3.2303 \(\int \frac{1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx\)

Optimal. Leaf size=253 \[ -\frac{4}{7 \sqrt{2 x+1}}-\frac{1}{7} \sqrt{\frac{1}{434} \left (178+35 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{7} \sqrt{\frac{1}{434} \left (178+35 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{7} \sqrt{\frac{2}{217} \left (35 \sqrt{35}-178\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{7} \sqrt{\frac{2}{217} \left (35 \sqrt{35}-178\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

[Out]

-4/(7*Sqrt[1 + 2*x]) + (Sqrt[(2*(-178 + 35*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 +
Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/7 - (Sqrt[(2*(-178 + 3
5*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-
2 + Sqrt[35])]])/7 - (Sqrt[(178 + 35*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 +
Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/7 + (Sqrt[(178 + 35*Sqrt[35])/434]*Log[
Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/7

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Rubi [A]  time = 1.05798, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{4}{7 \sqrt{2 x+1}}-\frac{1}{7} \sqrt{\frac{1}{434} \left (178+35 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{7} \sqrt{\frac{1}{434} \left (178+35 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{7} \sqrt{\frac{2}{217} \left (35 \sqrt{35}-178\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{7} \sqrt{\frac{2}{217} \left (35 \sqrt{35}-178\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-4/(7*Sqrt[1 + 2*x]) + (Sqrt[(2*(-178 + 35*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 +
Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/7 - (Sqrt[(2*(-178 + 3
5*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-
2 + Sqrt[35])]])/7 - (Sqrt[(178 + 35*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 +
Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/7 + (Sqrt[(178 + 35*Sqrt[35])/434]*Log[
Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/7

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Rubi in Sympy [A]  time = 68.6642, size = 357, normalized size = 1.41 \[ - \frac{\sqrt{14} \left (4 + \sqrt{35}\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{98 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \left (4 + \sqrt{35}\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{98 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (8 + 2 \sqrt{35}\right )}{10} + \frac{8 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{49 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (8 + 2 \sqrt{35}\right )}{10} + \frac{8 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{49 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} - \frac{4}{7 \sqrt{2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1+2*x)**(3/2)/(5*x**2+3*x+2),x)

[Out]

-sqrt(14)*(4 + sqrt(35))*log(2*x - sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 +
1 + sqrt(35)/5)/(98*sqrt(2 + sqrt(35))) + sqrt(14)*(4 + sqrt(35))*log(2*x + sqr
t(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(98*sqrt(2 + sqrt(35)
)) + sqrt(35)*(-sqrt(10)*sqrt(2 + sqrt(35))*(8 + 2*sqrt(35))/10 + 8*sqrt(10)*sqr
t(2 + sqrt(35))/5)*atan(sqrt(10)*(sqrt(2*x + 1) - sqrt(20 + 10*sqrt(35))/10)/sqr
t(-2 + sqrt(35)))/(49*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(
10)*sqrt(2 + sqrt(35))*(8 + 2*sqrt(35))/10 + 8*sqrt(10)*sqrt(2 + sqrt(35))/5)*at
an(sqrt(10)*(sqrt(2*x + 1) + sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(49
*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35))) - 4/(7*sqrt(2*x + 1))

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Mathematica [C]  time = 0.420242, size = 130, normalized size = 0.51 \[ -\frac{4}{7 \sqrt{2 x+1}}-\frac{2 \left (\sqrt{31}+2 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{7 \sqrt{-\frac{31}{5} i \left (\sqrt{31}-2 i\right )}}-\frac{2 \left (\sqrt{31}-2 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{7 \sqrt{\frac{31}{5} i \left (\sqrt{31}+2 i\right )}} \]

Antiderivative was successfully verified.

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

[Out]

-4/(7*Sqrt[1 + 2*x]) - (2*(2*I + Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqr
t[31]]])/(7*Sqrt[((-31*I)/5)*(-2*I + Sqrt[31])]) - (2*(-2*I + Sqrt[31])*ArcTan[S
qrt[5 + 10*x]/Sqrt[-2 + I*Sqrt[31]]])/(7*Sqrt[((31*I)/5)*(2*I + Sqrt[31])])

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Maple [B]  time = 0.046, size = 616, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/217*ln(-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5^(1/2)*7^(1/2)+10*
x+5)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)-27/3038*ln(-(2*5^(1/2)*7^(1/2)+4)^(1/2)
*5^(1/2)*(1+2*x)^(1/2)+5^(1/2)*7^(1/2)+10*x+5)*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/
2)-10/217/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(
1/2)+10*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)-27/1
519/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)+1
0*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)*5^(1/2)*7^
(1/2)+16/49/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5
^(1/2)+10*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*5^(1/2)*7^(1/2)+1/217*ln
(5^(1/2)*7^(1/2)+10*x+5+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2))*(2*5^
(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)+27/3038*ln(5^(1/2)*7^(1/2)+10*x+5+(2*5^(1/2)*7^(1
/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2))*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-10/217/(
10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((10*(1+2*x)^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2
)*5^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)-27/1519/(10*5^(1
/2)*7^(1/2)-20)^(1/2)*arctan((10*(1+2*x)^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/
2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)*5^(1/2)*7^(1/2)+16/49/(
10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((10*(1+2*x)^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2
)*5^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*5^(1/2)*7^(1/2)-4/7/(1+2*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.269793, size = 1200, normalized size = 4.74 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5006973370*329623^(3/4)*sqrt(70)*sqrt(31)*(7*125^(1/4)*sqrt(31)*sqrt(2*x + 1)*
(178*sqrt(7) + 245*sqrt(5))*log(3100/49*sqrt(7)*(329623^(1/4)*125^(1/4)*sqrt(70)
*(60060861017813015719297999*sqrt(7)*sqrt(5) + 355324845622215676469207696)*sqrt
(2*x + 1)*sqrt((178*sqrt(7)*sqrt(5) + 1225)/(12460*sqrt(7)*sqrt(5) + 74559)) + 1
75*sqrt(7)*(6056915871103348083790300*sqrt(7)*sqrt(5)*(2*x + 1) + 71666395066799
246768273598*x + 35833197533399623384136799) + 245*sqrt(5)*(60569158711033480837
90300*sqrt(7)*sqrt(5) + 35833197533399623384136799))/(6056915871103348083790300*
sqrt(7)*sqrt(5) + 35833197533399623384136799)) - 7*125^(1/4)*sqrt(31)*sqrt(2*x +
1)*(178*sqrt(7) + 245*sqrt(5))*log(-3100/49*sqrt(7)*(329623^(1/4)*125^(1/4)*sqr
t(70)*(60060861017813015719297999*sqrt(7)*sqrt(5) + 355324845622215676469207696)
*sqrt(2*x + 1)*sqrt((178*sqrt(7)*sqrt(5) + 1225)/(12460*sqrt(7)*sqrt(5) + 74559)
) - 175*sqrt(7)*(6056915871103348083790300*sqrt(7)*sqrt(5)*(2*x + 1) + 716663950
66799246768273598*x + 35833197533399623384136799) - 245*sqrt(5)*(605691587110334
8083790300*sqrt(7)*sqrt(5) + 35833197533399623384136799))/(605691587110334808379
0300*sqrt(7)*sqrt(5) + 35833197533399623384136799)) - 4*329623^(1/4)*sqrt(70)*sq
rt(31)*(178*sqrt(7) + 245*sqrt(5))*sqrt((178*sqrt(7)*sqrt(5) + 1225)/(12460*sqrt
(7)*sqrt(5) + 74559)) + 16492*125^(1/4)*sqrt(7)*sqrt(2*x + 1)*arctan(343*125^(1/
4)*sqrt(31)*(10*sqrt(7) + 27*sqrt(5))/(35*329623^(1/4)*sqrt(70)*sqrt(2*x + 1)*(1
78*sqrt(7) + 245*sqrt(5))*sqrt((178*sqrt(7)*sqrt(5) + 1225)/(12460*sqrt(7)*sqrt(
5) + 74559)) + 329623^(1/4)*sqrt(70)*(178*sqrt(7) + 245*sqrt(5))*sqrt(sqrt(7)*(3
29623^(1/4)*125^(1/4)*sqrt(70)*(60060861017813015719297999*sqrt(7)*sqrt(5) + 355
324845622215676469207696)*sqrt(2*x + 1)*sqrt((178*sqrt(7)*sqrt(5) + 1225)/(12460
*sqrt(7)*sqrt(5) + 74559)) + 175*sqrt(7)*(6056915871103348083790300*sqrt(7)*sqrt
(5)*(2*x + 1) + 71666395066799246768273598*x + 35833197533399623384136799) + 245
*sqrt(5)*(6056915871103348083790300*sqrt(7)*sqrt(5) + 35833197533399623384136799
))/(6056915871103348083790300*sqrt(7)*sqrt(5) + 35833197533399623384136799))*sqr
t((178*sqrt(7)*sqrt(5) + 1225)/(12460*sqrt(7)*sqrt(5) + 74559)) + 10633*125^(1/4
)*(5*sqrt(7) + 4*sqrt(5)))) + 16492*125^(1/4)*sqrt(7)*sqrt(2*x + 1)*arctan(343*1
25^(1/4)*sqrt(31)*(10*sqrt(7) + 27*sqrt(5))/(35*329623^(1/4)*sqrt(70)*sqrt(2*x +
1)*(178*sqrt(7) + 245*sqrt(5))*sqrt((178*sqrt(7)*sqrt(5) + 1225)/(12460*sqrt(7)
*sqrt(5) + 74559)) + 329623^(1/4)*sqrt(70)*(178*sqrt(7) + 245*sqrt(5))*sqrt(-sqr
t(7)*(329623^(1/4)*125^(1/4)*sqrt(70)*(60060861017813015719297999*sqrt(7)*sqrt(5
) + 355324845622215676469207696)*sqrt(2*x + 1)*sqrt((178*sqrt(7)*sqrt(5) + 1225)
/(12460*sqrt(7)*sqrt(5) + 74559)) - 175*sqrt(7)*(6056915871103348083790300*sqrt(
7)*sqrt(5)*(2*x + 1) + 71666395066799246768273598*x + 35833197533399623384136799
) - 245*sqrt(5)*(6056915871103348083790300*sqrt(7)*sqrt(5) + 3583319753339962338
4136799))/(6056915871103348083790300*sqrt(7)*sqrt(5) + 3583319753339962338413679
9))*sqrt((178*sqrt(7)*sqrt(5) + 1225)/(12460*sqrt(7)*sqrt(5) + 74559)) - 10633*1
25^(1/4)*(5*sqrt(7) + 4*sqrt(5)))))/(sqrt(2*x + 1)*(178*sqrt(7) + 245*sqrt(5))*s
qrt((178*sqrt(7)*sqrt(5) + 1225)/(12460*sqrt(7)*sqrt(5) + 74559)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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