\(\int \frac {(1+2 x)^{5/2}}{2+3 x+5 x^2} \, dx\) [2310]

Optimal result

Integrand size = 22, antiderivative size = 266 \[ \int \frac {(1+2 x)^{5/2}}{2+3 x+5 x^2} \, dx=\frac {16}{25} \sqrt {1+2 x}+\frac {4}{15} (1+2 x)^{3/2}+\frac {1}{25} \sqrt {\frac {2}{155} \left (7162+1225 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{25} \sqrt {\frac {2}{155} \left (7162+1225 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{25} \sqrt {\frac {1}{310} \left (-7162+1225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )-\frac {1}{25} \sqrt {\frac {1}{310} \left (-7162+1225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \]

[Out]

4/15*(1+2*x)^(3/2)+16/25*(1+2*x)^(1/2)+1/7750*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-22202
20+379750*35^(1/2))^(1/2)-1/7750*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-2220220+379750*35^
(1/2))^(1/2)+1/3875*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(2220220+379750
*35^(1/2))^(1/2)-1/3875*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(2220220+379
750*35^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 266, 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 = {717, 838, 840, 1183, 648, 632, 210, 642} \[ \int \frac {(1+2 x)^{5/2}}{2+3 x+5 x^2} \, dx=\frac {1}{25} \sqrt {\frac {2}{155} \left (7162+1225 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{25} \sqrt {\frac {2}{155} \left (7162+1225 \sqrt {35}\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {4}{15} (2 x+1)^{3/2}+\frac {16}{25} \sqrt {2 x+1}+\frac {1}{25} \sqrt {\frac {1}{310} \left (1225 \sqrt {35}-7162\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{25} \sqrt {\frac {1}{310} \left (1225 \sqrt {35}-7162\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right ) \]

[In]

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

[Out]

(16*Sqrt[1 + 2*x])/25 + (4*(1 + 2*x)^(3/2))/15 + (Sqrt[(2*(7162 + 1225*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sq
rt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/25 - (Sqrt[(2*(7162 + 1225*Sqrt[35]))/155]*ArcTan[(Sqr
t[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/25 + (Sqrt[(-7162 + 1225*Sqrt[35])/310]*Lo
g[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/25 - (Sqrt[(-7162 + 1225*Sqrt[35])/310]*Log
[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/25

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 717

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m - 1)/(c*(
m - 1))), x] + Dist[1/c, Int[(d + e*x)^(m - 2)*(Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, x]/(a + b*x + c*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] && NeQ[2*c*d - b*
e, 0] && GtQ[m, 1]

Rule 838

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*
((d + e*x)^m/(c*m)), x] + Dist[1/c, Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/
(a + b*x + c*x^2)), 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] && FractionQ[m] && GtQ[m, 0]

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 {4}{15} (1+2 x)^{3/2}+\frac {1}{5} \int \frac {\sqrt {1+2 x} (-3+8 x)}{2+3 x+5 x^2} \, dx \\ & = \frac {16}{25} \sqrt {1+2 x}+\frac {4}{15} (1+2 x)^{3/2}+\frac {1}{25} \int \frac {-47-38 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx \\ & = \frac {16}{25} \sqrt {1+2 x}+\frac {4}{15} (1+2 x)^{3/2}+\frac {2}{25} \text {Subst}\left (\int \frac {-56-38 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right ) \\ & = \frac {16}{25} \sqrt {1+2 x}+\frac {4}{15} (1+2 x)^{3/2}+\frac {\text {Subst}\left (\int \frac {-56 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-56+38 \sqrt {\frac {7}{5}}\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 )}{25 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {-56 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-56+38 \sqrt {\frac {7}{5}}\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 )}{25 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = \frac {16}{25} \sqrt {1+2 x}+\frac {4}{15} (1+2 x)^{3/2}-\frac {1}{125} \sqrt {921+152 \sqrt {35}} \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}{125} \sqrt {921+152 \sqrt {35}} \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}{25} \sqrt {\frac {1}{310} \left (-7162+1225 \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}{25} \sqrt {\frac {1}{310} \left (-7162+1225 \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 {16}{25} \sqrt {1+2 x}+\frac {4}{15} (1+2 x)^{3/2}+\frac {1}{25} \sqrt {\frac {1}{310} \left (-7162+1225 \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}{25} \sqrt {\frac {1}{310} \left (-7162+1225 \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}{125} \left (2 \sqrt {921+152 \sqrt {35}}\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 )+\frac {1}{125} \left (2 \sqrt {921+152 \sqrt {35}}\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 ) \\ & = \frac {16}{25} \sqrt {1+2 x}+\frac {4}{15} (1+2 x)^{3/2}+\frac {1}{25} \sqrt {\frac {2}{155} \left (7162+1225 \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}{25} \sqrt {\frac {2}{155} \left (7162+1225 \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}{25} \sqrt {\frac {1}{310} \left (-7162+1225 \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}{25} \sqrt {\frac {1}{310} \left (-7162+1225 \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 = 0.44 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.45 \[ \int \frac {(1+2 x)^{5/2}}{2+3 x+5 x^2} \, dx=\frac {620 \sqrt {1+2 x} (17+10 x)-6 \sqrt {155 \left (7162+199 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-6 \sqrt {155 \left (7162-199 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{11625} \]

[In]

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

[Out]

(620*Sqrt[1 + 2*x]*(17 + 10*x) - 6*Sqrt[155*(7162 + (199*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1
+ 2*x]] - 6*Sqrt[155*(7162 - (199*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]])/11625

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(-\frac {8 \left (-\frac {89 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (\sqrt {5}-\frac {135 \sqrt {7}}{178}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{1240}+\frac {89 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (\sqrt {5}-\frac {135 \sqrt {7}}{178}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{1240}-\frac {5 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (x +\frac {17}{10}\right ) \sqrt {1+2 x}}{3}+\left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right ) \left (\sqrt {5}\, \sqrt {7}+\frac {19}{4}\right )\right )}{25 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(276\)
derivativedivides	\(\frac {4 \left (1+2 x \right )^{\frac {3}{2}}}{15}+\frac {16 \sqrt {1+2 x}}{25}-\frac {\left (135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-178 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{7750}-\frac {2 \left (248 \sqrt {5}\, \sqrt {7}+\frac {\left (135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-178 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-178 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{7750}+\frac {2 \left (-248 \sqrt {5}\, \sqrt {7}-\frac {\left (135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-178 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(398\)
default	\(\frac {4 \left (1+2 x \right )^{\frac {3}{2}}}{15}+\frac {16 \sqrt {1+2 x}}{25}-\frac {\left (135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-178 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{7750}-\frac {2 \left (248 \sqrt {5}\, \sqrt {7}+\frac {\left (135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-178 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-178 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{7750}+\frac {2 \left (-248 \sqrt {5}\, \sqrt {7}-\frac {\left (135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-178 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(398\)
trager	\(\left (\frac {8 x}{15}+\frac {68}{75}\right ) \sqrt {1+2 x}+\frac {\operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right ) \ln \left (\frac {1537600 x \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{5}[phone] \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{3} x -15792640 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{3}-16193625 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {1+2 x}[phone] \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right ) x -700258712 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )-1711544275 \sqrt {1+2 x}}{155 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{2} x +7759 x +796}\right )}{25}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{2}+2220220\right ) \ln \left (-\frac {307520 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{2}+2220220\right ) \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{4} x [phone] \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{2}+2220220\right ) x +3158528 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{2}+2220220\right )-502002375 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {1+2 x}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{2}+2220220\right ) x [phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{2}+2220220\right )[phone] \sqrt {1+2 x}}{155 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}[phone]\right )^{2} x +6565 x -796}\right )}{3875}\)	\(435\)
risch	\(\frac {4 \left (10 x +17\right ) \sqrt {1+2 x}}{75}-\frac {27 \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}}{1550}+\frac {89 \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}}{3875}-\frac {27 \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}}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {178 \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 )}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {16 \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}}{25 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {27 \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}}{1550}-\frac {89 \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}}{3875}-\frac {27 \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}}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {178 \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 )}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {16 \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}}{25 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(621\)

[In]

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

[Out]

-8/25/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(-89/1240*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(5^(1/
2)-135/178*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)+89/1240*(10*5
^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(5^(1/2)-135/178*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)-5/3*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(x+17/10)*(1+2*x)^(1/2)+(arcta
n((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^(1/2)*7^(1/2)+19/4))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.73 \[ \int \frac {(1+2 x)^{5/2}}{2+3 x+5 x^2} \, dx=\frac {1}{7750} \, \sqrt {155} \sqrt {796 i \, \sqrt {31} - 28648} \log \left (\sqrt {155} \sqrt {796 i \, \sqrt {31} - 28648} {\left (27 i \, \sqrt {31} - 124\right )} + 379750 \, \sqrt {2 \, x + 1}\right ) - \frac {1}{7750} \, \sqrt {155} \sqrt {796 i \, \sqrt {31} - 28648} \log \left (\sqrt {155} \sqrt {796 i \, \sqrt {31} - 28648} {\left (-27 i \, \sqrt {31} + 124\right )} + 379750 \, \sqrt {2 \, x + 1}\right ) - \frac {1}{7750} \, \sqrt {155} \sqrt {-796 i \, \sqrt {31} - 28648} \log \left (\sqrt {155} {\left (27 i \, \sqrt {31} + 124\right )} \sqrt {-796 i \, \sqrt {31} - 28648} + 379750 \, \sqrt {2 \, x + 1}\right ) + \frac {1}{7750} \, \sqrt {155} \sqrt {-796 i \, \sqrt {31} - 28648} \log \left (\sqrt {155} {\left (-27 i \, \sqrt {31} - 124\right )} \sqrt {-796 i \, \sqrt {31} - 28648} + 379750 \, \sqrt {2 \, x + 1}\right ) + \frac {4}{75} \, {\left (10 \, x + 17\right )} \sqrt {2 \, x + 1} \]

[In]

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

[Out]

1/7750*sqrt(155)*sqrt(796*I*sqrt(31) - 28648)*log(sqrt(155)*sqrt(796*I*sqrt(31) - 28648)*(27*I*sqrt(31) - 124)
+ 379750*sqrt(2*x + 1)) - 1/7750*sqrt(155)*sqrt(796*I*sqrt(31) - 28648)*log(sqrt(155)*sqrt(796*I*sqrt(31) - 2
8648)*(-27*I*sqrt(31) + 124) + 379750*sqrt(2*x + 1)) - 1/7750*sqrt(155)*sqrt(-796*I*sqrt(31) - 28648)*log(sqrt
(155)*(27*I*sqrt(31) + 124)*sqrt(-796*I*sqrt(31) - 28648) + 379750*sqrt(2*x + 1)) + 1/7750*sqrt(155)*sqrt(-796
*I*sqrt(31) - 28648)*log(sqrt(155)*(-27*I*sqrt(31) - 124)*sqrt(-796*I*sqrt(31) - 28648) + 379750*sqrt(2*x + 1)
) + 4/75*(10*x + 17)*sqrt(2*x + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate((2*x + 1)^(5/2)/(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. 605 vs. \(2 (179) = 358\).

Time = 0.71 (sec) , antiderivative size = 605, normalized size of antiderivative = 2.27 \[ \int \frac {(1+2 x)^{5/2}}{2+3 x+5 x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/930387500*sqrt(31)*(3990*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 19*sqrt(31)*(7
/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 38*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 7980*(7/5)^(3/4)*sqrt(14
0*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 137200*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 274400*(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)) - 1/930387500*sqrt(31)*(3990*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqr
t(35) + 2450) - 19*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 38*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3
/2) + 7980*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 137200*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqr
t(35) + 2450) + 274400*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(-5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*s
qrt(35) + 1/2) - sqrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) - 1/1860775000*sqrt(31)*(19*sqrt(31)*(7/5)^(3/4)*(
140*sqrt(35) + 2450)^(3/2) + 3990*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 7980*(7/5
)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 38*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 137200*sq
rt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 274400*(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) + 1/1860775000*sqrt(31)*(19*sqrt(31)*(7/5)^(3
/4)*(140*sqrt(35) + 2450)^(3/2) + 3990*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 7980
*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 38*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 1372
00*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 274400*(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) + 4/15*(2*x + 1)^(3/2) + 16/25*sqrt(2*x
+ 1)

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.72 \[ \int \frac {(1+2 x)^{5/2}}{2+3 x+5 x^2} \, dx=\frac {16\,\sqrt {2\,x+1}}{25}+\frac {4\,{\left (2\,x+1\right )}^{3/2}}{15}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}\,25472{}\mathrm {i}}{48828125\,\left (\frac {4814208}{9765625}+\frac {\sqrt {31}\,713216{}\mathrm {i}}{9765625}\right )}+\frac {50944\,\sqrt {31}\,\sqrt {155}\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}}{1513671875\,\left (\frac {4814208}{9765625}+\frac {\sqrt {31}\,713216{}\mathrm {i}}{9765625}\right )}\right )\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,2{}\mathrm {i}}{3875}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}\,25472{}\mathrm {i}}{48828125\,\left (-\frac {4814208}{9765625}+\frac {\sqrt {31}\,713216{}\mathrm {i}}{9765625}\right )}-\frac {50944\,\sqrt {31}\,\sqrt {155}\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}}{1513671875\,\left (-\frac {4814208}{9765625}+\frac {\sqrt {31}\,713216{}\mathrm {i}}{9765625}\right )}\right )\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,2{}\mathrm {i}}{3875} \]

[In]

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

[Out]

(16*(2*x + 1)^(1/2))/25 + (4*(2*x + 1)^(3/2))/15 - (155^(1/2)*atan((155^(1/2)*(- 31^(1/2)*199i - 7162)^(1/2)*(
2*x + 1)^(1/2)*25472i)/(48828125*((31^(1/2)*713216i)/9765625 + 4814208/9765625)) + (50944*31^(1/2)*155^(1/2)*(
- 31^(1/2)*199i - 7162)^(1/2)*(2*x + 1)^(1/2))/(1513671875*((31^(1/2)*713216i)/9765625 + 4814208/9765625)))*(-
31^(1/2)*199i - 7162)^(1/2)*2i)/3875 + (155^(1/2)*atan((155^(1/2)*(31^(1/2)*199i - 7162)^(1/2)*(2*x + 1)^(1/2
)*25472i)/(48828125*((31^(1/2)*713216i)/9765625 - 4814208/9765625)) - (50944*31^(1/2)*155^(1/2)*(31^(1/2)*199i
- 7162)^(1/2)*(2*x + 1)^(1/2))/(1513671875*((31^(1/2)*713216i)/9765625 - 4814208/9765625)))*(31^(1/2)*199i -
7162)^(1/2)*2i)/3875