Integrand size = 29, antiderivative size = 67 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \sqrt {2+3 x^2}} \, dx=\frac {2}{3} \sqrt {2+3 x^2}+\frac {\text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}-\frac {\text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {2+3 x^2}}\right )}{2 \sqrt {11}} \] Output:
2/3*(3*x^2+2)^(1/2)+1/6*arcsinh(1/2*x*6^(1/2))*3^(1/2)-1/22*arctanh(1/11*( 4-3*x)*11^(1/2)/(3*x^2+2)^(1/2))*11^(1/2)
Time = 0.41 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.33 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \sqrt {2+3 x^2}} \, dx=\frac {2}{3} \sqrt {2+3 x^2}+\frac {\text {arctanh}\left (\sqrt {\frac {3}{11}}+2 \sqrt {\frac {3}{11}} x-\frac {2 \sqrt {2+3 x^2}}{\sqrt {11}}\right )}{\sqrt {11}}-\frac {\log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{2 \sqrt {3}} \] Input:
Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)*Sqrt[2 + 3*x^2]),x]
Output:
(2*Sqrt[2 + 3*x^2])/3 + ArcTanh[Sqrt[3/11] + 2*Sqrt[3/11]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[11]]/Sqrt[11] - Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]]/(2*Sqrt[3 ])
Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2185, 27, 719, 222, 488, 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.
\(\displaystyle \int \frac {4 x^2+3 x+1}{(2 x+1) \sqrt {3 x^2+2}} \, dx\) |
\(\Big \downarrow \) 2185 |
\(\displaystyle \frac {1}{12} \int \frac {12 (x+1)}{(2 x+1) \sqrt {3 x^2+2}}dx+\frac {2}{3} \sqrt {3 x^2+2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {x+1}{(2 x+1) \sqrt {3 x^2+2}}dx+\frac {2}{3} \sqrt {3 x^2+2}\) |
\(\Big \downarrow \) 719 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt {3 x^2+2}}dx+\frac {1}{2} \int \frac {1}{(2 x+1) \sqrt {3 x^2+2}}dx+\frac {2}{3} \sqrt {3 x^2+2}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{2} \int \frac {1}{(2 x+1) \sqrt {3 x^2+2}}dx+\frac {\text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}+\frac {2}{3} \sqrt {3 x^2+2}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{11-\frac {(4-3 x)^2}{3 x^2+2}}d\frac {4-3 x}{\sqrt {3 x^2+2}}+\frac {\text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}+\frac {2}{3} \sqrt {3 x^2+2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}-\frac {\text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {3 x^2+2}}\right )}{2 \sqrt {11}}+\frac {2}{3} \sqrt {3 x^2+2}\) |
Input:
Int[(1 + 3*x + 4*x^2)/((1 + 2*x)*Sqrt[2 + 3*x^2]),x]
Output:
(2*Sqrt[2 + 3*x^2])/3 + ArcSinh[Sqrt[3/2]*x]/(2*Sqrt[3]) - ArcTanh[(4 - 3* x)/(Sqrt[11]*Sqrt[2 + 3*x^2])]/(2*Sqrt[11])
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/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[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
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 = 0.94 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.82
method | result | size |
default | \(\frac {\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{6}+\frac {2 \sqrt {3 x^{2}+2}}{3}-\frac {\sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-12 x}}\right )}{22}\) | \(55\) |
risch | \(\frac {\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{6}+\frac {2 \sqrt {3 x^{2}+2}}{3}-\frac {\sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-12 x}}\right )}{22}\) | \(55\) |
trager | \(\frac {2 \sqrt {3 x^{2}+2}}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) x +11 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right )}{1+2 x}\right )}{22}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{6}\) | \(89\) |
Input:
int((4*x^2+3*x+1)/(1+2*x)/(3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/6*arcsinh(1/2*6^(1/2)*x)*3^(1/2)+2/3*(3*x^2+2)^(1/2)-1/22*11^(1/2)*arcta nh(2/11*(4-3*x)*11^(1/2)/(12*(1/2+x)^2+5-12*x)^(1/2))
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.31 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \sqrt {2+3 x^2}} \, dx=\frac {1}{12} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + \frac {1}{44} \, \sqrt {11} \log \left (-\frac {\sqrt {11} \sqrt {3 \, x^{2} + 2} {\left (3 \, x - 4\right )} + 21 \, x^{2} - 12 \, x + 19}{4 \, x^{2} + 4 \, x + 1}\right ) + \frac {2}{3} \, \sqrt {3 \, x^{2} + 2} \] Input:
integrate((4*x^2+3*x+1)/(1+2*x)/(3*x^2+2)^(1/2),x, algorithm="fricas")
Output:
1/12*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 1/44*sqrt(11)*l og(-(sqrt(11)*sqrt(3*x^2 + 2)*(3*x - 4) + 21*x^2 - 12*x + 19)/(4*x^2 + 4*x + 1)) + 2/3*sqrt(3*x^2 + 2)
\[ \int \frac {1+3 x+4 x^2}{(1+2 x) \sqrt {2+3 x^2}} \, dx=\int \frac {4 x^{2} + 3 x + 1}{\left (2 x + 1\right ) \sqrt {3 x^{2} + 2}}\, dx \] Input:
integrate((4*x**2+3*x+1)/(1+2*x)/(3*x**2+2)**(1/2),x)
Output:
Integral((4*x**2 + 3*x + 1)/((2*x + 1)*sqrt(3*x**2 + 2)), x)
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.87 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \sqrt {2+3 x^2}} \, dx=\frac {1}{6} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {1}{22} \, \sqrt {11} \operatorname {arsinh}\left (\frac {\sqrt {6} x}{2 \, {\left | 2 \, x + 1 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {2}{3} \, \sqrt {3 \, x^{2} + 2} \] Input:
integrate((4*x^2+3*x+1)/(1+2*x)/(3*x^2+2)^(1/2),x, algorithm="maxima")
Output:
1/6*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 1/22*sqrt(11)*arcsinh(1/2*sqrt(6)*x/a bs(2*x + 1) - 2/3*sqrt(6)/abs(2*x + 1)) + 2/3*sqrt(3*x^2 + 2)
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (49) = 98\).
Time = 0.25 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.48 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \sqrt {2+3 x^2}} \, dx=-\frac {1}{6} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {1}{22} \, \sqrt {11} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {11} - \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {11} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {2}{3} \, \sqrt {3 \, x^{2} + 2} \] Input:
integrate((4*x^2+3*x+1)/(1+2*x)/(3*x^2+2)^(1/2),x, algorithm="giac")
Output:
-1/6*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 1/22*sqrt(11)*log(-abs(-2 *sqrt(3)*x - sqrt(11) - sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(1 1) + sqrt(3) - 2*sqrt(3*x^2 + 2))) + 2/3*sqrt(3*x^2 + 2)
Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \sqrt {2+3 x^2}} \, dx=\frac {\sqrt {11}\,\left (2\,\ln \left (x+\frac {1}{2}\right )-2\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {11}\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {4}{3}\right )\right )}{44}+\frac {2\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{3}+\frac {\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{6} \] Input:
int((3*x + 4*x^2 + 1)/((2*x + 1)*(3*x^2 + 2)^(1/2)),x)
Output:
(11^(1/2)*(2*log(x + 1/2) - 2*log(x - (3^(1/2)*11^(1/2)*(x^2 + 2/3)^(1/2)) /3 - 4/3)))/44 + (2*3^(1/2)*(x^2 + 2/3)^(1/2))/3 + (3^(1/2)*asinh((2^(1/2) *3^(1/2)*x)/2))/6
Time = 0.20 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.87 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \sqrt {2+3 x^2}} \, dx=\frac {\sqrt {11}\, \mathit {atan} \left (\frac {2 \sqrt {3 x^{2}+2}\, i +2 \sqrt {3}\, i x}{\sqrt {11}-\sqrt {3}}\right ) i}{22}+\frac {2 \sqrt {3 x^{2}+2}}{3}+\frac {\sqrt {11}\, \mathrm {log}\left (4 \sqrt {3 x^{2}+2}\, \sqrt {3}\, x +\sqrt {33}+12 x^{2}-3\right )}{44}-\frac {\sqrt {11}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}+2}+\sqrt {11}+2 \sqrt {3}\, x +\sqrt {3}}{\sqrt {2}}\right )}{22}+\frac {\sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )}{6} \] Input:
int((4*x^2+3*x+1)/(1+2*x)/(3*x^2+2)^(1/2),x)
Output:
(6*sqrt(11)*atan((2*sqrt(3*x**2 + 2)*i + 2*sqrt(3)*i*x)/(sqrt(11) - sqrt(3 )))*i + 88*sqrt(3*x**2 + 2) + 3*sqrt(11)*log(4*sqrt(3*x**2 + 2)*sqrt(3)*x + sqrt(33) + 12*x**2 - 3) - 6*sqrt(11)*log((2*sqrt(3*x**2 + 2) + sqrt(11) + 2*sqrt(3)*x + sqrt(3))/sqrt(2)) + 22*sqrt(3)*log((sqrt(3*x**2 + 2) + sqr t(3)*x)/sqrt(2)))/132