3.2669 \(\int \frac{\sqrt{1-2 x}}{\sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0876837, 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 \[ 2 \sqrt{\frac{7}{5}} E\left (\sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )|\frac{33}{35}\right )-\frac{2 \sqrt{1-2 x} \sqrt{3 x+2}}{\sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.61802, size = 54, normalized size = 0.89 \[ - \frac{2 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{\sqrt{5 x + 3}} + \frac{2 \sqrt{35} E\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)*sqrt(3*x + 2)/sqrt(5*x + 3) + 2*sqrt(35)*elliptic_e(asin(sqrt(
55)*sqrt(-2*x + 1)/11), 33/35)/5

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Mathematica [A]  time = 0.150551, size = 61, normalized size = 1. \[ -\frac{2 \sqrt{1-2 x} \sqrt{3 x+2}}{\sqrt{5 x+3}}-\frac{2}{5} \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.024, size = 104, normalized size = 1.7 \[{\frac{2}{150\,{x}^{3}+115\,{x}^{2}-35\,x-30}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( \sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ({\frac{\sqrt{11}\sqrt{2}}{11}\sqrt{3+5\,x}},{\frac{i}{2}}\sqrt{11}\sqrt{3}\sqrt{2} \right ) -30\,{x}^{2}-5\,x+10 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(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^2-5*x+10)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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