Integrand size = 29, antiderivative size = 71 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \sqrt {2+3 x^2}} \, dx=-\frac {\sqrt {2+3 x^2}}{11 (1+2 x)}+\frac {\text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}+\frac {4 \text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {2+3 x^2}}\right )}{11 \sqrt {11}} \] Output:
-1/11*(3*x^2+2)^(1/2)/(1+2*x)+1/3*arcsinh(1/2*x*6^(1/2))*3^(1/2)+4/121*arc tanh(1/11*(4-3*x)*11^(1/2)/(3*x^2+2)^(1/2))*11^(1/2)
Time = 0.57 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.30 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \sqrt {2+3 x^2}} \, dx=-\frac {\sqrt {2+3 x^2}}{11+22 x}-\frac {8 \text {arctanh}\left (\frac {\sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {11}}\right )}{11 \sqrt {11}}-\frac {\log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{\sqrt {3}} \] Input:
Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^2*Sqrt[2 + 3*x^2]),x]
Output:
-(Sqrt[2 + 3*x^2]/(11 + 22*x)) - (8*ArcTanh[(Sqrt[3] + 2*Sqrt[3]*x - 2*Sqr t[2 + 3*x^2])/Sqrt[11]])/(11*Sqrt[11]) - Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2 ]]/Sqrt[3]
Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2182, 25, 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.
\(\displaystyle \int \frac {4 x^2+3 x+1}{(2 x+1)^2 \sqrt {3 x^2+2}} \, dx\) |
\(\Big \downarrow \) 2182 |
\(\displaystyle -\frac {1}{11} \int -\frac {22 x+7}{(2 x+1) \sqrt {3 x^2+2}}dx-\frac {\sqrt {3 x^2+2}}{11 (2 x+1)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{11} \int \frac {22 x+7}{(2 x+1) \sqrt {3 x^2+2}}dx-\frac {\sqrt {3 x^2+2}}{11 (2 x+1)}\) |
\(\Big \downarrow \) 719 |
\(\displaystyle \frac {1}{11} \left (11 \int \frac {1}{\sqrt {3 x^2+2}}dx-4 \int \frac {1}{(2 x+1) \sqrt {3 x^2+2}}dx\right )-\frac {\sqrt {3 x^2+2}}{11 (2 x+1)}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{11} \left (\frac {11 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}-4 \int \frac {1}{(2 x+1) \sqrt {3 x^2+2}}dx\right )-\frac {\sqrt {3 x^2+2}}{11 (2 x+1)}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {1}{11} \left (4 \int \frac {1}{11-\frac {(4-3 x)^2}{3 x^2+2}}d\frac {4-3 x}{\sqrt {3 x^2+2}}+\frac {11 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}\right )-\frac {\sqrt {3 x^2+2}}{11 (2 x+1)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{11} \left (\frac {11 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}+\frac {4 \text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {3 x^2+2}}\right )}{\sqrt {11}}\right )-\frac {\sqrt {3 x^2+2}}{11 (2 x+1)}\) |
Input:
Int[(1 + 3*x + 4*x^2)/((1 + 2*x)^2*Sqrt[2 + 3*x^2]),x]
Output:
-1/11*Sqrt[2 + 3*x^2]/(1 + 2*x) + ((11*ArcSinh[Sqrt[3/2]*x])/Sqrt[3] + (4* ArcTanh[(4 - 3*x)/(Sqrt[11]*Sqrt[2 + 3*x^2])])/Sqrt[11])/11
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[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b *e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
Time = 0.96 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.87
method | result | size |
risch | \(-\frac {\sqrt {3 x^{2}+2}}{11 \left (1+2 x \right )}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{3}+\frac {4 \sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-12 x}}\right )}{121}\) | \(62\) |
default | \(\frac {\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{3}-\frac {\sqrt {3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x}}{22 \left (\frac {1}{2}+x \right )}+\frac {4 \sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-12 x}}\right )}{121}\) | \(65\) |
trager | \(-\frac {\sqrt {3 x^{2}+2}}{11 \left (1+2 x \right )}-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) x +11 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right )}{1+2 x}\right )}{121}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{3}\) | \(96\) |
Input:
int((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/11*(3*x^2+2)^(1/2)/(1+2*x)+1/3*arcsinh(1/2*6^(1/2)*x)*3^(1/2)+4/121*11^ (1/2)*arctanh(2/11*(4-3*x)*11^(1/2)/(12*(1/2+x)^2+5-12*x)^(1/2))
Time = 0.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.49 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \sqrt {2+3 x^2}} \, dx=\frac {121 \, \sqrt {3} {\left (2 \, x + 1\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 12 \, \sqrt {11} {\left (2 \, x + 1\right )} \log \left (\frac {\sqrt {11} \sqrt {3 \, x^{2} + 2} {\left (3 \, x - 4\right )} - 21 \, x^{2} + 12 \, x - 19}{4 \, x^{2} + 4 \, x + 1}\right ) - 66 \, \sqrt {3 \, x^{2} + 2}}{726 \, {\left (2 \, x + 1\right )}} \] Input:
integrate((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(1/2),x, algorithm="fricas")
Output:
1/726*(121*sqrt(3)*(2*x + 1)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 12*sqrt(11)*(2*x + 1)*log((sqrt(11)*sqrt(3*x^2 + 2)*(3*x - 4) - 21*x^2 + 12*x - 19)/(4*x^2 + 4*x + 1)) - 66*sqrt(3*x^2 + 2))/(2*x + 1)
\[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \sqrt {2+3 x^2}} \, dx=\int \frac {4 x^{2} + 3 x + 1}{\left (2 x + 1\right )^{2} \sqrt {3 x^{2} + 2}}\, dx \] Input:
integrate((4*x**2+3*x+1)/(1+2*x)**2/(3*x**2+2)**(1/2),x)
Output:
Integral((4*x**2 + 3*x + 1)/((2*x + 1)**2*sqrt(3*x**2 + 2)), x)
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.92 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \sqrt {2+3 x^2}} \, dx=\frac {1}{3} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {4}{121} \, \sqrt {11} \operatorname {arsinh}\left (\frac {\sqrt {6} x}{2 \, {\left | 2 \, x + 1 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 1 \right |}}\right ) - \frac {\sqrt {3 \, x^{2} + 2}}{11 \, {\left (2 \, x + 1\right )}} \] Input:
integrate((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(1/2),x, algorithm="maxima")
Output:
1/3*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 4/121*sqrt(11)*arcsinh(1/2*sqrt(6)*x/ abs(2*x + 1) - 2/3*sqrt(6)/abs(2*x + 1)) - 1/11*sqrt(3*x^2 + 2)/(2*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (56) = 112\).
Time = 0.38 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.69 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \sqrt {2+3 x^2}} \, dx=\frac {4 \, \sqrt {11} \log \left (\sqrt {11} {\left (\sqrt {-\frac {6}{2 \, x + 1} + \frac {11}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {11}}{2 \, x + 1}\right )} - 3\right )}{121 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} - \frac {\sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {6}{2 \, x + 1} + \frac {11}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {2 \, \sqrt {11}}{2 \, x + 1} \right |}}{2 \, {\left (\sqrt {3} + \sqrt {-\frac {6}{2 \, x + 1} + \frac {11}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {11}}{2 \, x + 1}\right )}}\right )}{3 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} - \frac {\sqrt {-\frac {6}{2 \, x + 1} + \frac {11}{{\left (2 \, x + 1\right )}^{2}} + 3}}{22 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} \] Input:
integrate((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(1/2),x, algorithm="giac")
Output:
4/121*sqrt(11)*log(sqrt(11)*(sqrt(-6/(2*x + 1) + 11/(2*x + 1)^2 + 3) + sqr t(11)/(2*x + 1)) - 3)/sgn(1/(2*x + 1)) - 1/3*sqrt(3)*log(1/2*abs(-2*sqrt(3 ) + 2*sqrt(-6/(2*x + 1) + 11/(2*x + 1)^2 + 3) + 2*sqrt(11)/(2*x + 1))/(sqr t(3) + sqrt(-6/(2*x + 1) + 11/(2*x + 1)^2 + 3) + sqrt(11)/(2*x + 1)))/sgn( 1/(2*x + 1)) - 1/22*sqrt(-6/(2*x + 1) + 11/(2*x + 1)^2 + 3)/sgn(1/(2*x + 1 ))
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.96 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \sqrt {2+3 x^2}} \, dx=\frac {\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{3}-\frac {4\,\sqrt {11}\,\ln \left (x+\frac {1}{2}\right )}{121}+\frac {4\,\sqrt {11}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {11}\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {4}{3}\right )}{121}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{22\,\left (x+\frac {1}{2}\right )} \] Input:
int((3*x + 4*x^2 + 1)/((2*x + 1)^2*(3*x^2 + 2)^(1/2)),x)
Output:
(3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/3 - (4*11^(1/2)*log(x + 1/2))/121 + (4*11^(1/2)*log(x - (3^(1/2)*11^(1/2)*(x^2 + 2/3)^(1/2))/3 - 4/3))/121 - (3^(1/2)*(x^2 + 2/3)^(1/2))/(22*(x + 1/2))
Time = 0.20 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.27 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \sqrt {2+3 x^2}} \, dx=\frac {-66 \sqrt {3 x^{2}+2}+48 \sqrt {11}\, \mathrm {log}\left (-\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right ) x +24 \sqrt {11}\, \mathrm {log}\left (-\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right )-48 \sqrt {11}\, \mathrm {log}\left (2 x +1\right ) x -24 \sqrt {11}\, \mathrm {log}\left (2 x +1\right )-242 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right ) x -121 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}-\sqrt {3}\, x \right )+242 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right ) x +121 \sqrt {3}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}+\sqrt {3}\, x \right )}{1452 x +726} \] Input:
int((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(1/2),x)
Output:
( - 66*sqrt(3*x**2 + 2) + 48*sqrt(11)*log( - sqrt(3*x**2 + 2)*sqrt(11) + 3 *x - 4)*x + 24*sqrt(11)*log( - sqrt(3*x**2 + 2)*sqrt(11) + 3*x - 4) - 48*s qrt(11)*log(2*x + 1)*x - 24*sqrt(11)*log(2*x + 1) - 242*sqrt(3)*log(sqrt(3 *x**2 + 2) - sqrt(3)*x)*x - 121*sqrt(3)*log(sqrt(3*x**2 + 2) - sqrt(3)*x) + 242*sqrt(3)*log(sqrt(3*x**2 + 2) + sqrt(3)*x)*x + 121*sqrt(3)*log(sqrt(3 *x**2 + 2) + sqrt(3)*x))/(726*(2*x + 1))