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

Optimal. Leaf size=218 \[ -\frac {\log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{\sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{\sqrt {14 \left (2+\sqrt {35}\right )}}-\sqrt {\frac {2}{217} \left (2+\sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\sqrt {\frac {2}{217} \left (2+\sqrt {35}\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right ) \]

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Rubi [A]  time = 0.27, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {707, 1094, 634, 618, 204, 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 {14 \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 {14 \left (2+\sqrt {35}\right )}}-\sqrt {\frac {2}{217} \left (2+\sqrt {35}\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}{217} \left (2+\sqrt {35}\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[1/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)),x]

[Out]

-(Sqrt[(2*(2 + Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])
+ Sqrt[(2*(2 + Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]] -
Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)]/Sqrt[14*(2 + Sqrt[35])] + Log[Sqrt[35] +
Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)]/Sqrt[14*(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 634

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 707

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(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 1094

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx &=4 \operatorname {Subst}\left (\int \frac {1}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=\sqrt {\frac {2}{7 \left (2+\sqrt {35}\right )}} \operatorname {Subst}\left (\int \frac {\sqrt {\frac {2}{5} \left (2+\sqrt {35}\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 )+\sqrt {\frac {2}{7 \left (2+\sqrt {35}\right )}} \operatorname {Subst}\left (\int \frac {\sqrt {\frac {2}{5} \left (2+\sqrt {35}\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 )\\ &=\frac {\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 )}{\sqrt {35}}+\frac {\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 )}{\sqrt {35}}-\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 {14 \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 {14 \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 {14 \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 {14 \left (2+\sqrt {35}\right )}}-\frac {2 \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 {35}}-\frac {2 \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 {35}}\\ &=-\sqrt {\frac {2}{7 \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}{7 \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 {14 \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 {14 \left (2+\sqrt {35}\right )}}\\ \end {align*}

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Mathematica [C]  time = 0.24, size = 112, normalized size = 0.51 \begin {gather*} \frac {2 \left (\sqrt {2-i \sqrt {31}} \left (\sqrt {31}-2 i\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )+\sqrt {2+i \sqrt {31}} \left (\sqrt {31}+2 i\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )\right )}{7 \sqrt {155}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(Sqrt[2 - I*Sqrt[31]]*(-2*I + Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + Sqrt[2 + I*Sqrt[31]]
*(2*I + Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]]))/(7*Sqrt[155])

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IntegrateAlgebraic [C]  time = 0.29, size = 103, normalized size = 0.47 \begin {gather*} 2 \sqrt {\frac {1}{217} \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}{217} \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[1/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)),x]

[Out]

2*Sqrt[(2 + I*Sqrt[31])/217]*ArcTan[Sqrt[-2/7 - (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]] + 2*Sqrt[(2 - I*Sqrt[31])/217]*
ArcTan[Sqrt[-2/7 + (I/7)*Sqrt[31]]*Sqrt[1 + 2*x]]

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fricas [B]  time = 0.44, size = 365, normalized size = 1.67 \begin {gather*} -\frac {1}{470890} \, \sqrt {217} 35^{\frac {1}{4}} {\left (2 \, \sqrt {35} \sqrt {31} - 35 \, \sqrt {31}\right )} \sqrt {4 \, \sqrt {35} + 70} \log \left (4340 \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {31} \sqrt {2 \, x + 1} \sqrt {4 \, \sqrt {35} + 70} + 9417800 \, x + 941780 \, \sqrt {35} + 4708900\right ) + \frac {1}{470890} \, \sqrt {217} 35^{\frac {1}{4}} {\left (2 \, \sqrt {35} \sqrt {31} - 35 \, \sqrt {31}\right )} \sqrt {4 \, \sqrt {35} + 70} \log \left (-4340 \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {31} \sqrt {2 \, x + 1} \sqrt {4 \, \sqrt {35} + 70} + 9417800 \, x + 941780 \, \sqrt {35} + 4708900\right ) - \frac {2}{7595} \, \sqrt {217} 35^{\frac {3}{4}} \sqrt {4 \, \sqrt {35} + 70} \arctan \left (\frac {1}{235445} \, \sqrt {1085} \sqrt {217} 35^{\frac {1}{4}} \sqrt {\sqrt {217} 35^{\frac {1}{4}} \sqrt {31} \sqrt {2 \, x + 1} \sqrt {4 \, \sqrt {35} + 70} + 2170 \, x + 217 \, \sqrt {35} + 1085} \sqrt {4 \, \sqrt {35} + 70} - \frac {1}{217} \, \sqrt {217} 35^{\frac {1}{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}{7595} \, \sqrt {217} 35^{\frac {3}{4}} \sqrt {4 \, \sqrt {35} + 70} \arctan \left (\frac {1}{470890} \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {-4340 \, \sqrt {217} 35^{\frac {1}{4}} \sqrt {31} \sqrt {2 \, x + 1} \sqrt {4 \, \sqrt {35} + 70} + 9417800 \, x + 941780 \, \sqrt {35} + 4708900} \sqrt {4 \, \sqrt {35} + 70} - \frac {1}{217} \, \sqrt {217} 35^{\frac {1}{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/(1+2*x)^(1/2)/(5*x^2+3*x+2),x, algorithm="fricas")

[Out]

-1/470890*sqrt(217)*35^(1/4)*(2*sqrt(35)*sqrt(31) - 35*sqrt(31))*sqrt(4*sqrt(35) + 70)*log(4340*sqrt(217)*35^(
1/4)*sqrt(31)*sqrt(2*x + 1)*sqrt(4*sqrt(35) + 70) + 9417800*x + 941780*sqrt(35) + 4708900) + 1/470890*sqrt(217
)*35^(1/4)*(2*sqrt(35)*sqrt(31) - 35*sqrt(31))*sqrt(4*sqrt(35) + 70)*log(-4340*sqrt(217)*35^(1/4)*sqrt(31)*sqr
t(2*x + 1)*sqrt(4*sqrt(35) + 70) + 9417800*x + 941780*sqrt(35) + 4708900) - 2/7595*sqrt(217)*35^(3/4)*sqrt(4*s
qrt(35) + 70)*arctan(1/235445*sqrt(1085)*sqrt(217)*35^(1/4)*sqrt(sqrt(217)*35^(1/4)*sqrt(31)*sqrt(2*x + 1)*sqr
t(4*sqrt(35) + 70) + 2170*x + 217*sqrt(35) + 1085)*sqrt(4*sqrt(35) + 70) - 1/217*sqrt(217)*35^(1/4)*sqrt(2*x +
1)*sqrt(4*sqrt(35) + 70) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) - 2/7595*sqrt(217)*35^(3/4)*sqrt(4*sqrt(35
) + 70)*arctan(1/470890*sqrt(217)*35^(1/4)*sqrt(-4340*sqrt(217)*35^(1/4)*sqrt(31)*sqrt(2*x + 1)*sqrt(4*sqrt(35
) + 70) + 9417800*x + 941780*sqrt(35) + 4708900)*sqrt(4*sqrt(35) + 70) - 1/217*sqrt(217)*35^(1/4)*sqrt(2*x + 1
)*sqrt(4*sqrt(35) + 70) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31))

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giac [A]  time = 0.63, size = 279, normalized size = 1.28 \begin {gather*} \frac {1}{7595} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450}\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}{7595} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450}\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}{15190} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450}\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}{15190} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450}\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/(1+2*x)^(1/2)/(5*x^2+3*x+2),x, algorithm="giac")

[Out]

1/7595*sqrt(31)*(sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 2*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*ar
ctan(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)) + 1/7
595*sqrt(31)*(sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 2*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arcta
n(-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)) + 1/151
90*sqrt(31)*(sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 2*(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) - 1/15190*sqrt(31)*(sqrt(31)*(7/5)^
(1/4)*sqrt(140*sqrt(35) + 2450) - 2*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450))*log(-2*(7/5)^(1/4)*sqrt(2*x + 1)*s
qrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1)

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maple [B]  time = 0.34, size = 607, normalized size = 2.78 \begin {gather*} -\frac {5 \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 {2 \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 )}{217 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {4 \sqrt {5}\, \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 )}{7 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {5 \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 {2 \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 )}{217 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {4 \sqrt {5}\, \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 )}{7 \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 )}{62}+\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 )}{217}+\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 )}{62}-\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 )}{217} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/62*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)-1/217*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)-5/31/(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))+2/217/(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))+4/7/(10*5^(1/2)*7^(1/2)-20)^(1/2)*5^(1/2)*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))-1/62*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)+1/217*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)-5/31/(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))+2/217/(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))+4/7/(10*5^(1/2)*7^(1/2)-20)^(1/2)*5^(1/2)*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 {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {2 \, x + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.02, size = 167, normalized size = 0.77 \begin {gather*} \frac {\sqrt {217}\,\mathrm {atan}\left (\frac {256\,\sqrt {7}\,\sqrt {-2-\sqrt {31}\,1{}\mathrm {i}}\,\sqrt {2\,x+1}}{6125\,\left (\frac {256}{875}+\frac {\sqrt {31}\,128{}\mathrm {i}}{875}\right )}+\frac {\sqrt {217}\,\sqrt {-2-\sqrt {31}\,1{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{6125\,\left (\frac {256}{875}+\frac {\sqrt {31}\,128{}\mathrm {i}}{875}\right )}\right )\,\sqrt {-2-\sqrt {31}\,1{}\mathrm {i}}\,2{}\mathrm {i}}{217}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {256\,\sqrt {7}\,\sqrt {-2+\sqrt {31}\,1{}\mathrm {i}}\,\sqrt {2\,x+1}}{6125\,\left (-\frac {256}{875}+\frac {\sqrt {31}\,128{}\mathrm {i}}{875}\right )}-\frac {\sqrt {217}\,\sqrt {-2+\sqrt {31}\,1{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{6125\,\left (-\frac {256}{875}+\frac {\sqrt {31}\,128{}\mathrm {i}}{875}\right )}\right )\,\sqrt {-2+\sqrt {31}\,1{}\mathrm {i}}\,2{}\mathrm {i}}{217} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(217^(1/2)*atan((256*7^(1/2)*(- 31^(1/2)*1i - 2)^(1/2)*(2*x + 1)^(1/2))/(6125*((31^(1/2)*128i)/875 + 256/875))
+ (217^(1/2)*(- 31^(1/2)*1i - 2)^(1/2)*(2*x + 1)^(1/2)*128i)/(6125*((31^(1/2)*128i)/875 + 256/875)))*(- 31^(1
/2)*1i - 2)^(1/2)*2i)/217 + (217^(1/2)*atan((256*7^(1/2)*(31^(1/2)*1i - 2)^(1/2)*(2*x + 1)^(1/2))/(6125*((31^(
1/2)*128i)/875 - 256/875)) - (217^(1/2)*(31^(1/2)*1i - 2)^(1/2)*(2*x + 1)^(1/2)*128i)/(6125*((31^(1/2)*128i)/8
75 - 256/875)))*(31^(1/2)*1i - 2)^(1/2)*2i)/217

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {2 x + 1} \left (5 x^{2} + 3 x + 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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