3.2300 \(\int \frac{(1+2 x)^{3/2}}{2+3 x+5 x^2} \, dx\)

Optimal. Leaf size=253 \[ \frac{4}{5} \sqrt{2 x+1}+\frac{1}{5} \sqrt{\frac{1}{310} \left (178+35 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{5} \sqrt{\frac{1}{310} \left (178+35 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{5} \sqrt{\frac{2}{155} \left (35 \sqrt{35}-178\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{5} \sqrt{\frac{2}{155} \left (35 \sqrt{35}-178\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

[Out]

(4*Sqrt[1 + 2*x])/5 + (Sqrt[(2*(-178 + 35*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + S
qrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/5 - (Sqrt[(2*(-178 + 35
*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2
+ Sqrt[35])]])/5 + (Sqrt[(178 + 35*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + S
qrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/5 - (Sqrt[(178 + 35*Sqrt[35])/310]*Log[S
qrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/5

_______________________________________________________________________________________

Rubi [A]  time = 1.11378, 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 \[ \frac{4}{5} \sqrt{2 x+1}+\frac{1}{5} \sqrt{\frac{1}{310} \left (178+35 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{5} \sqrt{\frac{1}{310} \left (178+35 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{5} \sqrt{\frac{2}{155} \left (35 \sqrt{35}-178\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{5} \sqrt{\frac{2}{155} \left (35 \sqrt{35}-178\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[1 + 2*x])/5 + (Sqrt[(2*(-178 + 35*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + S
qrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/5 - (Sqrt[(2*(-178 + 35
*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2
+ Sqrt[35])]])/5 + (Sqrt[(178 + 35*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + S
qrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/5 - (Sqrt[(178 + 35*Sqrt[35])/310]*Log[S
qrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/5

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 68.0279, size = 367, normalized size = 1.45 \[ \frac{4 \sqrt{2 x + 1}}{5} + \frac{\sqrt{14} \left (\frac{4 \sqrt{35}}{5} + 7\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{70 \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{14} \left (\frac{4 \sqrt{35}}{5} + 7\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{70 \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (\frac{8 \sqrt{35}}{5} + 14\right )}{10} + \frac{14 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{35 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (\frac{8 \sqrt{35}}{5} + 14\right )}{10} + \frac{14 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{35 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(2*x + 1)/5 + sqrt(14)*(4*sqrt(35)/5 + 7)*log(2*x - sqrt(10)*sqrt(2 + sqrt
(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(70*sqrt(2 + sqrt(35))) - sqrt(14)*(4*sq
rt(35)/5 + 7)*log(2*x + sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(3
5)/5)/(70*sqrt(2 + sqrt(35))) - sqrt(35)*(-sqrt(10)*sqrt(2 + sqrt(35))*(8*sqrt(3
5)/5 + 14)/10 + 14*sqrt(10)*sqrt(2 + sqrt(35))/5)*atan(sqrt(10)*(sqrt(2*x + 1) -
sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(35*sqrt(-2 + sqrt(35))*sqrt(2
+ sqrt(35))) - sqrt(35)*(-sqrt(10)*sqrt(2 + sqrt(35))*(8*sqrt(35)/5 + 14)/10 + 1
4*sqrt(10)*sqrt(2 + sqrt(35))/5)*atan(sqrt(10)*(sqrt(2*x + 1) + sqrt(20 + 10*sqr
t(35))/10)/sqrt(-2 + sqrt(35)))/(35*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35)))

_______________________________________________________________________________________

Mathematica [C]  time = 0.346904, size = 130, normalized size = 0.51 \[ \frac{4}{5} \sqrt{2 x+1}+\frac{2 \left (4 \sqrt{31}+27 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{5 \sqrt{-155 i \left (\sqrt{31}-2 i\right )}}+\frac{2 \left (4 \sqrt{31}-27 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{5 \sqrt{155 i \left (\sqrt{31}+2 i\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[1 + 2*x])/5 + (2*(27*I + 4*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*S
qrt[31]]])/(5*Sqrt[(-155*I)*(-2*I + Sqrt[31])]) + (2*(-27*I + 4*Sqrt[31])*ArcTan
[Sqrt[5 + 10*x]/Sqrt[-2 + I*Sqrt[31]]])/(5*Sqrt[(155*I)*(2*I + Sqrt[31])])

_______________________________________________________________________________________

Maple [B]  time = 0.047, 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+2*x)^(3/2)/(5*x^2+3*x+2),x)

[Out]

4/5*(1+2*x)^(1/2)+27/1550*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)+1/155*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)+27/155/(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)+2/155/(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)*5^(1/2)*7^(1/2)-4/5/(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)-27/1550*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)-1/155*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)+27/155/(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)+2/1
55/(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)-4/5/(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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.271912, size = 1179, normalized size = 4.66 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1303356250*120125^(3/4)*sqrt(70)*sqrt(31)*(4*120125^(1/4)*sqrt(70)*sqrt(31)*sq
rt(2*x + 1)*(175*sqrt(7) + 178*sqrt(5))*sqrt((178*sqrt(7)*sqrt(5) + 1225)/(12460
*sqrt(7)*sqrt(5) + 74559)) - 5*343^(1/4)*sqrt(31)*(175*sqrt(7) + 178*sqrt(5))*lo
g(124/125*sqrt(5)*(120125^(1/4)*343^(1/4)*sqrt(70)*(355324845622215676469207696*
sqrt(7)*sqrt(5) + 2102130135623455550175429965)*sqrt(2*x + 1)*sqrt((178*sqrt(7)*
sqrt(5) + 1225)/(12460*sqrt(7)*sqrt(5) + 74559)) + 1225*sqrt(5)*(605691587110334
8083790300*sqrt(7)*sqrt(5)*(2*x + 1) + 71666395066799246768273598*x + 3583319753
3399623384136799) + 1225*sqrt(7)*(6056915871103348083790300*sqrt(7)*sqrt(5) + 35
833197533399623384136799))/(6056915871103348083790300*sqrt(7)*sqrt(5) + 35833197
533399623384136799)) + 5*343^(1/4)*sqrt(31)*(175*sqrt(7) + 178*sqrt(5))*log(-124
/125*sqrt(5)*(120125^(1/4)*343^(1/4)*sqrt(70)*(355324845622215676469207696*sqrt(
7)*sqrt(5) + 2102130135623455550175429965)*sqrt(2*x + 1)*sqrt((178*sqrt(7)*sqrt(
5) + 1225)/(12460*sqrt(7)*sqrt(5) + 74559)) - 1225*sqrt(5)*(60569158711033480837
90300*sqrt(7)*sqrt(5)*(2*x + 1) + 71666395066799246768273598*x + 358331975333996
23384136799) - 1225*sqrt(7)*(6056915871103348083790300*sqrt(7)*sqrt(5) + 3583319
7533399623384136799))/(6056915871103348083790300*sqrt(7)*sqrt(5) + 3583319753339
9623384136799)) + 11780*343^(1/4)*sqrt(5)*arctan(5425*343^(1/4)*sqrt(31)*(10*sqr
t(7) + 27*sqrt(5))/(120125^(1/4)*sqrt(70)*sqrt(31)*sqrt(31/5)*(175*sqrt(7) + 178
*sqrt(5))*sqrt(sqrt(5)*(120125^(1/4)*343^(1/4)*sqrt(70)*(35532484562221567646920
7696*sqrt(7)*sqrt(5) + 2102130135623455550175429965)*sqrt(2*x + 1)*sqrt((178*sqr
t(7)*sqrt(5) + 1225)/(12460*sqrt(7)*sqrt(5) + 74559)) + 1225*sqrt(5)*(6056915871
103348083790300*sqrt(7)*sqrt(5)*(2*x + 1) + 71666395066799246768273598*x + 35833
197533399623384136799) + 1225*sqrt(7)*(6056915871103348083790300*sqrt(7)*sqrt(5)
+ 35833197533399623384136799))/(6056915871103348083790300*sqrt(7)*sqrt(5) + 358
33197533399623384136799))*sqrt((178*sqrt(7)*sqrt(5) + 1225)/(12460*sqrt(7)*sqrt(
5) + 74559)) + 1085*120125^(1/4)*sqrt(70)*sqrt(2*x + 1)*(175*sqrt(7) + 178*sqrt(
5))*sqrt((178*sqrt(7)*sqrt(5) + 1225)/(12460*sqrt(7)*sqrt(5) + 74559)) + 168175*
343^(1/4)*(5*sqrt(7) + 4*sqrt(5)))) + 11780*343^(1/4)*sqrt(5)*arctan(5425*343^(1
/4)*sqrt(31)*(10*sqrt(7) + 27*sqrt(5))/(120125^(1/4)*sqrt(70)*sqrt(31)*sqrt(31/5
)*(175*sqrt(7) + 178*sqrt(5))*sqrt(-sqrt(5)*(120125^(1/4)*343^(1/4)*sqrt(70)*(35
5324845622215676469207696*sqrt(7)*sqrt(5) + 2102130135623455550175429965)*sqrt(2
*x + 1)*sqrt((178*sqrt(7)*sqrt(5) + 1225)/(12460*sqrt(7)*sqrt(5) + 74559)) - 122
5*sqrt(5)*(6056915871103348083790300*sqrt(7)*sqrt(5)*(2*x + 1) + 716663950667992
46768273598*x + 35833197533399623384136799) - 1225*sqrt(7)*(60569158711033480837
90300*sqrt(7)*sqrt(5) + 35833197533399623384136799))/(6056915871103348083790300*
sqrt(7)*sqrt(5) + 35833197533399623384136799))*sqrt((178*sqrt(7)*sqrt(5) + 1225)
/(12460*sqrt(7)*sqrt(5) + 74559)) + 1085*120125^(1/4)*sqrt(70)*sqrt(2*x + 1)*(17
5*sqrt(7) + 178*sqrt(5))*sqrt((178*sqrt(7)*sqrt(5) + 1225)/(12460*sqrt(7)*sqrt(5
) + 74559)) - 168175*343^(1/4)*(5*sqrt(7) + 4*sqrt(5)))))/((175*sqrt(7) + 178*sq
rt(5))*sqrt((178*sqrt(7)*sqrt(5) + 1225)/(12460*sqrt(7)*sqrt(5) + 74559)))

_______________________________________________________________________________________

Sympy [A]  time = 22.8477, size = 85, normalized size = 0.34 \[ \frac{4 \sqrt{2 x + 1}}{5} + \frac{16 \operatorname{RootSum}{\left (1230080 t^{4} + 1984 t^{2} + 7, \left ( t \mapsto t \log{\left (9920 t^{3} + 8 t + \sqrt{2 x + 1} \right )} \right )\right )}}{5} - \frac{28 \operatorname{RootSum}{\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log{\left (- \frac{27776 t^{3}}{5} + \frac{108 t}{5} + \sqrt{2 x + 1} \right )} \right )\right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(2*x + 1)/5 + 16*RootSum(1230080*_t**4 + 1984*_t**2 + 7, Lambda(_t, _t*log
(9920*_t**3 + 8*_t + sqrt(2*x + 1))))/5 - 28*RootSum(1722112*_t**4 + 1984*_t**2
+ 5, Lambda(_t, _t*log(-27776*_t**3/5 + 108*_t/5 + sqrt(2*x + 1))))/5

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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