\(\int \frac {1+3 x+4 x^2}{(1+2 x)^3 (2+3 x^2)^{5/2}} \, dx\) [135]

Optimal result

Integrand size = 29, antiderivative size = 117 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {358+351 x}{7986 \left (2+3 x^2\right )^{3/2}}+\frac {1216+2133 x}{29282 \sqrt {2+3 x^2}}-\frac {8 \sqrt {2+3 x^2}}{1331 (1+2 x)^2}-\frac {8 \sqrt {2+3 x^2}}{1331 (1+2 x)}-\frac {1216 \text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {2+3 x^2}}\right )}{14641 \sqrt {11}} \] Output:

1/7986*(358+351*x)/(3*x^2+2)^(3/2)+1/29282*(1216+2133*x)/(3*x^2+2)^(1/2)-8 
/1331*(3*x^2+2)^(1/2)/(1+2*x)^2-8*(3*x^2+2)^(1/2)/(1331+2662*x)-1216/16105 
1*arctanh(1/11*(4-3*x)*11^(1/2)/(3*x^2+2)^(1/2))*11^(1/2)
 

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.78 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {\frac {11 \left (7010+57371 x+109844 x^2+116937 x^3+111060 x^4+67284 x^5\right )}{(1+2 x)^2 \left (2+3 x^2\right )^{3/2}}+14592 \sqrt {11} \text {arctanh}\left (\frac {\sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {11}}\right )}{966306} \] Input:

Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^3*(2 + 3*x^2)^(5/2)),x]
 

Output:

((11*(7010 + 57371*x + 109844*x^2 + 116937*x^3 + 111060*x^4 + 67284*x^5))/ 
((1 + 2*x)^2*(2 + 3*x^2)^(3/2)) + 14592*Sqrt[11]*ArcTanh[(Sqrt[3] + 2*Sqrt 
[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[11]])/966306
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2178, 27, 2178, 27, 2182, 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.

\(\displaystyle \int \frac {4 x^2+3 x+1}{(2 x+1)^3 \left (3 x^2+2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 2178

\(\displaystyle \frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}-\frac {1}{18} \int -\frac {18 \left (936 x^3+2836 x^2+1914 x+607\right )}{1331 (2 x+1)^3 \left (3 x^2+2\right )^{3/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {936 x^3+2836 x^2+1914 x+607}{(2 x+1)^3 \left (3 x^2+2\right )^{3/2}}dx}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 2178

\(\displaystyle \frac {\frac {2133 x+1216}{22 \sqrt {3 x^2+2}}-\frac {1}{6} \int -\frac {96 \left (304 x^2+315 x+142\right )}{11 (2 x+1)^3 \sqrt {3 x^2+2}}dx}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {16}{11} \int \frac {304 x^2+315 x+142}{(2 x+1)^3 \sqrt {3 x^2+2}}dx+\frac {2133 x+1216}{22 \sqrt {3 x^2+2}}}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {\frac {16}{11} \left (-\frac {1}{22} \int -\frac {11 (271 x+196)}{(2 x+1)^2 \sqrt {3 x^2+2}}dx-\frac {11 \sqrt {3 x^2+2}}{2 (2 x+1)^2}\right )+\frac {2133 x+1216}{22 \sqrt {3 x^2+2}}}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {16}{11} \left (\frac {1}{2} \int \frac {271 x+196}{(2 x+1)^2 \sqrt {3 x^2+2}}dx-\frac {11 \sqrt {3 x^2+2}}{2 (2 x+1)^2}\right )+\frac {2133 x+1216}{22 \sqrt {3 x^2+2}}}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {\frac {16}{11} \left (\frac {1}{2} \left (152 \int \frac {1}{(2 x+1) \sqrt {3 x^2+2}}dx-\frac {11 \sqrt {3 x^2+2}}{2 x+1}\right )-\frac {11 \sqrt {3 x^2+2}}{2 (2 x+1)^2}\right )+\frac {2133 x+1216}{22 \sqrt {3 x^2+2}}}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {16}{11} \left (\frac {1}{2} \left (-152 \int \frac {1}{11-\frac {(4-3 x)^2}{3 x^2+2}}d\frac {4-3 x}{\sqrt {3 x^2+2}}-\frac {11 \sqrt {3 x^2+2}}{2 x+1}\right )-\frac {11 \sqrt {3 x^2+2}}{2 (2 x+1)^2}\right )+\frac {2133 x+1216}{22 \sqrt {3 x^2+2}}}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {16}{11} \left (\frac {1}{2} \left (-\frac {152 \text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {3 x^2+2}}\right )}{\sqrt {11}}-\frac {11 \sqrt {3 x^2+2}}{2 x+1}\right )-\frac {11 \sqrt {3 x^2+2}}{2 (2 x+1)^2}\right )+\frac {2133 x+1216}{22 \sqrt {3 x^2+2}}}{1331}+\frac {351 x+358}{7986 \left (3 x^2+2\right )^{3/2}}\)

Input:

Int[(1 + 3*x + 4*x^2)/((1 + 2*x)^3*(2 + 3*x^2)^(5/2)),x]
 

Output:

(358 + 351*x)/(7986*(2 + 3*x^2)^(3/2)) + ((1216 + 2133*x)/(22*Sqrt[2 + 3*x 
^2]) + (16*((-11*Sqrt[2 + 3*x^2])/(2*(1 + 2*x)^2) + ((-11*Sqrt[2 + 3*x^2]) 
/(1 + 2*x) - (152*ArcTanh[(4 - 3*x)/(Sqrt[11]*Sqrt[2 + 3*x^2])])/Sqrt[11]) 
/2))/11)/1331
 

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]
 

rule 2182

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]
 

Maple [A] (verified)

Time = 0.96 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.64

method	result	size
risch	\(\frac {67284 x^{5}+111060 x^{4}+116937 x^{3}+109844 x^{2}+57371 x +7010}{87846 \left (1+2 x \right )^{2} \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {1216 \sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-12 x}}\right )}{161051}\)	\(75\)
trager	\(\frac {\left (67284 x^{5}+111060 x^{4}+116937 x^{3}+109844 x^{2}+57371 x +7010\right ) \sqrt {3 x^{2}+2}}{87846 \left (6 x^{3}+3 x^{2}+4 x +2\right )^{2}}-\frac {1216 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right )-11 \sqrt {3 x^{2}+2}}{1+2 x}\right )}{161051}\)	\(102\)
default	\(\frac {152}{3993 \left (3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x \right )^{\frac {3}{2}}}+\frac {87 x}{2662 \left (3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x \right )^{\frac {3}{2}}}+\frac {1869 x}{29282 \sqrt {3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x}}+\frac {608}{14641 \sqrt {3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x}}-\frac {1216 \sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-12 x}}\right )}{161051}-\frac {1}{88 \left (\frac {1}{2}+x \right )^{2} \left (3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x \right )^{\frac {3}{2}}}+\frac {1}{484 \left (\frac {1}{2}+x \right ) \left (3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x \right )^{\frac {3}{2}}}\)	\(140\)

Input:

int((4*x^2+3*x+1)/(1+2*x)^3/(3*x^2+2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/87846*(67284*x^5+111060*x^4+116937*x^3+109844*x^2+57371*x+7010)/(1+2*x)^ 
2/(3*x^2+2)^(3/2)-1216/161051*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 = 0.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.27 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {3648 \, \sqrt {11} {\left (36 \, x^{6} + 36 \, x^{5} + 57 \, x^{4} + 48 \, x^{3} + 28 \, x^{2} + 16 \, x + 4\right )} \log \left (-\frac {\sqrt {11} \sqrt {3 \, x^{2} + 2} {\left (3 \, x - 4\right )} + 21 \, x^{2} - 12 \, x + 19}{4 \, x^{2} + 4 \, x + 1}\right ) + 11 \, {\left (67284 \, x^{5} + 111060 \, x^{4} + 116937 \, x^{3} + 109844 \, x^{2} + 57371 \, x + 7010\right )} \sqrt {3 \, x^{2} + 2}}{966306 \, {\left (36 \, x^{6} + 36 \, x^{5} + 57 \, x^{4} + 48 \, x^{3} + 28 \, x^{2} + 16 \, x + 4\right )}} \] Input:

integrate((4*x^2+3*x+1)/(1+2*x)^3/(3*x^2+2)^(5/2),x, algorithm="fricas")
 

Output:

1/966306*(3648*sqrt(11)*(36*x^6 + 36*x^5 + 57*x^4 + 48*x^3 + 28*x^2 + 16*x 
 + 4)*log(-(sqrt(11)*sqrt(3*x^2 + 2)*(3*x - 4) + 21*x^2 - 12*x + 19)/(4*x^ 
2 + 4*x + 1)) + 11*(67284*x^5 + 111060*x^4 + 116937*x^3 + 109844*x^2 + 573 
71*x + 7010)*sqrt(3*x^2 + 2))/(36*x^6 + 36*x^5 + 57*x^4 + 48*x^3 + 28*x^2 
+ 16*x + 4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((4*x**2+3*x+1)/(1+2*x)**3/(3*x**2+2)**(5/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.26 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {1216}{161051} \, \sqrt {11} \operatorname {arsinh}\left (\frac {\sqrt {6} x}{2 \, {\left | 2 \, x + 1 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {1869 \, x}{29282 \, \sqrt {3 \, x^{2} + 2}} + \frac {608}{14641 \, \sqrt {3 \, x^{2} + 2}} + \frac {87 \, x}{2662 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {1}{22 \, {\left (4 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{2} + 4 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} + \frac {1}{242 \, {\left (2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} + \frac {152}{3993 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \] Input:

integrate((4*x^2+3*x+1)/(1+2*x)^3/(3*x^2+2)^(5/2),x, algorithm="maxima")
 

Output:

1216/161051*sqrt(11)*arcsinh(1/2*sqrt(6)*x/abs(2*x + 1) - 2/3*sqrt(6)/abs( 
2*x + 1)) + 1869/29282*x/sqrt(3*x^2 + 2) + 608/14641/sqrt(3*x^2 + 2) + 87/ 
2662*x/(3*x^2 + 2)^(3/2) - 1/22/(4*(3*x^2 + 2)^(3/2)*x^2 + 4*(3*x^2 + 2)^( 
3/2)*x + (3*x^2 + 2)^(3/2)) + 1/242/(2*(3*x^2 + 2)^(3/2)*x + (3*x^2 + 2)^( 
3/2)) + 152/3993/(3*x^2 + 2)^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.56 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {1216}{161051} \, \sqrt {11} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {11} - \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {11} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {9 \, {\left ({\left (2133 \, x + 1216\right )} x + 1851\right )} x + 11234}{87846 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {4 \, {\left (\sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 24 \, \sqrt {3} x - 8 \, \sqrt {3} - 24 \, \sqrt {3 \, x^{2} + 2}\right )}}{1331 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{2}} \] Input:

integrate((4*x^2+3*x+1)/(1+2*x)^3/(3*x^2+2)^(5/2),x, algorithm="giac")
 

Output:

1216/161051*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))) + 1/878 
46*(9*((2133*x + 1216)*x + 1851)*x + 11234)/(3*x^2 + 2)^(3/2) + 4/1331*(sq 
rt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 24*sqrt(3)*x - 8*sqrt(3) - 24*sqrt 
(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + sqrt(3)*(sqrt(3)*x - sqrt( 
3*x^2 + 2)) - 2)^2
 

Mupad [B] (verification not implemented)

Time = 17.29 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.57 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {1216\,\sqrt {11}\,\ln \left (x+\frac {1}{2}\right )}{161051}-\frac {1216\,\sqrt {11}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {11}\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {4}{3}\right )}{161051}-\frac {179\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{95832\,\left (x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}-\frac {2}{3}\right )}+\frac {711\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{58564\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {711\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{58564\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {2\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1331\,\left (x^2+x+\frac {1}{4}\right )}+\frac {179\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{95832\,\left (-x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}+\frac {2}{3}\right )}-\frac {4\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1331\,\left (x+\frac {1}{2}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,13{}\mathrm {i}}{21296\,\left (x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}-\frac {2}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,9265{}\mathrm {i}}{2108304\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,9265{}\mathrm {i}}{2108304\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,13{}\mathrm {i}}{21296\,\left (-x^2+\frac {2{}\mathrm {i}\,\sqrt {6}\,x}{3}+\frac {2}{3}\right )} \] Input:

int((3*x + 4*x^2 + 1)/((2*x + 1)^3*(3*x^2 + 2)^(5/2)),x)
 

Output:

(1216*11^(1/2)*log(x + 1/2))/161051 - (1216*11^(1/2)*log(x - (3^(1/2)*11^( 
1/2)*(x^2 + 2/3)^(1/2))/3 - 4/3))/161051 - (179*3^(1/2)*(x^2 + 2/3)^(1/2)) 
/(95832*((6^(1/2)*x*2i)/3 + x^2 - 2/3)) + (711*3^(1/2)*(x^2 + 2/3)^(1/2))/ 
(58564*(x - (6^(1/2)*1i)/3)) + (711*3^(1/2)*(x^2 + 2/3)^(1/2))/(58564*(x + 
 (6^(1/2)*1i)/3)) - (2*3^(1/2)*(x^2 + 2/3)^(1/2))/(1331*(x + x^2 + 1/4)) + 
 (179*3^(1/2)*(x^2 + 2/3)^(1/2))/(95832*((6^(1/2)*x*2i)/3 - x^2 + 2/3)) - 
(4*3^(1/2)*(x^2 + 2/3)^(1/2))/(1331*(x + 1/2)) + (3^(1/2)*6^(1/2)*(x^2 + 2 
/3)^(1/2)*13i)/(21296*((6^(1/2)*x*2i)/3 + x^2 - 2/3)) - (3^(1/2)*6^(1/2)*( 
x^2 + 2/3)^(1/2)*9265i)/(2108304*(x - (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2)* 
(x^2 + 2/3)^(1/2)*9265i)/(2108304*(x + (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2) 
*(x^2 + 2/3)^(1/2)*13i)/(21296*((6^(1/2)*x*2i)/3 - x^2 + 2/3))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 356, normalized size of antiderivative = 3.04 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {740124 \sqrt {3 x^{2}+2}\, x^{5}+1221660 \sqrt {3 x^{2}+2}\, x^{4}+1286307 \sqrt {3 x^{2}+2}\, x^{3}+1208284 \sqrt {3 x^{2}+2}\, x^{2}+631081 \sqrt {3 x^{2}+2}\, x +77110 \sqrt {3 x^{2}+2}+262656 \sqrt {11}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right ) x^{6}+262656 \sqrt {11}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right ) x^{5}+415872 \sqrt {11}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right ) x^{4}+350208 \sqrt {11}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right ) x^{3}+204288 \sqrt {11}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right ) x^{2}+116736 \sqrt {11}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right ) x +29184 \sqrt {11}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right )-262656 \sqrt {11}\, \mathrm {log}\left (2 x +1\right ) x^{6}-262656 \sqrt {11}\, \mathrm {log}\left (2 x +1\right ) x^{5}-415872 \sqrt {11}\, \mathrm {log}\left (2 x +1\right ) x^{4}-350208 \sqrt {11}\, \mathrm {log}\left (2 x +1\right ) x^{3}-204288 \sqrt {11}\, \mathrm {log}\left (2 x +1\right ) x^{2}-116736 \sqrt {11}\, \mathrm {log}\left (2 x +1\right ) x -29184 \sqrt {11}\, \mathrm {log}\left (2 x +1\right )}{34787016 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x +3865224} \] Input:

int((4*x^2+3*x+1)/(1+2*x)^3/(3*x^2+2)^(5/2),x)
 

Output:

(740124*sqrt(3*x**2 + 2)*x**5 + 1221660*sqrt(3*x**2 + 2)*x**4 + 1286307*sq 
rt(3*x**2 + 2)*x**3 + 1208284*sqrt(3*x**2 + 2)*x**2 + 631081*sqrt(3*x**2 + 
 2)*x + 77110*sqrt(3*x**2 + 2) + 262656*sqrt(11)*log(sqrt(3*x**2 + 2)*sqrt 
(11) + 3*x - 4)*x**6 + 262656*sqrt(11)*log(sqrt(3*x**2 + 2)*sqrt(11) + 3*x 
 - 4)*x**5 + 415872*sqrt(11)*log(sqrt(3*x**2 + 2)*sqrt(11) + 3*x - 4)*x**4 
 + 350208*sqrt(11)*log(sqrt(3*x**2 + 2)*sqrt(11) + 3*x - 4)*x**3 + 204288* 
sqrt(11)*log(sqrt(3*x**2 + 2)*sqrt(11) + 3*x - 4)*x**2 + 116736*sqrt(11)*l 
og(sqrt(3*x**2 + 2)*sqrt(11) + 3*x - 4)*x + 29184*sqrt(11)*log(sqrt(3*x**2 
 + 2)*sqrt(11) + 3*x - 4) - 262656*sqrt(11)*log(2*x + 1)*x**6 - 262656*sqr 
t(11)*log(2*x + 1)*x**5 - 415872*sqrt(11)*log(2*x + 1)*x**4 - 350208*sqrt( 
11)*log(2*x + 1)*x**3 - 204288*sqrt(11)*log(2*x + 1)*x**2 - 116736*sqrt(11 
)*log(2*x + 1)*x - 29184*sqrt(11)*log(2*x + 1))/(966306*(36*x**6 + 36*x**5 
 + 57*x**4 + 48*x**3 + 28*x**2 + 16*x + 4))