ln \left (1+2 x +2 \sqrt {x^{2}+x +1}\right )}{2}\) | \(40\) |
default | \(\frac {x^{2}}{\sqrt {x^{2}+x +1}}+\frac {3 x}{2 \sqrt {x^{2}+x +1}}+\frac {5}{4 \sqrt {x^{2}+x +1}}+\frac {\frac {5}{12}+\frac {5 x}{6}}{\sqrt {x^{2}+x +1}}-\frac {3 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{2}\) | \(61\) |
Input:
int(x^3/(x^2+x+1)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/3*(3*x^2+7*x+5)/(x^2+x+1)^(1/2)-3/2*arcsinh(2/3*3^(1/2)*(x+1/2))
Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.14 \[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=\frac {19 \, x^{2} + 18 \, {\left (x^{2} + x + 1\right )} \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) + 4 \, {\left (3 \, x^{2} + 7 \, x + 5\right )} \sqrt {x^{2} + x + 1} + 19 \, x + 19}{12 \, {\left (x^{2} + x + 1\right )}} \] Input:
integrate(x^3/(x^2+x+1)^(3/2),x, algorithm="fricas")
Output:
1/12*(19*x^2 + 18*(x^2 + x + 1)*log(-2*x + 2*sqrt(x^2 + x + 1) - 1) + 4*(3 *x^2 + 7*x + 5)*sqrt(x^2 + x + 1) + 19*x + 19)/(x^2 + x + 1)
\[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=\int \frac {x^{3}}{\left (x^{2} + x + 1\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**3/(x**2+x+1)**(3/2),x)
Output:
Integral(x**3/(x**2 + x + 1)**(3/2), x)
Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.84 \[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=\frac {x^{2}}{\sqrt {x^{2} + x + 1}} + \frac {7 \, x}{3 \, \sqrt {x^{2} + x + 1}} + \frac {5}{3 \, \sqrt {x^{2} + x + 1}} - \frac {3}{2} \, \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) \] Input:
integrate(x^3/(x^2+x+1)^(3/2),x, algorithm="maxima")
Output:
x^2/sqrt(x^2 + x + 1) + 7/3*x/sqrt(x^2 + x + 1) + 5/3/sqrt(x^2 + x + 1) - 3/2*arcsinh(1/3*sqrt(3)*(2*x + 1))
Time = 0.12 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.68 \[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=\frac {{\left (3 \, x + 7\right )} x + 5}{3 \, \sqrt {x^{2} + x + 1}} + \frac {3}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \] Input:
integrate(x^3/(x^2+x+1)^(3/2),x, algorithm="giac")
Output:
1/3*((3*x + 7)*x + 5)/sqrt(x^2 + x + 1) + 3/2*log(-2*x + 2*sqrt(x^2 + x + 1) - 1)
Timed out. \[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=\int \frac {x^3}{{\left (x^2+x+1\right )}^{3/2}} \,d x \] Input:
int(x^3/(x + x^2 + 1)^(3/2),x)
Output:
int(x^3/(x + x^2 + 1)^(3/2), x)
Time = 0.15 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.21 \[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=\frac {6 \sqrt {x^{2}+x +1}\, x^{2}+14 \sqrt {x^{2}+x +1}\, x +10 \sqrt {x^{2}+x +1}-9 \,\mathrm {log}\left (\frac {2 \sqrt {x^{2}+x +1}+2 x +1}{\sqrt {3}}\right ) x^{2}-9 \,\mathrm {log}\left (\frac {2 \sqrt {x^{2}+x +1}+2 x +1}{\sqrt {3}}\right ) x -9 \,\mathrm {log}\left (\frac {2 \sqrt {x^{2}+x +1}+2 x +1}{\sqrt {3}}\right )+8 x^{2}+8 x +8}{6 x^{2}+6 x +6} \] Input:
int(x^3/(x^2+x+1)^(3/2),x)
Output:
(6*sqrt(x**2 + x + 1)*x**2 + 14*sqrt(x**2 + x + 1)*x + 10*sqrt(x**2 + x + 1) - 9*log((2*sqrt(x**2 + x + 1) + 2*x + 1)/sqrt(3))*x**2 - 9*log((2*sqrt( x**2 + x + 1) + 2*x + 1)/sqrt(3))*x - 9*log((2*sqrt(x**2 + x + 1) + 2*x + 1)/sqrt(3)) + 8*x**2 + 8*x + 8)/(6*(x**2 + x + 1))