3.2692 \(\int (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=218 \[ \frac{2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac{62 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{2475}-\frac{23 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{9625}-\frac{40703 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{433125}-\frac{5442127 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{7796250}-\frac{5442127 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3543750 \sqrt{33}}-\frac{90397364 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1771875 \sqrt{33}} \]

[Out]

(-5442127*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/7796250 - (40703*Sqrt[1 - 2
*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/433125 - (23*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 +
5*x)^(5/2))/9625 + (62*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/2475 + (2
*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/55 - (90397364*EllipticE[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1771875*Sqrt[33]) - (5442127*EllipticF[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3543750*Sqrt[33])

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Rubi [A]  time = 0.484428, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac{62 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{2475}-\frac{23 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{9625}-\frac{40703 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{433125}-\frac{5442127 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{7796250}-\frac{5442127 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3543750 \sqrt{33}}-\frac{90397364 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1771875 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-5442127*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/7796250 - (40703*Sqrt[1 - 2
*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/433125 - (23*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 +
5*x)^(5/2))/9625 + (62*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/2475 + (2
*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/55 - (90397364*EllipticE[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1771875*Sqrt[33]) - (5442127*EllipticF[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3543750*Sqrt[33])

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Rubi in Sympy [A]  time = 45.7725, size = 201, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{33} - \frac{37 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{297} + \frac{6436 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{31185} - \frac{110519 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{779625} - \frac{5199979 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{7796250} - \frac{90397364 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{58471875} - \frac{5442127 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{124031250} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(3/2)*(3*x + 2)**(5/2)*(5*x + 3)**(3/2)/33 - 37*(-2*x + 1)**(3/2)*
(3*x + 2)**(5/2)*sqrt(5*x + 3)/297 + 6436*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5
*x + 3)/31185 - 110519*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/779625 - 51
99979*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/7796250 - 90397364*sqrt(33)*ell
iptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/58471875 - 5442127*sqrt(35)*elli
ptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/124031250

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Mathematica [A]  time = 0.391771, size = 110, normalized size = 0.5 \[ \frac{361589456 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (42525000 x^4+43470000 x^3-17237250 x^2-27227430 x-810641\right )[phone] F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{116943750 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(361589456*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*(3*Sqrt[2 - 4*
x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-810641 - 27227430*x - 17237250*x^2 + 43470000*x
^3 + 42525000*x^4) + 36399853*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]
))/(116943750*Sqrt[2])

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Maple [C]  time = 0.022, size = 184, normalized size = 0.8 \[{\frac{1}{7016625000\,{x}^{3}[phone]\,{x}^{2}-1637212500\,x-1403325000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -38272500000\,{x}^{7}-68465250000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -361589456\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -5550525000\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-8261123010\,{x}^{2}-5071172010\,x-145915380 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/233887500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-38272500000*x^7-68465250
000*x^6+181999265*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))-361589456*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))-5550525000*x^5+53181589500*x^4+23721
281100*x^3-8261123010*x^2-5071172010*x-145915380)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(30*x^3 + 23*x^2 - 7*x - 6)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)
, 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)*(2+3*x)**(3/2)*(3+5*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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