\(\int \frac {1}{\sqrt {1+2 x} (2+3 x+5 x^2)^3} \, dx\) [2328]

Optimal result

Integrand size = 22, antiderivative size = 314 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (2+\sqrt {35}\right )} \left (7379+264 \sqrt {35}\right ) \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{47089}+\frac {3 \sqrt {\frac {1}{434} \left (2+\sqrt {35}\right )} \left (7379+264 \sqrt {35}\right ) \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{47089}-\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178}+\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178} \]

[Out]

1/434*(37+20*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)^2+1/94178*(9227+7920*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)-3/20436626*arc
tan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(7379+264*35^(1/2))*(868+434*35^(1/2))
^(1/2)+3/20436626*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(7379+264*35^(1/2)
)*(868+434*35^(1/2))^(1/2)-3/40873252*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-108561594148+
28071651650*35^(1/2))^(1/2)+3/40873252*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-108561594148
[phone]*35^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 314, 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 = {754, 836, 840, 1183, 648, 632, 210, 642} \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx=-\frac {3 \sqrt {\frac {1}{434} \left (2+\sqrt {35}\right )} \left (7379+264 \sqrt {35}\right ) \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{47089}+\frac {3 \sqrt {\frac {1}{434} \left (2+\sqrt {35}\right )} \left (7379+264 \sqrt {35}\right ) \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{47089}+\frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}+\frac {\sqrt {2 x+1} (7920 x+9227)}{94178 \left (5 x^2+3 x+2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (64681225 \sqrt {35}-250141922\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{94178}+\frac {3 \sqrt {\frac {1}{434} \left (64681225 \sqrt {35}-250141922\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{94178} \]

[In]

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

[Out]

(Sqrt[1 + 2*x]*(37 + 20*x))/(434*(2 + 3*x + 5*x^2)^2) + (Sqrt[1 + 2*x]*(9227 + 7920*x))/(94178*(2 + 3*x + 5*x^
2)) - (3*Sqrt[(2 + Sqrt[35])/434]*(7379 + 264*Sqrt[35])*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sq
rt[10*(-2 + Sqrt[35])]])/47089 + (3*Sqrt[(2 + Sqrt[35])/434]*(7379 + 264*Sqrt[35])*ArcTan[(Sqrt[10*(2 + Sqrt[3
5])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/47089 - (3*Sqrt[(-250141922 + 64681225*Sqrt[35])/434]*Log[
Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/94178 + (3*Sqrt[(-250141922 + 64681225*Sqrt[3
5])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/94178

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 836

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

Rule 840

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

Rule 1183

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

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {1}{434} \int \frac {271+100 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx \\ & = \frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {26097+7920 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{94178} \\ & = \frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {44274+7920 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{47089} \\ & = \frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {44274 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (44274-1584 \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 )}{94178 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {44274 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (44274-1584 \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 )}{94178 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = \frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}+\frac {\left (3 \left (9240+7379 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{3296230}+\frac {\left (3 \left (9240+7379 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{3296230}-\frac {\left (3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )}\right ) \text {Subst}\left (\int \frac {-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{94178}+\frac {\left (3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )}\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{94178} \\ & = \frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178}+\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178}-\frac {\left (3 \left (9240+7379 \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 )}{1648115}-\frac {\left (3 \left (9240+7379 \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 )}{1648115} \\ & = \frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (250141922+64681225 \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 )}{47089}+\frac {3 \sqrt {\frac {1}{434} \left (250141922+64681225 \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 )}{47089}-\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178}+\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{94178} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.72 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {217 \sqrt {1+2 x} \left (26483+47861 x+69895 x^2+39600 x^3\right )}{2 \left (2+3 x+5 x^2\right )^2}+3 \sqrt {217 \left (250141922+52010281 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+3 \sqrt {217 \left (250141922-52010281 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{10218313} \]

[In]

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

[Out]

((217*Sqrt[1 + 2*x]*(26483 + 47861*x + 69895*x^2 + 39600*x^3))/(2*(2 + 3*x + 5*x^2)^2) + 3*Sqrt[217*(250141922
+ (52010281*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] + 3*Sqrt[217*(250141922 - (52010281
*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]])/10218313

Maple [A] (verified)

Time = 1.70 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.03

method	result	size
pseudoelliptic	\(-\frac {7830000 \left (-\frac {2464 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (x^{3}+\frac {13979}{7920} x^{2}+\frac {4351}{3600} x +\frac {26483}{39600}\right ) \left (\sqrt {5}\, \sqrt {7}-\frac {39}{4}\right ) \sqrt {1+2 x}}{83607}+\left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (-\frac {\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (2830555 \sqrt {5}-2042902 \sqrt {7}\right ) \left (\ln \left (5+10 x +\sqrt {35}-\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )-\ln \left (5+10 x +\sqrt {35}+\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{31101804}+\left (\sqrt {5}\, \sqrt {7}-\frac {700}{261}\right ) \left (\arctan \left (\frac {10 \sqrt {1+2 x}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )+\arctan \left (\frac {-\sqrt {20+10 \sqrt {35}}+10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )\right )\right )\right )}{343 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (4 \sqrt {5}\, \sqrt {7}-39\right ) \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )^{2} \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )^{2}}\)	\(324\)
trager	\(\frac {\left (39600 x^{3}+69895 x^{2}+47861 x +26483\right ) \sqrt {1+2 x}}{94178 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{2}[phone]\right ) \ln \left (\frac {71188665856 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{2}[phone]\right ) \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{4} x +21852150152006112 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{2}[phone]\right ) x +8531203952040704 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{2}[phone]\right )+217250156179051311120 \sqrt {1+2 x}\, \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{2}+1637878602708070755480 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{2}[phone]\right ) x +1109535657155862471360 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{2}[phone]\right )-135134838449389184174582135 \sqrt {1+2 x}}{3472 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{2} x [phone] x -208041124}\right )}{10218313}-\frac {6 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right ) \ln \left (\frac {996641321984 x \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{5}-18716074679124032 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{3} x -119436855328569856 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{3}-7008069554162945520 \sqrt {1+2 x}\, \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{2}-458990644030336842880 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right ) x -1676273005765163471872 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )-5368987869213016478877125 \sqrt {1+2 x}}{3472 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+20918304337503125\right )^{2} x [phone] x [phone]}\right )}{47089}\)	\(457\)
risch	\(\frac {\left (39600 x^{3}+69895 x^{2}+47861 x +26483\right ) \sqrt {1+2 x}}{94178 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {35997 \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}}{20436626}-\frac {23721 \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}}{5839036}+\frac {35997 \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}}{10218313 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {118605 \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 )}{2919518 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {44274 \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}}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {35997 \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}}{20436626}+\frac {23721 \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}}{5839036}+\frac {35997 \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}}{10218313 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {118605 \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 )}{2919518 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {44274 \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}}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(643\)
derivativedivides	\(\frac {\frac {3 \left (-6045943503600+620096769600 \sqrt {5}\, \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{2765126589365 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-91048526818200 \sqrt {5}[phone] \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{27651265893650 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-59423591568600 \sqrt {5}\, \sqrt {7}[phone]\right ) \sqrt {1+2 x}}{13825632946825 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-123371070933600 \sqrt {7}[phone] \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{55302531787300 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}}{\left (\frac {\sqrt {5}\, \sqrt {7}}{5}-\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}+1+2 x \right )^{2}}+\frac {\frac {3 \left (14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}-10214510 \sqrt {7}\, \sqrt {2 \sqrt {35}+4}\right ) \ln \left (5+10 x +\sqrt {35}-\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{40873252}+\frac {15 \left (-17842422 \sqrt {35}[phone]+\frac {\left (14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}-10214510 \sqrt {7}\, \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 )}{10218313 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}+\frac {\frac {3 \left (-6045943503600+620096769600 \sqrt {5}\, \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{2765126589365 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-91048526818200 \sqrt {5}[phone] \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{20436626 \left (-2638398750+270605000 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-59423591568600 \sqrt {5}\, \sqrt {7}[phone]\right ) \sqrt {1+2 x}}{13825632946825 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-123371070933600 \sqrt {7}[phone] \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{20436626 \left (-5276797500+541210000 \sqrt {5}\, \sqrt {7}\right )}}{\left (\frac {\sqrt {5}\, \sqrt {7}}{5}+\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}+1+2 x \right )^{2}}+\frac {\frac {3 \left (-14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}[phone] \sqrt {7}\, \sqrt {2 \sqrt {35}+4}\right ) \ln \left (5+10 x +\sqrt {35}+\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{40873252}+\frac {15 \left (-17842422 \sqrt {35}[phone]-\frac {\left (-14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}[phone] \sqrt {7}\, \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 )}{10218313 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}\)	\(691\)
default	\(\frac {\frac {3 \left (-6045943503600+620096769600 \sqrt {5}\, \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{2765126589365 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-91048526818200 \sqrt {5}[phone] \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{27651265893650 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-59423591568600 \sqrt {5}\, \sqrt {7}[phone]\right ) \sqrt {1+2 x}}{13825632946825 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-123371070933600 \sqrt {7}[phone] \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{55302531787300 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}}{\left (\frac {\sqrt {5}\, \sqrt {7}}{5}-\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}+1+2 x \right )^{2}}+\frac {\frac {3 \left (14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}-10214510 \sqrt {7}\, \sqrt {2 \sqrt {35}+4}\right ) \ln \left (5+10 x +\sqrt {35}-\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{40873252}+\frac {15 \left (-17842422 \sqrt {35}[phone]+\frac {\left (14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}-10214510 \sqrt {7}\, \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 )}{10218313 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}+\frac {\frac {3 \left (-6045943503600+620096769600 \sqrt {5}\, \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{2765126589365 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-91048526818200 \sqrt {5}[phone] \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{20436626 \left (-2638398750+270605000 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-59423591568600 \sqrt {5}\, \sqrt {7}[phone]\right ) \sqrt {1+2 x}}{13825632946825 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-123371070933600 \sqrt {7}[phone] \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{20436626 \left (-5276797500+541210000 \sqrt {5}\, \sqrt {7}\right )}}{\left (\frac {\sqrt {5}\, \sqrt {7}}{5}+\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}+1+2 x \right )^{2}}+\frac {\frac {3 \left (-14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}[phone] \sqrt {7}\, \sqrt {2 \sqrt {35}+4}\right ) \ln \left (5+10 x +\sqrt {35}+\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{40873252}+\frac {15 \left (-17842422 \sqrt {35}[phone]-\frac {\left (-14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}[phone] \sqrt {7}\, \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 )}{10218313 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}\)	\(691\)

[In]

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

[Out]

-7830000/343/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(-2464/83607*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(x^3+13979/7920*x^2+4351
/3600*x+26483/39600)*(5^(1/2)*7^(1/2)-39/4)*(1+2*x)^(1/2)+(x^2+3/5*x+2/5)^2*(-1/31101804*(10*5^(1/2)*7^(1/2)-2
0)^(1/2)*(2830555*5^(1/2)-2042902*7^(1/2))*(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)+(5^(1/2)*7^(1/2)-700/261)*(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)))))/(4*5^(1/2)*7^(1/2)-39)/(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^(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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx=-\frac {\sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {468092529 i \, \sqrt {31} - 2251277298} \log \left (\sqrt {217} \sqrt {468092529 i \, \sqrt {31} - 2251277298} {\left (23998 i \, \sqrt {31} - 228749\right )} + 210537387375 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {468092529 i \, \sqrt {31} - 2251277298} \log \left (\sqrt {217} \sqrt {468092529 i \, \sqrt {31} - 2251277298} {\left (-23998 i \, \sqrt {31} + 228749\right )} + 210537387375 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-468092529 i \, \sqrt {31} - 2251277298} \log \left (\sqrt {217} {\left (23998 i \, \sqrt {31} + 228749\right )} \sqrt {-468092529 i \, \sqrt {31} - 2251277298} + 210537387375 \, \sqrt {2 \, x + 1}\right ) + \sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-468092529 i \, \sqrt {31} - 2251277298} \log \left (\sqrt {217} {\left (-23998 i \, \sqrt {31} - 228749\right )} \sqrt {-468092529 i \, \sqrt {31} - 2251277298} + 210537387375 \, \sqrt {2 \, x + 1}\right ) - 217 \, {\left (39600 \, x^{3} + 69895 \, x^{2} + 47861 \, x + 26483\right )} \sqrt {2 \, x + 1}}{20436626 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]

[In]

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

[Out]

-1/20436626*(sqrt(217)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(468092529*I*sqrt(31) - 2251277298)*log(sqrt(
217)*sqrt(468092529*I*sqrt(31) - 2251277298)*(23998*I*sqrt(31) - 228749) + 210537387375*sqrt(2*x + 1)) - sqrt(
217)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(468092529*I*sqrt(31) - 2251277298)*log(sqrt(217)*sqrt(46809252
9*I*sqrt(31) - 2251277298)*(-23998*I*sqrt(31) + 228749) + 210537387375*sqrt(2*x + 1)) - sqrt(217)*(25*x^4 + 30
*x^3 + 29*x^2 + 12*x + 4)*sqrt(-468092529*I*sqrt(31) - 2251277298)*log(sqrt(217)*(23998*I*sqrt(31) + 228749)*s
qrt(-468092529*I*sqrt(31) - 2251277298) + 210537387375*sqrt(2*x + 1)) + sqrt(217)*(25*x^4 + 30*x^3 + 29*x^2 +
12*x + 4)*sqrt(-468092529*I*sqrt(31) - 2251277298)*log(sqrt(217)*(-23998*I*sqrt(31) - 228749)*sqrt(-468092529*
I*sqrt(31) - 2251277298) + 210537387375*sqrt(2*x + 1)) - 217*(39600*x^3 + 69895*x^2 + 47861*x + 26483)*sqrt(2*
x + 1))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (227) = 454\).

Time = 0.53 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.04 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

3/175244067950*sqrt(31)*(13860*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 66*sqrt(31)
*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 132*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 27720*(7/5)^(3/4)*sq
rt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 1807855*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 3615710*
(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*sqrt(35) + 1/2) + sqrt(2*
x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) + 3/175244067950*sqrt(31)*(13860*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sq
rt(-140*sqrt(35) + 2450) - 66*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 132*(7/5)^(3/4)*(140*sqrt(35
) + 2450)^(3/2) + 27720*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 1807855*sqrt(31)*(7/5)^(1/4)
*sqrt(-140*sqrt(35) + 2450) + 3615710*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(-5/7*(7/5)^(3/4)*((7/5)^(1
/4)*sqrt(1/35*sqrt(35) + 1/2) - sqrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) + 3/350488135900*sqrt(31)*(66*sqrt(
31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 13860*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35)
- 35) - 27720*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 132*(7/5)^(3/4)*(-140*sqrt(35) + 245
0)^(3/2) + 1807855*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 3615710*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2
450))*log(2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) - 3/350488135900*sqrt(3
1)*(66*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 13860*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)
*(2*sqrt(35) - 35) - 27720*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 132*(7/5)^(3/4)*(-140*sq
rt(35) + 2450)^(3/2) + 1807855*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 3615710*(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) + 2/47089*
(19800*(2*x + 1)^(7/2) + 10495*(2*x + 1)^(5/2) + 15332*(2*x + 1)^(3/2) + 60305*sqrt(2*x + 1))/(5*(2*x + 1)^2 -
8*x + 3)^2

Mupad [B] (verification not implemented)

Time = 10.11 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx=-\frac {\frac {3446\,\sqrt {2\,x+1}}{33635}+\frac {30664\,{\left (2\,x+1\right )}^{3/2}}{1177225}+\frac {4198\,{\left (2\,x+1\right )}^{5/2}}{235445}+\frac {1584\,{\left (2\,x+1\right )}^{7/2}}{47089}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}\,23380272{}\mathrm {i}}{665489348040125\,\left (\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}+\frac {46760544\,\sqrt {31}\,\sqrt {217}\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}}{20630169789243875\,\left (\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}\right )\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,3{}\mathrm {i}}{10218313}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}\,23380272{}\mathrm {i}}{665489348040125\,\left (-\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}-\frac {46760544\,\sqrt {31}\,\sqrt {217}\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}}{20630169789243875\,\left (-\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}\right )\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,3{}\mathrm {i}}{10218313} \]

[In]

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

[Out]

(217^(1/2)*atan((217^(1/2)*(- 31^(1/2)*52010281i - 250141922)^(1/2)*(2*x + 1)^(1/2)*23380272i)/(66548934804012
5*((31^(1/2)*172523027088i)/95069906862875 + 561079767456/95069906862875)) + (46760544*31^(1/2)*217^(1/2)*(- 3
1^(1/2)*52010281i - 250141922)^(1/2)*(2*x + 1)^(1/2))/(20630169789243875*((31^(1/2)*172523027088i)/95069906862
875 + 561079767456/95069906862875)))*(- 31^(1/2)*52010281i - 250141922)^(1/2)*3i)/10218313 - ((3446*(2*x + 1)^
(1/2))/33635 + (30664*(2*x + 1)^(3/2))/1177225 + (4198*(2*x + 1)^(5/2))/235445 + (1584*(2*x + 1)^(7/2))/47089)
/((112*x)/25 - (86*(2*x + 1)^2)/25 + (8*(2*x + 1)^3)/5 - (2*x + 1)^4 + 7/25) - (217^(1/2)*atan((217^(1/2)*(31^
(1/2)*52010281i - 250141922)^(1/2)*(2*x + 1)^(1/2)*23380272i)/(665489348040125*((31^(1/2)*172523027088i)/95069
906862875 - 561079767456/95069906862875)) - (46760544*31^(1/2)*217^(1/2)*(31^(1/2)*52010281i - 250141922)^(1/2
)*(2*x + 1)^(1/2))/(20630169789243875*((31^(1/2)*172523027088i)/95069906862875 - 561079767456/95069906862875))
)*(31^(1/2)*52010281i - 250141922)^(1/2)*3i)/10218313