3.2662 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^{3/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=61 \[ \frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{\sqrt{3 x+2}}-2 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/Sqrt[2 + 3*x] - 2*Sqrt[11/3]*EllipticE[ArcSin[Sq
rt[3/7]*Sqrt[1 - 2*x]], 35/33]

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Rubi [A]  time = 0.0868808, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{\sqrt{3 x+2}}-2 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/Sqrt[2 + 3*x] - 2*Sqrt[11/3]*EllipticE[ArcSin[Sq
rt[3/7]*Sqrt[1 - 2*x]], 35/33]

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Rubi in Sympy [A]  time = 9.42041, size = 54, normalized size = 0.89 \[ \frac{2 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{\sqrt{3 x + 2}} - \frac{2 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3} \]

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)**(1/2),x)

[Out]

2*sqrt(-2*x + 1)*sqrt(5*x + 3)/sqrt(3*x + 2) - 2*sqrt(33)*elliptic_e(asin(sqrt(2
1)*sqrt(-2*x + 1)/7), 35/33)/3

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Mathematica [A]  time = 0.151324, size = 106, normalized size = 1.74 \[ \frac{6 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-2 \sqrt{2} (3 x+2) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+2 \sqrt{2} (3 x+2) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{9 x+6} \]

Antiderivative was successfully verified.

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

[Out]

(6*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + 2*Sqrt[2]*(2 + 3*x)*EllipticE[Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2*Sqrt[2]*(2 + 3*x)*EllipticF[ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]], -33/2])/(6 + 9*x)

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Maple [C]  time = 0.024, size = 158, normalized size = 2.6 \[{\frac{2}{90\,{x}^{3}+69\,{x}^{2}-21\,x-18}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( \sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ({\frac{\sqrt{11}\sqrt{2}}{11}\sqrt{3+5\,x}},{\frac{i}{2}}\sqrt{11}\sqrt{3}\sqrt{2} \right ) -\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ({\frac{\sqrt{11}\sqrt{2}}{11}\sqrt{3+5\,x}},{\frac{i}{2}}\sqrt{11}\sqrt{3}\sqrt{2} \right ) +30\,{x}^{2}+3\,x-9 \right ) } \]

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)^(1/2),x)

[Out]

2/3*(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^(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))+30*x^2+3*x-9)/(30
*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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