\(\int \frac {(1+2 x)^{3/2}}{(2+3 x+5 x^2)^2} \, dx\) [2318]

Optimal result

Integrand size = 22, antiderivative size = 270 \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=-\frac {(5-4 x) \sqrt {1+2 x}}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{31} \sqrt {\frac {2}{155} \left (218+47 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{31} \sqrt {\frac {2}{155} \left (218+47 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{31} \sqrt {\frac {1}{310} \left (-218+47 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{31} \sqrt {\frac {1}{310} \left (-218+47 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \]

[Out]

-1/31*(5-4*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)-1/9610*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-67
580+14570*35^(1/2))^(1/2)+1/9610*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-67580+14570*35^(1/
2))^(1/2)-1/4805*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(67580+14570*35^(1
/2))^(1/2)+1/4805*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(67580+14570*35^(1
/2))^(1/2)

Rubi [A] (verified)

Time = 0.26 (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 = {752, 840, 1183, 648, 632, 210, 642} \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=-\frac {1}{31} \sqrt {\frac {2}{155} \left (218+47 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {1}{31} \sqrt {\frac {2}{155} \left (218+47 \sqrt {35}\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {\sqrt {2 x+1} (5-4 x)}{31 \left (5 x^2+3 x+2\right )}-\frac {1}{31} \sqrt {\frac {1}{310} \left (47 \sqrt {35}-218\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{31} \sqrt {\frac {1}{310} \left (47 \sqrt {35}-218\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right ) \]

[In]

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

[Out]

-1/31*((5 - 4*x)*Sqrt[1 + 2*x])/(2 + 3*x + 5*x^2) - (Sqrt[(2*(218 + 47*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sq
rt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/31 + (Sqrt[(2*(218 + 47*Sqrt[35]))/155]*ArcTan[(Sqrt[1
0*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/31 - (Sqrt[(-218 + 47*Sqrt[35])/310]*Log[Sqrt
[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/31 + (Sqrt[(-218 + 47*Sqrt[35])/310]*Log[Sqrt[35]
+ Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/31

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 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}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{31} \int \frac {9+4 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{31 \left (2+3 x+5 x^2\right )}+\frac {2}{31} \text {Subst}\left (\int \frac {14+4 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right ) \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{31 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {14 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (14-4 \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 )}{31 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {14 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (14-4 \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 )}{31 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{155} \left (2+\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 )+\frac {1}{155} \left (2+\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 )-\frac {1}{31} \sqrt {\frac {1}{310} \left (-218+47 \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}{31} \sqrt {\frac {1}{310} \left (-218+47 \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 {(5-4 x) \sqrt {1+2 x}}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{31} \sqrt {\frac {1}{310} \left (-218+47 \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}{31} \sqrt {\frac {1}{310} \left (-218+47 \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}{155} \left (2 \left (2+\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 )-\frac {1}{155} \left (2 \left (2+\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 ) \\ & = -\frac {(5-4 x) \sqrt {1+2 x}}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{31} \sqrt {\frac {2}{155} \left (218+47 \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}{31} \sqrt {\frac {2}{155} \left (218+47 \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}{31} \sqrt {\frac {1}{310} \left (-218+47 \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}{31} \sqrt {\frac {1}{310} \left (-218+47 \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 = 0.84 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.48 \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {2 \left (\frac {155 \sqrt {1+2 x} (-5+4 x)}{4+6 x+10 x^2}+\sqrt {155 \left (218+31 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+\sqrt {155 \left (218-31 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{4805} \]

[In]

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

[Out]

(2*((155*Sqrt[1 + 2*x]*(-5 + 4*x))/(4 + 6*x + 10*x^2) + Sqrt[155*(218 + (31*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*
Sqrt[31])/7]*Sqrt[1 + 2*x]] + Sqrt[155*(218 - (31*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2
*x]]))/4805

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.16

method	result	size
pseudoelliptic	\(-\frac {3100 \left (\frac {39 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \left (\sqrt {5}-\frac {20 \sqrt {7}}{39}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{620}-\frac {39 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \left (\sqrt {5}-\frac {20 \sqrt {7}}{39}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{620}-\frac {2 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (-\frac {5}{4}+x \right ) \sqrt {1+2 x}}{5}+\left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right ) \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \left (\sqrt {5}\, \sqrt {7}+2\right )\right )}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (48050 x^{2}+28830 x +19220\right )}\)	\(312\)
derivativedivides	\(\frac {\frac {8 \left (1+2 x \right )^{\frac {3}{2}}}{155}-\frac {28 \sqrt {1+2 x}}{155}}{\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}}+\frac {\left (-39 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+20 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{9610}+\frac {2 \left (62 \sqrt {5}\, \sqrt {7}+\frac {\left (-39 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+20 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (39 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-20 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{9610}+\frac {2 \left (62 \sqrt {5}\, \sqrt {7}-\frac {\left (39 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-20 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(415\)
default	\(\frac {\frac {8 \left (1+2 x \right )^{\frac {3}{2}}}{155}-\frac {28 \sqrt {1+2 x}}{155}}{\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}}+\frac {\left (-39 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+20 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{9610}+\frac {2 \left (62 \sqrt {5}\, \sqrt {7}+\frac {\left (-39 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+20 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (39 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-20 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{9610}+\frac {2 \left (62 \sqrt {5}\, \sqrt {7}-\frac {\left (39 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-20 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(415\)
trager	\(\frac {\left (-5+4 x \right ) \sqrt {1+2 x}}{155 x^{2}+93 x +62}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{2}+16895\right ) \ln \left (\frac {12147040 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{2}+16895\right ) \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{4} x [phone] \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{2}+16895\right ) x +4858816 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{2}+16895\right )-84010620 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{2} \sqrt {1+2 x}+4334340 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{2}+16895\right ) x +2471072 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{2}+16895\right )[phone] \sqrt {1+2 x}}{620 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{2} x +125 x -124}\right )}{4805}+\frac {2 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right ) \ln \left (\frac {60735200 x \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{5}[phone] x \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{3}-24294080 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{3}+2710020 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{2} \sqrt {1+2 x}+19068 x \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )-4728864 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )+7231091 \sqrt {1+2 x}}{620 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+15463\right )^{2} x +311 x +124}\right )}{31}\)	\(447\)
risch	\(\frac {\left (-5+4 x \right ) \sqrt {1+2 x}}{155 x^{2}+93 x +62}-\frac {39 \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}}{9610}+\frac {2 \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}}{961}-\frac {39 \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 )}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {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 ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {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 ) \sqrt {5}\, \sqrt {7}}{31 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {39 \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}}{9610}-\frac {2 \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}}{961}-\frac {39 \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 )}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {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 ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {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 ) \sqrt {5}\, \sqrt {7}}{31 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(633\)

[In]

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

[Out]

-3100*(39/620*(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)*(5^(1/2)-20/39*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)-39/620*(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)*(5^(1/2)-20/39*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)-2/5*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(-5/4+x)*(1+2*x)^(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))-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)))*(x^2+3/5*x+2/5)*(5^(1/2)*7^(1/2)+2))/(10*5^(
1/2)*7^(1/2)-20)^(1/2)/(48050*x^2+28830*x+19220)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.38 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.91 \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=-\frac {\sqrt {155} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {124 i \, \sqrt {31} - 872} \log \left (\sqrt {155} \sqrt {124 i \, \sqrt {31} - 872} {\left (4 i \, \sqrt {31} - 31\right )} + 14570 \, \sqrt {2 \, x + 1}\right ) - \sqrt {155} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {124 i \, \sqrt {31} - 872} \log \left (\sqrt {155} \sqrt {124 i \, \sqrt {31} - 872} {\left (-4 i \, \sqrt {31} + 31\right )} + 14570 \, \sqrt {2 \, x + 1}\right ) - \sqrt {155} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-124 i \, \sqrt {31} - 872} \log \left (\sqrt {155} {\left (4 i \, \sqrt {31} + 31\right )} \sqrt {-124 i \, \sqrt {31} - 872} + 14570 \, \sqrt {2 \, x + 1}\right ) + \sqrt {155} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-124 i \, \sqrt {31} - 872} \log \left (\sqrt {155} {\left (-4 i \, \sqrt {31} - 31\right )} \sqrt {-124 i \, \sqrt {31} - 872} + 14570 \, \sqrt {2 \, x + 1}\right ) - 310 \, {\left (4 \, x - 5\right )} \sqrt {2 \, x + 1}}{9610 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \]

[In]

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

[Out]

-1/9610*(sqrt(155)*(5*x^2 + 3*x + 2)*sqrt(124*I*sqrt(31) - 872)*log(sqrt(155)*sqrt(124*I*sqrt(31) - 872)*(4*I*
sqrt(31) - 31) + 14570*sqrt(2*x + 1)) - sqrt(155)*(5*x^2 + 3*x + 2)*sqrt(124*I*sqrt(31) - 872)*log(sqrt(155)*s
qrt(124*I*sqrt(31) - 872)*(-4*I*sqrt(31) + 31) + 14570*sqrt(2*x + 1)) - sqrt(155)*(5*x^2 + 3*x + 2)*sqrt(-124*
I*sqrt(31) - 872)*log(sqrt(155)*(4*I*sqrt(31) + 31)*sqrt(-124*I*sqrt(31) - 872) + 14570*sqrt(2*x + 1)) + sqrt(
155)*(5*x^2 + 3*x + 2)*sqrt(-124*I*sqrt(31) - 872)*log(sqrt(155)*(-4*I*sqrt(31) - 31)*sqrt(-124*I*sqrt(31) - 8
72) + 14570*sqrt(2*x + 1)) - 310*(4*x - 5)*sqrt(2*x + 1))/(5*x^2 + 3*x + 2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate((2*x + 1)^(3/2)/(5*x^2 + 3*x + 2)^2, 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 = 0.67 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.30 \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/576840250*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) + 17150*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 34300*(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)) + 1/576840250*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) + 17150*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) +
34300*(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)) + 1/1153680500*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) + 17150*sqrt(31)*(7/5)^(1/4)*sqrt(
140*sqrt(35) + 2450) - 34300*(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) - 1/1153680500*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) + 17150*sqrt(31)*(7/5)^(1/4)*sqrt(140
*sqrt(35) + 2450) - 34300*(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) + 4/31*(2*(2*x + 1)^(3/2) - 7*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)

Mupad [B] (verification not implemented)

Time = 9.86 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.77 \[ \int \frac {(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=-\frac {\frac {28\,\sqrt {2\,x+1}}{155}-\frac {8\,{\left (2\,x+1\right )}^{3/2}}{155}}{{\left (2\,x+1\right )}^2-\frac {8\,x}{5}+\frac {3}{5}}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{3003125\,\left (\frac {3584}{600625}+\frac {\sqrt {31}\,896{}\mathrm {i}}{600625}\right )}+\frac {256\,\sqrt {31}\,\sqrt {155}\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}}{93096875\,\left (\frac {3584}{600625}+\frac {\sqrt {31}\,896{}\mathrm {i}}{600625}\right )}\right )\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,2{}\mathrm {i}}{4805}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{3003125\,\left (-\frac {3584}{600625}+\frac {\sqrt {31}\,896{}\mathrm {i}}{600625}\right )}-\frac {256\,\sqrt {31}\,\sqrt {155}\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}}{93096875\,\left (-\frac {3584}{600625}+\frac {\sqrt {31}\,896{}\mathrm {i}}{600625}\right )}\right )\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,2{}\mathrm {i}}{4805} \]

[In]

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

[Out]

(155^(1/2)*atan((155^(1/2)*(- 31^(1/2)*31i - 218)^(1/2)*(2*x + 1)^(1/2)*128i)/(3003125*((31^(1/2)*896i)/600625
+ 3584/600625)) + (256*31^(1/2)*155^(1/2)*(- 31^(1/2)*31i - 218)^(1/2)*(2*x + 1)^(1/2))/(93096875*((31^(1/2)*
896i)/600625 + 3584/600625)))*(- 31^(1/2)*31i - 218)^(1/2)*2i)/4805 - ((28*(2*x + 1)^(1/2))/155 - (8*(2*x + 1)
^(3/2))/155)/((2*x + 1)^2 - (8*x)/5 + 3/5) - (155^(1/2)*atan((155^(1/2)*(31^(1/2)*31i - 218)^(1/2)*(2*x + 1)^(
1/2)*128i)/(3003125*((31^(1/2)*896i)/600625 - 3584/600625)) - (256*31^(1/2)*155^(1/2)*(31^(1/2)*31i - 218)^(1/
2)*(2*x + 1)^(1/2))/(93096875*((31^(1/2)*896i)/600625 - 3584/600625)))*(31^(1/2)*31i - 218)^(1/2)*2i)/4805