\(\int \frac {1+3 x+4 x^2}{(1+2 x)^2 (2+3 x^2)^{3/2}} \, dx\) [128]

Optimal result

Integrand size = 29, antiderivative size = 75 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {-10+97 x}{242 \sqrt {2+3 x^2}}-\frac {4 \sqrt {2+3 x^2}}{121 (1+2 x)}+\frac {4 \text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {2+3 x^2}}\right )}{121 \sqrt {11}} \] Output:

1/242*(-10+97*x)/(3*x^2+2)^(1/2)-4*(3*x^2+2)^(1/2)/(121+242*x)+4/1331*arct 
anh(1/11*(4-3*x)*11^(1/2)/(3*x^2+2)^(1/2))*11^(1/2)
 

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {11 \left (-26+77 x+170 x^2\right )+8 (1+2 x) \sqrt {22+33 x^2} \text {arctanh}\left (\frac {4-3 x}{\sqrt {22+33 x^2}}\right )}{2662 (1+2 x) \sqrt {2+3 x^2}} \] Input:

Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^2*(2 + 3*x^2)^(3/2)),x]
 

Output:

(11*(-26 + 77*x + 170*x^2) + 8*(1 + 2*x)*Sqrt[22 + 33*x^2]*ArcTanh[(4 - 3* 
x)/Sqrt[22 + 33*x^2]])/(2662*(1 + 2*x)*Sqrt[2 + 3*x^2])
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2178, 27, 679, 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.

\(\displaystyle \int \frac {4 x^2+3 x+1}{(2 x+1)^2 \left (3 x^2+2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2178

\(\displaystyle -\frac {1}{6} \int -\frac {24 (3-5 x)}{121 (2 x+1)^2 \sqrt {3 x^2+2}}dx-\frac {10-97 x}{242 \sqrt {3 x^2+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{121} \int \frac {3-5 x}{(2 x+1)^2 \sqrt {3 x^2+2}}dx-\frac {10-97 x}{242 \sqrt {3 x^2+2}}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {4}{121} \left (-\int \frac {1}{(2 x+1) \sqrt {3 x^2+2}}dx-\frac {\sqrt {3 x^2+2}}{2 x+1}\right )-\frac {10-97 x}{242 \sqrt {3 x^2+2}}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {4}{121} \left (\int \frac {1}{11-\frac {(4-3 x)^2}{3 x^2+2}}d\frac {4-3 x}{\sqrt {3 x^2+2}}-\frac {\sqrt {3 x^2+2}}{2 x+1}\right )-\frac {10-97 x}{242 \sqrt {3 x^2+2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {4}{121} \left (\frac {\text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {3 x^2+2}}\right )}{\sqrt {11}}-\frac {\sqrt {3 x^2+2}}{2 x+1}\right )-\frac {10-97 x}{242 \sqrt {3 x^2+2}}\)

Input:

Int[(1 + 3*x + 4*x^2)/((1 + 2*x)^2*(2 + 3*x^2)^(3/2)),x]
 

Output:

-1/242*(10 - 97*x)/Sqrt[2 + 3*x^2] + (4*(-(Sqrt[2 + 3*x^2]/(1 + 2*x)) + Ar 
cTanh[(4 - 3*x)/(Sqrt[11]*Sqrt[2 + 3*x^2])]/Sqrt[11]))/121
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 488

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]
 

rule 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 2178

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]
 

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.80

method	result	size
risch	\(\frac {170 x^{2}+77 x -26}{242 \left (1+2 x \right ) \sqrt {3 x^{2}+2}}+\frac {4 \sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-12 x}}\right )}{1331}\)	\(60\)
trager	\(\frac {\left (170 x^{2}+77 x -26\right ) \sqrt {3 x^{2}+2}}{1452 x^{3}+726 x^{2}+968 x +484}-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) x +11 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right )}{1+2 x}\right )}{1331}\)	\(86\)
default	\(\frac {x}{2 \sqrt {3 x^{2}+2}}-\frac {1}{22 \left (\frac {1}{2}+x \right ) \sqrt {3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x}}-\frac {2}{121 \sqrt {3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x}}-\frac {18 x}{121 \sqrt {3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x}}+\frac {4 \sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-12 x}}\right )}{1331}\)	\(98\)

Input:

int((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/242*(170*x^2+77*x-26)/(1+2*x)/(3*x^2+2)^(1/2)+4/1331*11^(1/2)*arctanh(2/ 
11*(4-3*x)*11^(1/2)/(12*(1/2+x)^2+5-12*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.37 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {4 \, \sqrt {11} {\left (6 \, x^{3} + 3 \, x^{2} + 4 \, x + 2\right )} \log \left (\frac {\sqrt {11} \sqrt {3 \, x^{2} + 2} {\left (3 \, x - 4\right )} - 21 \, x^{2} + 12 \, x - 19}{4 \, x^{2} + 4 \, x + 1}\right ) + 11 \, {\left (170 \, x^{2} + 77 \, x - 26\right )} \sqrt {3 \, x^{2} + 2}}{2662 \, {\left (6 \, x^{3} + 3 \, x^{2} + 4 \, x + 2\right )}} \] Input:

integrate((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(3/2),x, algorithm="fricas")
 

Output:

1/2662*(4*sqrt(11)*(6*x^3 + 3*x^2 + 4*x + 2)*log((sqrt(11)*sqrt(3*x^2 + 2) 
*(3*x - 4) - 21*x^2 + 12*x - 19)/(4*x^2 + 4*x + 1)) + 11*(170*x^2 + 77*x - 
 26)*sqrt(3*x^2 + 2))/(6*x^3 + 3*x^2 + 4*x + 2)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\int \frac {4 x^{2} + 3 x + 1}{\left (2 x + 1\right )^{2} \left (3 x^{2} + 2\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((4*x**2+3*x+1)/(1+2*x)**2/(3*x**2+2)**(3/2),x)
 

Output:

Integral((4*x**2 + 3*x + 1)/((2*x + 1)**2*(3*x**2 + 2)**(3/2)), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.12 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=-\frac {4}{1331} \, \sqrt {11} \operatorname {arsinh}\left (\frac {\sqrt {6} x}{2 \, {\left | 2 \, x + 1 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {85 \, x}{242 \, \sqrt {3 \, x^{2} + 2}} - \frac {2}{121 \, \sqrt {3 \, x^{2} + 2}} - \frac {1}{11 \, {\left (2 \, \sqrt {3 \, x^{2} + 2} x + \sqrt {3 \, x^{2} + 2}\right )}} \] Input:

integrate((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(3/2),x, algorithm="maxima")
 

Output:

-4/1331*sqrt(11)*arcsinh(1/2*sqrt(6)*x/abs(2*x + 1) - 2/3*sqrt(6)/abs(2*x 
+ 1)) + 85/242*x/sqrt(3*x^2 + 2) - 2/121/sqrt(3*x^2 + 2) - 1/11/(2*sqrt(3* 
x^2 + 2)*x + sqrt(3*x^2 + 2))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (60) = 120\).

Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.24 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=-\frac {1}{7986} \, \sqrt {11} {\left (85 \, \sqrt {11} \sqrt {3} + 24 \, \log \left (\sqrt {11} \sqrt {3} - 3\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - \frac {\frac {\frac {93}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} + \frac {44}{{\left (2 \, x + 1\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}}{2 \, x + 1} - \frac {85}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}}{242 \, \sqrt {-\frac {6}{2 \, x + 1} + \frac {11}{{\left (2 \, x + 1\right )}^{2}} + 3}} + \frac {4 \, \sqrt {11} \log \left (\sqrt {11} {\left (\sqrt {-\frac {6}{2 \, x + 1} + \frac {11}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {11}}{2 \, x + 1}\right )} - 3\right )}{1331 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} \] Input:

integrate((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(3/2),x, algorithm="giac")
 

Output:

-1/7986*sqrt(11)*(85*sqrt(11)*sqrt(3) + 24*log(sqrt(11)*sqrt(3) - 3))*sgn( 
1/(2*x + 1)) - 1/242*((93/sgn(1/(2*x + 1)) + 44/((2*x + 1)*sgn(1/(2*x + 1) 
)))/(2*x + 1) - 85/sgn(1/(2*x + 1)))/sqrt(-6/(2*x + 1) + 11/(2*x + 1)^2 + 
3) + 4/1331*sqrt(11)*log(sqrt(11)*(sqrt(-6/(2*x + 1) + 11/(2*x + 1)^2 + 3) 
 + sqrt(11)/(2*x + 1)) - 3)/sgn(1/(2*x + 1))
 

Mupad [B] (verification not implemented)

Time = 16.83 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.09 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {4\,\sqrt {11}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {11}\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {4}{3}\right )}{1331}-\frac {4\,\sqrt {11}\,\ln \left (x+\frac {1}{2}\right )}{1331}+\frac {97\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1452\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {97\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1452\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {2\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{121\,\left (x+\frac {1}{2}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,5{}\mathrm {i}}{1452\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,5{}\mathrm {i}}{1452\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:

int((3*x + 4*x^2 + 1)/((2*x + 1)^2*(3*x^2 + 2)^(3/2)),x)
 

Output:

(4*11^(1/2)*log(x - (3^(1/2)*11^(1/2)*(x^2 + 2/3)^(1/2))/3 - 4/3))/1331 - 
(4*11^(1/2)*log(x + 1/2))/1331 + (97*3^(1/2)*(x^2 + 2/3)^(1/2))/(1452*(x - 
 (6^(1/2)*1i)/3)) + (97*3^(1/2)*(x^2 + 2/3)^(1/2))/(1452*(x + (6^(1/2)*1i) 
/3)) - (2*3^(1/2)*(x^2 + 2/3)^(1/2))/(121*(x + 1/2)) + (3^(1/2)*6^(1/2)*(x 
^2 + 2/3)^(1/2)*5i)/(1452*(x - (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(x^2 + 
2/3)^(1/2)*5i)/(1452*(x + (6^(1/2)*1i)/3))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.60 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx=\frac {1870 \sqrt {3 x^{2}+2}\, x^{2}+847 \sqrt {3 x^{2}+2}\, x -286 \sqrt {3 x^{2}+2}+48 \sqrt {11}\, \mathrm {log}\left (-\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right ) x^{3}+24 \sqrt {11}\, \mathrm {log}\left (-\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right ) x^{2}+32 \sqrt {11}\, \mathrm {log}\left (-\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right ) x +16 \sqrt {11}\, \mathrm {log}\left (-\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right )-48 \sqrt {11}\, \mathrm {log}\left (2 x +1\right ) x^{3}-24 \sqrt {11}\, \mathrm {log}\left (2 x +1\right ) x^{2}-32 \sqrt {11}\, \mathrm {log}\left (2 x +1\right ) x -16 \sqrt {11}\, \mathrm {log}\left (2 x +1\right )}{15972 x^{3}+7986 x^{2}+10648 x +5324} \] Input:

int((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(3/2),x)
 

Output:

(1870*sqrt(3*x**2 + 2)*x**2 + 847*sqrt(3*x**2 + 2)*x - 286*sqrt(3*x**2 + 2 
) + 48*sqrt(11)*log( - sqrt(3*x**2 + 2)*sqrt(11) + 3*x - 4)*x**3 + 24*sqrt 
(11)*log( - sqrt(3*x**2 + 2)*sqrt(11) + 3*x - 4)*x**2 + 32*sqrt(11)*log( - 
 sqrt(3*x**2 + 2)*sqrt(11) + 3*x - 4)*x + 16*sqrt(11)*log( - sqrt(3*x**2 + 
 2)*sqrt(11) + 3*x - 4) - 48*sqrt(11)*log(2*x + 1)*x**3 - 24*sqrt(11)*log( 
2*x + 1)*x**2 - 32*sqrt(11)*log(2*x + 1)*x - 16*sqrt(11)*log(2*x + 1))/(26 
62*(6*x**3 + 3*x**2 + 4*x + 2))