Integrand size = 29, antiderivative size = 53 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=\frac {-38+21 x}{66 \sqrt {2+3 x^2}}-\frac {2 \text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {2+3 x^2}}\right )}{11 \sqrt {11}} \] Output:
1/66*(-38+21*x)/(3*x^2+2)^(1/2)-2/121*arctanh(1/11*(4-3*x)*11^(1/2)/(3*x^2 +2)^(1/2))*11^(1/2)
Time = 0.76 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=\frac {-418+231 x-12 \sqrt {22+33 x^2} \text {arctanh}\left (\frac {4-3 x}{\sqrt {22+33 x^2}}\right )}{726 \sqrt {2+3 x^2}} \] Input:
Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)*(2 + 3*x^2)^(3/2)),x]
Output:
(-418 + 231*x - 12*Sqrt[22 + 33*x^2]*ArcTanh[(4 - 3*x)/Sqrt[22 + 33*x^2]]) /(726*Sqrt[2 + 3*x^2])
Time = 0.23 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2178, 27, 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.
\(\displaystyle \int \frac {4 x^2+3 x+1}{(2 x+1) \left (3 x^2+2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2178 |
\(\displaystyle -\frac {1}{6} \int -\frac {12}{11 (2 x+1) \sqrt {3 x^2+2}}dx-\frac {38-21 x}{66 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{11} \int \frac {1}{(2 x+1) \sqrt {3 x^2+2}}dx-\frac {38-21 x}{66 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle -\frac {2}{11} \int \frac {1}{11-\frac {(4-3 x)^2}{3 x^2+2}}d\frac {4-3 x}{\sqrt {3 x^2+2}}-\frac {38-21 x}{66 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2 \text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {3 x^2+2}}\right )}{11 \sqrt {11}}-\frac {38-21 x}{66 \sqrt {3 x^2+2}}\) |
Input:
Int[(1 + 3*x + 4*x^2)/((1 + 2*x)*(2 + 3*x^2)^(3/2)),x]
Output:
-1/66*(38 - 21*x)/Sqrt[2 + 3*x^2] - (2*ArcTanh[(4 - 3*x)/(Sqrt[11]*Sqrt[2 + 3*x^2])])/(11*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/(((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[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(d + e*x )^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x ] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
Time = 0.92 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {-38+21 x}{66 \sqrt {3 x^{2}+2}}-\frac {2 \sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-12 x}}\right )}{121}\) | \(48\) |
trager | \(\frac {-38+21 x}{66 \sqrt {3 x^{2}+2}}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) x +11 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right )}{1+2 x}\right )}{121}\) | \(64\) |
default | \(\frac {x}{4 \sqrt {3 x^{2}+2}}+\frac {1}{11 \sqrt {3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x}}+\frac {3 x}{44 \sqrt {3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x}}-\frac {2 \sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-12 x}}\right )}{121}-\frac {2}{3 \sqrt {3 x^{2}+2}}\) | \(88\) |
Input:
int((4*x^2+3*x+1)/(1+2*x)/(3*x^2+2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/66*(-38+21*x)/(3*x^2+2)^(1/2)-2/121*11^(1/2)*arctanh(2/11*(4-3*x)*11^(1/ 2)/(12*(1/2+x)^2+5-12*x)^(1/2))
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.57 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=\frac {6 \, \sqrt {11} {\left (3 \, x^{2} + 2\right )} \log \left (-\frac {\sqrt {11} \sqrt {3 \, x^{2} + 2} {\left (3 \, x - 4\right )} + 21 \, x^{2} - 12 \, x + 19}{4 \, x^{2} + 4 \, x + 1}\right ) + 11 \, \sqrt {3 \, x^{2} + 2} {\left (21 \, x - 38\right )}}{726 \, {\left (3 \, x^{2} + 2\right )}} \] Input:
integrate((4*x^2+3*x+1)/(1+2*x)/(3*x^2+2)^(3/2),x, algorithm="fricas")
Output:
1/726*(6*sqrt(11)*(3*x^2 + 2)*log(-(sqrt(11)*sqrt(3*x^2 + 2)*(3*x - 4) + 2 1*x^2 - 12*x + 19)/(4*x^2 + 4*x + 1)) + 11*sqrt(3*x^2 + 2)*(21*x - 38))/(3 *x^2 + 2)
\[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=\int \frac {4 x^{2} + 3 x + 1}{\left (2 x + 1\right ) \left (3 x^{2} + 2\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((4*x**2+3*x+1)/(1+2*x)/(3*x**2+2)**(3/2),x)
Output:
Integral((4*x**2 + 3*x + 1)/((2*x + 1)*(3*x**2 + 2)**(3/2)), x)
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.09 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=\frac {2}{121} \, \sqrt {11} \operatorname {arsinh}\left (\frac {\sqrt {6} x}{2 \, {\left | 2 \, x + 1 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {7 \, x}{22 \, \sqrt {3 \, x^{2} + 2}} - \frac {19}{33 \, \sqrt {3 \, x^{2} + 2}} \] Input:
integrate((4*x^2+3*x+1)/(1+2*x)/(3*x^2+2)^(3/2),x, algorithm="maxima")
Output:
2/121*sqrt(11)*arcsinh(1/2*sqrt(6)*x/abs(2*x + 1) - 2/3*sqrt(6)/abs(2*x + 1)) + 7/22*x/sqrt(3*x^2 + 2) - 19/33/sqrt(3*x^2 + 2)
Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.55 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=\frac {2}{121} \, \sqrt {11} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {11} - \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {11} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {21 \, x - 38}{66 \, \sqrt {3 \, x^{2} + 2}} \] Input:
integrate((4*x^2+3*x+1)/(1+2*x)/(3*x^2+2)^(3/2),x, algorithm="giac")
Output:
2/121*sqrt(11)*log(-abs(-2*sqrt(3)*x - sqrt(11) - sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(11) + sqrt(3) - 2*sqrt(3*x^2 + 2))) + 1/66*(21*x - 38)/sqrt(3*x^2 + 2)
Time = 17.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.00 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=\frac {\sqrt {11}\,\left (2\,\ln \left (x+\frac {1}{2}\right )-2\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {11}\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {4}{3}\right )\right )}{121}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-114+\sqrt {6}\,21{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2376\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (114+\sqrt {6}\,21{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2376\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:
int((3*x + 4*x^2 + 1)/((2*x + 1)*(3*x^2 + 2)^(3/2)),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)))/121 - (3^(1/2)*6^(1/2)*(6^(1/2)*21i - 114)*(x^2 + 2/3)^(1/2)*1 i)/(2376*(x - (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*21i + 114)*(x^2 + 2/3)^(1/2)*1i)/(2376*(x + (6^(1/2)*1i)/3))
Time = 0.19 (sec) , antiderivative size = 234, normalized size of antiderivative = 4.42 \[ \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2+3 x^2\right )^{3/2}} \, dx=\frac {36 \sqrt {11}\, \mathit {atan} \left (\frac {2 \sqrt {3 x^{2}+2}\, i +2 \sqrt {3}\, i x}{\sqrt {11}-\sqrt {3}}\right ) i \,x^{2}+24 \sqrt {11}\, \mathit {atan} \left (\frac {2 \sqrt {3 x^{2}+2}\, i +2 \sqrt {3}\, i x}{\sqrt {11}-\sqrt {3}}\right ) i +231 \sqrt {3 x^{2}+2}\, x -418 \sqrt {3 x^{2}+2}+18 \sqrt {11}\, \mathrm {log}\left (4 \sqrt {3 x^{2}+2}\, \sqrt {3}\, x +\sqrt {33}+12 x^{2}-3\right ) x^{2}+12 \sqrt {11}\, \mathrm {log}\left (4 \sqrt {3 x^{2}+2}\, \sqrt {3}\, x +\sqrt {33}+12 x^{2}-3\right )-36 \sqrt {11}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}+2}+\sqrt {11}+2 \sqrt {3}\, x +\sqrt {3}}{\sqrt {2}}\right ) x^{2}-24 \sqrt {11}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}+2}+\sqrt {11}+2 \sqrt {3}\, x +\sqrt {3}}{\sqrt {2}}\right )+231 \sqrt {3}\, x^{2}+154 \sqrt {3}}{2178 x^{2}+1452} \] Input:
int((4*x^2+3*x+1)/(1+2*x)/(3*x^2+2)^(3/2),x)
Output:
(36*sqrt(11)*atan((2*sqrt(3*x**2 + 2)*i + 2*sqrt(3)*i*x)/(sqrt(11) - sqrt( 3)))*i*x**2 + 24*sqrt(11)*atan((2*sqrt(3*x**2 + 2)*i + 2*sqrt(3)*i*x)/(sqr t(11) - sqrt(3)))*i + 231*sqrt(3*x**2 + 2)*x - 418*sqrt(3*x**2 + 2) + 18*s qrt(11)*log(4*sqrt(3*x**2 + 2)*sqrt(3)*x + sqrt(33) + 12*x**2 - 3)*x**2 + 12*sqrt(11)*log(4*sqrt(3*x**2 + 2)*sqrt(3)*x + sqrt(33) + 12*x**2 - 3) - 3 6*sqrt(11)*log((2*sqrt(3*x**2 + 2) + sqrt(11) + 2*sqrt(3)*x + sqrt(3))/sqr t(2))*x**2 - 24*sqrt(11)*log((2*sqrt(3*x**2 + 2) + sqrt(11) + 2*sqrt(3)*x + sqrt(3))/sqrt(2)) + 231*sqrt(3)*x**2 + 154*sqrt(3))/(726*(3*x**2 + 2))