\(\int \frac {\sqrt {1+2 x}}{2+3 x+5 x^2} \, dx\) [2312]

Optimal result

Integrand size = 22, antiderivative size = 222 \[ \int \frac {\sqrt {1+2 x}}{2+3 x+5 x^2} \, dx=-\sqrt {\frac {2}{5 \left (-2+\sqrt {35}\right )}} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\sqrt {\frac {2}{5 \left (-2+\sqrt {35}\right )}} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {\log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}}-\frac {\log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}} \]

[Out]

-1/5*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*10^(1/2)/(-2+35^(1/2))^(1/2)+1
/5*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*10^(1/2)/(-2+35^(1/2))^(1/2)+ln(5
+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(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))/(20+10*35^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {713, 1141, 1175, 632, 210, 1178, 642} \[ \int \frac {\sqrt {1+2 x}}{2+3 x+5 x^2} \, dx=-\sqrt {\frac {2}{5 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\sqrt {\frac {2}{5 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {\log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}}-\frac {\log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}} \]

[In]

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

[Out]

-(Sqrt[2/(5*(-2 + Sqrt[35]))]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]]) +
Sqrt[2/(5*(-2 + Sqrt[35]))]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]] + L
og[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)]/Sqrt[10*(2 + Sqrt[35])] - Log[Sqrt[35] + Sq
rt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)]/Sqrt[10*(2 + Sqrt[35])]

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 713

Int[Sqrt[(d_.) + (e_.)*(x_)]/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[2*e, Subst[Int[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}, 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]

Rule 1141

Int[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, Dist[1/2, Int[(q + x^2)/(
a + b*x^2 + c*x^4), x], x] - Dist[1/2, Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && Lt
Q[b^2 - 4*a*c, 0] && PosQ[a*c]

Rule 1175

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

Rule 1178

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

Rubi steps \begin{align*} \text {integral}& = 4 \text {Subst}\left (\int \frac {x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {\sqrt {\frac {7}{5}}-x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\right )+2 \text {Subst}\left (\int \frac {\sqrt {\frac {7}{5}}+x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right ) \\ & = \frac {1}{5} \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 )+\frac {1}{5} \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 )+\frac {\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 )}{\sqrt {10 \left (2+\sqrt {35}\right )}}+\frac {\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 )}{\sqrt {10 \left (2+\sqrt {35}\right )}} \\ & = \frac {\log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}}-\frac {\log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}}-\frac {2}{5} \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 )-\frac {2}{5} \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 ) \\ & = -\sqrt {\frac {2}{5 \left (-2+\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 )+\sqrt {\frac {2}{5 \left (-2+\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 {\log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}}-\frac {\log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.45 \[ \int \frac {\sqrt {1+2 x}}{2+3 x+5 x^2} \, dx=\frac {2 \left (\sqrt {2-i \sqrt {31}} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+\sqrt {2+i \sqrt {31}} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{\sqrt {155}} \]

[In]

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

[Out]

(2*(Sqrt[2 - I*Sqrt[31]]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] + Sqrt[2 + I*Sqrt[31]]*ArcTan[Sqrt[(I
/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]]))/Sqrt[155]

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.09

method	result	size
pseudoelliptic	\(-\frac {2 \left (\frac {\left (\sqrt {5}-\frac {5 \sqrt {7}}{2}\right ) \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{310}-\frac {\left (\sqrt {5}-\frac {5 \sqrt {7}}{2}\right ) \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{310}+\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right )}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(241\)
derivativedivides	\(-\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (2 \sqrt {5}-5 \sqrt {7}\right ) \left (\frac {\ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{10}+\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{5 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{31}+\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (2 \sqrt {5}-5 \sqrt {7}\right ) \left (\frac {\ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{10}-\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{5 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{31}\)	\(270\)
default	\(-\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (2 \sqrt {5}-5 \sqrt {7}\right ) \left (\frac {\ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{10}+\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{5 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{31}+\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (2 \sqrt {5}-5 \sqrt {7}\right ) \left (\frac {\ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{10}-\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{5 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{31}\)	\(270\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{2}+620\right ) \ln \left (\frac {57660 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{2}+620\right ) \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{4} x -155 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{2}+620\right ) x -2976 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{2}+620\right )-14415 \sqrt {1+2 x}\, \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{2}-55 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{2}+620\right ) x -88 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{2}+620\right )-4495 \sqrt {1+2 x}}{155 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{2} x +5 x +4}\right )}{155}-\operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right ) \ln \left (\frac {288300 x \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{5}+15655 x \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{3}+14880 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{3}+465 \sqrt {1+2 x}\, \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{2}-63 x \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )-56 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )-133 \sqrt {1+2 x}}{155 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+124 \textit {\_Z}^{2}+7\right )^{2} x -x -4}\right )\)	\(421\)

[In]

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

[Out]

-2*(1/310*(5^(1/2)-5/2*7^(1/2))*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(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)-1/310*(5^(1/2)-5/2*7^(1/2))*(10*5^(1/2)*7^(1/2)-20)^(
1/2)*(2*5^(1/2)*7^(1/2)+4)^(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)+
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))-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)))/(10*5^(1/2)*7^(1/2)-20)^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {1+2 x}}{2+3 x+5 x^2} \, dx=\frac {1}{310} \, \sqrt {155} \sqrt {4 i \, \sqrt {31} - 8} \log \left (i \, \sqrt {155} \sqrt {31} \sqrt {4 i \, \sqrt {31} - 8} + 310 \, \sqrt {2 \, x + 1}\right ) - \frac {1}{310} \, \sqrt {155} \sqrt {4 i \, \sqrt {31} - 8} \log \left (-i \, \sqrt {155} \sqrt {31} \sqrt {4 i \, \sqrt {31} - 8} + 310 \, \sqrt {2 \, x + 1}\right ) - \frac {1}{310} \, \sqrt {155} \sqrt {-4 i \, \sqrt {31} - 8} \log \left (i \, \sqrt {155} \sqrt {31} \sqrt {-4 i \, \sqrt {31} - 8} + 310 \, \sqrt {2 \, x + 1}\right ) + \frac {1}{310} \, \sqrt {155} \sqrt {-4 i \, \sqrt {31} - 8} \log \left (-i \, \sqrt {155} \sqrt {31} \sqrt {-4 i \, \sqrt {31} - 8} + 310 \, \sqrt {2 \, x + 1}\right ) \]

[In]

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

[Out]

1/310*sqrt(155)*sqrt(4*I*sqrt(31) - 8)*log(I*sqrt(155)*sqrt(31)*sqrt(4*I*sqrt(31) - 8) + 310*sqrt(2*x + 1)) -
1/310*sqrt(155)*sqrt(4*I*sqrt(31) - 8)*log(-I*sqrt(155)*sqrt(31)*sqrt(4*I*sqrt(31) - 8) + 310*sqrt(2*x + 1)) -
1/310*sqrt(155)*sqrt(-4*I*sqrt(31) - 8)*log(I*sqrt(155)*sqrt(31)*sqrt(-4*I*sqrt(31) - 8) + 310*sqrt(2*x + 1))
+ 1/310*sqrt(155)*sqrt(-4*I*sqrt(31) - 8)*log(-I*sqrt(155)*sqrt(31)*sqrt(-4*I*sqrt(31) - 8) + 310*sqrt(2*x +
1))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (162) = 324\).

Time = 0.70 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.08 \[ \int \frac {\sqrt {1+2 x}}{2+3 x+5 x^2} \, dx=\frac {1}{37215500} \, \sqrt {31} {\left (210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )}\right )} \arctan \left (\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) + \frac {1}{37215500} \, \sqrt {31} {\left (210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )}\right )} \arctan \left (-\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} - \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) + \frac {1}{74431000} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}}\right )} \log \left (2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) - \frac {1}{74431000} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}}\right )} \log \left (-2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) \]

[In]

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

[Out]

1/37215500*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(3
5) + 2450)*(2*sqrt(35) - 35))*arctan(5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*sqrt(35) + 1/2) + sqrt(2*x + 1))/s
qrt(-1/35*sqrt(35) + 1/2)) + 1/37215500*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) + 42
0*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35))*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)) + 1/74431000*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))*log(2*(7/5)^(1/4)*sqr
t(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) - 1/74431000*sqrt(31)*(sqrt(31)*(7/5)^(3/4)*(140*s
qrt(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))*log(-2*(7/5)^(1/4)
*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {1+2 x}}{2+3 x+5 x^2} \, dx=-\frac {2\,\sqrt {155}\,\mathrm {atanh}\left (\sqrt {155}\,\sqrt {-2-\sqrt {31}\,1{}\mathrm {i}}\,\left (\frac {2\,\left (\frac {2}{155}+\frac {\sqrt {31}\,1{}\mathrm {i}}{155}\right )\,\sqrt {2\,x+1}}{7}+\frac {27\,\sqrt {2\,x+1}}{1085}\right )\right )\,\sqrt {-2-\sqrt {31}\,1{}\mathrm {i}}}{155}-\frac {2\,\sqrt {155}\,\mathrm {atanh}\left (-\sqrt {155}\,\sqrt {-2+\sqrt {31}\,1{}\mathrm {i}}\,\left (\frac {2\,\left (-\frac {2}{155}+\frac {\sqrt {31}\,1{}\mathrm {i}}{155}\right )\,\sqrt {2\,x+1}}{7}-\frac {27\,\sqrt {2\,x+1}}{1085}\right )\right )\,\sqrt {-2+\sqrt {31}\,1{}\mathrm {i}}}{155} \]

[In]

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

[Out]

- (2*155^(1/2)*atanh(155^(1/2)*(- 31^(1/2)*1i - 2)^(1/2)*((2*((31^(1/2)*1i)/155 + 2/155)*(2*x + 1)^(1/2))/7 +
(27*(2*x + 1)^(1/2))/1085))*(- 31^(1/2)*1i - 2)^(1/2))/155 - (2*155^(1/2)*atanh(-155^(1/2)*(31^(1/2)*1i - 2)^(
1/2)*((2*((31^(1/2)*1i)/155 - 2/155)*(2*x + 1)^(1/2))/7 - (27*(2*x + 1)^(1/2))/1085))*(31^(1/2)*1i - 2)^(1/2))
/155