\(\int \frac {\sqrt {1+2 x}}{(2+3 x+5 x^2)^3} \, dx\) [2327]

Optimal result

Integrand size = 22, antiderivative size = 300 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}-\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{6727}+\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{6727}+\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454}-\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454} \]

[Out]

1/62*(3+10*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)^2+1/13454*(599+1790*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)+1/5839036*ln(5+10
*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-4188560908+784183750*35^(1/2))^(1/2)-1/5839036*ln(5+10*x+3
5^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-4188560908+784183750*35^(1/2))^(1/2)-1/2919518*arctan((-10*(1+
2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(4188560908+784183750*35^(1/2))^(1/2)+1/2919518*ar
ctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(4188560908+784183750*35^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 300, 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 = {750, 836, 840, 1183, 648, 632, 210, 642} \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx=-\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{6727}+\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \sqrt {35}\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{6727}+\frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}+\frac {\sqrt {2 x+1} (1790 x+599)}{13454 \left (5 x^2+3 x+2\right )}+\frac {\sqrt {\frac {1}{434} \left (1806875 \sqrt {35}-9651062\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{13454}-\frac {\sqrt {\frac {1}{434} \left (1806875 \sqrt {35}-9651062\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{13454} \]

[In]

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

[Out]

(Sqrt[1 + 2*x]*(3 + 10*x))/(62*(2 + 3*x + 5*x^2)^2) + (Sqrt[1 + 2*x]*(599 + 1790*x))/(13454*(2 + 3*x + 5*x^2))
- (Sqrt[(9651062 + 1806875*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + S
qrt[35])]])/6727 + (Sqrt[(9651062 + 1806875*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])
/Sqrt[10*(-2 + Sqrt[35])]])/6727 + (Sqrt[(-9651062 + 1806875*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[3
5])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/13454 - (Sqrt[(-9651062 + 1806875*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2
+ Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/13454

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 750

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(b + 2*c
*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
- 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*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
, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && 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} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}-\frac {1}{62} \int \frac {-27-50 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx \\ & = \frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-1439-1790 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{13454} \\ & = \frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}-\frac {\text {Subst}\left (\int \frac {-1088-1790 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{6727} \\ & = \frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}-\frac {\text {Subst}\left (\int \frac {-1088 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-1088+358 \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 )}{13454 \sqrt {14 \left (2+\sqrt {35}\right )}}-\frac {\text {Subst}\left (\int \frac {-1088 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-1088+358 \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 )}{13454 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = \frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}+\frac {\left (6265+544 \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 )}{470890}+\frac {\left (6265+544 \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 )}{470890}+\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \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 )}{13454}-\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \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 )}{13454} \\ & = \frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454}-\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454}-\frac {\left (6265+544 \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 )}{235445}-\frac {\left (6265+544 \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 )}{235445} \\ & = \frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}-\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \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 )}{6727}+\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \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 )}{6727}+\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454}-\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{13454} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.69 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {217 \sqrt {1+2 x} \left (1849+7547 x+8365 x^2+8950 x^3\right )}{2 \left (2+3 x+5 x^2\right )^2}+\sqrt {217 \left (9651062-825499 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+\sqrt {217 \left (9651062+825499 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{1459759} \]

[In]

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

[Out]

((217*Sqrt[1 + 2*x]*(1849 + 7547*x + 8365*x^2 + 8950*x^3))/(2*(2 + 3*x + 5*x^2)^2) + Sqrt[217*(9651062 - (8254
99*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] + Sqrt[217*(9651062 + (825499*I)*Sqrt[31])]*A
rcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]])/1459759

Maple [A] (verified)

Time = 1.51 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.08

method	result	size
pseudoelliptic	\(\frac {\frac {7160000 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (x^{3}+\frac {1673}{1790} x^{2}+\frac {7547}{8950} x +\frac {1849}{8950}\right ) \left (\sqrt {5}\, \sqrt {7}-\frac {39}{4}\right ) \sqrt {1+2 x}}{6727}+\frac {40000 \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (\frac {\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (452130 \sqrt {5}-413047 \sqrt {7}\right ) \left (\ln \left (5+10 x +\sqrt {35}-\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )-\ln \left (5+10 x +\sqrt {35}+\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{476656}+\left (\sqrt {5}\, \sqrt {7}-\frac {175}{4}\right ) \left (\arctan \left (\frac {10 \sqrt {1+2 x}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )+\arctan \left (\frac {-\sqrt {20+10 \sqrt {35}}+10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )\right )\right )}{49}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (4 \sqrt {5}\, \sqrt {7}-39\right ) \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )^{2} \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )^{2}}\)	\(324\)
trager	\(\frac {\left (8950 x^{3}+8365 x^{2}+7547 x +1849\right ) \sqrt {1+2 x}}{13454 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {2 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right ) \ln \left (\frac {-202713247744 x \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{5}-1452607421494208 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{3} x +3107245926918000 \sqrt {1+2 x}\, \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}-385575076726784 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{3}-2451977189563195620 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right ) x -25592768729287571875 \sqrt {1+2 x}-1049468876066602528 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )}{3472 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} x +7174565 x -3301996}\right )}{6727}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}[phone]\right ) \ln \left (-\frac {14479517696 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}[phone]\right ) \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{4} x [phone] \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}[phone]\right ) x -27541076909056 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}[phone]\right )+96324623734458000 \sqrt {1+2 x}\, \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}+45826377334291750 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}[phone]\right ) x -78148905624314000 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}[phone]\right )+1328879814356719851625 \sqrt {1+2 x}}{3472 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} x [phone] x +3301996}\right )}{1459759}\)	\(458\)
risch	\(\frac {\left (8950 x^{3}+8365 x^{2}+7547 x +1849\right ) \sqrt {1+2 x}}{13454 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {451 \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}}{417074}+\frac {7353 \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}}{5839036}-\frac {2255 \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 )}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {7353 \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}}{2919518 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {1088 \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}}{47089 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {451 \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}}{417074}-\frac {7353 \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}}{5839036}-\frac {2255 \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 )}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {7353 \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}}{2919518 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {1088 \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}}{47089 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(643\)
derivativedivides	\(\frac {\frac {\sqrt {5}\, \left (-13012793430 \sqrt {5}[phone] \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{6269664905 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-214587133600 \sqrt {5}[phone] \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{62696649050 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-141628999400 \sqrt {5}\, \sqrt {7}[phone]\right ) \sqrt {1+2 x}}{31348324525 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-76332028500 \sqrt {7}[phone] \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{62696649050 \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 {\left (-2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+2065235 \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 )}{5839036}-\frac {5 \left (1315392 \sqrt {35}-4721920+\frac {\left (-2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+2065235 \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 )}{1459759 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}+\frac {\frac {\sqrt {5}\, \left (-13012793430 \sqrt {5}[phone] \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{6269664905 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-214587133600 \sqrt {5}[phone] \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{2919518 \left (-41876250+4295000 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-141628999400 \sqrt {5}\, \sqrt {7}[phone]\right ) \sqrt {1+2 x}}{31348324525 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-76332028500 \sqrt {7}[phone] \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{2919518 \left (-41876250+4295000 \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 {\left (-2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+2065235 \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 )}{5839036}+\frac {5 \left (-1315392 \sqrt {35}+4721920-\frac {\left (-2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+2065235 \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 )}{1459759 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}\)	\(699\)
default	\(\frac {\frac {\sqrt {5}\, \left (-13012793430 \sqrt {5}[phone] \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{6269664905 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-214587133600 \sqrt {5}[phone] \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{62696649050 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-141628999400 \sqrt {5}\, \sqrt {7}[phone]\right ) \sqrt {1+2 x}}{31348324525 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-76332028500 \sqrt {7}[phone] \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{62696649050 \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 {\left (-2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+2065235 \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 )}{5839036}-\frac {5 \left (1315392 \sqrt {35}-4721920+\frac {\left (-2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+2065235 \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 )}{1459759 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}+\frac {\frac {\sqrt {5}\, \left (-13012793430 \sqrt {5}[phone] \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{6269664905 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-214587133600 \sqrt {5}[phone] \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{2919518 \left (-41876250+4295000 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-141628999400 \sqrt {5}\, \sqrt {7}[phone]\right ) \sqrt {1+2 x}}{31348324525 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-76332028500 \sqrt {7}[phone] \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{2919518 \left (-41876250+4295000 \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 {\left (-2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+2065235 \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 )}{5839036}+\frac {5 \left (-1315392 \sqrt {35}+4721920-\frac {\left (-2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+2065235 \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 )}{1459759 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}\)	\(699\)

[In]

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

[Out]

40000/49/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(1253/961*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(x^3+1673/1790*x^2+7547/8950*x+
1849/8950)*(5^(1/2)*7^(1/2)-39/4)*(1+2*x)^(1/2)+(x^2+3/5*x+2/5)^2*(1/476656*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(452
130*5^(1/2)-413047*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)-175/4)*(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*3
5^(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 = 1.02 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {825499 i \, \sqrt {31} - 9651062} \log \left (\sqrt {217} \sqrt {825499 i \, \sqrt {31} - 9651062} {\left (7353 i \, \sqrt {31} + 16864\right )} + 1960459375 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {825499 i \, \sqrt {31} - 9651062} \log \left (\sqrt {217} \sqrt {825499 i \, \sqrt {31} - 9651062} {\left (-7353 i \, \sqrt {31} - 16864\right )} + 1960459375 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-825499 i \, \sqrt {31} - 9651062} \log \left (\sqrt {217} {\left (7353 i \, \sqrt {31} - 16864\right )} \sqrt {-825499 i \, \sqrt {31} - 9651062} + 1960459375 \, \sqrt {2 \, x + 1}\right ) + \sqrt {217} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-825499 i \, \sqrt {31} - 9651062} \log \left (\sqrt {217} {\left (-7353 i \, \sqrt {31} + 16864\right )} \sqrt {-825499 i \, \sqrt {31} - 9651062} + 1960459375 \, \sqrt {2 \, x + 1}\right ) + 217 \, {\left (8950 \, x^{3} + 8365 \, x^{2} + 7547 \, x + 1849\right )} \sqrt {2 \, x + 1}}{2919518 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]

[In]

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

[Out]

1/2919518*(sqrt(217)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(825499*I*sqrt(31) - 9651062)*log(sqrt(217)*sqr
t(825499*I*sqrt(31) - 9651062)*(7353*I*sqrt(31) + 16864) + 1960459375*sqrt(2*x + 1)) - sqrt(217)*(25*x^4 + 30*
x^3 + 29*x^2 + 12*x + 4)*sqrt(825499*I*sqrt(31) - 9651062)*log(sqrt(217)*sqrt(825499*I*sqrt(31) - 9651062)*(-7
353*I*sqrt(31) - 16864) + 1960459375*sqrt(2*x + 1)) - sqrt(217)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(-82
5499*I*sqrt(31) - 9651062)*log(sqrt(217)*(7353*I*sqrt(31) - 16864)*sqrt(-825499*I*sqrt(31) - 9651062) + 196045
9375*sqrt(2*x + 1)) + sqrt(217)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(-825499*I*sqrt(31) - 9651062)*log(s
qrt(217)*(-7353*I*sqrt(31) + 16864)*sqrt(-825499*I*sqrt(31) - 9651062) + 1960459375*sqrt(2*x + 1)) + 217*(8950
*x^3 + 8365*x^2 + 7547*x + 1849)*sqrt(2*x + 1))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (213) = 426\).

Time = 0.71 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.14 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/100139467400*sqrt(31)*(37590*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 179*sqrt(31
)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 358*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 75180*(7/5)^(3/4)*s
qrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 533120*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 1066240*
(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)) + 1/100139467400*sqrt(31)*(37590*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sq
rt(-140*sqrt(35) + 2450) - 179*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 358*(7/5)^(3/4)*(140*sqrt(3
5) + 2450)^(3/2) + 75180*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 533120*sqrt(31)*(7/5)^(1/4)
*sqrt(-140*sqrt(35) + 2450) + 1066240*(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)) + 1/200278934800*sqrt(31)*(179*sqrt
(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 37590*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35
) - 35) - 75180*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 358*(7/5)^(3/4)*(-140*sqrt(35) + 24
50)^(3/2) + 533120*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 1066240*(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) - 1/200278934800*sqrt(3
1)*(179*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 37590*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450
)*(2*sqrt(35) - 35) - 75180*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 358*(7/5)^(3/4)*(-140*s
qrt(35) + 2450)^(3/2) + 533120*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 1066240*(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) + 2/6727*(
4475*(2*x + 1)^(7/2) - 5060*(2*x + 1)^(5/2) + 11789*(2*x + 1)^(3/2) - 3808*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x
+ 3)^2

Mupad [B] (verification not implemented)

Time = 10.17 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {1088\,\sqrt {2\,x+1}}{24025}-\frac {23578\,{\left (2\,x+1\right )}^{3/2}}{168175}+\frac {2024\,{\left (2\,x+1\right )}^{5/2}}{33635}-\frac {358\,{\left (2\,x+1\right )}^{7/2}}{6727}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}\,13744{}\mathrm {i}}{1940202180875\,\left (-\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}-\frac {27488\,\sqrt {31}\,\sqrt {217}\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}}{60146267607125\,\left (-\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}\right )\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,1{}\mathrm {i}}{1459759}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}\,13744{}\mathrm {i}}{1940202180875\,\left (\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}+\frac {27488\,\sqrt {31}\,\sqrt {217}\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}}{60146267607125\,\left (\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}\right )\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,1{}\mathrm {i}}{1459759} \]

[In]

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

[Out]

((1088*(2*x + 1)^(1/2))/24025 - (23578*(2*x + 1)^(3/2))/168175 + (2024*(2*x + 1)^(5/2))/33635 - (358*(2*x + 1)
^(7/2))/6727)/((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((2
17^(1/2)*(- 31^(1/2)*825499i - 9651062)^(1/2)*(2*x + 1)^(1/2)*13744i)/(1940202180875*((31^(1/2)*7476736i)/2771
71740125 - 101059632/277171740125)) - (27488*31^(1/2)*217^(1/2)*(- 31^(1/2)*825499i - 9651062)^(1/2)*(2*x + 1)
^(1/2))/(60146267607125*((31^(1/2)*7476736i)/277171740125 - 101059632/277171740125)))*(- 31^(1/2)*825499i - 96
51062)^(1/2)*1i)/1459759 + (217^(1/2)*atan((217^(1/2)*(31^(1/2)*825499i - 9651062)^(1/2)*(2*x + 1)^(1/2)*13744
i)/(1940202180875*((31^(1/2)*7476736i)/277171740125 + 101059632/277171740125)) + (27488*31^(1/2)*217^(1/2)*(31
^(1/2)*825499i - 9651062)^(1/2)*(2*x + 1)^(1/2))/(60146267607125*((31^(1/2)*7476736i)/277171740125 + 101059632
/277171740125)))*(31^(1/2)*825499i - 9651062)^(1/2)*1i)/1459759