3.2767 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx\)

Optimal. Leaf size=280 \[ \frac{60080 \sqrt{1-2 x} (5 x+3)^{5/2}}{34749 (3 x+2)^{9/2}}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{1287 (3 x+2)^{11/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}-\frac{2622980 \sqrt{1-2 x} (5 x+3)^{3/2}}{1702701 (3 x+2)^{7/2}}+\frac{129922578224 \sqrt{1-2 x} \sqrt{5 x+3}}{5256237987 \sqrt{3 x+2}}+\frac{1876198516 \sqrt{1-2 x} \sqrt{5 x+3}}{750891141 (3 x+2)^{3/2}}-\frac{54281308 \sqrt{1-2 x} \sqrt{5 x+3}}{35756721 (3 x+2)^{5/2}}-\frac{3894280616 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{477839817 \sqrt{33}}-\frac{129922578224 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{477839817 \sqrt{33}} \]

[Out]

(-54281308*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35756721*(2 + 3*x)^(5/2)) + (1876198516
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(750891141*(2 + 3*x)^(3/2)) + (129922578224*Sqrt[1
- 2*x]*Sqrt[3 + 5*x])/(5256237987*Sqrt[2 + 3*x]) - (2622980*Sqrt[1 - 2*x]*(3 +
5*x)^(3/2))/(1702701*(2 + 3*x)^(7/2)) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(39*
(2 + 3*x)^(13/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(1287*(2 + 3*x)^(11/2)
) + (60080*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(34749*(2 + 3*x)^(9/2)) - (12992257822
4*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(477839817*Sqrt[33]) - (389
4280616*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(477839817*Sqrt[33])

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Rubi [A]  time = 0.681419, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{60080 \sqrt{1-2 x} (5 x+3)^{5/2}}{34749 (3 x+2)^{9/2}}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{1287 (3 x+2)^{11/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}-\frac{2622980 \sqrt{1-2 x} (5 x+3)^{3/2}}{1702701 (3 x+2)^{7/2}}+\frac{129922578224 \sqrt{1-2 x} \sqrt{5 x+3}}{5256237987 \sqrt{3 x+2}}+\frac{1876198516 \sqrt{1-2 x} \sqrt{5 x+3}}{750891141 (3 x+2)^{3/2}}-\frac{54281308 \sqrt{1-2 x} \sqrt{5 x+3}}{35756721 (3 x+2)^{5/2}}-\frac{3894280616 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{477839817 \sqrt{33}}-\frac{129922578224 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{477839817 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(15/2),x]

[Out]

(-54281308*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35756721*(2 + 3*x)^(5/2)) + (1876198516
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(750891141*(2 + 3*x)^(3/2)) + (129922578224*Sqrt[1
- 2*x]*Sqrt[3 + 5*x])/(5256237987*Sqrt[2 + 3*x]) - (2622980*Sqrt[1 - 2*x]*(3 +
5*x)^(3/2))/(1702701*(2 + 3*x)^(7/2)) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(39*
(2 + 3*x)^(13/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(1287*(2 + 3*x)^(11/2)
) + (60080*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(34749*(2 + 3*x)^(9/2)) - (12992257822
4*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(477839817*Sqrt[33]) - (389
4280616*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(477839817*Sqrt[33])

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Rubi in Sympy [A]  time = 66.3478, size = 258, normalized size = 0.92 \[ - \frac{49810 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{1702701 \left (3 x + 2\right )^{\frac{9}{2}}} - \frac{370 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{9009 \left (3 x + 2\right )^{\frac{11}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{39 \left (3 x + 2\right )^{\frac{13}{2}}} + \frac{610730 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3972969 \left (3 x + 2\right )^{\frac{7}{2}}} + \frac{129922578224 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{5256237987 \sqrt{3 x + 2}} + \frac{1876198516 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{750891141 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{11823632 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{35756721 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{129922578224 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{15768713961} - \frac{3894280616 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{16724393595} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**(15/2),x)

[Out]

-49810*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(1702701*(3*x + 2)**(9/2)) - 370*(-2*x +
1)**(5/2)*(5*x + 3)**(3/2)/(9009*(3*x + 2)**(11/2)) - 2*(-2*x + 1)**(5/2)*(5*x +
3)**(5/2)/(39*(3*x + 2)**(13/2)) + 610730*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(3972
969*(3*x + 2)**(7/2)) + 129922578224*sqrt(-2*x + 1)*sqrt(5*x + 3)/(5256237987*sq
rt(3*x + 2)) + 1876198516*sqrt(-2*x + 1)*sqrt(5*x + 3)/(750891141*(3*x + 2)**(3/
2)) + 11823632*sqrt(-2*x + 1)*sqrt(5*x + 3)/(35756721*(3*x + 2)**(5/2)) - 129922
578224*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/15768713961 -
3894280616*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/1672439
3595

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Mathematica [A]  time = 0.473664, size = 117, normalized size = 0.42 \[ \frac{\frac{48 \sqrt{2-4 x} \sqrt{5 x+3} \left (47356779762648 x^6+191022825888450 x^5+321056742490902 x^4+287874442427697 x^3+145238558453649 x^2+39086872650957 x+4382625184685\right )}{(3 x+2)^{13/2}}-1050671168960 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{126149711688 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(15/2),x]

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(4382625184685 + 39086872650957*x + 14523855845
3649*x^2 + 287874442427697*x^3 + 321056742490902*x^4 + 191022825888450*x^5 + 473
56779762648*x^6))/(2 + 3*x)^(13/2) + 2078761251584*EllipticE[ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]], -33/2] - 1050671168960*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]
, -33/2])/(126149711688*Sqrt[2])

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Maple [C]  time = 0.032, size = 862, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2),x)

[Out]

2/15768713961*(23935602567870*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^6*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2
)-47356779762648*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^6*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]
480*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)-189427119050592*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)-315711865084320*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)-280632768963840*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)+1420
703392879440*x^8+70920303904800*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)-140316384481920*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]
15941444*x^7+18912081041280*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)-37
417702528512*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)+9778559734528578*
x^6+2101342337920*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))-4157522503168*
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))+7880198067307566*x^5+23312693984
74443*x^4-982548126959616*x^3-1058407652589420*x^2-338633978304558*x-39443626662
165)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(13/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{15}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(15/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(15/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(15/2),x, algorithm="fricas")

[Out]

integral((100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((21
87*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128
)*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)**(5/2)/(2+3*x)**(15/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{15}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(15/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(15/2), x)