Integrand size = 27, antiderivative size = 66 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=-\frac {1}{9} \sqrt {2+3 x^2}+\frac {5}{3} x \sqrt {2+3 x^2}+\frac {8}{27} \left (2+3 x^2\right )^{3/2}-\frac {7 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \] Output:
-1/9*(3*x^2+2)^(1/2)+5/3*x*(3*x^2+2)^(1/2)+8/27*(3*x^2+2)^(3/2)-7/9*arcsin h(1/2*x*6^(1/2))*3^(1/2)
Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.85 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {1}{27} \sqrt {2+3 x^2} \left (13+45 x+24 x^2\right )+\frac {7 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{3 \sqrt {3}} \] Input:
Integrate[((1 + 2*x)*(1 + 3*x + 4*x^2))/Sqrt[2 + 3*x^2],x]
Output:
(Sqrt[2 + 3*x^2]*(13 + 45*x + 24*x^2))/27 + (7*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(3*Sqrt[3])
Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2185, 27, 676, 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.
\(\displaystyle \int \frac {(2 x+1) \left (4 x^2+3 x+1\right )}{\sqrt {3 x^2+2}} \, dx\) |
\(\Big \downarrow \) 2185 |
\(\displaystyle \frac {1}{36} \int -\frac {28 (1-3 x) (2 x+1)}{\sqrt {3 x^2+2}}dx+\frac {2}{9} \sqrt {3 x^2+2} (2 x+1)^2\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{9} (2 x+1)^2 \sqrt {3 x^2+2}-\frac {7}{9} \int \frac {(1-3 x) (2 x+1)}{\sqrt {3 x^2+2}}dx\) |
\(\Big \downarrow \) 676 |
\(\displaystyle \frac {2}{9} (2 x+1)^2 \sqrt {3 x^2+2}-\frac {7}{9} \left (3 \int \frac {1}{\sqrt {3 x^2+2}}dx-\sqrt {3 x^2+2} x-\frac {1}{3} \sqrt {3 x^2+2}\right )\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {2}{9} (2 x+1)^2 \sqrt {3 x^2+2}-\frac {7}{9} \left (\sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\sqrt {3 x^2+2} x-\frac {1}{3} \sqrt {3 x^2+2}\right )\) |
Input:
Int[((1 + 2*x)*(1 + 3*x + 4*x^2))/Sqrt[2 + 3*x^2],x]
Output:
(2*(1 + 2*x)^2*Sqrt[2 + 3*x^2])/9 - (7*(-1/3*Sqrt[2 + 3*x^2] - x*Sqrt[2 + 3*x^2] + Sqrt[3]*ArcSinh[Sqrt[3/2]*x]))/9
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[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)) Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g , p}, x] && !LeQ[p, -1]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 0.76 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.53
method | result | size |
risch | \(\frac {\left (24 x^{2}+45 x +13\right ) \sqrt {3 x^{2}+2}}{27}-\frac {7 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{9}\) | \(35\) |
default | \(-\frac {7 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{9}+\frac {13 \sqrt {3 x^{2}+2}}{27}+\frac {5 x \sqrt {3 x^{2}+2}}{3}+\frac {8 x^{2} \sqrt {3 x^{2}+2}}{9}\) | \(51\) |
trager | \(\left (\frac {8}{9} x^{2}+\frac {5}{3} x +\frac {13}{27}\right ) \sqrt {3 x^{2}+2}+\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{9}\) | \(52\) |
meijerg | \(\frac {\sqrt {3}\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{3}+\frac {10 \sqrt {3}\, \left (\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \sqrt {\frac {3 x^{2}}{2}+1}}{2}-\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )\right )}{9 \sqrt {\pi }}+\frac {5 \sqrt {2}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {\frac {3 x^{2}}{2}+1}\right )}{6 \sqrt {\pi }}+\frac {8 \sqrt {2}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-6 x^{2}+8\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{6}\right )}{9 \sqrt {\pi }}\) | \(125\) |
Input:
int((1+2*x)*(4*x^2+3*x+1)/(3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/27*(24*x^2+45*x+13)*(3*x^2+2)^(1/2)-7/9*arcsinh(1/2*6^(1/2)*x)*3^(1/2)
Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.74 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {1}{27} \, {\left (24 \, x^{2} + 45 \, x + 13\right )} \sqrt {3 \, x^{2} + 2} + \frac {7}{18} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \] Input:
integrate((1+2*x)*(4*x^2+3*x+1)/(3*x^2+2)^(1/2),x, algorithm="fricas")
Output:
1/27*(24*x^2 + 45*x + 13)*sqrt(3*x^2 + 2) + 7/18*sqrt(3)*log(sqrt(3)*sqrt( 3*x^2 + 2)*x - 3*x^2 - 1)
Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {8 x^{2} \sqrt {3 x^{2} + 2}}{9} + \frac {5 x \sqrt {3 x^{2} + 2}}{3} + \frac {13 \sqrt {3 x^{2} + 2}}{27} - \frac {7 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{9} \] Input:
integrate((1+2*x)*(4*x**2+3*x+1)/(3*x**2+2)**(1/2),x)
Output:
8*x**2*sqrt(3*x**2 + 2)/9 + 5*x*sqrt(3*x**2 + 2)/3 + 13*sqrt(3*x**2 + 2)/2 7 - 7*sqrt(3)*asinh(sqrt(6)*x/2)/9
Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.76 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {8}{9} \, \sqrt {3 \, x^{2} + 2} x^{2} + \frac {5}{3} \, \sqrt {3 \, x^{2} + 2} x - \frac {7}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {13}{27} \, \sqrt {3 \, x^{2} + 2} \] Input:
integrate((1+2*x)*(4*x^2+3*x+1)/(3*x^2+2)^(1/2),x, algorithm="maxima")
Output:
8/9*sqrt(3*x^2 + 2)*x^2 + 5/3*sqrt(3*x^2 + 2)*x - 7/9*sqrt(3)*arcsinh(1/2* sqrt(6)*x) + 13/27*sqrt(3*x^2 + 2)
Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.67 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {1}{27} \, {\left (3 \, {\left (8 \, x + 15\right )} x + 13\right )} \sqrt {3 \, x^{2} + 2} + \frac {7}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \] Input:
integrate((1+2*x)*(4*x^2+3*x+1)/(3*x^2+2)^(1/2),x, algorithm="giac")
Output:
1/27*(3*(8*x + 15)*x + 13)*sqrt(3*x^2 + 2) + 7/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))
Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.53 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {8\,x^2}{3}+5\,x+\frac {13}{9}\right )}{3}-\frac {7\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{9} \] Input:
int(((2*x + 1)*(3*x + 4*x^2 + 1))/(3*x^2 + 2)^(1/2),x)
Output:
(3^(1/2)*(x^2 + 2/3)^(1/2)*(5*x + (8*x^2)/3 + 13/9))/3 - (7*3^(1/2)*asinh( (6^(1/2)*x)/2))/9
Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {8 \sqrt {3 x^{2}+2}\, x^{2}}{9}+\frac {5 \sqrt {3 x^{2}+2}\, x}{3}+\frac {13 \sqrt {3 x^{2}+2}}{27}-\frac {7 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )}{9} \] Input:
int((1+2*x)*(4*x^2+3*x+1)/(3*x^2+2)^(1/2),x)
Output:
(24*sqrt(3*x**2 + 2)*x**2 + 45*sqrt(3*x**2 + 2)*x + 13*sqrt(3*x**2 + 2) - 21*sqrt(3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2)))/27