\(\int \frac {(1+2 x)^{9/2}}{(2+3 x+5 x^2)^3} \, dx\) [2323]

Optimal result

Integrand size = 22, antiderivative size = 313 \[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=-\frac {1584 \sqrt {1+2 x}}{24025}-\frac {(5-4 x) (1+2 x)^{7/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(1143-1088 x) (1+2 x)^{3/2}}{9610 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{310} \left (250141922+64681225 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{24025}+\frac {3 \sqrt {\frac {1}{310} \left (250141922+64681225 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{24025}+\frac {3 \sqrt {\frac {1}{310} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{48050}-\frac {3 \sqrt {\frac {1}{310} \left (-250141922+64681225 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{48050} \]

[Out]

-1/62*(5-4*x)*(1+2*x)^(7/2)/(5*x^2+3*x+2)^2-1/9610*(1143-1088*x)*(1+2*x)^(3/2)/(5*x^2+3*x+2)-1584/24025*(1+2*x
)^(1/2)+3/14895500*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-77543995820+20051179750*35^(1/2)
)^(1/2)-3/14895500*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-77543995820+20051179750*35^(1/2)
)^(1/2)-3/7447750*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(77543995820+2005
1179750*35^(1/2))^(1/2)+3/7447750*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(7
7543995820+20051179750*35^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {752, 832, 838, 840, 1183, 648, 632, 210, 642} \[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=-\frac {3 \sqrt {\frac {1}{310} \left (250141922+64681225 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{24025}+\frac {3 \sqrt {\frac {1}{310} \left (250141922+64681225 \sqrt {35}\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{24025}-\frac {(5-4 x) (2 x+1)^{7/2}}{62 \left (5 x^2+3 x+2\right )^2}-\frac {(1143-1088 x) (2 x+1)^{3/2}}{9610 \left (5 x^2+3 x+2\right )}-\frac {1584 \sqrt {2 x+1}}{24025}+\frac {3 \sqrt {\frac {1}{310} \left (64681225 \sqrt {35}-250141922\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{48050}-\frac {3 \sqrt {\frac {1}{310} \left (64681225 \sqrt {35}-250141922\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{48050} \]

[In]

Int[(1 + 2*x)^(9/2)/(2 + 3*x + 5*x^2)^3,x]

[Out]

(-1584*Sqrt[1 + 2*x])/24025 - ((5 - 4*x)*(1 + 2*x)^(7/2))/(62*(2 + 3*x + 5*x^2)^2) - ((1143 - 1088*x)*(1 + 2*x
)^(3/2))/(9610*(2 + 3*x + 5*x^2)) - (3*Sqrt[(250141922 + 64681225*Sqrt[35])/310]*ArcTan[(Sqrt[10*(2 + Sqrt[35]
)] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/24025 + (3*Sqrt[(250141922 + 64681225*Sqrt[35])/310]*ArcTan[
(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/24025 + (3*Sqrt[(-250141922 + 64681225
*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/48050 - (3*Sqrt[(-2501419
22 + 64681225*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/48050

Rule 210

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

Rule 632

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

Rule 642

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

Rule 648

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

Rule 752

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

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2)^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*
g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2
*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m
+ 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], 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] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &
& RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

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 {(5-4 x) (1+2 x)^{7/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {1}{62} \int \frac {(47-4 x) (1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx \\ & = -\frac {(5-4 x) (1+2 x)^{7/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(1143-1088 x) (1+2 x)^{3/2}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {\sqrt {1+2 x} (-4269+1584 x)}{2+3 x+5 x^2} \, dx}{9610} \\ & = -\frac {1584 \sqrt {1+2 x}}{24025}-\frac {(5-4 x) (1+2 x)^{7/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(1143-1088 x) (1+2 x)^{3/2}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-27681-44274 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{48050} \\ & = -\frac {1584 \sqrt {1+2 x}}{24025}-\frac {(5-4 x) (1+2 x)^{7/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(1143-1088 x) (1+2 x)^{3/2}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\text {Subst}\left (\int \frac {-11088-44274 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{24025} \\ & = -\frac {1584 \sqrt {1+2 x}}{24025}-\frac {(5-4 x) (1+2 x)^{7/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(1143-1088 x) (1+2 x)^{3/2}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\text {Subst}\left (\int \frac {-11088 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-11088+44274 \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 )}{48050 \sqrt {14 \left (2+\sqrt {35}\right )}}-\frac {\text {Subst}\left (\int \frac {-11088 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-11088+44274 \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 )}{48050 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = -\frac {1584 \sqrt {1+2 x}}{24025}-\frac {(5-4 x) (1+2 x)^{7/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(1143-1088 x) (1+2 x)^{3/2}}{9610 \left (2+3 x+5 x^2\right )}-\frac {\left (3 \left (9240-7379 \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 )}{240250 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\left (3 \left (9240-7379 \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 )}{240250 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\left (3 \left (7379+264 \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 )}{240250}+\frac {\left (3 \left (7379+264 \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 )}{240250} \\ & = -\frac {1584 \sqrt {1+2 x}}{24025}-\frac {(5-4 x) (1+2 x)^{7/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(1143-1088 x) (1+2 x)^{3/2}}{9610 \left (2+3 x+5 x^2\right )}+\frac {3 \left (7379-264 \sqrt {35}\right ) \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{48050 \sqrt {10 \left (2+\sqrt {35}\right )}}-\frac {3 \left (7379-264 \sqrt {35}\right ) \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{48050 \sqrt {10 \left (2+\sqrt {35}\right )}}-\frac {\left (3 \left (7379+264 \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 )}{120125}-\frac {\left (3 \left (7379+264 \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 )}{120125} \\ & = -\frac {1584 \sqrt {1+2 x}}{24025}-\frac {(5-4 x) (1+2 x)^{7/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {(1143-1088 x) (1+2 x)^{3/2}}{9610 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{310} \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 )}{24025}+\frac {3 \sqrt {\frac {1}{310} \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 )}{24025}+\frac {3 \left (7379-264 \sqrt {35}\right ) \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{48050 \sqrt {10 \left (2+\sqrt {35}\right )}}-\frac {3 \left (7379-264 \sqrt {35}\right ) \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{48050 \sqrt {10 \left (2+\sqrt {35}\right )}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.87 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.46 \[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {-\frac {155 \sqrt {1+2 x} \left (27977+87291 x+144557 x^2+86150 x^3\right )}{2 \left (2+3 x+5 x^2\right )^2}+3 \sqrt {155 \left (250141922-52010281 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+3 \sqrt {155 \left (250141922+52010281 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{3723875} \]

[In]

Integrate[(1 + 2*x)^(9/2)/(2 + 3*x + 5*x^2)^3,x]

[Out]

((-155*Sqrt[1 + 2*x]*(27977 + 87291*x + 144557*x^2 + 86150*x^3))/(2*(2 + 3*x + 5*x^2)^2) + 3*Sqrt[155*(2501419
22 - (52010281*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] + 3*Sqrt[155*(250141922 + (520102
81*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]])/3723875

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.05

method	result	size
pseudoelliptic	\(-\frac {792 \left (\frac {11999 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (\sqrt {5}-\frac {39535 \sqrt {7}}{23998}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{81840}-\frac {11999 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (\sqrt {5}-\frac {39535 \sqrt {7}}{23998}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{81840}+\frac {1723 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (x^{3}+\frac {144557}{86150} x^{2}+\frac {87291}{86150} x +\frac {27977}{86150}\right ) \sqrt {1+2 x}}{792}+\left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right ) \left (\sqrt {5}\, \sqrt {7}+\frac {7379}{264}\right )\right )}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (5 x^{2}+3 x +2\right )^{2}}\)	\(328\)
derivativedivides	\(\frac {-\frac {3446 \left (1+2 x \right )^{\frac {7}{2}}}{961}-\frac {30664 \left (1+2 x \right )^{\frac {5}{2}}}{24025}-\frac {29386 \left (1+2 x \right )^{\frac {3}{2}}}{24025}-\frac {77616 \sqrt {1+2 x}}{24025}}{\left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}+\frac {3 \left (-23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{14895500}+\frac {3 \left (16368 \sqrt {5}\, \sqrt {7}+\frac {\left (-23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{744775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3 \left (23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{14895500}+\frac {3 \left (16368 \sqrt {5}\, \sqrt {7}-\frac {\left (23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{744775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(435\)
default	\(\frac {-\frac {3446 \left (1+2 x \right )^{\frac {7}{2}}}{961}-\frac {30664 \left (1+2 x \right )^{\frac {5}{2}}}{24025}-\frac {29386 \left (1+2 x \right )^{\frac {3}{2}}}{24025}-\frac {77616 \sqrt {1+2 x}}{24025}}{\left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}+\frac {3 \left (-23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{14895500}+\frac {3 \left (16368 \sqrt {5}\, \sqrt {7}+\frac {\left (-23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{744775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3 \left (23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{14895500}+\frac {3 \left (16368 \sqrt {5}\, \sqrt {7}-\frac {\left (23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{744775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(435\)
trager	\(-\frac {\left (86150 x^{3}+144557 x^{2}+87291 x +27977\right ) \sqrt {1+2 x}}{48050 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {6 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right ) \ln \left (\frac {362233958400 x \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{5}+115421533122664640 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{3} x +60773838593955840 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{3}+3575545690899462000 \sqrt {1+2 x}\, \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2}+6367063751256090807636 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right ) x +4313191619771312411552 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )-3113705955055050326603275 \sqrt {1+2 x}}{2480 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2} x [phone] x -208041124}\right )}{24025}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2}[phone]\right ) \ln \left (\frac {36223395840 \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2}[phone]\right ) \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{4} x +3072346456740640 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2}[phone]\right ) x -6077383859395584 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2}[phone]\right )-110841916417883322000 \sqrt {1+2 x}\, \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2}-217589498184877183750 \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2}[phone]\right ) x -794655418120440646000 \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2}[phone]\right )-118884731389716793460850625 \sqrt {1+2 x}}{2480 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2} x [phone] x [phone]}\right )}{3723875}\)	\(457\)
risch	\(-\frac {\left (86150 x^{3}+144557 x^{2}+87291 x +27977\right ) \sqrt {1+2 x}}{48050 \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 {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{7447750}+\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 {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{2979100}-\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 ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{744775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {23721 \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}}{1489550 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {1584 \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}}{24025 \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 {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{7447750}-\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 {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{2979100}-\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 ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{744775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {23721 \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}}{1489550 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {1584 \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}}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(643\)

[In]

int((1+2*x)^(9/2)/(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)

[Out]

-792/961*(11999/81840*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(x^2+3/5*x+2/5)^2*(5^(1/2)-395
35/23998*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)-11999/81840*(10
*5^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(x^2+3/5*x+2/5)^2*(5^(1/2)-39535/23998*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)+1723/792*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(
x^3+144557/86150*x^2+87291/86150*x+27977/86150)*(1+2*x)^(1/2)+(x^2+3/5*x+2/5)^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)))*(5^(1/2)*7^(1/2)+7379/264))/(10*5^(1/2)*7^(1/2)-20)^(1/2)/(5*x
^2+3*x+2)^2

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.98 \[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\sqrt {155} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {468092529 i \, \sqrt {31} - 2251277298} \log \left (\sqrt {155} \sqrt {468092529 i \, \sqrt {31} - 2251277298} {\left (7907 i \, \sqrt {31} + 8184\right )} + 30076769625 \, \sqrt {2 \, x + 1}\right ) - \sqrt {155} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {468092529 i \, \sqrt {31} - 2251277298} \log \left (\sqrt {155} \sqrt {468092529 i \, \sqrt {31} - 2251277298} {\left (-7907 i \, \sqrt {31} - 8184\right )} + 30076769625 \, \sqrt {2 \, x + 1}\right ) - \sqrt {155} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-468092529 i \, \sqrt {31} - 2251277298} \log \left (\sqrt {155} {\left (7907 i \, \sqrt {31} - 8184\right )} \sqrt {-468092529 i \, \sqrt {31} - 2251277298} + 30076769625 \, \sqrt {2 \, x + 1}\right ) + \sqrt {155} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-468092529 i \, \sqrt {31} - 2251277298} \log \left (\sqrt {155} {\left (-7907 i \, \sqrt {31} + 8184\right )} \sqrt {-468092529 i \, \sqrt {31} - 2251277298} + 30076769625 \, \sqrt {2 \, x + 1}\right ) - 155 \, {\left (86150 \, x^{3} + 144557 \, x^{2} + 87291 \, x + 27977\right )} \sqrt {2 \, x + 1}}{7447750 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]

[In]

integrate((1+2*x)^(9/2)/(5*x^2+3*x+2)^3,x, algorithm="fricas")

[Out]

1/7447750*(sqrt(155)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(468092529*I*sqrt(31) - 2251277298)*log(sqrt(15
5)*sqrt(468092529*I*sqrt(31) - 2251277298)*(7907*I*sqrt(31) + 8184) + 30076769625*sqrt(2*x + 1)) - sqrt(155)*(
25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(468092529*I*sqrt(31) - 2251277298)*log(sqrt(155)*sqrt(468092529*I*sq
rt(31) - 2251277298)*(-7907*I*sqrt(31) - 8184) + 30076769625*sqrt(2*x + 1)) - sqrt(155)*(25*x^4 + 30*x^3 + 29*
x^2 + 12*x + 4)*sqrt(-468092529*I*sqrt(31) - 2251277298)*log(sqrt(155)*(7907*I*sqrt(31) - 8184)*sqrt(-46809252
9*I*sqrt(31) - 2251277298) + 30076769625*sqrt(2*x + 1)) + sqrt(155)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt
(-468092529*I*sqrt(31) - 2251277298)*log(sqrt(155)*(-7907*I*sqrt(31) + 8184)*sqrt(-468092529*I*sqrt(31) - 2251
277298) + 30076769625*sqrt(2*x + 1)) - 155*(86150*x^3 + 144557*x^2 + 87291*x + 27977)*sqrt(2*x + 1))/(25*x^4 +
30*x^3 + 29*x^2 + 12*x + 4)

Sympy [F(-1)]

Timed out. \[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((1+2*x)**(9/2)/(5*x**2+3*x+2)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {{\left (2 \, x + 1\right )}^{\frac {9}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}} \,d x } \]

[In]

integrate((1+2*x)^(9/2)/(5*x^2+3*x+2)^3,x, algorithm="maxima")

[Out]

integrate((2*x + 1)^(9/2)/(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. \(2 (222) = 444\).

Time = 0.70 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.05 \[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((1+2*x)^(9/2)/(5*x^2+3*x+2)^3,x, algorithm="giac")

[Out]

3/1788204775000*sqrt(31)*(1549590*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 7379*sqr
t(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 14758*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 3099180*(7/5)
^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 9055200*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) +
18110400*(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)) + 3/1788204775000*sqrt(31)*(1549590*sqrt(31)*(7/5)^(3/4)*(2*sqrt
(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 7379*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 14758*(7/5)^(
3/4)*(140*sqrt(35) + 2450)^(3/2) + 3099180*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 9055200*s
qrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 18110400*(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)) + 3/35764095500
00*sqrt(31)*(7379*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 1549590*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqr
t(35) + 2450)*(2*sqrt(35) - 35) - 3099180*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 14758*(7/
5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 9055200*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 18110400*(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) - 3/3576409550000*sqrt(31)*(7379*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 1549590*sqrt(31)*(7
/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 3099180*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(3
5) + 2450) + 14758*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 9055200*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) +
2450) - 18110400*(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) - 2/24025*(43075*(2*x + 1)^(7/2) + 15332*(2*x + 1)^(5/2) + 14693*(2*x + 1)^(3/2)
+ 38808*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)^2

Mupad [B] (verification not implemented)

Time = 10.02 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.78 \[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {77616\,\sqrt {2\,x+1}}{600625}+\frac {29386\,{\left (2\,x+1\right )}^{3/2}}{600625}+\frac {30664\,{\left (2\,x+1\right )}^{5/2}}{600625}+\frac {3446\,{\left (2\,x+1\right )}^{7/2}}{24025}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}\,23380272{}\mathrm {i}}{45093798828125\,\left (-\frac {1294074674928}{9018759765625}+\frac {\sqrt {31}\,43206742656{}\mathrm {i}}{9018759765625}\right )}-\frac {46760544\,\sqrt {31}\,\sqrt {155}\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}}{1397907763671875\,\left (-\frac {1294074674928}{9018759765625}+\frac {\sqrt {31}\,43206742656{}\mathrm {i}}{9018759765625}\right )}\right )\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,3{}\mathrm {i}}{3723875}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}\,23380272{}\mathrm {i}}{45093798828125\,\left (\frac {1294074674928}{9018759765625}+\frac {\sqrt {31}\,43206742656{}\mathrm {i}}{9018759765625}\right )}+\frac {46760544\,\sqrt {31}\,\sqrt {155}\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}}{1397907763671875\,\left (\frac {1294074674928}{9018759765625}+\frac {\sqrt {31}\,43206742656{}\mathrm {i}}{9018759765625}\right )}\right )\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,3{}\mathrm {i}}{3723875} \]

[In]

int((2*x + 1)^(9/2)/(3*x + 5*x^2 + 2)^3,x)

[Out]

((77616*(2*x + 1)^(1/2))/600625 + (29386*(2*x + 1)^(3/2))/600625 + (30664*(2*x + 1)^(5/2))/600625 + (3446*(2*x
+ 1)^(7/2))/24025)/((112*x)/25 - (86*(2*x + 1)^2)/25 + (8*(2*x + 1)^3)/5 - (2*x + 1)^4 + 7/25) - (155^(1/2)*a
tan((155^(1/2)*(- 31^(1/2)*52010281i - 250141922)^(1/2)*(2*x + 1)^(1/2)*23380272i)/(45093798828125*((31^(1/2)*
43206742656i)/9018759765625 - 1294074674928/9018759765625)) - (46760544*31^(1/2)*155^(1/2)*(- 31^(1/2)*5201028
1i - 250141922)^(1/2)*(2*x + 1)^(1/2))/(1397907763671875*((31^(1/2)*43206742656i)/9018759765625 - 129407467492
8/9018759765625)))*(- 31^(1/2)*52010281i - 250141922)^(1/2)*3i)/3723875 + (155^(1/2)*atan((155^(1/2)*(31^(1/2)
*52010281i - 250141922)^(1/2)*(2*x + 1)^(1/2)*23380272i)/(45093798828125*((31^(1/2)*43206742656i)/901875976562
5 + 1294074674928/9018759765625)) + (46760544*31^(1/2)*155^(1/2)*(31^(1/2)*52010281i - 250141922)^(1/2)*(2*x +
1)^(1/2))/(1397907763671875*((31^(1/2)*43206742656i)/9018759765625 + 1294074674928/9018759765625)))*(31^(1/2)
*52010281i - 250141922)^(1/2)*3i)/3723875