\(\int \frac {(1+2 x)^{3/2}}{(2+3 x+5 x^2)^3} \, dx\) [2326]

Optimal result

Integrand size = 22, antiderivative size = 300 \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922} \]

[Out]

-1/62*(5-4*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)^2+1/1922*(67+120*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)-3/834148*ln(5+10*x+3
5^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-6545588+1173970*35^(1/2))^(1/2)+3/834148*ln(5+10*x+35^(1/2)+(1
+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-6545588+1173970*35^(1/2))^(1/2)-3/417074*arctan((-10*(1+2*x)^(1/2)+(20+1
0*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(6545588+1173970*35^(1/2))^(1/2)+3/417074*arctan((10*(1+2*x)^(1/2)
+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(6545588+1173970*35^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {752, 836, 840, 1183, 648, 632, 210, 642} \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=-\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {\sqrt {2 x+1} (5-4 x)}{62 \left (5 x^2+3 x+2\right )^2}+\frac {\sqrt {2 x+1} (120 x+67)}{1922 \left (5 x^2+3 x+2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922} \]

[In]

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

[Out]

-1/62*((5 - 4*x)*Sqrt[1 + 2*x])/(2 + 3*x + 5*x^2)^2 + (Sqrt[1 + 2*x]*(67 + 120*x))/(1922*(2 + 3*x + 5*x^2)) -
(3*Sqrt[(15082 + 2705*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35
])]])/961 + (3*Sqrt[(15082 + 2705*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(
-2 + Sqrt[35])]])/961 - (3*Sqrt[(-15082 + 2705*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 +
2*x] + 5*(1 + 2*x)])/1922 + (3*Sqrt[(-15082 + 2705*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[
1 + 2*x] + 5*(1 + 2*x)])/1922

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 840

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1183

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = -\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {1}{62} \int \frac {17+20 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {1239+840 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{13454} \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {1638+840 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{6727} \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {1638 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (1638-168 \sqrt {35}\right ) x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{13454 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {1638 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (1638-168 \sqrt {35}\right ) x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{13454 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\left (3 \left (140+39 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{67270}+\frac {\left (3 \left (140+39 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{67270}-\frac {\left (3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )}\right ) \text {Subst}\left (\int \frac {-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{1922}+\frac {\left (3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )}\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{1922} \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}-\frac {\left (3 \left (140+39 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )}{33635}-\frac {\left (3 \left (140+39 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )}{33635} \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-2 \sqrt {1+2 x}\right )\right )+\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )\right )-\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.40 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.48 \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {217 \sqrt {1+2 x} \left (-21+565 x+695 x^2+600 x^3\right )}{2 \left (2+3 x+5 x^2\right )^2}+3 \sqrt {217 \left (15082+961 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+3 \sqrt {217 \left (15082-961 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{208537} \]

[In]

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

[Out]

((217*Sqrt[1 + 2*x]*(-21 + 565*x + 695*x^2 + 600*x^3))/(2*(2 + 3*x + 5*x^2)^2) + 3*Sqrt[217*(15082 + (961*I)*S
qrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] + 3*Sqrt[217*(15082 - (961*I)*Sqrt[31])]*ArcTan[Sqrt
[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]])/208537

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.09

method	result	size
pseudoelliptic	\(-\frac {2925 \left (\frac {329 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (\sqrt {5}-\frac {218 \sqrt {7}}{329}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4836}-\frac {329 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (\sqrt {5}-\frac {218 \sqrt {7}}{329}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4836}-\frac {28 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (x^{3}+\frac {139}{120} x^{2}+\frac {113}{120} x -\frac {7}{200}\right ) \sqrt {1+2 x}}{39}+\left (\sqrt {5}\, \sqrt {7}+\frac {140}{39}\right ) \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right )\right )}{6727 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (5 x^{2}+3 x +2\right )^{2}}\)	\(328\)
derivativedivides	\(\frac {\frac {600 \left (1+2 x \right )^{\frac {7}{2}}}{961}-\frac {410 \left (1+2 x \right )^{\frac {5}{2}}}{961}+\frac {1280 \left (1+2 x \right )^{\frac {3}{2}}}{961}-\frac {1638 \sqrt {1+2 x}}{961}}{\left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}+\frac {3 \left (-1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4170740}+\frac {3 \left (2418 \sqrt {5}\, \sqrt {7}+\frac {\left (-1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3 \left (1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4170740}+\frac {3 \left (2418 \sqrt {5}\, \sqrt {7}-\frac {\left (1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(435\)
default	\(\frac {\frac {600 \left (1+2 x \right )^{\frac {7}{2}}}{961}-\frac {410 \left (1+2 x \right )^{\frac {5}{2}}}{961}+\frac {1280 \left (1+2 x \right )^{\frac {3}{2}}}{961}-\frac {1638 \sqrt {1+2 x}}{961}}{\left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}+\frac {3 \left (-1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4170740}+\frac {3 \left (2418 \sqrt {5}\, \sqrt {7}+\frac {\left (-1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3 \left (1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4170740}+\frac {3 \left (2418 \sqrt {5}\, \sqrt {7}-\frac {\left (1645 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-1090 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(435\)
trager	\(\frac {\left (600 x^{3}+695 x^{2}+565 x -21\right ) \sqrt {1+2 x}}{1922 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {6 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right ) \ln \left (\frac {7377527808 x \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{5}[phone] \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{3} x -16335954432 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{3}-5415288816 \sqrt {1+2 x}\, \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}[phone] \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right ) x -61800326272 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )-92971296325 \sqrt {1+2 x}}{3472 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} x +17965 x +3844}\right )}{961}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}+1636397\right ) \ln \left (-\frac {526966272 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}+1636397\right ) \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{4} x [phone] \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}+1636397\right ) x [phone] \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}+1636397\right )-167873953296 \sqrt {1+2 x}\, \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}+1636397\right ) x [phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}+1636397\right )[phone] \sqrt {1+2 x}}{3472 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} x +12199 x -3844}\right )}{208537}\)	\(458\)
risch	\(\frac {\left (600 x^{3}+695 x^{2}+565 x -21\right ) \sqrt {1+2 x}}{1922 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {141 \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{119164}+\frac {327 \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{417074}-\frac {705 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{59582 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {327 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {234 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \sqrt {7}}{6727 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {141 \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{119164}-\frac {327 \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{417074}-\frac {705 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{59582 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {327 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {234 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \sqrt {7}}{6727 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(643\)

[In]

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

[Out]

-2925/6727/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(329/4836*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(
x^2+3/5*x+2/5)^2*(5^(1/2)-218/329*7^(1/2))*ln(5^(1/2)*7^(1/2)-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2
)+5+10*x)-329/4836*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(x^2+3/5*x+2/5)^2*(5^(1/2)-218/32
9*7^(1/2))*ln(5^(1/2)*7^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5+10*x)-28/39*(10*5^(1/2)*7^(1
/2)-20)^(1/2)*(x^3+139/120*x^2+113/120*x-7/200)*(1+2*x)^(1/2)+(5^(1/2)*7^(1/2)+140/39)*(x^2+3/5*x+2/5)^2*(arct
an((5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-10*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))-arctan((5^(1/2)*(2*5^
(1/2)*7^(1/2)+4)^(1/2)+10*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))))/(5*x^2+3*x+2)^2

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.02 \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=-\frac {\sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {8649 i \, \sqrt {31} - 135738} \log \left (\sqrt {217} \sqrt {8649 i \, \sqrt {31} - 135738} {\left (218 i \, \sqrt {31} - 1209\right )} + 8804775 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {8649 i \, \sqrt {31} - 135738} \log \left (\sqrt {217} \sqrt {8649 i \, \sqrt {31} - 135738} {\left (-218 i \, \sqrt {31} + 1209\right )} + 8804775 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-8649 i \, \sqrt {31} - 135738} \log \left (\sqrt {217} {\left (218 i \, \sqrt {31} + 1209\right )} \sqrt {-8649 i \, \sqrt {31} - 135738} + 8804775 \, \sqrt {2 \, x + 1}\right ) + \sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-8649 i \, \sqrt {31} - 135738} \log \left (\sqrt {217} {\left (-218 i \, \sqrt {31} - 1209\right )} \sqrt {-8649 i \, \sqrt {31} - 135738} + 8804775 \, \sqrt {2 \, x + 1}\right ) - 217 \, {\left (600 \, x^{3} + 695 \, x^{2} + 565 \, x - 21\right )} \sqrt {2 \, x + 1}}{417074 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]

[In]

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

[Out]

-1/417074*(sqrt(217)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(8649*I*sqrt(31) - 135738)*log(sqrt(217)*sqrt(8
649*I*sqrt(31) - 135738)*(218*I*sqrt(31) - 1209) + 8804775*sqrt(2*x + 1)) - sqrt(217)*(25*x^4 + 30*x^3 + 29*x^
2 + 12*x + 4)*sqrt(8649*I*sqrt(31) - 135738)*log(sqrt(217)*sqrt(8649*I*sqrt(31) - 135738)*(-218*I*sqrt(31) + 1
209) + 8804775*sqrt(2*x + 1)) - sqrt(217)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(-8649*I*sqrt(31) - 135738
)*log(sqrt(217)*(218*I*sqrt(31) + 1209)*sqrt(-8649*I*sqrt(31) - 135738) + 8804775*sqrt(2*x + 1)) + sqrt(217)*(
25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(-8649*I*sqrt(31) - 135738)*log(sqrt(217)*(-218*I*sqrt(31) - 1209)*sq
rt(-8649*I*sqrt(31) - 135738) + 8804775*sqrt(2*x + 1)) - 217*(600*x^3 + 695*x^2 + 565*x - 21)*sqrt(2*x + 1))/(
25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)

Sympy [F]

\[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {\left (2 x + 1\right )^{\frac {3}{2}}}{\left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \]

[In]

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

[Out]

Integral((2*x + 1)**(3/2)/(5*x**2 + 3*x + 2)**3, x)

Maxima [F]

\[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {{\left (2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (213) = 426\).

Time = 0.70 (sec) , antiderivative size = 640, normalized size of antiderivative = 2.13 \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

3/3576409550*sqrt(31)*(210*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - sqrt(31)*(7/5)^
(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 2*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 420*(7/5)^(3/4)*sqrt(140*sqrt
(35) + 2450)*(2*sqrt(35) - 35) + 9555*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 19110*(7/5)^(1/4)*sqrt
(140*sqrt(35) + 2450))*arctan(5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*sqrt(35) + 1/2) + sqrt(2*x + 1))/sqrt(-1/
35*sqrt(35) + 1/2)) + 3/3576409550*sqrt(31)*(210*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2
450) - sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 2*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 420*(7/
5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 9555*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) +
19110*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(-5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*sqrt(35) + 1/2) -
sqrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) + 3/7152819100*sqrt(31)*(sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)
^(3/2) + 210*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 420*(7/5)^(3/4)*(2*sqrt(35) +
35)*sqrt(-140*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 9555*sqrt(31)*(7/5)^(1/4)*sqrt(1
40*sqrt(35) + 2450) - 19110*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450))*log(2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*
sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) - 3/7152819100*sqrt(31)*(sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/
2) + 210*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 420*(7/5)^(3/4)*(2*sqrt(35) + 35)*
sqrt(-140*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 9555*sqrt(31)*(7/5)^(1/4)*sqrt(140*s
qrt(35) + 2450) - 19110*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450))*log(-2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqr
t(35) + 1/2) + 2*x + sqrt(7/5) + 1) + 2/961*(300*(2*x + 1)^(7/2) - 205*(2*x + 1)^(5/2) + 640*(2*x + 1)^(3/2) -
819*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)^2

Mupad [B] (verification not implemented)

Time = 10.44 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.82 \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {1638\,\sqrt {2\,x+1}}{24025}-\frac {256\,{\left (2\,x+1\right )}^{3/2}}{4805}+\frac {82\,{\left (2\,x+1\right )}^{5/2}}{4805}-\frac {24\,{\left (2\,x+1\right )}^{7/2}}{961}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}\,432{}\mathrm {i}}{5656566125\,\left (\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}+\frac {864\,\sqrt {31}\,\sqrt {217}\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}}{175353549875\,\left (\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}\right )\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,3{}\mathrm {i}}{208537}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}\,432{}\mathrm {i}}{5656566125\,\left (-\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}-\frac {864\,\sqrt {31}\,\sqrt {217}\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}}{175353549875\,\left (-\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}\right )\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,3{}\mathrm {i}}{208537} \]

[In]

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

[Out]

((1638*(2*x + 1)^(1/2))/24025 - (256*(2*x + 1)^(3/2))/4805 + (82*(2*x + 1)^(5/2))/4805 - (24*(2*x + 1)^(7/2))/
961)/((112*x)/25 - (86*(2*x + 1)^2)/25 + (8*(2*x + 1)^3)/5 - (2*x + 1)^4 + 7/25) + (217^(1/2)*atan((217^(1/2)*
(- 31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2)*432i)/(5656566125*((31^(1/2)*16848i)/808080875 + 94176/8080808
75)) + (864*31^(1/2)*217^(1/2)*(- 31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2))/(175353549875*((31^(1/2)*16848
i)/808080875 + 94176/808080875)))*(- 31^(1/2)*961i - 15082)^(1/2)*3i)/208537 - (217^(1/2)*atan((217^(1/2)*(31^
(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2)*432i)/(5656566125*((31^(1/2)*16848i)/808080875 - 94176/808080875)) -
(864*31^(1/2)*217^(1/2)*(31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2))/(175353549875*((31^(1/2)*16848i)/80808
0875 - 94176/808080875)))*(31^(1/2)*961i - 15082)^(1/2)*3i)/208537