\(\int \frac {(1+2 x)^{5/2}}{(2+3 x+5 x^2)^2} \, dx\) [2317]

Optimal result

Integrand size = 22, antiderivative size = 283 \[ \int \frac {(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=-\frac {8}{155} \sqrt {1+2 x}-\frac {(5-4 x) (1+2 x)^{3/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{155} \sqrt {\frac {2}{155} \left (32678+10325 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{155} \sqrt {\frac {2}{155} \left (32678+10325 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{155} \sqrt {\frac {1}{310} \left (-32678+10325 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )-\frac {1}{155} \sqrt {\frac {1}{310} \left (-32678+10325 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \]

[Out]

-1/31*(5-4*x)*(1+2*x)^(3/2)/(5*x^2+3*x+2)-8/155*(1+2*x)^(1/2)+1/48050*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*
35^(1/2))^(1/2))*(-10130180+3200750*35^(1/2))^(1/2)-1/48050*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^
(1/2))*(-10130180+3200750*35^(1/2))^(1/2)-1/24025*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35
^(1/2))^(1/2))*(10130180+3200750*35^(1/2))^(1/2)+1/24025*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20
+10*35^(1/2))^(1/2))*(10130180+3200750*35^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 283, 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, 838, 840, 1183, 648, 632, 210, 642} \[ \int \frac {(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=-\frac {1}{155} \sqrt {\frac {2}{155} \left (32678+10325 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {1}{155} \sqrt {\frac {2}{155} \left (32678+10325 \sqrt {35}\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {(5-4 x) (2 x+1)^{3/2}}{31 \left (5 x^2+3 x+2\right )}-\frac {8}{155} \sqrt {2 x+1}+\frac {1}{155} \sqrt {\frac {1}{310} \left (10325 \sqrt {35}-32678\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{155} \sqrt {\frac {1}{310} \left (10325 \sqrt {35}-32678\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right ) \]

[In]

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

[Out]

(-8*Sqrt[1 + 2*x])/155 - ((5 - 4*x)*(1 + 2*x)^(3/2))/(31*(2 + 3*x + 5*x^2)) - (Sqrt[(2*(32678 + 10325*Sqrt[35]
))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/155 + (Sqrt[(2*(32678 +
10325*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/155 + (S
qrt[(-32678 + 10325*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/155 -
(Sqrt[(-32678 + 10325*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/155

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 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)^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{31} \int \frac {(19-4 x) \sqrt {1+2 x}}{2+3 x+5 x^2} \, dx \\ & = -\frac {8}{155} \sqrt {1+2 x}-\frac {(5-4 x) (1+2 x)^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{155} \int \frac {111+194 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx \\ & = -\frac {8}{155} \sqrt {1+2 x}-\frac {(5-4 x) (1+2 x)^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {2}{155} \text {Subst}\left (\int \frac {28+194 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right ) \\ & = -\frac {8}{155} \sqrt {1+2 x}-\frac {(5-4 x) (1+2 x)^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {28 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (28-194 \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 )}{155 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {28 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (28-194 \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 )}{155 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = -\frac {8}{155} \sqrt {1+2 x}-\frac {(5-4 x) (1+2 x)^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{775} \left (97+2 \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}{775} \left (97+2 \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}{155} \sqrt {\frac {1}{310} \left (-32678+10325 \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}{155} \sqrt {\frac {1}{310} \left (-32678+10325 \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 {8}{155} \sqrt {1+2 x}-\frac {(5-4 x) (1+2 x)^{3/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{155} \sqrt {\frac {1}{310} \left (-32678+10325 \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}{155} \sqrt {\frac {1}{310} \left (-32678+10325 \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}{775} \left (2 \left (97+2 \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}{775} \left (2 \left (97+2 \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 {8}{155} \sqrt {1+2 x}-\frac {(5-4 x) (1+2 x)^{3/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{155} \sqrt {\frac {2}{155} \left (32678+10325 \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}{155} \sqrt {\frac {2}{155} \left (32678+10325 \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}{155} \sqrt {\frac {1}{310} \left (-32678+10325 \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}{155} \sqrt {\frac {1}{310} \left (-32678+10325 \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 = 1.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.46 \[ \int \frac {(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {2 \left (-\frac {155 \sqrt {1+2 x} (41+54 x)}{4+6 x+10 x^2}+\sqrt {155 \left (32678-9269 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+\sqrt {155 \left (32678+9269 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{24025} \]

[In]

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

[Out]

(2*((-155*Sqrt[1 + 2*x]*(41 + 54*x))/(4 + 6*x + 10*x^2) + Sqrt[155*(32678 - (9269*I)*Sqrt[31])]*ArcTan[Sqrt[(-
2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] + Sqrt[155*(32678 + (9269*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*
Sqrt[1 + 2*x]]))/24025

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.10

method	result	size
pseudoelliptic	\(\frac {-1320 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \left (\sqrt {5}-\frac {505 \sqrt {7}}{264}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )+1320 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \left (\sqrt {5}-\frac {505 \sqrt {7}}{264}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )-16740 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (x +\frac {41}{54}\right ) \sqrt {1+2 x}+6200 \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right ) \left (\sqrt {5}\, \sqrt {7}+\frac {97}{2}\right ) \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (240250 x^{2}+144150 x +96100\right )}\)	\(312\)
derivativedivides	\(\frac {-\frac {108 \left (1+2 x \right )^{\frac {3}{2}}}{775}-\frac {56 \sqrt {1+2 x}}{775}}{\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}}-\frac {\left (264 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-505 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{48050}-\frac {2 \left (-124 \sqrt {5}\, \sqrt {7}+\frac {\left (264 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-505 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{4805 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (264 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-505 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{48050}+\frac {2 \left (124 \sqrt {5}\, \sqrt {7}-\frac {\left (264 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-505 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{4805 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(415\)
default	\(\frac {-\frac {108 \left (1+2 x \right )^{\frac {3}{2}}}{775}-\frac {56 \sqrt {1+2 x}}{775}}{\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}}-\frac {\left (264 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-505 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{48050}-\frac {2 \left (-124 \sqrt {5}\, \sqrt {7}+\frac {\left (264 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-505 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{4805 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (264 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-505 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{48050}+\frac {2 \left (124 \sqrt {5}\, \sqrt {7}-\frac {\left (264 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-505 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{4805 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(415\)
trager	\(-\frac {\left (41+54 x \right ) \sqrt {1+2 x}}{155 \left (5 x^{2}+3 x +2\right )}+\frac {2 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right ) \ln \left (-\frac {10763200 x \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{5}[phone] \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{3} x [phone] \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{3}-1956115500 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {1+2 x}[phone] \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right ) x [phone] \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )[phone] \sqrt {1+2 x}}{620 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2} x +4871 x -37076}\right )}{155}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2}+2532545\right ) \ln \left (-\frac {2152640 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2}+2532545\right ) \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{4} x [phone] \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2}+2532545\right ) x -257455744 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2}+2532545\right )[phone] \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {1+2 x}-9025451250 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2}+2532545\right ) x -20781098000 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2}+2532545\right )[phone] \sqrt {1+2 x}}{620 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}+8104144 \textit {\_Z}^{2}[phone]\right )^{2} x +60485 x +37076}\right )}{24025}\)	\(449\)
risch	\(-\frac {\left (41+54 x \right ) \sqrt {1+2 x}}{155 \left (5 x^{2}+3 x +2\right )}-\frac {132 \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}}{24025}+\frac {101 \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}}{9610}-\frac {264 \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 )}{4805 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {101 \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}}{4805 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {8 \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}}{155 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {132 \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}}{24025}-\frac {101 \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}}{9610}-\frac {264 \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 )}{4805 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {101 \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}}{4805 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {8 \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}}{155 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(633\)

[In]

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

[Out]

6200*(-33/155*(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)*(5^(1/2)-505/264*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)+33/155*(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)*(5^(1/2)-505/264*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)-27/10*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(x+41/54)*(1+2*x)^(1/2)+(arct
an((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)+97/2)*(x^2+3/5*x+2/5
))/(10*5^(1/2)*7^(1/2)-20)^(1/2)/(240250*x^2+144150*x+96100)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.45 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.87 \[ \int \frac {(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {\sqrt {155} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {37076 i \, \sqrt {31} - 130712} \log \left (\sqrt {155} \sqrt {37076 i \, \sqrt {31} - 130712} {\left (101 i \, \sqrt {31} + 62\right )} + 3200750 \, \sqrt {2 \, x + 1}\right ) - \sqrt {155} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {37076 i \, \sqrt {31} - 130712} \log \left (\sqrt {155} \sqrt {37076 i \, \sqrt {31} - 130712} {\left (-101 i \, \sqrt {31} - 62\right )} + 3200750 \, \sqrt {2 \, x + 1}\right ) - \sqrt {155} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-37076 i \, \sqrt {31} - 130712} \log \left (\sqrt {155} {\left (101 i \, \sqrt {31} - 62\right )} \sqrt {-37076 i \, \sqrt {31} - 130712} + 3200750 \, \sqrt {2 \, x + 1}\right ) + \sqrt {155} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-37076 i \, \sqrt {31} - 130712} \log \left (\sqrt {155} {\left (-101 i \, \sqrt {31} + 62\right )} \sqrt {-37076 i \, \sqrt {31} - 130712} + 3200750 \, \sqrt {2 \, x + 1}\right ) - 310 \, {\left (54 \, x + 41\right )} \sqrt {2 \, x + 1}}{48050 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \]

[In]

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

[Out]

1/48050*(sqrt(155)*(5*x^2 + 3*x + 2)*sqrt(37076*I*sqrt(31) - 130712)*log(sqrt(155)*sqrt(37076*I*sqrt(31) - 130
712)*(101*I*sqrt(31) + 62) + 3200750*sqrt(2*x + 1)) - sqrt(155)*(5*x^2 + 3*x + 2)*sqrt(37076*I*sqrt(31) - 1307
12)*log(sqrt(155)*sqrt(37076*I*sqrt(31) - 130712)*(-101*I*sqrt(31) - 62) + 3200750*sqrt(2*x + 1)) - sqrt(155)*
(5*x^2 + 3*x + 2)*sqrt(-37076*I*sqrt(31) - 130712)*log(sqrt(155)*(101*I*sqrt(31) - 62)*sqrt(-37076*I*sqrt(31)
- 130712) + 3200750*sqrt(2*x + 1)) + sqrt(155)*(5*x^2 + 3*x + 2)*sqrt(-37076*I*sqrt(31) - 130712)*log(sqrt(155
)*(-101*I*sqrt(31) + 62)*sqrt(-37076*I*sqrt(31) - 130712) + 3200750*sqrt(2*x + 1)) - 310*(54*x + 41)*sqrt(2*x
+ 1))/(5*x^2 + 3*x + 2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (196) = 392\).

Time = 0.68 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.20 \[ \int \frac {(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/5768402500*sqrt(31)*(20370*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 97*sqrt(31)*(
7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 194*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 40740*(7/5)^(3/4)*sqrt
(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 68600*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 137200*(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)) + 1/5768402500*sqrt(31)*(20370*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140
*sqrt(35) + 2450) - 97*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 194*(7/5)^(3/4)*(140*sqrt(35) + 245
0)^(3/2) + 40740*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 68600*sqrt(31)*(7/5)^(1/4)*sqrt(-14
0*sqrt(35) + 2450) + 137200*(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)) + 1/11536805000*sqrt(31)*(97*sqrt(31)*(7/5)^(
3/4)*(140*sqrt(35) + 2450)^(3/2) + 20370*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 40
740*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 194*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) +
68600*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 137200*(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) - 1/11536805000*sqrt(31)*(97*sqrt(31)
*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 20370*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) -
35) - 40740*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 194*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^
(3/2) + 68600*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 137200*(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) - 4/155*(27*(2*x + 1)^(3/2)
+ 14*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.73 \[ \int \frac {(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=-\frac {\frac {56\,\sqrt {2\,x+1}}{775}+\frac {108\,{\left (2\,x+1\right )}^{3/2}}{775}}{{\left (2\,x+1\right )}^2-\frac {8\,x}{5}+\frac {3}{5}}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-32678-\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}\,38272{}\mathrm {i}}{375390625\,\left (-\frac {27058304}{75078125}+\frac {\sqrt {31}\,535808{}\mathrm {i}}{75078125}\right )}-\frac {76544\,\sqrt {31}\,\sqrt {155}\,\sqrt {-32678-\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}}{11637109375\,\left (-\frac {27058304}{75078125}+\frac {\sqrt {31}\,535808{}\mathrm {i}}{75078125}\right )}\right )\,\sqrt {-32678-\sqrt {31}\,9269{}\mathrm {i}}\,2{}\mathrm {i}}{24025}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-32678+\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}\,38272{}\mathrm {i}}{375390625\,\left (\frac {27058304}{75078125}+\frac {\sqrt {31}\,535808{}\mathrm {i}}{75078125}\right )}+\frac {76544\,\sqrt {31}\,\sqrt {155}\,\sqrt {-32678+\sqrt {31}\,9269{}\mathrm {i}}\,\sqrt {2\,x+1}}{11637109375\,\left (\frac {27058304}{75078125}+\frac {\sqrt {31}\,535808{}\mathrm {i}}{75078125}\right )}\right )\,\sqrt {-32678+\sqrt {31}\,9269{}\mathrm {i}}\,2{}\mathrm {i}}{24025} \]

[In]

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

[Out]

(155^(1/2)*atan((155^(1/2)*(31^(1/2)*9269i - 32678)^(1/2)*(2*x + 1)^(1/2)*38272i)/(375390625*((31^(1/2)*535808
i)/75078125 + 27058304/75078125)) + (76544*31^(1/2)*155^(1/2)*(31^(1/2)*9269i - 32678)^(1/2)*(2*x + 1)^(1/2))/
(11637109375*((31^(1/2)*535808i)/75078125 + 27058304/75078125)))*(31^(1/2)*9269i - 32678)^(1/2)*2i)/24025 - (1
55^(1/2)*atan((155^(1/2)*(- 31^(1/2)*9269i - 32678)^(1/2)*(2*x + 1)^(1/2)*38272i)/(375390625*((31^(1/2)*535808
i)/75078125 - 27058304/75078125)) - (76544*31^(1/2)*155^(1/2)*(- 31^(1/2)*9269i - 32678)^(1/2)*(2*x + 1)^(1/2)
)/(11637109375*((31^(1/2)*535808i)/75078125 - 27058304/75078125)))*(- 31^(1/2)*9269i - 32678)^(1/2)*2i)/24025
- ((56*(2*x + 1)^(1/2))/775 + (108*(2*x + 1)^(3/2))/775)/((2*x + 1)^2 - (8*x)/5 + 3/5)