\(\int \frac {1}{(1+2 x)^{5/2} (2+3 x+5 x^2)} \, dx\) [2315]

Optimal result

Integrand size = 22, antiderivative size = 266 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx=-\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \]

[Out]

-4/21/(1+2*x)^(3/2)-16/49/(1+2*x)^(1/2)-1/21266*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-310
8308+531650*35^(1/2))^(1/2)+1/21266*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-3108308+531650*
35^(1/2))^(1/2)+1/10633*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(3108308+53
1650*35^(1/2))^(1/2)-1/10633*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(310830
8+531650*35^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {723, 842, 840, 1183, 648, 632, 210, 642} \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {16}{49 \sqrt {2 x+1}}-\frac {4}{21 (2 x+1)^{3/2}}-\frac {1}{49} \sqrt {\frac {1}{434} \left (1225 \sqrt {35}-7162\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{49} \sqrt {\frac {1}{434} \left (1225 \sqrt {35}-7162\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right ) \]

[In]

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

[Out]

-4/(21*(1 + 2*x)^(3/2)) - 16/(49*Sqrt[1 + 2*x]) + (Sqrt[(2*(7162 + 1225*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + S
qrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/49 - (Sqrt[(2*(7162 + 1225*Sqrt[35]))/217]*ArcTan[(Sq
rt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/49 - (Sqrt[(-7162 + 1225*Sqrt[35])/434]*L
og[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/49 + (Sqrt[(-7162 + 1225*Sqrt[35])/434]*Lo
g[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/49

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 723

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

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 842

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

Rule 1183

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

Rubi steps \begin{align*} \text {integral}& = -\frac {4}{21 (1+2 x)^{3/2}}+\frac {1}{7} \int \frac {-1-10 x}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx \\ & = -\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {1}{49} \int \frac {-39-40 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx \\ & = -\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {2}{49} \text {Subst}\left (\int \frac {-38-40 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right ) \\ & = -\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {\text {Subst}\left (\int \frac {-38 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-38+8 \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 )}{49 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {-38 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-38+8 \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 )}{49 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = -\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}-\frac {\left (140+19 \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 )}{1715}-\frac {\left (140+19 \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 )}{1715}-\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \text {Subst}\left (\int \frac {-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )+\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \text {Subst}\left (\int \frac {\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right ) \\ & = -\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}-\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {\left (2 \left (140+19 \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 )}{1715}+\frac {\left (2 \left (140+19 \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 )}{1715} \\ & = -\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-2 \sqrt {1+2 x}\right )\right )-\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )\right )-\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{49} \sqrt {\frac {1}{434} \left (-7162+1225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.69 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.45 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\frac {2 \left (-\frac {434 (19+24 x)}{(1+2 x)^{3/2}}-3 \sqrt {217 \left (7162-199 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-3 \sqrt {217 \left (7162+199 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{31899} \]

[In]

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

[Out]

(2*((-434*(19 + 24*x))/(1 + 2*x)^(3/2) - 3*Sqrt[217*(7162 - (199*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7
]*Sqrt[1 + 2*x]] - 3*Sqrt[217*(7162 + (199*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]]))/
31899

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.15

method	result	size
pseudoelliptic	\(-\frac {76 \left (-\frac {189 \left (\sqrt {5}-\frac {178 \sqrt {7}}{189}\right ) \left (x +\frac {1}{2}\right ) \sqrt {1+2 x}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{2356}+\frac {189 \left (\sqrt {5}-\frac {178 \sqrt {7}}{189}\right ) \left (x +\frac {1}{2}\right ) \sqrt {1+2 x}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{2356}+\left (\sqrt {5}\, \sqrt {7}+\frac {140}{19}\right ) \left (x +\frac {1}{2}\right ) \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right ) \sqrt {1+2 x}+\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (\frac {56 x}{19}+\frac {7}{3}\right )\right )}{343 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (1+2 x \right )^{\frac {3}{2}}}\)	\(307\)
derivativedivides	\(-\frac {4}{21 \left (1+2 x \right )^{\frac {3}{2}}}-\frac {16}{49 \sqrt {1+2 x}}-\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{106330}-\frac {2 \left (1178 \sqrt {5}\, \sqrt {7}+\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{106330}+\frac {2 \left (-1178 \sqrt {5}\, \sqrt {7}-\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(398\)
default	\(-\frac {4}{21 \left (1+2 x \right )^{\frac {3}{2}}}-\frac {16}{49 \sqrt {1+2 x}}-\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{106330}-\frac {2 \left (1178 \sqrt {5}\, \sqrt {7}+\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{106330}+\frac {2 \left (-1178 \sqrt {5}\, \sqrt {7}-\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(398\)
trager	\(-\frac {4 \left (24 x +19\right )}{147 \left (1+2 x \right )^{\frac {3}{2}}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right ) \ln \left (-\frac {733243 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right ) \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{4} x [phone] \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right ) x [phone] \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} \sqrt {1+2 x}-5379368 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right )[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right ) x [phone] \sqrt {1+2 x}-174737920 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right )}{217 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} x +7759 x +796}\right )}{10633}-\frac {\operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right ) \ln \left (\frac {-5132701 x \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{5}-374431547 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{3} x [phone] \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} \sqrt {1+2 x}-37655576 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{3}-6784080780 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right ) x -301062125 \sqrt {1+2 x}-1262449632 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )}{217 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} x +6565 x -796}\right )}{49}\)	\(436\)

[In]

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

[Out]

-76/343/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(-189/2356*(5^(1/2)-178/189*7^(1/2))*(x+1/2)*(1+2*x)^(1/2)*(2*5^(1/2)*7^
(1/2)+4)^(1/2)*(10*5^(1/2)*7^(1/2)-20)^(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)+189/2356*(5^(1/2)-178/189*7^(1/2))*(x+1/2)*(1+2*x)^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(10*5^(1/2)*7
^(1/2)-20)^(1/2)*ln(5^(1/2)*7^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5+10*x)+(5^(1/2)*7^(1/2)
+140/19)*(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)))*(1+2*x)^(1/2)+(
10*5^(1/2)*7^(1/2)-20)^(1/2)*(56/19*x+7/3))/(1+2*x)^(3/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.41 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx=-\frac {3 \, \sqrt {217} {\left (4 \, x^{2} + 4 \, x + 1\right )} \sqrt {796 i \, \sqrt {31} - 28648} \log \left (\sqrt {217} \sqrt {796 i \, \sqrt {31} - 28648} {\left (178 i \, \sqrt {31} + 589\right )} + 2658250 \, \sqrt {2 \, x + 1}\right ) - 3 \, \sqrt {217} {\left (4 \, x^{2} + 4 \, x + 1\right )} \sqrt {796 i \, \sqrt {31} - 28648} \log \left (\sqrt {217} \sqrt {796 i \, \sqrt {31} - 28648} {\left (-178 i \, \sqrt {31} - 589\right )} + 2658250 \, \sqrt {2 \, x + 1}\right ) - 3 \, \sqrt {217} {\left (4 \, x^{2} + 4 \, x + 1\right )} \sqrt {-796 i \, \sqrt {31} - 28648} \log \left (\sqrt {217} {\left (178 i \, \sqrt {31} - 589\right )} \sqrt {-796 i \, \sqrt {31} - 28648} + 2658250 \, \sqrt {2 \, x + 1}\right ) + 3 \, \sqrt {217} {\left (4 \, x^{2} + 4 \, x + 1\right )} \sqrt {-796 i \, \sqrt {31} - 28648} \log \left (\sqrt {217} {\left (-178 i \, \sqrt {31} + 589\right )} \sqrt {-796 i \, \sqrt {31} - 28648} + 2658250 \, \sqrt {2 \, x + 1}\right ) + 1736 \, {\left (24 \, x + 19\right )} \sqrt {2 \, x + 1}}{63798 \, {\left (4 \, x^{2} + 4 \, x + 1\right )}} \]

[In]

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

[Out]

-1/63798*(3*sqrt(217)*(4*x^2 + 4*x + 1)*sqrt(796*I*sqrt(31) - 28648)*log(sqrt(217)*sqrt(796*I*sqrt(31) - 28648
)*(178*I*sqrt(31) + 589) + 2658250*sqrt(2*x + 1)) - 3*sqrt(217)*(4*x^2 + 4*x + 1)*sqrt(796*I*sqrt(31) - 28648)
*log(sqrt(217)*sqrt(796*I*sqrt(31) - 28648)*(-178*I*sqrt(31) - 589) + 2658250*sqrt(2*x + 1)) - 3*sqrt(217)*(4*
x^2 + 4*x + 1)*sqrt(-796*I*sqrt(31) - 28648)*log(sqrt(217)*(178*I*sqrt(31) - 589)*sqrt(-796*I*sqrt(31) - 28648
) + 2658250*sqrt(2*x + 1)) + 3*sqrt(217)*(4*x^2 + 4*x + 1)*sqrt(-796*I*sqrt(31) - 28648)*log(sqrt(217)*(-178*I
*sqrt(31) + 589)*sqrt(-796*I*sqrt(31) - 28648) + 2658250*sqrt(2*x + 1)) + 1736*(24*x + 19)*sqrt(2*x + 1))/(4*x
^2 + 4*x + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (179) = 358\).

Time = 0.51 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.25 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/91177975*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) + 4655*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 9310*(7/5)^(1/4)*sqrt(1
40*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)) - 1/91177975*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) + 4655*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 9310
*(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)) - 1/182355950*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)*sq
rt(-140*sqrt(35) + 2450) - 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 4655*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqr
t(35) + 2450) + 9310*(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) + 1/182355950*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(-14
0*sqrt(35) + 2450) - 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 4655*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35)
+ 2450) + 9310*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450))*log(-2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1
/2) + 2*x + sqrt(7/5) + 1) - 4/147*(24*x + 19)/(2*x + 1)^(3/2)

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx=-\frac {\frac {32\,x}{49}+\frac {76}{147}}{{\left (2\,x+1\right )}^{3/2}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}\,25472{}\mathrm {i}}{720600125\,\left (-\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}-\frac {50944\,\sqrt {31}\,\sqrt {217}\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}}{22338603875\,\left (-\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}\right )\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,2{}\mathrm {i}}{10633}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}\,25472{}\mathrm {i}}{720600125\,\left (\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}+\frac {50944\,\sqrt {31}\,\sqrt {217}\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}}{22338603875\,\left (\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}\right )\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,2{}\mathrm {i}}{10633} \]

[In]

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

[Out]

(217^(1/2)*atan((217^(1/2)*(- 31^(1/2)*199i - 7162)^(1/2)*(2*x + 1)^(1/2)*25472i)/(720600125*((31^(1/2)*483968
i)/102942875 - 4534016/102942875)) - (50944*31^(1/2)*217^(1/2)*(- 31^(1/2)*199i - 7162)^(1/2)*(2*x + 1)^(1/2))
/(22338603875*((31^(1/2)*483968i)/102942875 - 4534016/102942875)))*(- 31^(1/2)*199i - 7162)^(1/2)*2i)/10633 -
((32*x)/49 + 76/147)/(2*x + 1)^(3/2) - (217^(1/2)*atan((217^(1/2)*(31^(1/2)*199i - 7162)^(1/2)*(2*x + 1)^(1/2)
*25472i)/(720600125*((31^(1/2)*483968i)/102942875 + 4534016/102942875)) + (50944*31^(1/2)*217^(1/2)*(31^(1/2)*
199i - 7162)^(1/2)*(2*x + 1)^(1/2))/(22338603875*((31^(1/2)*483968i)/102942875 + 4534016/102942875)))*(31^(1/2
)*199i - 7162)^(1/2)*2i)/10633