3.2311 \(\int \frac{1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx\)

Optimal. Leaf size=296 \[ \frac{20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}-\frac{4680}{10633 \sqrt{2 x+1}}-\frac{820}{4557 (2 x+1)^{3/2}}-\frac{5 \sqrt{\frac{1}{434} \left (2632525 \sqrt{35}-12504542\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{10633}+\frac{5 \sqrt{\frac{1}{434} \left (2632525 \sqrt{35}-12504542\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{10633}+\frac{5 \sqrt{\frac{2}{217} \left (12504542+2632525 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{10633}-\frac{5 \sqrt{\frac{2}{217} \left (12504542+2632525 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{10633} \]

[Out]

-820/(4557*(1 + 2*x)^(3/2)) - 4680/(10633*Sqrt[1 + 2*x]) + (37 + 20*x)/(217*(1 +
2*x)^(3/2)*(2 + 3*x + 5*x^2)) + (5*Sqrt[(2*(12504542 + 2632525*Sqrt[35]))/217]*
ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/1
0633 - (5*Sqrt[(2*(12504542 + 2632525*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[
35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/10633 - (5*Sqrt[(-12504542
+ 2632525*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] +
5*(1 + 2*x)])/10633 + (5*Sqrt[(-12504542 + 2632525*Sqrt[35])/434]*Log[Sqrt[35] +
Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/10633

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Rubi [A]  time = 1.376, antiderivative size = 296, 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 \[ \frac{20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}-\frac{4680}{10633 \sqrt{2 x+1}}-\frac{820}{4557 (2 x+1)^{3/2}}-\frac{5 \sqrt{\frac{1}{434} \left (2632525 \sqrt{35}-12504542\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{10633}+\frac{5 \sqrt{\frac{1}{434} \left (2632525 \sqrt{35}-12504542\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{10633}+\frac{5 \sqrt{\frac{2}{217} \left (12504542+2632525 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{10633}-\frac{5 \sqrt{\frac{2}{217} \left (12504542+2632525 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{10633} \]

Antiderivative was successfully verified.

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

[Out]

-820/(4557*(1 + 2*x)^(3/2)) - 4680/(10633*Sqrt[1 + 2*x]) + (37 + 20*x)/(217*(1 +
2*x)^(3/2)*(2 + 3*x + 5*x^2)) + (5*Sqrt[(2*(12504542 + 2632525*Sqrt[35]))/217]*
ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/1
0633 - (5*Sqrt[(2*(12504542 + 2632525*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[
35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/10633 - (5*Sqrt[(-12504542
+ 2632525*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] +
5*(1 + 2*x)])/10633 + (5*Sqrt[(-12504542 + 2632525*Sqrt[35])/434]*Log[Sqrt[35] +
Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/10633

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Rubi in Sympy [A]  time = 85.7388, size = 394, normalized size = 1.33 \[ \frac{\sqrt{14} \left (- 1170 \sqrt{35} + 2495\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{148862 \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{14} \left (- 1170 \sqrt{35} + 2495\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{148862 \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- 2340 \sqrt{35} + 4990\right )}{10} + 998 \sqrt{10} \sqrt{2 + \sqrt{35}}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{74431 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- 2340 \sqrt{35} + 4990\right )}{10} + 998 \sqrt{10} \sqrt{2 + \sqrt{35}}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{74431 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} - \frac{4680}{10633 \sqrt{2 x + 1}} + \frac{20 x + 37}{217 \left (2 x + 1\right )^{\frac{3}{2}} \left (5 x^{2} + 3 x + 2\right )} - \frac{820}{4557 \left (2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(14)*(-1170*sqrt(35) + 2495)*log(2*x - sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x
+ 1)/5 + 1 + sqrt(35)/5)/(148862*sqrt(2 + sqrt(35))) - sqrt(14)*(-1170*sqrt(35)
+ 2495)*log(2*x + sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/
(148862*sqrt(2 + sqrt(35))) - sqrt(35)*(-sqrt(10)*sqrt(2 + sqrt(35))*(-2340*sqrt
(35) + 4990)/10 + 998*sqrt(10)*sqrt(2 + sqrt(35)))*atan(sqrt(10)*(sqrt(2*x + 1)
- sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(74431*sqrt(-2 + sqrt(35))*sqr
t(2 + sqrt(35))) - sqrt(35)*(-sqrt(10)*sqrt(2 + sqrt(35))*(-2340*sqrt(35) + 4990
)/10 + 998*sqrt(10)*sqrt(2 + sqrt(35)))*atan(sqrt(10)*(sqrt(2*x + 1) + sqrt(20 +
10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(74431*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(
35))) - 4680/(10633*sqrt(2*x + 1)) + (20*x + 37)/(217*(2*x + 1)**(3/2)*(5*x**2 +
3*x + 2)) - 820/(4557*(2*x + 1)**(3/2))

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Mathematica [C]  time = 1.39035, size = 165, normalized size = 0.56 \[ \frac{2 \left (-\frac{31 \left (140400 x^3+183140 x^2+112560 x+34121\right )}{2 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}+\frac{15 i \left (967 \sqrt{31}+7254 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{\sqrt{-\frac{1}{5} i \left (\sqrt{31}-2 i\right )}}-\frac{15 i \left (967 \sqrt{31}-7254 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{\sqrt{\frac{1}{5} i \left (\sqrt{31}+2 i\right )}}\right )}{988869} \]

Antiderivative was successfully verified.

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

[Out]

(2*((-31*(34121 + 112560*x + 183140*x^2 + 140400*x^3))/(2*(1 + 2*x)^(3/2)*(2 + 3
*x + 5*x^2)) + ((15*I)*(7254*I + 967*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I
*Sqrt[31]]])/Sqrt[(-I/5)*(-2*I + Sqrt[31])] - ((15*I)*(-7254*I + 967*Sqrt[31])*A
rcTan[Sqrt[5 + 10*x]/Sqrt[-2 + I*Sqrt[31]]])/Sqrt[(I/5)*(2*I + Sqrt[31])]))/9888
69

_______________________________________________________________________________________

Maple [B]  time = 0.042, size = 503, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16/147/(1+2*x)^(3/2)-128/343/(1+2*x)^(1/2)-16/343*(89/62*(1+2*x)^(3/2)+233/620*
(1+2*x)^(1/2))/((1+2*x)^2-8/5*x+3/5)-4594/2307361*ln(5+10*x+35^(1/2)-(1+2*x)^(1/
2)*(20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))^(1/2)*35^(1/2)+4835/659246*ln(5+10*x
+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))^(1/2)-9188/2307
361/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)-(20+10*35^(1/2))^(1/2))/(-2
0+10*35^(1/2))^(1/2))*(20+10*35^(1/2))*35^(1/2)+4835/329623/(-20+10*35^(1/2))^(1
/2)*arctan((10*(1+2*x)^(1/2)-(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(2
0+10*35^(1/2))-9980/74431/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)-(20+1
0*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*35^(1/2)+4594/2307361*ln(5+10*x+35^(
1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))^(1/2)*35^(1/2)-4835/
659246*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))
^(1/2)-9188/2307361/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)+(20+10*35^(
1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(20+10*35^(1/2))*35^(1/2)+4835/329623/(-20
+10*35^(1/2))^(1/2)*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^
(1/2))^(1/2))*(20+10*35^(1/2))-9980/74431/(-20+10*35^(1/2))^(1/2)*arctan((10*(1+
2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*35^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.281245, size = 1418, normalized size = 4.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1400950147531326*329623^(3/4)*sqrt(4298)*sqrt(31)*(21710316180*471245^(1/4)*s
qrt(7)*(10*x^3 + 11*x^2 + 7*x + 2)*sqrt(2*x + 1)*arctan(3264331*471245^(1/4)*sqr
t(31)*(4835*sqrt(7) - 9188*sqrt(5))/(329623^(1/4)*sqrt(47585)*sqrt(4298)*sqrt(31
)*(12504542*sqrt(7) - 18427675*sqrt(5))*sqrt(sqrt(7)*(471245^(1/4)*329623^(1/4)*
sqrt(4298)*(48489836974248471885409979605113532913794279891997072727389865924013
618909066*sqrt(7)*sqrt(5) - 2868697441433726740636862260853177699509222235364553
49810870402261352054617301)*sqrt(2*x + 1)*sqrt((12504542*sqrt(7)*sqrt(5) - 92138
375)/(65837038857100*sqrt(7)*sqrt(5) - 398920146276639)) + 10745*sqrt(7)*(257464
15041928598100512814456614948400249993937736407650668550790798205500*sqrt(7)*sqr
t(5)*(2*x + 1) - 304635691182274371224393890523612595402303700145868421450480846
832524054398*x - 152317845591137185612196945261806297701151850072934210725240423
416262027199) + 15043*sqrt(5)*(2574641504192859810051281445661494840024999393773
6407650668550790798205500*sqrt(7)*sqrt(5) - 152317845591137185612196945261806297
701151850072934210725240423416262027199))/(2574641504192859810051281445661494840
0249993937736407650668550790798205500*sqrt(7)*sqrt(5) - 152317845591137185612196
945261806297701151850072934210725240423416262027199))*sqrt((12504542*sqrt(7)*sqr
t(5) - 92138375)/(65837038857100*sqrt(7)*sqrt(5) - 398920146276639)) + 333095*32
9623^(1/4)*sqrt(4298)*sqrt(2*x + 1)*(12504542*sqrt(7) - 18427675*sqrt(5))*sqrt((
12504542*sqrt(7)*sqrt(5) - 92138375)/(65837038857100*sqrt(7)*sqrt(5) - 398920146
276639)) + 101194261*471245^(1/4)*(1170*sqrt(7) - 499*sqrt(5)))) + 21710316180*4
71245^(1/4)*sqrt(7)*(10*x^3 + 11*x^2 + 7*x + 2)*sqrt(2*x + 1)*arctan(3264331*471
245^(1/4)*sqrt(31)*(4835*sqrt(7) - 9188*sqrt(5))/(329623^(1/4)*sqrt(47585)*sqrt(
4298)*sqrt(31)*(12504542*sqrt(7) - 18427675*sqrt(5))*sqrt(-sqrt(7)*(471245^(1/4)
*329623^(1/4)*sqrt(4298)*(484898369742484718854099796051135329137942798919970727
27389865924013618909066*sqrt(7)*sqrt(5) - 28686974414337267406368622608531776995
0922223536455349810870402261352054617301)*sqrt(2*x + 1)*sqrt((12504542*sqrt(7)*s
qrt(5) - 92138375)/(65837038857100*sqrt(7)*sqrt(5) - 398920146276639)) - 10745*s
qrt(7)*(257464150419285981005128144566149484002499939377364076506685507907982055
00*sqrt(7)*sqrt(5)*(2*x + 1) - 3046356911822743712243938905236125954023037001458
68421450480846832524054398*x - 1523178455911371856121969452618062977011518500729
34210725240423416262027199) - 15043*sqrt(5)*(25746415041928598100512814456614948
400249993937736407650668550790798205500*sqrt(7)*sqrt(5) - 1523178455911371856121
96945261806297701151850072934210725240423416262027199))/(25746415041928598100512
814456614948400249993937736407650668550790798205500*sqrt(7)*sqrt(5) - 1523178455
91137185612196945261806297701151850072934210725240423416262027199))*sqrt((125045
42*sqrt(7)*sqrt(5) - 92138375)/(65837038857100*sqrt(7)*sqrt(5) - 398920146276639
)) + 333095*329623^(1/4)*sqrt(4298)*sqrt(2*x + 1)*(12504542*sqrt(7) - 18427675*s
qrt(5))*sqrt((12504542*sqrt(7)*sqrt(5) - 92138375)/(65837038857100*sqrt(7)*sqrt(
5) - 398920146276639)) - 101194261*471245^(1/4)*(1170*sqrt(7) - 499*sqrt(5)))) +
105*471245^(1/4)*sqrt(31)*(12504542*sqrt(7)*(10*x^3 + 11*x^2 + 7*x + 2) - 18427
675*sqrt(5)*(10*x^3 + 11*x^2 + 7*x + 2))*sqrt(2*x + 1)*log(2974062500/49*sqrt(7)
*(471245^(1/4)*329623^(1/4)*sqrt(4298)*(4848983697424847188540997960511353291379
4279891997072727389865924013618909066*sqrt(7)*sqrt(5) - 286869744143372674063686
226085317769950922223536455349810870402261352054617301)*sqrt(2*x + 1)*sqrt((1250
4542*sqrt(7)*sqrt(5) - 92138375)/(65837038857100*sqrt(7)*sqrt(5) - 3989201462766
39)) + 10745*sqrt(7)*(2574641504192859810051281445661494840024999393773640765066
8550790798205500*sqrt(7)*sqrt(5)*(2*x + 1) - 30463569118227437122439389052361259
5402303700145868421450480846832524054398*x - 15231784559113718561219694526180629
7701151850072934210725240423416262027199) + 15043*sqrt(5)*(257464150419285981005
12814456614948400249993937736407650668550790798205500*sqrt(7)*sqrt(5) - 15231784
5591137185612196945261806297701151850072934210725240423416262027199))/(257464150
41928598100512814456614948400249993937736407650668550790798205500*sqrt(7)*sqrt(5
) - 152317845591137185612196945261806297701151850072934210725240423416262027199)
) - 105*471245^(1/4)*sqrt(31)*(12504542*sqrt(7)*(10*x^3 + 11*x^2 + 7*x + 2) - 18
427675*sqrt(5)*(10*x^3 + 11*x^2 + 7*x + 2))*sqrt(2*x + 1)*log(-2974062500/49*sqr
t(7)*(471245^(1/4)*329623^(1/4)*sqrt(4298)*(484898369742484718854099796051135329
13794279891997072727389865924013618909066*sqrt(7)*sqrt(5) - 28686974414337267406
3686226085317769950922223536455349810870402261352054617301)*sqrt(2*x + 1)*sqrt((
12504542*sqrt(7)*sqrt(5) - 92138375)/(65837038857100*sqrt(7)*sqrt(5) - 398920146
276639)) - 10745*sqrt(7)*(257464150419285981005128144566149484002499939377364076
50668550790798205500*sqrt(7)*sqrt(5)*(2*x + 1) - 3046356911822743712243938905236
12595402303700145868421450480846832524054398*x - 1523178455911371856121969452618
06297701151850072934210725240423416262027199) - 15043*sqrt(5)*(25746415041928598
100512814456614948400249993937736407650668550790798205500*sqrt(7)*sqrt(5) - 1523
17845591137185612196945261806297701151850072934210725240423416262027199))/(25746
415041928598100512814456614948400249993937736407650668550790798205500*sqrt(7)*sq
rt(5) - 152317845591137185612196945261806297701151850072934210725240423416262027
199)) + 329623^(1/4)*sqrt(4298)*sqrt(31)*(12504542*sqrt(7)*(140400*x^3 + 183140*
x^2 + 112560*x + 34121) - 18427675*sqrt(5)*(140400*x^3 + 183140*x^2 + 112560*x +
34121))*sqrt((12504542*sqrt(7)*sqrt(5) - 92138375)/(65837038857100*sqrt(7)*sqrt
(5) - 398920146276639)))/((12504542*sqrt(7)*(10*x^3 + 11*x^2 + 7*x + 2) - 184276
75*sqrt(5)*(10*x^3 + 11*x^2 + 7*x + 2))*sqrt(2*x + 1)*sqrt((12504542*sqrt(7)*sqr
t(5) - 92138375)/(65837038857100*sqrt(7)*sqrt(5) - 398920146276639)))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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