Integrand size = 29, antiderivative size = 95 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {-10+97 x}{726 \left (2+3 x^2\right )^{3/2}}+\frac {24+887 x}{7986 \sqrt {2+3 x^2}}-\frac {16 \sqrt {2+3 x^2}}{1331 (1+2 x)}-\frac {32 \text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {2+3 x^2}}\right )}{1331 \sqrt {11}} \] Output:
1/726*(-10+97*x)/(3*x^2+2)^(3/2)+1/7986*(24+887*x)/(3*x^2+2)^(1/2)-16*(3*x ^2+2)^(1/2)/(1331+2662*x)-32/14641*arctanh(1/11*(4-3*x)*11^(1/2)/(3*x^2+2) ^(1/2))*11^(1/2)
Time = 1.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.96 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {11 \left (-446+2717 x+4602 x^2+2805 x^3+4458 x^4\right )-192 \sqrt {22+33 x^2} \left (2+4 x+3 x^2+6 x^3\right ) \text {arctanh}\left (\frac {4-3 x}{\sqrt {22+33 x^2}}\right )}{87846 (1+2 x) \left (2+3 x^2\right )^{3/2}} \] Input:
Integrate[(1 + 3*x + 4*x^2)/((1 + 2*x)^2*(2 + 3*x^2)^(5/2)),x]
Output:
(11*(-446 + 2717*x + 4602*x^2 + 2805*x^3 + 4458*x^4) - 192*Sqrt[22 + 33*x^ 2]*(2 + 4*x + 3*x^2 + 6*x^3)*ArcTanh[(4 - 3*x)/Sqrt[22 + 33*x^2]])/(87846* (1 + 2*x)*(2 + 3*x^2)^(3/2))
Time = 0.34 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2178, 27, 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.
| \(\displaystyle \int \frac {4 x^2+3 x+1}{(2 x+1)^2 \left (3 x^2+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle -\frac {1}{18} \int -\frac {6 \left (388 x^2+328 x+133\right )}{121 (2 x+1)^2 \left (3 x^2+2\right )^{3/2}}dx-\frac {10-97 x}{726 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{363} \int \frac {388 x^2+328 x+133}{(2 x+1)^2 \left (3 x^2+2\right )^{3/2}}dx-\frac {10-97 x}{726 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle \frac {1}{363} \left (\frac {887 x+24}{22 \sqrt {3 x^2+2}}-\frac {1}{6} \int -\frac {288 (x+6)}{11 (2 x+1)^2 \sqrt {3 x^2+2}}dx\right )-\frac {10-97 x}{726 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{363} \left (\frac {48}{11} \int \frac {x+6}{(2 x+1)^2 \sqrt {3 x^2+2}}dx+\frac {887 x+24}{22 \sqrt {3 x^2+2}}\right )-\frac {10-97 x}{726 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {1}{363} \left (\frac {48}{11} \left (2 \int \frac {1}{(2 x+1) \sqrt {3 x^2+2}}dx-\frac {\sqrt {3 x^2+2}}{2 x+1}\right )+\frac {887 x+24}{22 \sqrt {3 x^2+2}}\right )-\frac {10-97 x}{726 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {1}{363} \left (\frac {48}{11} \left (-2 \int \frac {1}{11-\frac {(4-3 x)^2}{3 x^2+2}}d\frac {4-3 x}{\sqrt {3 x^2+2}}-\frac {\sqrt {3 x^2+2}}{2 x+1}\right )+\frac {887 x+24}{22 \sqrt {3 x^2+2}}\right )-\frac {10-97 x}{726 \left (3 x^2+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{363} \left (\frac {48}{11} \left (-\frac {2 \text {arctanh}\left (\frac {4-3 x}{\sqrt {11} \sqrt {3 x^2+2}}\right )}{\sqrt {11}}-\frac {\sqrt {3 x^2+2}}{2 x+1}\right )+\frac {887 x+24}{22 \sqrt {3 x^2+2}}\right )-\frac {10-97 x}{726 \left (3 x^2+2\right )^{3/2}}\) |
Input:
Int[(1 + 3*x + 4*x^2)/((1 + 2*x)^2*(2 + 3*x^2)^(5/2)),x]
Output:
-1/726*(10 - 97*x)/(2 + 3*x^2)^(3/2) + ((24 + 887*x)/(22*Sqrt[2 + 3*x^2]) + (48*(-(Sqrt[2 + 3*x^2]/(1 + 2*x)) - (2*ArcTanh[(4 - 3*x)/(Sqrt[11]*Sqrt[ 2 + 3*x^2])])/Sqrt[11]))/11)/363
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[((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]
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 = 0.94 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(\frac {4458 x^{4}+2805 x^{3}+4602 x^{2}+2717 x -446}{7986 \left (3 x^{2}+2\right )^{\frac {3}{2}} \left (1+2 x \right )}-\frac {32 \sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-12 x}}\right )}{14641}\) | \(70\) |
| trager | \(\frac {4458 x^{4}+2805 x^{3}+4602 x^{2}+2717 x -446}{7986 \left (3 x^{2}+2\right )^{\frac {3}{2}} \left (1+2 x \right )}-\frac {32 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11\right )-11 \sqrt {3 x^{2}+2}}{1+2 x}\right )}{14641}\) | \(87\) |
| default | \(\frac {x}{6 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {x}{6 \sqrt {3 x^{2}+2}}-\frac {1}{22 \left (\frac {1}{2}+x \right ) \left (3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x \right )^{\frac {3}{2}}}+\frac {4}{363 \left (3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x \right )^{\frac {3}{2}}}-\frac {10 x}{121 \left (3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x \right )^{\frac {3}{2}}}-\frac {98 x}{1331 \sqrt {3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x}}+\frac {16}{1331 \sqrt {3 \left (\frac {1}{2}+x \right )^{2}+\frac {5}{4}-3 x}}-\frac {32 \sqrt {11}\, \operatorname {arctanh}\left (\frac {2 \left (4-3 x \right ) \sqrt {11}}{11 \sqrt {12 \left (\frac {1}{2}+x \right )^{2}+5-12 x}}\right )}{14641}\) | \(143\) |
Input:
int((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/7986*(4458*x^4+2805*x^3+4602*x^2+2717*x-446)/(3*x^2+2)^(3/2)/(1+2*x)-32/ 14641*11^(1/2)*arctanh(2/11*(4-3*x)*11^(1/2)/(12*(1/2+x)^2+5-12*x)^(1/2))
Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.41 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {96 \, \sqrt {11} {\left (18 \, x^{5} + 9 \, x^{4} + 24 \, x^{3} + 12 \, x^{2} + 8 \, x + 4\right )} \log \left (-\frac {\sqrt {11} \sqrt {3 \, x^{2} + 2} {\left (3 \, x - 4\right )} + 21 \, x^{2} - 12 \, x + 19}{4 \, x^{2} + 4 \, x + 1}\right ) + 11 \, {\left (4458 \, x^{4} + 2805 \, x^{3} + 4602 \, x^{2} + 2717 \, x - 446\right )} \sqrt {3 \, x^{2} + 2}}{87846 \, {\left (18 \, x^{5} + 9 \, x^{4} + 24 \, x^{3} + 12 \, x^{2} + 8 \, x + 4\right )}} \] Input:
integrate((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(5/2),x, algorithm="fricas")
Output:
1/87846*(96*sqrt(11)*(18*x^5 + 9*x^4 + 24*x^3 + 12*x^2 + 8*x + 4)*log(-(sq rt(11)*sqrt(3*x^2 + 2)*(3*x - 4) + 21*x^2 - 12*x + 19)/(4*x^2 + 4*x + 1)) + 11*(4458*x^4 + 2805*x^3 + 4602*x^2 + 2717*x - 446)*sqrt(3*x^2 + 2))/(18* x^5 + 9*x^4 + 24*x^3 + 12*x^2 + 8*x + 4)
Timed out. \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((4*x**2+3*x+1)/(1+2*x)**2/(3*x**2+2)**(5/2),x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.13 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {32}{14641} \, \sqrt {11} \operatorname {arsinh}\left (\frac {\sqrt {6} x}{2 \, {\left | 2 \, x + 1 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {743 \, x}{7986 \, \sqrt {3 \, x^{2} + 2}} + \frac {16}{1331 \, \sqrt {3 \, x^{2} + 2}} + \frac {61 \, x}{726 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {1}{11 \, {\left (2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} + \frac {4}{363 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \] Input:
integrate((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(5/2),x, algorithm="maxima")
Output:
32/14641*sqrt(11)*arcsinh(1/2*sqrt(6)*x/abs(2*x + 1) - 2/3*sqrt(6)/abs(2*x + 1)) + 743/7986*x/sqrt(3*x^2 + 2) + 16/1331/sqrt(3*x^2 + 2) + 61/726*x/( 3*x^2 + 2)^(3/2) - 1/11/(2*(3*x^2 + 2)^(3/2)*x + (3*x^2 + 2)^(3/2)) + 4/36 3/(3*x^2 + 2)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (76) = 152\).
Time = 0.26 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.45 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=-\frac {1}{263538} \, \sqrt {11} {\left (743 \, \sqrt {11} \sqrt {3} - 576 \, \log \left (\sqrt {11} \sqrt {3} - 3\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - \frac {32 \, \sqrt {11} \log \left (\sqrt {11} {\left (\sqrt {-\frac {6}{2 \, x + 1} + \frac {11}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {11}}{2 \, x + 1}\right )} - 3\right )}{14641 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} + \frac {\frac {\frac {\frac {11 \, {\left (\frac {731}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} + \frac {528}{{\left (2 \, x + 1\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}\right )}}{2 \, x + 1} - \frac {14163}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}}{2 \, x + 1} + \frac {6111}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}}{2 \, x + 1} - \frac {2229}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}}{7986 \, {\left (\frac {6}{2 \, x + 1} - \frac {11}{{\left (2 \, x + 1\right )}^{2}} - 3\right )} \sqrt {-\frac {6}{2 \, x + 1} + \frac {11}{{\left (2 \, x + 1\right )}^{2}} + 3}} \] Input:
integrate((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(5/2),x, algorithm="giac")
Output:
-1/263538*sqrt(11)*(743*sqrt(11)*sqrt(3) - 576*log(sqrt(11)*sqrt(3) - 3))* sgn(1/(2*x + 1)) - 32/14641*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)) + 1/7986*(((1 1*(731/sgn(1/(2*x + 1)) + 528/((2*x + 1)*sgn(1/(2*x + 1))))/(2*x + 1) - 14 163/sgn(1/(2*x + 1)))/(2*x + 1) + 6111/sgn(1/(2*x + 1)))/(2*x + 1) - 2229/ sgn(1/(2*x + 1)))/((6/(2*x + 1) - 11/(2*x + 1)^2 - 3)*sqrt(-6/(2*x + 1) + 11/(2*x + 1)^2 + 3))
Time = 0.25 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.84 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {\sqrt {11}\,\left (8\,\ln \left (x+\frac {1}{2}\right )-8\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {11}\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {4}{3}\right )\right )}{14641}+\frac {\sqrt {11}\,\left (\frac {48\,\ln \left (x+\frac {1}{2}\right )}{1331}-\frac {48\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {11}\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {4}{3}\right )}{1331}\right )}{22}-\frac {8\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1331\,\left (x+\frac {1}{2}\right )}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {291}{1936}+\frac {\sqrt {6}\,15{}\mathrm {i}}{1936}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (-\frac {97}{968}+\frac {\sqrt {6}\,5{}\mathrm {i}}{968}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {291}{1936}+\frac {\sqrt {6}\,15{}\mathrm {i}}{1936}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (\frac {97}{968}+\frac {\sqrt {6}\,5{}\mathrm {i}}{968}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-288+\sqrt {6}\,2481{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1149984\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (288+\sqrt {6}\,2481{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1149984\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:
int((3*x + 4*x^2 + 1)/((2*x + 1)^2*(3*x^2 + 2)^(5/2)),x)
Output:
(11^(1/2)*(8*log(x + 1/2) - 8*log(x - (3^(1/2)*11^(1/2)*(x^2 + 2/3)^(1/2)) /3 - 4/3)))/14641 + (11^(1/2)*((48*log(x + 1/2))/1331 - (48*log(x - (3^(1/ 2)*11^(1/2)*(x^2 + 2/3)^(1/2))/3 - 4/3))/1331))/22 - (8*3^(1/2)*(x^2 + 2/3 )^(1/2))/(1331*(x + 1/2)) - (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*15i)/193 6 - 291/1936)/(x + (6^(1/2)*1i)/3) + (6^(1/2)*((6^(1/2)*5i)/968 - 97/968)* 1i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27 + (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2 )*15i)/1936 + 291/1936)/(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2)*5i)/968 + 97/968)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*6^(1/2)*(6^(1/2)* 2481i - 288)*(x^2 + 2/3)^(1/2)*1i)/(1149984*(x + (6^(1/2)*1i)/3)) - (3^(1/ 2)*6^(1/2)*(6^(1/2)*2481i + 288)*(x^2 + 2/3)^(1/2)*1i)/(1149984*(x - (6^(1 /2)*1i)/3))
Time = 0.27 (sec) , antiderivative size = 301, normalized size of antiderivative = 3.17 \[ \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx=\frac {49038 \sqrt {3 x^{2}+2}\, x^{4}+30855 \sqrt {3 x^{2}+2}\, x^{3}+50622 \sqrt {3 x^{2}+2}\, x^{2}+29887 \sqrt {3 x^{2}+2}\, x -4906 \sqrt {3 x^{2}+2}+3456 \sqrt {11}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right ) x^{5}+1728 \sqrt {11}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right ) x^{4}+4608 \sqrt {11}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right ) x^{3}+2304 \sqrt {11}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right ) x^{2}+1536 \sqrt {11}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right ) x +768 \sqrt {11}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {11}+3 x -4\right )-3456 \sqrt {11}\, \mathrm {log}\left (2 x +1\right ) x^{5}-1728 \sqrt {11}\, \mathrm {log}\left (2 x +1\right ) x^{4}-4608 \sqrt {11}\, \mathrm {log}\left (2 x +1\right ) x^{3}-2304 \sqrt {11}\, \mathrm {log}\left (2 x +1\right ) x^{2}-1536 \sqrt {11}\, \mathrm {log}\left (2 x +1\right ) x -768 \sqrt {11}\, \mathrm {log}\left (2 x +1\right )}{1581228 x^{5}+790614 x^{4}+2108304 x^{3}+1054152 x^{2}+702768 x +351384} \] Input:
int((4*x^2+3*x+1)/(1+2*x)^2/(3*x^2+2)^(5/2),x)
Output:
(49038*sqrt(3*x**2 + 2)*x**4 + 30855*sqrt(3*x**2 + 2)*x**3 + 50622*sqrt(3* x**2 + 2)*x**2 + 29887*sqrt(3*x**2 + 2)*x - 4906*sqrt(3*x**2 + 2) + 3456*s qrt(11)*log(sqrt(3*x**2 + 2)*sqrt(11) + 3*x - 4)*x**5 + 1728*sqrt(11)*log( sqrt(3*x**2 + 2)*sqrt(11) + 3*x - 4)*x**4 + 4608*sqrt(11)*log(sqrt(3*x**2 + 2)*sqrt(11) + 3*x - 4)*x**3 + 2304*sqrt(11)*log(sqrt(3*x**2 + 2)*sqrt(11 ) + 3*x - 4)*x**2 + 1536*sqrt(11)*log(sqrt(3*x**2 + 2)*sqrt(11) + 3*x - 4) *x + 768*sqrt(11)*log(sqrt(3*x**2 + 2)*sqrt(11) + 3*x - 4) - 3456*sqrt(11) *log(2*x + 1)*x**5 - 1728*sqrt(11)*log(2*x + 1)*x**4 - 4608*sqrt(11)*log(2 *x + 1)*x**3 - 2304*sqrt(11)*log(2*x + 1)*x**2 - 1536*sqrt(11)*log(2*x + 1 )*x - 768*sqrt(11)*log(2*x + 1))/(87846*(18*x**5 + 9*x**4 + 24*x**3 + 12*x **2 + 8*x + 4))