3.2756 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac{230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{891 (3 x+2)^{9/2}}+\frac{12280 (5 x+3)^{3/2} \sqrt{1-2 x}}{6237 (3 x+2)^{7/2}}+\frac{780320008 \sqrt{5 x+3} \sqrt{1-2 x}}{19253619 \sqrt{3 x+2}}+\frac{11243972 \sqrt{5 x+3} \sqrt{1-2 x}}{2750517 (3 x+2)^{3/2}}-\frac{325796 \sqrt{5 x+3} \sqrt{1-2 x}}{130977 (3 x+2)^{5/2}}-\frac{23441272 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1750329 \sqrt{33}}-\frac{780320008 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1750329 \sqrt{33}} \]

[Out]

(-325796*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(130977*(2 + 3*x)^(5/2)) + (11243972*Sqrt[
1 - 2*x]*Sqrt[3 + 5*x])/(2750517*(2 + 3*x)^(3/2)) + (780320008*Sqrt[1 - 2*x]*Sqr
t[3 + 5*x])/(19253619*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(33*(
2 + 3*x)^(11/2)) + (230*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(891*(2 + 3*x)^(9/2)) +
(12280*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(6237*(2 + 3*x)^(7/2)) - (780320008*Ellip
ticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1750329*Sqrt[33]) - (23441272*Ell
ipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1750329*Sqrt[33])

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Rubi [A]  time = 0.582152, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac{230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{891 (3 x+2)^{9/2}}+\frac{12280 (5 x+3)^{3/2} \sqrt{1-2 x}}{6237 (3 x+2)^{7/2}}+\frac{780320008 \sqrt{5 x+3} \sqrt{1-2 x}}{19253619 \sqrt{3 x+2}}+\frac{11243972 \sqrt{5 x+3} \sqrt{1-2 x}}{2750517 (3 x+2)^{3/2}}-\frac{325796 \sqrt{5 x+3} \sqrt{1-2 x}}{130977 (3 x+2)^{5/2}}-\frac{23441272 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1750329 \sqrt{33}}-\frac{780320008 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1750329 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(13/2),x]

[Out]

(-325796*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(130977*(2 + 3*x)^(5/2)) + (11243972*Sqrt[
1 - 2*x]*Sqrt[3 + 5*x])/(2750517*(2 + 3*x)^(3/2)) + (780320008*Sqrt[1 - 2*x]*Sqr
t[3 + 5*x])/(19253619*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(33*(
2 + 3*x)^(11/2)) + (230*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(891*(2 + 3*x)^(9/2)) +
(12280*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(6237*(2 + 3*x)^(7/2)) - (780320008*Ellip
ticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1750329*Sqrt[33]) - (23441272*Ell
ipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1750329*Sqrt[33])

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Rubi in Sympy [A]  time = 58.5387, size = 230, normalized size = 0.92 \[ - \frac{230 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{6237 \left (3 x + 2\right )^{\frac{9}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{33 \left (3 x + 2\right )^{\frac{11}{2}}} + \frac{1810 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{14553 \left (3 x + 2\right )^{\frac{7}{2}}} + \frac{780320008 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{19253619 \sqrt{3 x + 2}} + \frac{11243972 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2750517 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{79444 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{130977 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{780320008 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{57760857} - \frac{23441272 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{61261515} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**(13/2),x)

[Out]

-230*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(6237*(3*x + 2)**(9/2)) - 2*(-2*x + 1)**(5/
2)*(5*x + 3)**(3/2)/(33*(3*x + 2)**(11/2)) + 1810*(-2*x + 1)**(3/2)*sqrt(5*x + 3
)/(14553*(3*x + 2)**(7/2)) + 780320008*sqrt(-2*x + 1)*sqrt(5*x + 3)/(19253619*sq
rt(3*x + 2)) + 11243972*sqrt(-2*x + 1)*sqrt(5*x + 3)/(2750517*(3*x + 2)**(3/2))
+ 79444*sqrt(-2*x + 1)*sqrt(5*x + 3)/(130977*(3*x + 2)**(5/2)) - 780320008*sqrt(
33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/57760857 - 23441272*sqrt(
35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/61261515

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Mathematica [A]  time = 0.447432, size = 115, normalized size = 0.46 \[ \frac{\frac{24 \sqrt{1-2 x} \sqrt{5 x+3} \left (94808880972 x^5+319217269302 x^4+429993423180 x^3+289719086787 x^2+97637232762 x+13163824553\right )}{(3 x+2)^{11/2}}+16 \sqrt{2} \left (195080002 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-98384755 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{231043428} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(13/2),x]

[Out]

((24*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(13163824553 + 97637232762*x + 289719086787*x^2
+ 429993423180*x^3 + 319217269302*x^4 + 94808880972*x^5))/(2 + 3*x)^(11/2) + 16
*Sqrt[2]*(195080002*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 9838475
5*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/231043428

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Maple [C]  time = 0.03, size = 743, normalized size = 3. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^(13/2),x)

[Out]

2/57760857*(47814990930*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/
2*I*11^(1/2)*3^(1/2)*2^(1/2))*x^5*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-9480
8880972*2^(1/2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(
1/2)*2^(1/2))*x^5*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)
*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x
^4*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-316029603240*2^(1/2)*EllipticE(1/11
*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x^4*(3+5*x)^(1/2
)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/
2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x^3*(1-2*x)^(1/2)*(3+5*x)^(1/2)
*(2+3*x)^(1/2)-421372804320*2^(1/2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2
),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x^3*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)+
141674047200*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2
)*3^(1/2)*2^(1/2))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-280915202880*2^
(1/2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/
2))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*x^7+47224682400*
2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(
1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-93638400960*2^(1/2)*EllipticE(
1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x*(3+5*x)^(1
/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*x^6+6296624320*2^(1/2)*(3+5*x)^(1/
2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2
*I*11^(1/2)*3^(1/2)*2^(1/2))-12485120128*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-
2*x)^(1/2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*
2^(1/2))[phone]*x^5+7108597449432*x^4-71666565399*x^3-1919645346207*x^2-
839243621199*x-118474420977)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(1
1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2),x, algorithm="fricas")

[Out]

integral((20*x^3 - 8*x^2 - 7*x + 3)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((729*x^6 + 291
6*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*sqrt(3*x + 2)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**(5/2)*(3+5*x)**(3/2)/(2+3*x)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(3/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2), x)