\(\int \frac {1}{\sqrt {1+2 x} (2+3 x+5 x^2)^2} \, dx\) [2320]

Optimal result

Integrand size = 22, antiderivative size = 270 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {\sqrt {1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}-\frac {1}{217} \sqrt {\frac {2}{217} \left (32678+10325 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{217} \sqrt {\frac {2}{217} \left (32678+10325 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{217} \sqrt {\frac {1}{434} \left (-32678+10325 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{217} \sqrt {\frac {1}{434} \left (-32678+10325 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \]

[Out]

1/217*(37+20*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)-1/94178*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(
-14182252+4481050*35^(1/2))^(1/2)+1/94178*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-14182252+
4481050*35^(1/2))^(1/2)-1/47089*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(14
182252+4481050*35^(1/2))^(1/2)+1/47089*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2
))*(14182252+4481050*35^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {754, 840, 1183, 648, 632, 210, 642} \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx=-\frac {1}{217} \sqrt {\frac {2}{217} \left (32678+10325 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {1}{217} \sqrt {\frac {2}{217} \left (32678+10325 \sqrt {35}\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {\sqrt {2 x+1} (20 x+37)}{217 \left (5 x^2+3 x+2\right )}-\frac {1}{217} \sqrt {\frac {1}{434} \left (10325 \sqrt {35}-32678\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{217} \sqrt {\frac {1}{434} \left (10325 \sqrt {35}-32678\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right ) \]

[In]

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

[Out]

(Sqrt[1 + 2*x]*(37 + 20*x))/(217*(2 + 3*x + 5*x^2)) - (Sqrt[(2*(32678 + 10325*Sqrt[35]))/217]*ArcTan[(Sqrt[10*
(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/217 + (Sqrt[(2*(32678 + 10325*Sqrt[35]))/217]*A
rcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/217 - (Sqrt[(-32678 + 10325*Sqrt
[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/217 + (Sqrt[(-32678 + 10325*Sq
rt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/217

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 754

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

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 {\sqrt {1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}+\frac {1}{217} \int \frac {107+20 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx \\ & = \frac {\sqrt {1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}+\frac {2}{217} \text {Subst}\left (\int \frac {194+20 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right ) \\ & = \frac {\sqrt {1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {194 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (194-4 \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 )}{217 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {194 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (194-4 \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 )}{217 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = \frac {\sqrt {1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}+\frac {\left (70+97 \sqrt {35}\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 )}{7595}+\frac {\left (70+97 \sqrt {35}\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 )}{7595}-\frac {1}{217} \sqrt {\frac {1}{434} \left (-32678+10325 \sqrt {35}\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 )+\frac {1}{217} \sqrt {\frac {1}{434} \left (-32678+10325 \sqrt {35}\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 ) \\ & = \frac {\sqrt {1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}-\frac {1}{217} \sqrt {\frac {1}{434} \left (-32678+10325 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{217} \sqrt {\frac {1}{434} \left (-32678+10325 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )-\frac {\left (2 \left (70+97 \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 )}{7595}-\frac {\left (2 \left (70+97 \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 )}{7595} \\ & = \frac {\sqrt {1+2 x} (37+20 x)}{217 \left (2+3 x+5 x^2\right )}-\frac {1}{217} \sqrt {\frac {2}{217} \left (32678+10325 \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 {1}{217} \sqrt {\frac {2}{217} \left (32678+10325 \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 {1}{217} \sqrt {\frac {1}{434} \left (-32678+10325 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{217} \sqrt {\frac {1}{434} \left (-32678+10325 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.59 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.48 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {2 \left (\frac {217 \sqrt {1+2 x} (37+20 x)}{4+6 x+10 x^2}+\sqrt {217 \left (32678+9269 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+\sqrt {217 \left (32678-9269 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{47089} \]

[In]

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

[Out]

(2*((217*Sqrt[1 + 2*x]*(37 + 20*x))/(4 + 6*x + 10*x^2) + Sqrt[217*(32678 + (9269*I)*Sqrt[31])]*ArcTan[Sqrt[(-2
- I*Sqrt[31])/7]*Sqrt[1 + 2*x]] + Sqrt[217*(32678 - (9269*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*S
qrt[1 + 2*x]]))/47089

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.92

method	result	size
pseudoelliptic	\(\frac {\frac {40 \left (x +\frac {37}{20}\right ) \left (\sqrt {5}-\frac {5 \sqrt {7}}{2}\right ) \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {1+2 x}}{217}+\frac {5 \left (\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (4063 \sqrt {5}\, \sqrt {7}-16310\right ) \left (\ln \left (5+10 x +\sqrt {35}-\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )-\ln \left (5+10 x +\sqrt {35}+\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-403620 \left (\arctan \left (\frac {10 \sqrt {1+2 x}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )+\arctan \left (\frac {-\sqrt {20+10 \sqrt {35}}+10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )\right ) \left (\sqrt {5}-\frac {4 \sqrt {7}}{21}\right )\right ) \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )}{94178}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (2 \sqrt {5}-5 \sqrt {7}\right ) \left (5 x^{2}+3 x +2\right )}\)	\(249\)
trager	\(\frac {\left (37+20 x \right ) \sqrt {1+2 x}}{1085 x^{2}+651 x +434}-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2}+3545563\right ) \ln \left (\frac {4090016 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2}+3545563\right ) \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{4} x [phone] \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2}+3545563\right ) x [phone] \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {1+2 x}[phone] \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2}+3545563\right )[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2}+3545563\right ) x -54331101492115 \sqrt {1+2 x}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2}+3545563\right )}{868 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2} x +4871 x -37076}\right )}{47089}-\frac {2 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right ) \ln \left (-\frac {-28630112 x \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{5}[phone] \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{3} x [phone] \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {1+2 x}[phone] \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{3}-11260542660 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right ) x [phone] \sqrt {1+2 x}-25927395104 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )}{868 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2} x +60485 x +37076}\right )}{217}\)	\(448\)
derivativedivides	\(\frac {\frac {2 \left (-3244150 \sqrt {5}\, \sqrt {7}+6488300\right ) \sqrt {5}\, \sqrt {1+2 x}}{351990275 \left (2 \sqrt {5}-5 \sqrt {7}\right )}-\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (-1946490 \sqrt {5}\, \sqrt {7}[phone]\right )}{70398055 \left (2 \sqrt {5}-5 \sqrt {7}\right )}}{\frac {\sqrt {5}\, \sqrt {7}}{5}-\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}+1+2 x}+\frac {-\frac {\left (-4063 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+16310 \sqrt {2 \sqrt {35}+4}\right ) \ln \left (5+10 x +\sqrt {35}-\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{94178}-\frac {10 \left (42098 \sqrt {5}-12028 \sqrt {7}+\frac {\left (-4063 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+16310 \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {-\sqrt {20+10 \sqrt {35}}+10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )}{47089 \sqrt {-20+10 \sqrt {35}}}}{2 \sqrt {5}-5 \sqrt {7}}+\frac {\frac {2 \left (-3244150 \sqrt {5}\, \sqrt {7}+6488300\right ) \sqrt {5}\, \sqrt {1+2 x}}{351990275 \left (2 \sqrt {5}-5 \sqrt {7}\right )}+\frac {5 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (-1946490 \sqrt {5}\, \sqrt {7}[phone]\right )}{47089 \left (14950 \sqrt {5}-37375 \sqrt {7}\right )}}{\frac {\sqrt {5}\, \sqrt {7}}{5}+\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}+1+2 x}+\frac {\frac {\left (-4063 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+16310 \sqrt {2 \sqrt {35}+4}\right ) \ln \left (5+10 x +\sqrt {35}+\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{94178}+\frac {10 \left (-42098 \sqrt {5}+12028 \sqrt {7}-\frac {\left (-4063 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+16310 \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {10 \sqrt {1+2 x}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )}{47089 \sqrt {-20+10 \sqrt {35}}}}{2 \sqrt {5}-5 \sqrt {7}}\)	\(551\)
default	\(\frac {\frac {2 \left (-3244150 \sqrt {5}\, \sqrt {7}+6488300\right ) \sqrt {5}\, \sqrt {1+2 x}}{351990275 \left (2 \sqrt {5}-5 \sqrt {7}\right )}-\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (-1946490 \sqrt {5}\, \sqrt {7}[phone]\right )}{70398055 \left (2 \sqrt {5}-5 \sqrt {7}\right )}}{\frac {\sqrt {5}\, \sqrt {7}}{5}-\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}+1+2 x}+\frac {\frac {\left (4063 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}-16310 \sqrt {2 \sqrt {35}+4}\right ) \ln \left (5+10 x +\sqrt {35}-\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{94178}+\frac {10 \left (-42098 \sqrt {5}+12028 \sqrt {7}+\frac {\left (4063 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}-16310 \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {-\sqrt {20+10 \sqrt {35}}+10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )}{47089 \sqrt {-20+10 \sqrt {35}}}}{2 \sqrt {5}-5 \sqrt {7}}+\frac {\frac {2 \left (-3244150 \sqrt {5}\, \sqrt {7}+6488300\right ) \sqrt {5}\, \sqrt {1+2 x}}{351990275 \left (2 \sqrt {5}-5 \sqrt {7}\right )}+\frac {5 \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (-1946490 \sqrt {5}\, \sqrt {7}[phone]\right )}{47089 \left (14950 \sqrt {5}-37375 \sqrt {7}\right )}}{\frac {\sqrt {5}\, \sqrt {7}}{5}+\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}+1+2 x}+\frac {\frac {\left (-4063 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+16310 \sqrt {2 \sqrt {35}+4}\right ) \ln \left (5+10 x +\sqrt {35}+\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{94178}+\frac {10 \left (-42098 \sqrt {5}+12028 \sqrt {7}-\frac {\left (-4063 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+16310 \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {10 \sqrt {1+2 x}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )}{47089 \sqrt {-20+10 \sqrt {35}}}}{2 \sqrt {5}-5 \sqrt {7}}\)	\(551\)
risch	\(\frac {\left (37+20 x \right ) \sqrt {1+2 x}}{1085 x^{2}+651 x +434}-\frac {101 \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}}{13454}+\frac {132 \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}}{47089}-\frac {505 \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 )}{6727 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {264 \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}}{47089 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {388 \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}}{1519 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {101 \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}}{13454}-\frac {132 \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}}{47089}-\frac {505 \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 )}{6727 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {264 \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}}{47089 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {388 \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}}{1519 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(633\)

[In]

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

[Out]

5/94178/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(3472*(x+37/20)*(5^(1/2)-5/2*7^(1/2))*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(1+2
*x)^(1/2)+((10*5^(1/2)*7^(1/2)-20)^(1/2)*(4063*5^(1/2)*7^(1/2)-16310)*(ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10
*35^(1/2))^(1/2))-ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2)))*(2*5^(1/2)*7^(1/2)+4)^(1/2)-403620
*(arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))+arctan((-(20+10*35^(1/2))^(1/2)+10
*(1+2*x)^(1/2))/(-20+10*35^(1/2))^(1/2)))*(5^(1/2)-4/21*7^(1/2)))*(x^2+3/5*x+2/5))/(2*5^(1/2)-5*7^(1/2))/(5*x^
2+3*x+2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx=-\frac {\sqrt {217} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {37076 i \, \sqrt {31} - 130712} \log \left (\sqrt {217} \sqrt {37076 i \, \sqrt {31} - 130712} {\left (264 i \, \sqrt {31} - 3007\right )} + 22405250 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {37076 i \, \sqrt {31} - 130712} \log \left (\sqrt {217} \sqrt {37076 i \, \sqrt {31} - 130712} {\left (-264 i \, \sqrt {31} + 3007\right )} + 22405250 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-37076 i \, \sqrt {31} - 130712} \log \left (\sqrt {217} {\left (264 i \, \sqrt {31} + 3007\right )} \sqrt {-37076 i \, \sqrt {31} - 130712} + 22405250 \, \sqrt {2 \, x + 1}\right ) + \sqrt {217} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-37076 i \, \sqrt {31} - 130712} \log \left (\sqrt {217} {\left (-264 i \, \sqrt {31} - 3007\right )} \sqrt {-37076 i \, \sqrt {31} - 130712} + 22405250 \, \sqrt {2 \, x + 1}\right ) - 434 \, {\left (20 \, x + 37\right )} \sqrt {2 \, x + 1}}{94178 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \]

[In]

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

[Out]

-1/94178*(sqrt(217)*(5*x^2 + 3*x + 2)*sqrt(37076*I*sqrt(31) - 130712)*log(sqrt(217)*sqrt(37076*I*sqrt(31) - 13
0712)*(264*I*sqrt(31) - 3007) + 22405250*sqrt(2*x + 1)) - sqrt(217)*(5*x^2 + 3*x + 2)*sqrt(37076*I*sqrt(31) -
130712)*log(sqrt(217)*sqrt(37076*I*sqrt(31) - 130712)*(-264*I*sqrt(31) + 3007) + 22405250*sqrt(2*x + 1)) - sqr
t(217)*(5*x^2 + 3*x + 2)*sqrt(-37076*I*sqrt(31) - 130712)*log(sqrt(217)*(264*I*sqrt(31) + 3007)*sqrt(-37076*I*
sqrt(31) - 130712) + 22405250*sqrt(2*x + 1)) + sqrt(217)*(5*x^2 + 3*x + 2)*sqrt(-37076*I*sqrt(31) - 130712)*lo
g(sqrt(217)*(-264*I*sqrt(31) - 3007)*sqrt(-37076*I*sqrt(31) - 130712) + 22405250*sqrt(2*x + 1)) - 434*(20*x +
37)*sqrt(2*x + 1))/(5*x^2 + 3*x + 2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (187) = 374\).

Time = 0.50 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.30 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/807576350*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) + 47530*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 95060*(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)) + 1/807576350*sqrt(31)*(210*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 24
50) - 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) + 47530*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) +
95060*(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)) + 1/1615152700*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) + 47530*sqrt(31)*(7/5)^(1/4)*sqrt(
140*sqrt(35) + 2450) - 95060*(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) - 1/1615152700*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) + 47530*sqrt(31)*(7/5)^(1/4)*sqrt(140
*sqrt(35) + 2450) - 95060*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450))*log(-2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*s
qrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) + 4/217*(10*(2*x + 1)^(3/2) + 27*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3
)

Mupad [B] (verification not implemented)

Time = 9.98 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {\frac {108\,\sqrt {2\,x+1}}{1085}+\frac {8\,{\left (2\,x+1\right )}^{3/2}}{217}}{{\left (2\,x+1\right )}^2-\frac {8\,x}{5}+\frac {3}{5}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-32678-\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}\,38272{}\mathrm {i}}{2018940875\,\left (\frac {10103808}{288420125}+\frac {\sqrt {31}\,3712384{}\mathrm {i}}{288420125}\right )}+\frac {76544\,\sqrt {31}\,\sqrt {217}\,\sqrt {-32678-\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}}{62587167125\,\left (\frac {10103808}{288420125}+\frac {\sqrt {31}\,3712384{}\mathrm {i}}{288420125}\right )}\right )\,\sqrt {-32678-\sqrt {31}\,9269{}\mathrm {i}}\,2{}\mathrm {i}}{47089}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-32678+\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}\,38272{}\mathrm {i}}{2018940875\,\left (-\frac {10103808}{288420125}+\frac {\sqrt {31}\,3712384{}\mathrm {i}}{288420125}\right )}-\frac {76544\,\sqrt {31}\,\sqrt {217}\,\sqrt {-32678+\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}}{62587167125\,\left (-\frac {10103808}{288420125}+\frac {\sqrt {31}\,3712384{}\mathrm {i}}{288420125}\right )}\right )\,\sqrt {-32678+\sqrt {31}\,9269{}\mathrm {i}}\,2{}\mathrm {i}}{47089} \]

[In]

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

[Out]

((108*(2*x + 1)^(1/2))/1085 + (8*(2*x + 1)^(3/2))/217)/((2*x + 1)^2 - (8*x)/5 + 3/5) + (217^(1/2)*atan((217^(1
/2)*(- 31^(1/2)*9269i - 32678)^(1/2)*(2*x + 1)^(1/2)*38272i)/(2018940875*((31^(1/2)*3712384i)/288420125 + 1010
3808/288420125)) + (76544*31^(1/2)*217^(1/2)*(- 31^(1/2)*9269i - 32678)^(1/2)*(2*x + 1)^(1/2))/(62587167125*((
31^(1/2)*3712384i)/288420125 + 10103808/288420125)))*(- 31^(1/2)*9269i - 32678)^(1/2)*2i)/47089 - (217^(1/2)*a
tan((217^(1/2)*(31^(1/2)*9269i - 32678)^(1/2)*(2*x + 1)^(1/2)*38272i)/(2018940875*((31^(1/2)*3712384i)/2884201
25 - 10103808/288420125)) - (76544*31^(1/2)*217^(1/2)*(31^(1/2)*9269i - 32678)^(1/2)*(2*x + 1)^(1/2))/(6258716
7125*((31^(1/2)*3712384i)/288420125 - 10103808/288420125)))*(31^(1/2)*9269i - 32678)^(1/2)*2i)/47089