\(\int \frac {(1+2 x)^{5/2}}{(2+3 x+5 x^2)^3} \, dx\) [2325]

Optimal result

Integrand size = 22, antiderivative size = 300 \[ \int \frac {(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=-\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3}{961} \sqrt {\frac {1}{310} \left (15082+2705 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {3}{961} \sqrt {\frac {1}{310} \left (15082+2705 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}-\frac {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922} \]

[Out]

-1/62*(5-4*x)*(1+2*x)^(3/2)/(5*x^2+3*x+2)^2+3/1922*(11+78*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)+3/595820*ln(5+10*x+35
^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-4675420+838550*35^(1/2))^(1/2)-3/595820*ln(5+10*x+35^(1/2)+(1+2
*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-4675420+838550*35^(1/2))^(1/2)-3/297910*arctan((-10*(1+2*x)^(1/2)+(20+10*3
5^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(4675420+838550*35^(1/2))^(1/2)+3/297910*arctan((10*(1+2*x)^(1/2)+(20
+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(4675420+838550*35^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.31 (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 = {752, 834, 840, 1183, 648, 632, 210, 642} \[ \int \frac {(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=-\frac {3}{961} \sqrt {\frac {1}{310} \left (15082+2705 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {3}{961} \sqrt {\frac {1}{310} \left (15082+2705 \sqrt {35}\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {(5-4 x) (2 x+1)^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}+\frac {3 (78 x+11) \sqrt {2 x+1}}{1922 \left (5 x^2+3 x+2\right )}+\frac {3 \sqrt {\frac {1}{310} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922}-\frac {3 \sqrt {\frac {1}{310} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922} \]

[In]

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

[Out]

-1/62*((5 - 4*x)*(1 + 2*x)^(3/2))/(2 + 3*x + 5*x^2)^2 + (3*Sqrt[1 + 2*x]*(11 + 78*x))/(1922*(2 + 3*x + 5*x^2))
- (3*Sqrt[(15082 + 2705*Sqrt[35])/310]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt
[35])]])/961 + (3*Sqrt[(15082 + 2705*Sqrt[35])/310]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[1
0*(-2 + Sqrt[35])]])/961 + (3*Sqrt[(-15082 + 2705*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1
+ 2*x] + 5*(1 + 2*x)])/1922 - (3*Sqrt[(-15082 + 2705*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sq
rt[1 + 2*x] + 5*(1 + 2*x)])/1922

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 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*((f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/
((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*
(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*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, 0] && (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 {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {1}{62} \int \frac {\sqrt {1+2 x} (27+12 x)}{\left (2+3 x+5 x^2\right )^2} \, dx \\ & = -\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {201+234 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{1922} \\ & = -\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {1}{961} \text {Subst}\left (\int \frac {168+234 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right ) \\ & = -\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {168 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (168-234 \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 )}{1922 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {168 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (168-234 \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 )}{1922 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = -\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\left (3 \left (39+4 \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 )}{9610}+\frac {\left (3 \left (39+4 \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 )}{9610}+\frac {\left (3 \sqrt {\frac {1}{310} \left (-15082+2705 \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 )}{1922}-\frac {\left (3 \sqrt {\frac {1}{310} \left (-15082+2705 \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 )}{1922} \\ & = -\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}-\frac {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}-\frac {\left (3 \left (39+4 \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 )}{4805}-\frac {\left (3 \left (39+4 \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 )}{4805} \\ & = -\frac {(5-4 x) (1+2 x)^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 \sqrt {1+2 x} (11+78 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3}{961} \sqrt {\frac {7541}{155}+\frac {541 \sqrt {35}}{62}} \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 {3}{961} \sqrt {\frac {7541}{155}+\frac {541 \sqrt {35}}{62}} \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 {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}-\frac {3 \sqrt {\frac {1}{310} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.41 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.48 \[ \int \frac {(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {155 \sqrt {1+2 x} \left (-89+381 x+1115 x^2+1170 x^3\right )}{2 \left (2+3 x+5 x^2\right )^2}+3 \sqrt {155 \left (15082-961 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+3 \sqrt {155 \left (15082+961 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{148955} \]

[In]

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

[Out]

((155*Sqrt[1 + 2*x]*(-89 + 381*x + 1115*x^2 + 1170*x^3))/(2*(2 + 3*x + 5*x^2)^2) + 3*Sqrt[155*(15082 - (961*I)
*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] + 3*Sqrt[155*(15082 + (961*I)*Sqrt[31])]*ArcTan[Sq
rt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]])/148955

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.09

method	result	size
pseudoelliptic	\(\frac {-\frac {1635 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (\sqrt {5}-\frac {235 \sqrt {7}}{218}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{59582}+\frac {1635 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (\sqrt {5}-\frac {235 \sqrt {7}}{218}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{59582}+\frac {585 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (x^{3}+\frac {223}{234} x^{2}+\frac {127}{390} x -\frac {89}{1170}\right ) \sqrt {1+2 x}}{961}+\frac {300 \left (\sqrt {5}\, \sqrt {7}+\frac {39}{4}\right ) \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right )}{961}}{\left (5 x^{2}+3 x +2\right )^{2} \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(328\)
derivativedivides	\(\frac {\frac {1170 \left (1+2 x \right )^{\frac {7}{2}}}{961}-\frac {1280 \left (1+2 x \right )^{\frac {5}{2}}}{961}+\frac {574 \left (1+2 x \right )^{\frac {3}{2}}}{961}-\frac {1176 \sqrt {1+2 x}}{961}}{\left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}+\frac {3 \left (-218 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+235 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{595820}+\frac {3 \left (248 \sqrt {5}\, \sqrt {7}+\frac {\left (-218 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+235 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{29791 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3 \left (218 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-235 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{595820}+\frac {3 \left (248 \sqrt {5}\, \sqrt {7}-\frac {\left (218 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-235 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{29791 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(435\)
default	\(\frac {\frac {1170 \left (1+2 x \right )^{\frac {7}{2}}}{961}-\frac {1280 \left (1+2 x \right )^{\frac {5}{2}}}{961}+\frac {574 \left (1+2 x \right )^{\frac {3}{2}}}{961}-\frac {1176 \sqrt {1+2 x}}{961}}{\left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}+\frac {3 \left (-218 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+235 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{595820}+\frac {3 \left (248 \sqrt {5}\, \sqrt {7}+\frac {\left (-218 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+235 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{29791 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3 \left (218 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-235 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{595820}+\frac {3 \left (248 \sqrt {5}\, \sqrt {7}-\frac {\left (218 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-235 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{29791 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(435\)
trager	\(\frac {\left (1170 x^{3}+1115 x^{2}+381 x -89\right ) \sqrt {1+2 x}}{1922 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {6 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right ) \ln \left (-\frac {16925900800 x \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{5}[phone] \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{3} x [phone] \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{3}-19340317200 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {1+2 x}[phone] \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right ) x [phone] \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )[phone] \sqrt {1+2 x}}{2480 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} x +12199 x -3844}\right )}{961}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}+1168855\right ) \ln \left (\frac {1692590080 \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}+1168855\right ) \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{4} x [phone] \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}+1168855\right ) x -5247029248 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}+1168855\right )-599549833200 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {1+2 x}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}+1168855\right ) x -31817941200 \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}+1168855\right )-14410550930375 \sqrt {1+2 x}}{2480 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} x +17965 x +3844}\right )}{148955}\)	\(458\)
risch	\(\frac {\left (1170 x^{3}+1115 x^{2}+381 x -89\right ) \sqrt {1+2 x}}{1922 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {327 \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}}{297910}+\frac {141 \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}}{119164}-\frac {327 \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 )}{29791 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {141 \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}}{59582 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {24 \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}}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {327 \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}}{297910}-\frac {141 \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}}{119164}-\frac {327 \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 )}{29791 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {141 \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}}{59582 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {24 \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}}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(643\)

[In]

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

[Out]

300/961*(-109/1240*(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)-235/21
8*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)+109/1240*(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)-235/218*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)+39/20*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(x^3+223/234*x^2+1
27/390*x-89/1170)*(1+2*x)^(1/2)+(5^(1/2)*7^(1/2)+39/4)*(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))))/(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 = 0.28 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.02 \[ \int \frac {(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\sqrt {155} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {8649 i \, \sqrt {31} - 135738} \log \left (\sqrt {155} \sqrt {8649 i \, \sqrt {31} - 135738} {\left (47 i \, \sqrt {31} + 124\right )} + 1257825 \, \sqrt {2 \, x + 1}\right ) - \sqrt {155} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {8649 i \, \sqrt {31} - 135738} \log \left (\sqrt {155} \sqrt {8649 i \, \sqrt {31} - 135738} {\left (-47 i \, \sqrt {31} - 124\right )} + 1257825 \, \sqrt {2 \, x + 1}\right ) - \sqrt {155} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-8649 i \, \sqrt {31} - 135738} \log \left (\sqrt {155} {\left (47 i \, \sqrt {31} - 124\right )} \sqrt {-8649 i \, \sqrt {31} - 135738} + 1257825 \, \sqrt {2 \, x + 1}\right ) + \sqrt {155} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-8649 i \, \sqrt {31} - 135738} \log \left (\sqrt {155} {\left (-47 i \, \sqrt {31} + 124\right )} \sqrt {-8649 i \, \sqrt {31} - 135738} + 1257825 \, \sqrt {2 \, x + 1}\right ) + 155 \, {\left (1170 \, x^{3} + 1115 \, x^{2} + 381 \, x - 89\right )} \sqrt {2 \, x + 1}}{297910 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]

[In]

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

[Out]

1/297910*(sqrt(155)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(8649*I*sqrt(31) - 135738)*log(sqrt(155)*sqrt(86
49*I*sqrt(31) - 135738)*(47*I*sqrt(31) + 124) + 1257825*sqrt(2*x + 1)) - sqrt(155)*(25*x^4 + 30*x^3 + 29*x^2 +
12*x + 4)*sqrt(8649*I*sqrt(31) - 135738)*log(sqrt(155)*sqrt(8649*I*sqrt(31) - 135738)*(-47*I*sqrt(31) - 124)
+ 1257825*sqrt(2*x + 1)) - sqrt(155)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(-8649*I*sqrt(31) - 135738)*log
(sqrt(155)*(47*I*sqrt(31) - 124)*sqrt(-8649*I*sqrt(31) - 135738) + 1257825*sqrt(2*x + 1)) + sqrt(155)*(25*x^4
+ 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(-8649*I*sqrt(31) - 135738)*log(sqrt(155)*(-47*I*sqrt(31) + 124)*sqrt(-8649*
I*sqrt(31) - 135738) + 1257825*sqrt(2*x + 1)) + 155*(1170*x^3 + 1115*x^2 + 381*x - 89)*sqrt(2*x + 1))/(25*x^4
+ 30*x^3 + 29*x^2 + 12*x + 4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate((2*x + 1)^(5/2)/(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. \(2 (213) = 426\).

Time = 0.72 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.14 \[ \int \frac {(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

3/71528191000*sqrt(31)*(8190*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 39*sqrt(31)*(
7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 78*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 16380*(7/5)^(3/4)*sqrt(
140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 137200*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 274400*(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)) + 3/71528191000*sqrt(31)*(8190*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140
*sqrt(35) + 2450) - 39*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 78*(7/5)^(3/4)*(140*sqrt(35) + 2450
)^(3/2) + 16380*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 137200*sqrt(31)*(7/5)^(1/4)*sqrt(-14
0*sqrt(35) + 2450) + 274400*(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)) + 3/143056382000*sqrt(31)*(39*sqrt(31)*(7/5)^
(3/4)*(140*sqrt(35) + 2450)^(3/2) + 8190*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 16
380*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 78*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 1
37200*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 274400*(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) - 3/143056382000*sqrt(31)*(39*sqrt(31
)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 8190*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) -
35) - 16380*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 78*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(
3/2) + 137200*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 274400*(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) + 2/961*(585*(2*x + 1)^(7/2)
- 640*(2*x + 1)^(5/2) + 287*(2*x + 1)^(3/2) - 588*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)^2

Mupad [B] (verification not implemented)

Time = 10.56 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.82 \[ \int \frac {(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {1176\,\sqrt {2\,x+1}}{24025}-\frac {574\,{\left (2\,x+1\right )}^{3/2}}{24025}+\frac {256\,{\left (2\,x+1\right )}^{5/2}}{4805}-\frac {234\,{\left (2\,x+1\right )}^{7/2}}{4805}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}\,432{}\mathrm {i}}{2886003125\,\left (-\frac {142128}{577200625}+\frac {\sqrt {31}\,12096{}\mathrm {i}}{577200625}\right )}-\frac {864\,\sqrt {31}\,\sqrt {155}\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}}{89466096875\,\left (-\frac {142128}{577200625}+\frac {\sqrt {31}\,12096{}\mathrm {i}}{577200625}\right )}\right )\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,3{}\mathrm {i}}{148955}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}\,432{}\mathrm {i}}{2886003125\,\left (\frac {142128}{577200625}+\frac {\sqrt {31}\,12096{}\mathrm {i}}{577200625}\right )}+\frac {864\,\sqrt {31}\,\sqrt {155}\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}}{89466096875\,\left (\frac {142128}{577200625}+\frac {\sqrt {31}\,12096{}\mathrm {i}}{577200625}\right )}\right )\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,3{}\mathrm {i}}{148955} \]

[In]

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

[Out]

((1176*(2*x + 1)^(1/2))/24025 - (574*(2*x + 1)^(3/2))/24025 + (256*(2*x + 1)^(5/2))/4805 - (234*(2*x + 1)^(7/2
))/4805)/((112*x)/25 - (86*(2*x + 1)^2)/25 + (8*(2*x + 1)^3)/5 - (2*x + 1)^4 + 7/25) - (155^(1/2)*atan((155^(1
/2)*(- 31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2)*432i)/(2886003125*((31^(1/2)*12096i)/577200625 - 142128/57
7200625)) - (864*31^(1/2)*155^(1/2)*(- 31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2))/(89466096875*((31^(1/2)*1
2096i)/577200625 - 142128/577200625)))*(- 31^(1/2)*961i - 15082)^(1/2)*3i)/148955 + (155^(1/2)*atan((155^(1/2)
*(31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2)*432i)/(2886003125*((31^(1/2)*12096i)/577200625 + 142128/5772006
25)) + (864*31^(1/2)*155^(1/2)*(31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2))/(89466096875*((31^(1/2)*12096i)/
577200625 + 142128/577200625)))*(31^(1/2)*961i - 15082)^(1/2)*3i)/148955