3.20.81 \(\int \frac {\sqrt {1+2 x}}{2+3 x+5 x^2} \, dx\)

Optimal. Leaf size=222 \[ \frac {\log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}}-\frac {\log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}}-\sqrt {\frac {2}{5 \left (\sqrt {35}-2\right )}} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\sqrt {\frac {2}{5 \left (\sqrt {35}-2\right )}} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]

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Rubi [A]  time = 0.25, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {699, 1127, 1161, 618, 204, 1164, 628} \begin {gather*} \frac {\log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}}-\frac {\log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}}-\sqrt {\frac {2}{5 \left (\sqrt {35}-2\right )}} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\sqrt {\frac {2}{5 \left (\sqrt {35}-2\right )}} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2/(5*(-2 + Sqrt[35]))]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]]) +
Sqrt[2/(5*(-2 + Sqrt[35]))]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]] + L
og[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)]/Sqrt[10*(2 + Sqrt[35])] - Log[Sqrt[35] + Sq
rt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)]/Sqrt[10*(2 + Sqrt[35])]

Rule 204

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

Rule 618

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 628

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 699

Int[Sqrt[(d_.) + (e_.)*(x_)]/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[2*e, Subst[Int[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}, 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]

Rule 1127

Int[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, Dist[1/2, Int[(q + x^2)/(
a + b*x^2 + c*x^4), x], x] - Dist[1/2, Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && Lt
Q[b^2 - 4*a*c, 0] && PosQ[a*c]

Rule 1161

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

Rule 1164

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

Rubi steps

\begin {align*} \int \frac {\sqrt {1+2 x}}{2+3 x+5 x^2} \, dx &=4 \operatorname {Subst}\left (\int \frac {x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {\sqrt {\frac {7}{5}}-x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\right )+2 \operatorname {Subst}\left (\int \frac {\sqrt {\frac {7}{5}}+x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=\frac {1}{5} \operatorname {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}{5} \operatorname {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 {\operatorname {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 )}{\sqrt {10 \left (2+\sqrt {35}\right )}}+\frac {\operatorname {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 )}{\sqrt {10 \left (2+\sqrt {35}\right )}}\\ &=\frac {\log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}}-\frac {\log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}}-\frac {2}{5} \operatorname {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 {2}{5} \operatorname {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 )\\ &=-\sqrt {\frac {2}{5 \left (-2+\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 )+\sqrt {\frac {2}{5 \left (-2+\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 {\log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}}-\frac {\log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{\sqrt {10 \left (2+\sqrt {35}\right )}}\\ \end {align*}

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Mathematica [C]  time = 0.20, size = 112, normalized size = 0.50 \begin {gather*} \frac {2 \left (\sqrt {-2+i \sqrt {31}} \left (\sqrt {31}-2 i\right ) \tan ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {-2-i \sqrt {31}}}\right )+\sqrt {-2-i \sqrt {31}} \left (\sqrt {31}+2 i\right ) \tan ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {-2+i \sqrt {31}}}\right )\right )}{5 \sqrt {217}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(Sqrt[-2 + I*Sqrt[31]]*(-2*I + Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]] + Sqrt[-2 - I*Sqrt[31
]]*(2*I + Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 + I*Sqrt[31]]]))/(5*Sqrt[217])

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IntegrateAlgebraic [C]  time = 0.43, size = 103, normalized size = 0.46 \begin {gather*} 2 \sqrt {\frac {1}{155} \left (2-i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}-\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right )+2 \sqrt {\frac {1}{155} \left (2+i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {-\frac {2}{7}+\frac {i \sqrt {31}}{7}} \sqrt {2 x+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[(2 - I*Sqrt[31])/155]*ArcTan[Sqrt[-2/7 - (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]] + 2*Sqrt[(2 + I*Sqrt[31])/155]*
ArcTan[Sqrt[-2/7 + (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]]

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fricas [B]  time = 0.43, size = 371, normalized size = 1.67 \begin {gather*} \frac {1}{336350} \, \sqrt {155} 35^{\frac {1}{4}} {\left (2 \, \sqrt {35} \sqrt {31} - 35 \, \sqrt {31}\right )} \sqrt {4 \, \sqrt {35} + 70} \log \left (\frac {124}{7} \, \sqrt {155} 35^{\frac {3}{4}} \sqrt {31} \sqrt {2 \, x + 1} \sqrt {4 \, \sqrt {35} + 70} + 192200 \, x + 19220 \, \sqrt {35} + 96100\right ) - \frac {1}{336350} \, \sqrt {155} 35^{\frac {1}{4}} {\left (2 \, \sqrt {35} \sqrt {31} - 35 \, \sqrt {31}\right )} \sqrt {4 \, \sqrt {35} + 70} \log \left (-\frac {124}{7} \, \sqrt {155} 35^{\frac {3}{4}} \sqrt {31} \sqrt {2 \, x + 1} \sqrt {4 \, \sqrt {35} + 70} + 192200 \, x + 19220 \, \sqrt {35} + 96100\right ) - \frac {2}{5425} \, \sqrt {155} 35^{\frac {3}{4}} \sqrt {4 \, \sqrt {35} + 70} \arctan \left (\frac {1}{1177225} \, \sqrt {155} 35^{\frac {3}{4}} \sqrt {31} \sqrt {7} \sqrt {\sqrt {155} 35^{\frac {3}{4}} \sqrt {31} \sqrt {2 \, x + 1} \sqrt {4 \, \sqrt {35} + 70} + 10850 \, x + 1085 \, \sqrt {35} + 5425} \sqrt {4 \, \sqrt {35} + 70} - \frac {1}{1085} \, \sqrt {155} 35^{\frac {3}{4}} \sqrt {2 \, x + 1} \sqrt {4 \, \sqrt {35} + 70} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) - \frac {2}{5425} \, \sqrt {155} 35^{\frac {3}{4}} \sqrt {4 \, \sqrt {35} + 70} \arctan \left (\frac {1}{2354450} \, \sqrt {155} 35^{\frac {3}{4}} \sqrt {7} \sqrt {-124 \, \sqrt {155} 35^{\frac {3}{4}} \sqrt {31} \sqrt {2 \, x + 1} \sqrt {4 \, \sqrt {35} + 70} + 1345400 \, x + 134540 \, \sqrt {35} + 672700} \sqrt {4 \, \sqrt {35} + 70} - \frac {1}{1085} \, \sqrt {155} 35^{\frac {3}{4}} \sqrt {2 \, x + 1} \sqrt {4 \, \sqrt {35} + 70} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/336350*sqrt(155)*35^(1/4)*(2*sqrt(35)*sqrt(31) - 35*sqrt(31))*sqrt(4*sqrt(35) + 70)*log(124/7*sqrt(155)*35^(
3/4)*sqrt(31)*sqrt(2*x + 1)*sqrt(4*sqrt(35) + 70) + 192200*x + 19220*sqrt(35) + 96100) - 1/336350*sqrt(155)*35
^(1/4)*(2*sqrt(35)*sqrt(31) - 35*sqrt(31))*sqrt(4*sqrt(35) + 70)*log(-124/7*sqrt(155)*35^(3/4)*sqrt(31)*sqrt(2
*x + 1)*sqrt(4*sqrt(35) + 70) + 192200*x + 19220*sqrt(35) + 96100) - 2/5425*sqrt(155)*35^(3/4)*sqrt(4*sqrt(35)
+ 70)*arctan(1/1177225*sqrt(155)*35^(3/4)*sqrt(31)*sqrt(7)*sqrt(sqrt(155)*35^(3/4)*sqrt(31)*sqrt(2*x + 1)*sqr
t(4*sqrt(35) + 70) + 10850*x + 1085*sqrt(35) + 5425)*sqrt(4*sqrt(35) + 70) - 1/1085*sqrt(155)*35^(3/4)*sqrt(2*
x + 1)*sqrt(4*sqrt(35) + 70) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) - 2/5425*sqrt(155)*35^(3/4)*sqrt(4*sqrt
(35) + 70)*arctan(1/2354450*sqrt(155)*35^(3/4)*sqrt(7)*sqrt(-124*sqrt(155)*35^(3/4)*sqrt(31)*sqrt(2*x + 1)*sqr
t(4*sqrt(35) + 70) + 1345400*x + 134540*sqrt(35) + 672700)*sqrt(4*sqrt(35) + 70) - 1/1085*sqrt(155)*35^(3/4)*s
qrt(2*x + 1)*sqrt(4*sqrt(35) + 70) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31))

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giac [B]  time = 1.11, size = 461, normalized size = 2.08 \begin {gather*} \frac {1}{37215500} \, \sqrt {31} {\left (210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )}\right )} \arctan \left (\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) + \frac {1}{37215500} \, \sqrt {31} {\left (210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )}\right )} \arctan \left (-\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} - \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) + \frac {1}{74431000} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}}\right )} \log \left (2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) - \frac {1}{74431000} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}}\right )} \log \left (-2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/37215500*sqrt(31)*(210*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - sqrt(31)*(7/5)^(3
/4)*(-140*sqrt(35) + 2450)^(3/2) + 2*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 420*(7/5)^(3/4)*sqrt(140*sqrt(3
5) + 2450)*(2*sqrt(35) - 35))*arctan(5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*sqrt(35) + 1/2) + sqrt(2*x + 1))/s
qrt(-1/35*sqrt(35) + 1/2)) + 1/37215500*sqrt(31)*(210*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35
) + 2450) - sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 2*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 42
0*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35))*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)) + 1/74431000*sqrt(31)*(sqrt(31)*(7/5)^(3/4)*(140*sqrt
(35) + 2450)^(3/2) + 210*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 420*(7/5)^(3/4)*(2
*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2))*log(2*(7/5)^(1/4)*sqr
t(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) - 1/74431000*sqrt(31)*(sqrt(31)*(7/5)^(3/4)*(140*s
qrt(35) + 2450)^(3/2) + 210*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 420*(7/5)^(3/4)
*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2))*log(-2*(7/5)^(1/4)
*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1)

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maple [B]  time = 0.38, size = 486, normalized size = 2.19 \begin {gather*} -\frac {2 \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{31 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}\, \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{31 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {2 \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{31 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}\, \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {2 x +1}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{31 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{155}+\frac {\sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{62}+\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{155}-\frac {\sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \ln \left (10 x +\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {2 x +1}+5\right )}{62} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/155*5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*ln(10*x+5^(1/2)*7^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^
(1/2)+5)-2/31/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)*arctan((5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+
10*(2*x+1)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))-1/62*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*ln(10*x+5^(1/2)*7^(1
/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+5)+1/31/(10*5^(1/2)*7^(1/2)-20)^(1/2)*5^(1/2)*(2*5^(1/2)
*7^(1/2)+4)*7^(1/2)*arctan((5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(2*x+1)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2
))-1/155*5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*ln(10*x+5^(1/2)*7^(1/2)-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+
1)^(1/2)+5)-2/31/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)*arctan((-5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1
/2)+10*(2*x+1)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))+1/62*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*ln(10*x+5^(1/2)*
7^(1/2)-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2)+5)+1/31/(10*5^(1/2)*7^(1/2)-20)^(1/2)*5^(1/2)*(2*5^(
1/2)*7^(1/2)+4)*7^(1/2)*arctan((-5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(2*x+1)^(1/2))/(10*5^(1/2)*7^(1/2)-20)
^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {2 \, x + 1}}{5 \, x^{2} + 3 \, x + 2}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.13, size = 116, normalized size = 0.52 \begin {gather*} -\frac {2\,\sqrt {155}\,\mathrm {atanh}\left (\sqrt {155}\,\sqrt {-2-\sqrt {31}\,1{}\mathrm {i}}\,\left (\frac {2\,\left (\frac {2}{155}+\frac {\sqrt {31}\,1{}\mathrm {i}}{155}\right )\,\sqrt {2\,x+1}}{7}+\frac {27\,\sqrt {2\,x+1}}{1085}\right )\right )\,\sqrt {-2-\sqrt {31}\,1{}\mathrm {i}}}{155}-\frac {2\,\sqrt {155}\,\mathrm {atanh}\left (-\sqrt {155}\,\sqrt {-2+\sqrt {31}\,1{}\mathrm {i}}\,\left (\frac {2\,\left (-\frac {2}{155}+\frac {\sqrt {31}\,1{}\mathrm {i}}{155}\right )\,\sqrt {2\,x+1}}{7}-\frac {27\,\sqrt {2\,x+1}}{1085}\right )\right )\,\sqrt {-2+\sqrt {31}\,1{}\mathrm {i}}}{155} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (2*155^(1/2)*atanh(155^(1/2)*(- 31^(1/2)*1i - 2)^(1/2)*((2*((31^(1/2)*1i)/155 + 2/155)*(2*x + 1)^(1/2))/7 +
(27*(2*x + 1)^(1/2))/1085))*(- 31^(1/2)*1i - 2)^(1/2))/155 - (2*155^(1/2)*atanh(-155^(1/2)*(31^(1/2)*1i - 2)^(
1/2)*((2*((31^(1/2)*1i)/155 - 2/155)*(2*x + 1)^(1/2))/7 - (27*(2*x + 1)^(1/2))/1085))*(31^(1/2)*1i - 2)^(1/2))
/155

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sympy [A]  time = 4.57, size = 32, normalized size = 0.14 \begin {gather*} 4 \operatorname {RootSum} {\left (1230080 t^{4} + 1984 t^{2} + 7, \left (t \mapsto t \log {\left (9920 t^{3} + 8 t + \sqrt {2 x + 1} \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*RootSum(1230080*_t**4 + 1984*_t**2 + 7, Lambda(_t, _t*log(9920*_t**3 + 8*_t + sqrt(2*x + 1))))

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