3.2321 \(\int \frac {1}{(1+2 x)^{3/2} (2+3 x+5 x^2)^2} \, dx\)

Optimal. Leaf size=283 \[ \frac {20 x+37}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}-\frac {604}{1519 \sqrt {2 x+1}}-\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1519}+\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1519}+\frac {\sqrt {\frac {2}{217} \left (968975 \sqrt {35}-5682718\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{1519}-\frac {\sqrt {\frac {2}{217} \left (968975 \sqrt {35}-5682718\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{1519} \]

[Out]

-604/1519/(1+2*x)^(1/2)+1/217*(37+20*x)/(5*x^2+3*x+2)/(1+2*x)^(1/2)+1/329623*arctan((-10*(1+2*x)^(1/2)+(20+10*
35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(-2466299612+420535150*35^(1/2))^(1/2)-1/329623*arctan((10*(1+2*x)^(
1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(-2466299612+420535150*35^(1/2))^(1/2)-1/659246*ln(5+10*
x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(2466299612+420535150*35^(1/2))^(1/2)+1/659246*ln(5+10*x+35^(
1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(2466299612+420535150*35^(1/2))^(1/2)

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Rubi [A]  time = 0.37, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {740, 828, 826, 1169, 634, 618, 204, 628} \[ \frac {20 x+37}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}-\frac {604}{1519 \sqrt {2 x+1}}-\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1519}+\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1519}+\frac {\sqrt {\frac {2}{217} \left (968975 \sqrt {35}-5682718\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{1519}-\frac {\sqrt {\frac {2}{217} \left (968975 \sqrt {35}-5682718\right )} \tan ^{-1}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{1519} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-604/(1519*Sqrt[1 + 2*x]) + (37 + 20*x)/(217*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) + (Sqrt[(2*(-5682718 + 968975*Sq
rt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/1519 - (Sqrt[(2*(
-5682718 + 968975*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]
])/1519 - (Sqrt[(5682718 + 968975*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 +
2*x)])/1519 + (Sqrt[(5682718 + 968975*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5
*(1 + 2*x)])/1519

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 740

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

Rule 826

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 828

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

Rule 1169

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*} \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx &=\frac {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {1}{217} \int \frac {181+60 x}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx\\ &=-\frac {604}{1519 \sqrt {1+2 x}}+\frac {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\int \frac {59-1510 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{1519}\\ &=-\frac {604}{1519 \sqrt {1+2 x}}+\frac {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {2 \operatorname {Subst}\left (\int \frac {1628-1510 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{1519}\\ &=-\frac {604}{1519 \sqrt {1+2 x}}+\frac {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {1628 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (1628+302 \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 )}{1519 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {1628 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (1628+302 \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 )}{1519 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {604}{1519 \sqrt {1+2 x}}+\frac {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\left (-5285+814 \sqrt {35}\right ) \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 )}{53165}+\frac {\left (-5285+814 \sqrt {35}\right ) \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 )}{53165}-\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \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 )}{1519}+\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \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 )}{1519}\\ &=-\frac {604}{1519 \sqrt {1+2 x}}+\frac {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1519}+\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1519}+\frac {\left (2 \left (5285-814 \sqrt {35}\right )\right ) \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 )}{53165}+\frac {\left (2 \left (5285-814 \sqrt {35}\right )\right ) \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 )}{53165}\\ &=-\frac {604}{1519 \sqrt {1+2 x}}+\frac {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {2}{217} \left (-5682718+968975 \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 )}{1519}-\frac {\sqrt {\frac {2}{217} \left (-5682718+968975 \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 )}{1519}-\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1519}+\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1519}\\ \end {align*}

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Mathematica [C]  time = 0.39, size = 158, normalized size = 0.56 \[ \frac {1}{217} \left (\frac {20 x+37}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}-\frac {604}{7 \sqrt {2 x+1}}+\frac {2 \sqrt {10-5 i \sqrt {31}} \left (25234+3657 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2-i \sqrt {31}}}\right )+2 \sqrt {10+5 i \sqrt {31}} \left (25234-3657 i \sqrt {31}\right ) \tanh ^{-1}\left (\frac {\sqrt {10 x+5}}{\sqrt {2+i \sqrt {31}}}\right )}{7595}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-604/(7*Sqrt[1 + 2*x]) + (37 + 20*x)/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) + (2*Sqrt[10 - (5*I)*Sqrt[31]]*(25234
+ (3657*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 2*Sqrt[10 + (5*I)*Sqrt[31]]*(25234 - (3657
*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]])/7595)/217

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fricas [B]  time = 1.09, size = 576, normalized size = 2.04 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2682041589056045650*(16794436*21898835^(1/4)*sqrt(217)*sqrt(35)*(10*x^3 + 11*x^2 + 7*x + 2)*sqrt(-1101282334
8100*sqrt(35) + 65723878543750)*arctan(1/84465779527115502604115125*21898835^(3/4)*sqrt(4369)*sqrt(791)*sqrt(2
17)*sqrt(21898835^(1/4)*sqrt(217)*(151*sqrt(35)*sqrt(31) + 814*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sq
rt(35) + 65723878543750) + 8136694311550*x + 813669431155*sqrt(35) + 4068347155775)*(814*sqrt(35) + 5285)*sqrt
(-11012823348100*sqrt(35) + 65723878543750) - 1/22526438201526175*21898835^(3/4)*sqrt(217)*sqrt(2*x + 1)*(814*
sqrt(35) + 5285)*sqrt(-11012823348100*sqrt(35) + 65723878543750) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) + 1
6794436*21898835^(1/4)*sqrt(217)*sqrt(35)*(10*x^3 + 11*x^2 + 7*x + 2)*sqrt(-11012823348100*sqrt(35) + 65723878
543750)*arctan(1/206941159841432981380082056250*21898835^(3/4)*sqrt(4369)*sqrt(217)*sqrt(-4747977500*21898835^
(1/4)*sqrt(217)*(151*sqrt(35)*sqrt(31) + 814*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) + 657238785
43750) + 38632841515617390125000*x + 3863284151561739012500*sqrt(35) + 19316420757808695062500)*(814*sqrt(35)
+ 5285)*sqrt(-11012823348100*sqrt(35) + 65723878543750) - 1/22526438201526175*21898835^(3/4)*sqrt(217)*sqrt(2*
x + 1)*(814*sqrt(35) + 5285)*sqrt(-11012823348100*sqrt(35) + 65723878543750) + 1/31*sqrt(35)*sqrt(31) + 2/31*s
qrt(31)) + 21898835^(1/4)*sqrt(217)*(5682718*sqrt(35)*sqrt(31)*(10*x^3 + 11*x^2 + 7*x + 2) + 33914125*sqrt(31)
*(10*x^3 + 11*x^2 + 7*x + 2))*sqrt(-11012823348100*sqrt(35) + 65723878543750)*log(4747977500/4369*21898835^(1/
4)*sqrt(217)*(151*sqrt(35)*sqrt(31) + 814*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) + 657238785437
50) + 8842490619276125000*x + 884249061927612500*sqrt(35) + 4421245309638062500) - 21898835^(1/4)*sqrt(217)*(5
682718*sqrt(35)*sqrt(31)*(10*x^3 + 11*x^2 + 7*x + 2) + 33914125*sqrt(31)*(10*x^3 + 11*x^2 + 7*x + 2))*sqrt(-11
012823348100*sqrt(35) + 65723878543750)*log(-4747977500/4369*21898835^(1/4)*sqrt(217)*(151*sqrt(35)*sqrt(31) +
814*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) + 65723878543750) + 8842490619276125000*x + 8842490
61927612500*sqrt(35) + 4421245309638062500) - 1765662665606350*(3020*x^2 + 1672*x + 949)*sqrt(2*x + 1))/(10*x^
3 + 11*x^2 + 7*x + 2)

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giac [B]  time = 0.96, size = 633, normalized size = 2.24 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/11306068900*sqrt(31)*(31710*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 151*sqrt(31
)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 302*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 63420*(7/5)^(3/4)*s
qrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 797720*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 1595440*
(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)) - 1/11306068900*sqrt(31)*(31710*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqr
t(-140*sqrt(35) + 2450) - 151*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 302*(7/5)^(3/4)*(140*sqrt(35
) + 2450)^(3/2) + 63420*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 797720*sqrt(31)*(7/5)^(1/4)*
sqrt(-140*sqrt(35) + 2450) - 1595440*(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)) - 1/22612137800*sqrt(31)*(151*sqrt(3
1)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 31710*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35)
- 35) - 63420*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 302*(7/5)^(3/4)*(-140*sqrt(35) + 2450
)^(3/2) - 797720*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 1595440*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 245
0))*log(2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) + 1/22612137800*sqrt(31)*
(151*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 31710*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(
2*sqrt(35) - 35) - 63420*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 302*(7/5)^(3/4)*(-140*sqrt
(35) + 2450)^(3/2) - 797720*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 1595440*(7/5)^(1/4)*sqrt(-140*sqr
t(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) - 4/1519*(755
*(2*x + 1)^2 - 1348*x + 194)/(5*(2*x + 1)^(5/2) - 4*(2*x + 1)^(3/2) + 7*sqrt(2*x + 1))

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maple [B]  time = 0.17, size = 651, normalized size = 2.30 \[ -\frac {2560 \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 )}{47089 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {3657 \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 )}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3256 \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 )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {2560 \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 )}{47089 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {3657 \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 )}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3256 \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 )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {256 \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 )}{47089}-\frac {3657 \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 )}{659246}+\frac {256 \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 )}{47089}+\frac {3657 \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 )}{659246}-\frac {16 \left (\frac {27 \left (2 x +1\right )^{\frac {3}{2}}}{124}-\frac {89 \sqrt {2 x +1}}{310}\right )}{49 \left (-\frac {8 x}{5}+\left (2 x +1\right )^{2}+\frac {3}{5}\right )}-\frac {16}{49 \sqrt {2 x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/49*(27/124*(2*x+1)^(3/2)-89/310*(2*x+1)^(1/2))/(-8/5*x+(2*x+1)^2+3/5)+256/47089*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)+3657/659246*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)-2560/470
89/(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))-3657/329623/(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))+3256/1063
3/(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))-256/47089*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)-3657/659246*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)-2560/47089/(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))-3657
/329623/(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))+3256/10633/(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))-16/49/(2*
x+1)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2} {\left (2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.04, size = 217, normalized size = 0.77 \[ -\frac {\frac {604\,{\left (2\,x+1\right )}^2}{1519}-\frac {5392\,x}{7595}+\frac {776}{7595}}{\frac {7\,\sqrt {2\,x+1}}{5}-\frac {4\,{\left (2\,x+1\right )}^{3/2}}{5}+{\left (2\,x+1\right )}^{5/2}}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}\,559232{}\mathrm {i}}{692496720125\,\left (\frac {2045111424}{98928102875}+\frac {\sqrt {31}\,455214848{}\mathrm {i}}{98928102875}\right )}-\frac {1118464\,\sqrt {31}\,\sqrt {217}\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}}{21467398323875\,\left (\frac {2045111424}{98928102875}+\frac {\sqrt {31}\,455214848{}\mathrm {i}}{98928102875}\right )}\right )\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,2{}\mathrm {i}}{329623}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}\,559232{}\mathrm {i}}{692496720125\,\left (-\frac {2045111424}{98928102875}+\frac {\sqrt {31}\,455214848{}\mathrm {i}}{98928102875}\right )}+\frac {1118464\,\sqrt {31}\,\sqrt {217}\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}}{21467398323875\,\left (-\frac {2045111424}{98928102875}+\frac {\sqrt {31}\,455214848{}\mathrm {i}}{98928102875}\right )}\right )\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,2{}\mathrm {i}}{329623} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(217^(1/2)*atan((217^(1/2)*(31^(1/2)*135439i + 5682718)^(1/2)*(2*x + 1)^(1/2)*559232i)/(692496720125*((31^(1/2
)*455214848i)/98928102875 - 2045111424/98928102875)) + (1118464*31^(1/2)*217^(1/2)*(31^(1/2)*135439i + 5682718
)^(1/2)*(2*x + 1)^(1/2))/(21467398323875*((31^(1/2)*455214848i)/98928102875 - 2045111424/98928102875)))*(31^(1
/2)*135439i + 5682718)^(1/2)*2i)/329623 - (217^(1/2)*atan((217^(1/2)*(5682718 - 31^(1/2)*135439i)^(1/2)*(2*x +
1)^(1/2)*559232i)/(692496720125*((31^(1/2)*455214848i)/98928102875 + 2045111424/98928102875)) - (1118464*31^(
1/2)*217^(1/2)*(5682718 - 31^(1/2)*135439i)^(1/2)*(2*x + 1)^(1/2))/(21467398323875*((31^(1/2)*455214848i)/9892
8102875 + 2045111424/98928102875)))*(5682718 - 31^(1/2)*135439i)^(1/2)*2i)/329623 - ((604*(2*x + 1)^2)/1519 -
(5392*x)/7595 + 776/7595)/((7*(2*x + 1)^(1/2))/5 - (4*(2*x + 1)^(3/2))/5 + (2*x + 1)^(5/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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