3.2808 \(\int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=98 \[ -\frac{5}{9} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{1}{9} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{31}{9} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-5*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 - (31*Sqrt[11/3]*EllipticE[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/9 - (Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*
Sqrt[1 - 2*x]], 35/33])/9

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Rubi [A]  time = 0.193563, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{5}{9} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{1}{9} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{31}{9} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 - (31*Sqrt[11/3]*EllipticE[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/9 - (Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*
Sqrt[1 - 2*x]], 35/33])/9

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Rubi in Sympy [A]  time = 17.9311, size = 85, normalized size = 0.87 \[ - \frac{5 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{9} - \frac{31 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{27} - \frac{\sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/9 - 31*sqrt(33)*elliptic_e(asin(sq
rt(21)*sqrt(-2*x + 1)/7), 35/33)/27 - sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*
x + 1)/7), 35/33)/27

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Mathematica [A]  time = 0.084576, size = 92, normalized size = 0.94 \[ \frac{1}{54} \left (-30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-29 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+62 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + 62*Sqrt[2]*EllipticE[ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]], -33/2] - 29*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]], -33/2])/54

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Maple [C]  time = 0.02, size = 164, normalized size = 1.7 \[{\frac{1}{1620\,{x}^{3}+1242\,{x}^{2}-378\,x-324}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 29\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -62\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -900\,{x}^{3}-690\,{x}^{2}+210\,x+180 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/54*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(29*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))-62*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Elliptic
E(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))-900*x^3-69
0*x^2+210*x+180)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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