3.2670 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{20 \sqrt{1-2 x} \sqrt{3 x+2}}{\sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{\sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}+4 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*Sqrt[1 - 2*x])/(Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (20*Sqrt[1 - 2*x]*Sqrt[2 + 3*x
])/Sqrt[3 + 5*x] + 4*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]
+ (4*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/Sqrt[33]

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Rubi [A]  time = 0.266853, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{20 \sqrt{1-2 x} \sqrt{3 x+2}}{\sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{\sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}+4 \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x])/(Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (20*Sqrt[1 - 2*x]*Sqrt[2 + 3*x
])/Sqrt[3 + 5*x] + 4*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]
+ (4*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/Sqrt[33]

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Rubi in Sympy [A]  time = 25.4904, size = 109, normalized size = 0.93 \[ - \frac{20 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{\sqrt{5 x + 3}} + \frac{2 \sqrt{- 2 x + 1}}{\sqrt{3 x + 2} \sqrt{5 x + 3}} + 4 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right ) + \frac{4 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{35} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-20*sqrt(-2*x + 1)*sqrt(3*x + 2)/sqrt(5*x + 3) + 2*sqrt(-2*x + 1)/(sqrt(3*x + 2)
*sqrt(5*x + 3)) + 4*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)
+ 4*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/35

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Mathematica [A]  time = 0.229839, size = 128, normalized size = 1.09 \[ \frac{2 \sqrt{2} \left (15 x^2+19 x+6\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-4 \sqrt{2} \left (15 x^2+19 x+6\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} (30 x+19)}{(3 x+2) (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(19 + 30*x) - 4*Sqrt[2]*(6 + 19*x
+ 15*x^2)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 2*Sqrt[2]*(6 + 19
*x + 15*x^2)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/((2 + 3*x)*(3 +
5*x))

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Maple [C]  time = 0.027, size = 158, normalized size = 1.4 \[ -2\,{\frac{\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( \sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +60\,{x}^{2}+8\,x-19 \right ) }{30\,{x}^{3}+23\,{x}^{2}-7\,x-6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(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))-2*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))+60*x^2+8*x-19)/(
30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-2 \, x + 1}}{{\left (15 \, x^{2} + 19 \, x + 6\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(-2*x + 1)/((15*x^2 + 19*x + 6)*sqrt(5*x + 3)*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)**(1/2)/(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. \[ \int \frac{\sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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