\(\int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx\) [2309]

Optimal result

Integrand size = 22, antiderivative size = 279 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=-\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \]

[Out]

16/75*(1+2*x)^(3/2)+4/25*(1+2*x)^(5/2)-76/125*(1+2*x)^(1/2)+1/19375*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))
^(1/2))/(-20+10*35^(1/2))^(1/2))*(-52296380+13291250*35^(1/2))^(1/2)-1/19375*arctan((10*(1+2*x)^(1/2)+(20+10*3
5^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(-52296380+13291250*35^(1/2))^(1/2)-1/38750*ln(5+10*x+35^(1/2)-(1+2*x
)^(1/2)*(20+10*35^(1/2))^(1/2))*(52296380+13291250*35^(1/2))^(1/2)+1/38750*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(2
0+10*35^(1/2))^(1/2))*(52296380+13291250*35^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {717, 838, 840, 1183, 648, 632, 210, 642} \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\frac {1}{125} \sqrt {\frac {2}{155} \left (42875 \sqrt {35}-168698\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{125} \sqrt {\frac {2}{155} \left (42875 \sqrt {35}-168698\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {4}{25} (2 x+1)^{5/2}+\frac {16}{75} (2 x+1)^{3/2}-\frac {76}{125} \sqrt {2 x+1}-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )+\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right ) \]

[In]

Int[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2),x]

[Out]

(-76*Sqrt[1 + 2*x])/125 + (16*(1 + 2*x)^(3/2))/75 + (4*(1 + 2*x)^(5/2))/25 + (Sqrt[(2*(-168698 + 42875*Sqrt[35
]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/125 - (Sqrt[(2*(-16869
8 + 42875*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/125 -
(Sqrt[(168698 + 42875*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/125
+ (Sqrt[(168698 + 42875*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/1
25

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

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

Rule 717

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m - 1)/(c*(
m - 1))), x] + Dist[1/c, Int[(d + e*x)^(m - 2)*(Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, x]/(a + b*x + c*x^2)),
x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*
e, 0] && GtQ[m, 1]

Rule 838

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

Rule 840

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

Rule 1183

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

Rubi steps \begin{align*} \text {integral}& = \frac {4}{25} (1+2 x)^{5/2}+\frac {1}{5} \int \frac {(1+2 x)^{3/2} (-3+8 x)}{2+3 x+5 x^2} \, dx \\ & = \frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{25} \int \frac {(-47-38 x) \sqrt {1+2 x}}{2+3 x+5 x^2} \, dx \\ & = -\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{125} \int \frac {-83-432 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx \\ & = -\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {2}{125} \text {Subst}\left (\int \frac {266-432 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right ) \\ & = -\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {\text {Subst}\left (\int \frac {266 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (266+432 \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 )}{125 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {266 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (266+432 \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 )}{125 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = -\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{625} \left (-216+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 )+\frac {1}{625} \left (-216+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 )-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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 {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}-\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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}{625} \left (2 \left (216-19 \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 )+\frac {1}{625} \left (2 \left (216-19 \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 ) \\ & = -\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \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}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \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}{125} \sqrt {\frac {1}{310} \left (168698+42875 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {1}{125} \sqrt {\frac {1}{310} \left (168698+42875 \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 = 0.70 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.44 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\frac {1240 \sqrt {1+2 x} \left (-11+50 x+30 x^2\right )-6 \sqrt {155 \left (-168698-34021 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-6 \sqrt {155 \left (-168698+34021 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{58125} \]

[In]

Integrate[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2),x]

[Out]

(1240*Sqrt[1 + 2*x]*(-11 + 50*x + 30*x^2) - 6*Sqrt[155*(-168698 - (34021*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqr
t[31])/7]*Sqrt[1 + 2*x]] - 6*Sqrt[155*(-168698 + (34021*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt
[1 + 2*x]])/58125

Maple [A] (verified)

Time = 1.35 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.01

method	result	size
pseudoelliptic	\(-\frac {38 \left (\frac {233 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (\sqrt {5}+\frac {890 \sqrt {7}}{233}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{11780}-\frac {233 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (\sqrt {5}+\frac {890 \sqrt {7}}{233}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{11780}-\frac {40 \left (x^{2}+\frac {5}{3} x -\frac {11}{30}\right ) \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {1+2 x}}{19}+\left (\sqrt {5}\, \sqrt {7}-\frac {216}{19}\right ) \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right )\right )}{125 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(281\)
derivativedivides	\(\frac {4 \left (1+2 x \right )^{\frac {5}{2}}}{25}+\frac {16 \left (1+2 x \right )^{\frac {3}{2}}}{75}-\frac {76 \sqrt {1+2 x}}{125}+\frac {\left (-890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{38750}+\frac {2 \left (1178 \sqrt {5}\, \sqrt {7}+\frac {\left (-890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{38750}+\frac {2 \left (1178 \sqrt {5}\, \sqrt {7}-\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(407\)
default	\(\frac {4 \left (1+2 x \right )^{\frac {5}{2}}}{25}+\frac {16 \left (1+2 x \right )^{\frac {3}{2}}}{75}-\frac {76 \sqrt {1+2 x}}{125}+\frac {\left (-890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{38750}+\frac {2 \left (1178 \sqrt {5}\, \sqrt {7}+\frac {\left (-890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{38750}+\frac {2 \left (1178 \sqrt {5}\, \sqrt {7}-\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(407\)
trager	\(\left (\frac {16}{25} x^{2}+\frac {16}{15} x -\frac {88}{375}\right ) \sqrt {1+2 x}+\frac {\operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right ) \ln \left (\frac {408425 x \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{5}-990852845 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{3} x [phone] \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{3}[phone] \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {1+2 x}-525953389168 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right ) x -2061191406976 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )-83912677753625 \sqrt {1+2 x}}{155 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{2} x -270761 x -136084}\right )}{125}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{2}-52296380\right ) \ln \left (-\frac {81685 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{2}-52296380\right ) \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{4} x -157444815 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{2}-52296380\right ) x -143432536 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{2}-52296380\right )[phone] \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {1+2 x}-149515537500 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{2}-52296380\right ) x -100021740000 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{2}-52296380\right )+2098333883121875 \sqrt {1+2 x}}{155 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}[phone]\right )^{2} x -66635 x +136084}\right )}{19375}\)	\(440\)
risch	\(\frac {8 \left (30 x^{2}+50 x -11\right ) \sqrt {1+2 x}}{375}-\frac {89 \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{3875}-\frac {233 \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}}{38750}-\frac {178 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {233 \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 )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {76 \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}}{125 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {89 \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{3875}+\frac {233 \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}}{38750}-\frac {178 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {233 \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 )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {76 \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}}{125 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(626\)

[In]

int((1+2*x)^(7/2)/(5*x^2+3*x+2),x,method=_RETURNVERBOSE)

[Out]

-38/125*(233/11780*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(5^(1/2)+890/233*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)-233/11780*(10*5^(1/2)*7^(1/2)-20)^(1/2)*
(2*5^(1/2)*7^(1/2)+4)^(1/2)*(5^(1/2)+890/233*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)-40/19*(x^2+5/3*x-11/30)*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(1+2*x)^(1/2)+(5^(1/2)*7^(1/2)-216/
19)*(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))))/(10*5^(1/2)*7^(1/2)-20)^(1/
2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.72 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=-\frac {1}{38750} \, \sqrt {155} \sqrt {136084 i \, \sqrt {31} + 674792} \log \left (\sqrt {155} \sqrt {136084 i \, \sqrt {31} + 674792} {\left (178 i \, \sqrt {31} - 589\right )} + 13291250 \, \sqrt {2 \, x + 1}\right ) + \frac {1}{38750} \, \sqrt {155} \sqrt {136084 i \, \sqrt {31} + 674792} \log \left (\sqrt {155} \sqrt {136084 i \, \sqrt {31} + 674792} {\left (-178 i \, \sqrt {31} + 589\right )} + 13291250 \, \sqrt {2 \, x + 1}\right ) + \frac {1}{38750} \, \sqrt {155} \sqrt {-136084 i \, \sqrt {31} + 674792} \log \left (\sqrt {155} {\left (178 i \, \sqrt {31} + 589\right )} \sqrt {-136084 i \, \sqrt {31} + 674792} + 13291250 \, \sqrt {2 \, x + 1}\right ) - \frac {1}{38750} \, \sqrt {155} \sqrt {-136084 i \, \sqrt {31} + 674792} \log \left (\sqrt {155} {\left (-178 i \, \sqrt {31} - 589\right )} \sqrt {-136084 i \, \sqrt {31} + 674792} + 13291250 \, \sqrt {2 \, x + 1}\right ) + \frac {8}{375} \, {\left (30 \, x^{2} + 50 \, x - 11\right )} \sqrt {2 \, x + 1} \]

[In]

integrate((1+2*x)^(7/2)/(5*x^2+3*x+2),x, algorithm="fricas")

[Out]

-1/38750*sqrt(155)*sqrt(136084*I*sqrt(31) + 674792)*log(sqrt(155)*sqrt(136084*I*sqrt(31) + 674792)*(178*I*sqrt
(31) - 589) + 13291250*sqrt(2*x + 1)) + 1/38750*sqrt(155)*sqrt(136084*I*sqrt(31) + 674792)*log(sqrt(155)*sqrt(
136084*I*sqrt(31) + 674792)*(-178*I*sqrt(31) + 589) + 13291250*sqrt(2*x + 1)) + 1/38750*sqrt(155)*sqrt(-136084
*I*sqrt(31) + 674792)*log(sqrt(155)*(178*I*sqrt(31) + 589)*sqrt(-136084*I*sqrt(31) + 674792) + 13291250*sqrt(2
*x + 1)) - 1/38750*sqrt(155)*sqrt(-136084*I*sqrt(31) + 674792)*log(sqrt(155)*(-178*I*sqrt(31) - 589)*sqrt(-136
084*I*sqrt(31) + 674792) + 13291250*sqrt(2*x + 1)) + 8/375*(30*x^2 + 50*x - 11)*sqrt(2*x + 1)

Sympy [F]

\[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\int \frac {\left (2 x + 1\right )^{\frac {7}{2}}}{5 x^{2} + 3 x + 2}\, dx \]

[In]

integrate((1+2*x)**(7/2)/(5*x**2+3*x+2),x)

[Out]

Integral((2*x + 1)**(7/2)/(5*x**2 + 3*x + 2), x)

Maxima [F]

\[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\int { \frac {{\left (2 \, x + 1\right )}^{\frac {7}{2}}}{5 \, x^{2} + 3 \, x + 2} \,d x } \]

[In]

integrate((1+2*x)^(7/2)/(5*x^2+3*x+2),x, algorithm="maxima")

[Out]

integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (188) = 376\).

Time = 0.70 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.20 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\text {Too large to display} \]

[In]

integrate((1+2*x)^(7/2)/(5*x^2+3*x+2),x, algorithm="giac")

[Out]

4/25*(2*x + 1)^(5/2) - 1/1162984375*sqrt(31)*(11340*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35)
+ 2450) - 54*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 108*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) +
22680*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 162925*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35
) + 2450) - 325850*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*sqrt(3
5) + 1/2) + sqrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) - 1/1162984375*sqrt(31)*(11340*sqrt(31)*(7/5)^(3/4)*(2*
sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 54*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 108*(7/5)^(
3/4)*(140*sqrt(35) + 2450)^(3/2) + 22680*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 162925*sqrt
(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 325850*(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)) - 1/2325968750*sqrt(
31)*(54*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 11340*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450
)*(2*sqrt(35) - 35) - 22680*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 108*(7/5)^(3/4)*(-140*s
qrt(35) + 2450)^(3/2) - 162925*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 325850*(7/5)^(1/4)*sqrt(-140*s
qrt(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) + 1/23259687
50*sqrt(31)*(54*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 11340*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35
) + 2450)*(2*sqrt(35) - 35) - 22680*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 108*(7/5)^(3/4)
*(-140*sqrt(35) + 2450)^(3/2) - 162925*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 325850*(7/5)^(1/4)*sqr
t(-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
6/75*(2*x + 1)^(3/2) - 76/125*sqrt(2*x + 1)

Mupad [B] (verification not implemented)

Time = 9.90 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.72 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\frac {16\,{\left (2\,x+1\right )}^{3/2}}{75}-\frac {76\,\sqrt {2\,x+1}}{125}+\frac {4\,{\left (2\,x+1\right )}^{5/2}}{25}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {168698+\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}\,4354688{}\mathrm {i}}{6103515625\,\left (-\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}+\frac {8709376\,\sqrt {31}\,\sqrt {155}\,\sqrt {168698+\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}}{189208984375\,\left (-\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}\right )\,\sqrt {168698+\sqrt {31}\,34021{}\mathrm {i}}\,2{}\mathrm {i}}{19375}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {168698-\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}\,4354688{}\mathrm {i}}{6103515625\,\left (\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}-\frac {8709376\,\sqrt {31}\,\sqrt {155}\,\sqrt {168698-\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}}{189208984375\,\left (\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}\right )\,\sqrt {168698-\sqrt {31}\,34021{}\mathrm {i}}\,2{}\mathrm {i}}{19375} \]

[In]

int((2*x + 1)^(7/2)/(3*x + 5*x^2 + 2),x)

[Out]

(16*(2*x + 1)^(3/2))/75 - (76*(2*x + 1)^(1/2))/125 + (4*(2*x + 1)^(5/2))/25 + (155^(1/2)*atan((155^(1/2)*(31^(
1/2)*34021i + 168698)^(1/2)*(2*x + 1)^(1/2)*4354688i)/(6103515625*((31^(1/2)*579173504i)/1220703125 - 54259412
48/1220703125)) + (8709376*31^(1/2)*155^(1/2)*(31^(1/2)*34021i + 168698)^(1/2)*(2*x + 1)^(1/2))/(189208984375*
((31^(1/2)*579173504i)/1220703125 - 5425941248/1220703125)))*(31^(1/2)*34021i + 168698)^(1/2)*2i)/19375 - (155
^(1/2)*atan((155^(1/2)*(168698 - 31^(1/2)*34021i)^(1/2)*(2*x + 1)^(1/2)*4354688i)/(6103515625*((31^(1/2)*57917
3504i)/1220703125 + 5425941248/1220703125)) - (8709376*31^(1/2)*155^(1/2)*(168698 - 31^(1/2)*34021i)^(1/2)*(2*
x + 1)^(1/2))/(189208984375*((31^(1/2)*579173504i)/1220703125 + 5425941248/1220703125)))*(168698 - 31^(1/2)*34
021i)^(1/2)*2i)/19375