3.2700 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ \frac{74 \sqrt{1-2 x} (5 x+3)^{3/2}}{297 (3 x+2)^{9/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}+\frac{1446357824 \sqrt{1-2 x} \sqrt{5 x+3}}{74875185 \sqrt{3 x+2}}+\frac{20799916 \sqrt{1-2 x} \sqrt{5 x+3}}{10696455 (3 x+2)^{3/2}}+\frac{442076 \sqrt{1-2 x} \sqrt{5 x+3}}{1528065 (3 x+2)^{5/2}}-\frac{12872 \sqrt{1-2 x} \sqrt{5 x+3}}{43659 (3 x+2)^{7/2}}-\frac{43537016 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835 \sqrt{33}}-\frac{1446357824 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835 \sqrt{33}} \]

[Out]

(-12872*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43659*(2 + 3*x)^(7/2)) + (442076*Sqrt[1 -
2*x]*Sqrt[3 + 5*x])/(1528065*(2 + 3*x)^(5/2)) + (20799916*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(10696455*(2 + 3*x)^(3/2)) + (1446357824*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(74
875185*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(33*(2 + 3*x)^(11/2)
) + (74*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(297*(2 + 3*x)^(9/2)) - (1446357824*Ellip
ticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6806835*Sqrt[33]) - (43537016*Ell
ipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6806835*Sqrt[33])

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Rubi [A]  time = 0.587038, 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 \[ \frac{74 \sqrt{1-2 x} (5 x+3)^{3/2}}{297 (3 x+2)^{9/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}+\frac{1446357824 \sqrt{1-2 x} \sqrt{5 x+3}}{74875185 \sqrt{3 x+2}}+\frac{20799916 \sqrt{1-2 x} \sqrt{5 x+3}}{10696455 (3 x+2)^{3/2}}+\frac{442076 \sqrt{1-2 x} \sqrt{5 x+3}}{1528065 (3 x+2)^{5/2}}-\frac{12872 \sqrt{1-2 x} \sqrt{5 x+3}}{43659 (3 x+2)^{7/2}}-\frac{43537016 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835 \sqrt{33}}-\frac{1446357824 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-12872*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43659*(2 + 3*x)^(7/2)) + (442076*Sqrt[1 -
2*x]*Sqrt[3 + 5*x])/(1528065*(2 + 3*x)^(5/2)) + (20799916*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(10696455*(2 + 3*x)^(3/2)) + (1446357824*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(74
875185*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(33*(2 + 3*x)^(11/2)
) + (74*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(297*(2 + 3*x)^(9/2)) - (1446357824*Ellip
ticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6806835*Sqrt[33]) - (43537016*Ell
ipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6806835*Sqrt[33])

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Rubi in Sympy [A]  time = 53.1777, size = 230, normalized size = 0.92 \[ - \frac{74 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{2079 \left (3 x + 2\right )^{\frac{9}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{33 \left (3 x + 2\right )^{\frac{11}{2}}} + \frac{1446357824 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{74875185 \sqrt{3 x + 2}} + \frac{20799916 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{10696455 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{442076 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1528065 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{4222 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{43659 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{1446357824 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{224625555} - \frac{43537016 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{224625555} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-74*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(2079*(3*x + 2)**(9/2)) - 2*(-2*x + 1)**(3/2
)*(5*x + 3)**(3/2)/(33*(3*x + 2)**(11/2)) + 1446357824*sqrt(-2*x + 1)*sqrt(5*x +
3)/(74875185*sqrt(3*x + 2)) + 20799916*sqrt(-2*x + 1)*sqrt(5*x + 3)/(10696455*(
3*x + 2)**(3/2)) + 442076*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1528065*(3*x + 2)**(5/2)
) + 4222*sqrt(-2*x + 1)*sqrt(5*x + 3)/(43659*(3*x + 2)**(7/2)) - 1446357824*sqrt
(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/224625555 - 43537016*sqr
t(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/224625555

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Mathematica [A]  time = 0.441864, size = 112, normalized size = 0.45 \[ \frac{\frac{24 \sqrt{2-4 x} \sqrt{5 x+3} \left (175732475616 x^5+591671694906 x^4+797050394730 x^3+537061687749 x^2+180988667568 x+24398176891\right )}{(3 x+2)^{11/2}}-5823976480 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{898502220 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

((24*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(24398176891 + 180988667568*x + 537061687749*x^
2 + 797050394730*x^3 + 591671694906*x^4 + 175732475616*x^5))/(2 + 3*x)^(11/2) +
11570862592*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5823976480*Elli
pticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(898502220*Sqrt[2])

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

[Out]

2/224625555*(88451642790*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)-175
732475616*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)-585774918720*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)-781033224960*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
)[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^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-520688816640*
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+8735964720
0*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)-173562938880*2^(1/2)*Ellipti
cE(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+11647952960*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))-23141725184*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+13177956562506*x^4-132608462283*x^3-35586438803
07*x^2-1555703477439*x-219583592019)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2
+3*x)^(11/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(10*x^2 + x - 3)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((729*x^6 + 2916*x^5 + 4
860*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)**(3/2)*(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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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