3.2298 \(\int \frac{(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx\)

Optimal. Leaf size=279 \[ \frac{4}{25} (2 x+1)^{5/2}+\frac{16}{75} (2 x+1)^{3/2}-\frac{76}{125} \sqrt{2 x+1}-\frac{1}{125} \sqrt{\frac{1}{310} \left (168698+42875 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{125} \sqrt{\frac{1}{310} \left (168698+42875 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{125} \sqrt{\frac{2}{155} \left (42875 \sqrt{35}-168698\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{125} \sqrt{\frac{2}{155} \left (42875 \sqrt{35}-168698\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

[Out]

(-76*Sqrt[1 + 2*x])/125 + (16*(1 + 2*x)^(3/2))/75 + (4*(1 + 2*x)^(5/2))/25 + (Sq
rt[(2*(-168698 + 42875*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt
[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/125 - (Sqrt[(2*(-168698 + 42875*Sqrt[35]))
/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35]
)]])/125 - (Sqrt[(168698 + 42875*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt
[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/125 + (Sqrt[(168698 + 42875*Sqrt[35])/310]*
Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/125

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Rubi [A]  time = 1.46041, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{4}{25} (2 x+1)^{5/2}+\frac{16}{75} (2 x+1)^{3/2}-\frac{76}{125} \sqrt{2 x+1}-\frac{1}{125} \sqrt{\frac{1}{310} \left (168698+42875 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{125} \sqrt{\frac{1}{310} \left (168698+42875 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{125} \sqrt{\frac{2}{155} \left (42875 \sqrt{35}-168698\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{125} \sqrt{\frac{2}{155} \left (42875 \sqrt{35}-168698\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2),x]

[Out]

(-76*Sqrt[1 + 2*x])/125 + (16*(1 + 2*x)^(3/2))/75 + (4*(1 + 2*x)^(5/2))/25 + (Sq
rt[(2*(-168698 + 42875*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt
[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/125 - (Sqrt[(2*(-168698 + 42875*Sqrt[35]))
/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35]
)]])/125 - (Sqrt[(168698 + 42875*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt
[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/125 + (Sqrt[(168698 + 42875*Sqrt[35])/310]*
Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/125

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Rubi in Sympy [A]  time = 84.0495, size = 391, normalized size = 1.4 \[ \frac{4 \left (2 x + 1\right )^{\frac{5}{2}}}{25} + \frac{16 \left (2 x + 1\right )^{\frac{3}{2}}}{75} - \frac{76 \sqrt{2 x + 1}}{125} - \frac{\sqrt{14} \left (133 + \frac{216 \sqrt{35}}{5}\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{1750 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \left (133 + \frac{216 \sqrt{35}}{5}\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{1750 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (266 + \frac{432 \sqrt{35}}{5}\right )}{10} + \frac{266 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{875 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (266 + \frac{432 \sqrt{35}}{5}\right )}{10} + \frac{266 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{875 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+2*x)**(7/2)/(5*x**2+3*x+2),x)

[Out]

4*(2*x + 1)**(5/2)/25 + 16*(2*x + 1)**(3/2)/75 - 76*sqrt(2*x + 1)/125 - sqrt(14)
*(133 + 216*sqrt(35)/5)*log(2*x - sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 +
1 + sqrt(35)/5)/(1750*sqrt(2 + sqrt(35))) + sqrt(14)*(133 + 216*sqrt(35)/5)*log(
2*x + sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(1750*sqrt(2
+ sqrt(35))) + sqrt(35)*(-sqrt(10)*sqrt(2 + sqrt(35))*(266 + 432*sqrt(35)/5)/10
+ 266*sqrt(10)*sqrt(2 + sqrt(35))/5)*atan(sqrt(10)*(sqrt(2*x + 1) - sqrt(20 + 1
0*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(875*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35))
) + sqrt(35)*(-sqrt(10)*sqrt(2 + sqrt(35))*(266 + 432*sqrt(35)/5)/10 + 266*sqrt(
10)*sqrt(2 + sqrt(35))/5)*atan(sqrt(10)*(sqrt(2*x + 1) + sqrt(20 + 10*sqrt(35))/
10)/sqrt(-2 + sqrt(35)))/(875*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35)))

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Mathematica [C]  time = 0.720463, size = 142, normalized size = 0.51 \[ \frac{8}{375} \sqrt{2 x+1} \left (30 x^2+50 x-11\right )-\frac{2 i \left (233 \sqrt{31}-6696 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{3875 \sqrt{-10-5 i \sqrt{31}}}+\frac{2 i \left (233 \sqrt{31}+6696 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{3875 \sqrt{5 i \left (\sqrt{31}+2 i\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2),x]

[Out]

(8*Sqrt[1 + 2*x]*(-11 + 50*x + 30*x^2))/375 - (((2*I)/3875)*(-6696*I + 233*Sqrt[
31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]])/Sqrt[-10 - (5*I)*Sqrt[31]] +
(((2*I)/3875)*(6696*I + 233*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 + I*Sqrt[31]
]])/Sqrt[(5*I)*(2*I + Sqrt[31])]

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Maple [B]  time = 0.203, size = 634, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+2*x)^(7/2)/(5*x^2+3*x+2),x)

[Out]

4/25*(1+2*x)^(5/2)+16/75*(1+2*x)^(3/2)-76/125*(1+2*x)^(1/2)-233/38750*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)-89/3875*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)-233/3875/(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)-178/3875/(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)+76/125/(
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)+233/38750*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)+89/3875*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)-233/3875/(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)-178/3875/(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))/(1
0*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)*5^(1/2)*7^(1/2)+76/125/(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. \[ \int \frac{{\left (2 \, x + 1\right )}^{\frac{7}{2}}}{5 \, x^{2} + 3 \, x + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2),x, algorithm="maxima")

[Out]

integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.282612, size = 1206, normalized size = 4.32 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2),x, algorithm="fricas")

[Out]

1/97751718750*120125^(3/4)*sqrt(70)*sqrt(31)*(8*120125^(1/4)*sqrt(70)*sqrt(31)*(
214375*sqrt(7)*(30*x^2 + 50*x - 11) + 168698*sqrt(5)*(30*x^2 + 50*x - 11))*sqrt(
2*x + 1)*sqrt((168698*sqrt(7)*sqrt(5) + 1500625)/(14465853500*sqrt(7)*sqrt(5) +
92798312079)) + 15*343^(1/4)*sqrt(31)*(214375*sqrt(7) + 168698*sqrt(5))*log(2977
24/125*sqrt(5)*(120125^(1/4)*343^(1/4)*sqrt(70)*(2965545773805035055814396432958
1894624492439642765206829684*sqrt(7)*sqrt(5) + 175444041981990546642542429674142
329118053782694232680705335)*sqrt(2*x + 1)*sqrt((168698*sqrt(7)*sqrt(5) + 150062
5)/(14465853500*sqrt(7)*sqrt(5) + 92798312079)) + 1225*sqrt(5)*(1526370995419339
7316467080022375073516955615396805667500*sqrt(7)*sqrt(5)*(2*x + 1) + 18060269091
2145753173086630691689543849231068554531832798*x + 90301345456072876586543315345
844771924615534277265916399) + 1225*sqrt(7)*(15263709954193397316467080022375073
516955615396805667500*sqrt(7)*sqrt(5) + 9030134545607287658654331534584477192461
5534277265916399))/(15263709954193397316467080022375073516955615396805667500*sqr
t(7)*sqrt(5) + 90301345456072876586543315345844771924615534277265916399)) - 15*3
43^(1/4)*sqrt(31)*(214375*sqrt(7) + 168698*sqrt(5))*log(-297724/125*sqrt(5)*(120
125^(1/4)*343^(1/4)*sqrt(70)*(29655457738050350558143964329581894624492439642765
206829684*sqrt(7)*sqrt(5) + 1754440419819905466425424296741423291180537826942326
80705335)*sqrt(2*x + 1)*sqrt((168698*sqrt(7)*sqrt(5) + 1500625)/(14465853500*sqr
t(7)*sqrt(5) + 92798312079)) - 1225*sqrt(5)*(15263709954193397316467080022375073
516955615396805667500*sqrt(7)*sqrt(5)*(2*x + 1) + 180602690912145753173086630691
689543849231068554531832798*x + 903013454560728765865433153458447719246155342772
65916399) - 1225*sqrt(7)*(152637099541933973164670800223750735169556153968056675
00*sqrt(7)*sqrt(5) + 90301345456072876586543315345844771924615534277265916399))/
(15263709954193397316467080022375073516955615396805667500*sqrt(7)*sqrt(5) + 9030
1345456072876586543315345844771924615534277265916399)) + 63279060*343^(1/4)*sqrt
(5)*arctan(5425*343^(1/4)*sqrt(31)*(890*sqrt(7) + 233*sqrt(5))/(120125^(1/4)*sqr
t(70)*sqrt(31)*sqrt(31/5)*(214375*sqrt(7) + 168698*sqrt(5))*sqrt(sqrt(5)*(120125
^(1/4)*343^(1/4)*sqrt(70)*(29655457738050350558143964329581894624492439642765206
829684*sqrt(7)*sqrt(5) + 1754440419819905466425424296741423291180537826942326807
05335)*sqrt(2*x + 1)*sqrt((168698*sqrt(7)*sqrt(5) + 1500625)/(14465853500*sqrt(7
)*sqrt(5) + 92798312079)) + 1225*sqrt(5)*(15263709954193397316467080022375073516
955615396805667500*sqrt(7)*sqrt(5)*(2*x + 1) + 180602690912145753173086630691689
543849231068554531832798*x + 903013454560728765865433153458447719246155342772659
16399) + 1225*sqrt(7)*(15263709954193397316467080022375073516955615396805667500*
sqrt(7)*sqrt(5) + 90301345456072876586543315345844771924615534277265916399))/(15
263709954193397316467080022375073516955615396805667500*sqrt(7)*sqrt(5) + 9030134
5456072876586543315345844771924615534277265916399))*sqrt((168698*sqrt(7)*sqrt(5)
+ 1500625)/(14465853500*sqrt(7)*sqrt(5) + 92798312079)) + 1085*120125^(1/4)*sqr
t(70)*sqrt(2*x + 1)*(214375*sqrt(7) + 168698*sqrt(5))*sqrt((168698*sqrt(7)*sqrt(
5) + 1500625)/(14465853500*sqrt(7)*sqrt(5) + 92798312079)) + 168175*343^(1/4)*(9
5*sqrt(7) + 216*sqrt(5)))) + 63279060*343^(1/4)*sqrt(5)*arctan(5425*343^(1/4)*sq
rt(31)*(890*sqrt(7) + 233*sqrt(5))/(120125^(1/4)*sqrt(70)*sqrt(31)*sqrt(31/5)*(2
14375*sqrt(7) + 168698*sqrt(5))*sqrt(-sqrt(5)*(120125^(1/4)*343^(1/4)*sqrt(70)*(
29655457738050350558143964329581894624492439642765206829684*sqrt(7)*sqrt(5) + 17
5444041981990546642542429674142329118053782694232680705335)*sqrt(2*x + 1)*sqrt((
168698*sqrt(7)*sqrt(5) + 1500625)/(14465853500*sqrt(7)*sqrt(5) + 92798312079)) -
1225*sqrt(5)*(15263709954193397316467080022375073516955615396805667500*sqrt(7)*
sqrt(5)*(2*x + 1) + 180602690912145753173086630691689543849231068554531832798*x
+ 90301345456072876586543315345844771924615534277265916399) - 1225*sqrt(7)*(1526
3709954193397316467080022375073516955615396805667500*sqrt(7)*sqrt(5) + 903013454
56072876586543315345844771924615534277265916399))/(15263709954193397316467080022
375073516955615396805667500*sqrt(7)*sqrt(5) + 9030134545607287658654331534584477
1924615534277265916399))*sqrt((168698*sqrt(7)*sqrt(5) + 1500625)/(14465853500*sq
rt(7)*sqrt(5) + 92798312079)) + 1085*120125^(1/4)*sqrt(70)*sqrt(2*x + 1)*(214375
*sqrt(7) + 168698*sqrt(5))*sqrt((168698*sqrt(7)*sqrt(5) + 1500625)/(14465853500*
sqrt(7)*sqrt(5) + 92798312079)) - 168175*343^(1/4)*(95*sqrt(7) + 216*sqrt(5)))))
/((214375*sqrt(7) + 168698*sqrt(5))*sqrt((168698*sqrt(7)*sqrt(5) + 1500625)/(144
65853500*sqrt(7)*sqrt(5) + 92798312079)))

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Sympy [A]  time = 93.9397, size = 109, normalized size = 0.39 \[ \frac{4 \left (2 x + 1\right )^{\frac{5}{2}}}{25} + \frac{16 \left (2 x + 1\right )^{\frac{3}{2}}}{75} - \frac{76 \sqrt{2 x + 1}}{125} - \frac{864 \operatorname{RootSum}{\left (1230080 t^{4} + 1984 t^{2} + 7, \left ( t \mapsto t \log{\left (9920 t^{3} + 8 t + \sqrt{2 x + 1} \right )} \right )\right )}}{125} + \frac{532 \operatorname{RootSum}{\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log{\left (- \frac{27776 t^{3}}{5} + \frac{108 t}{5} + \sqrt{2 x + 1} \right )} \right )\right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+2*x)**(7/2)/(5*x**2+3*x+2),x)

[Out]

4*(2*x + 1)**(5/2)/25 + 16*(2*x + 1)**(3/2)/75 - 76*sqrt(2*x + 1)/125 - 864*Root
Sum(1230080*_t**4 + 1984*_t**2 + 7, Lambda(_t, _t*log(9920*_t**3 + 8*_t + sqrt(2
*x + 1))))/125 + 532*RootSum(1722112*_t**4 + 1984*_t**2 + 5, Lambda(_t, _t*log(-
27776*_t**3/5 + 108*_t/5 + sqrt(2*x + 1))))/125

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, x + 1\right )}^{\frac{7}{2}}}{5 \, x^{2} + 3 \, x + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2),x, algorithm="giac")

[Out]

integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2), x)