3.2802 \(\int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac{2 \sqrt{\frac{5}{7}} \sqrt{-5 x-3} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{3 x+2}\right )|\frac{2}{35}\right )}{3 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{7 \sqrt{3 x+2}} \]

[Out]

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

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Rubi [A]  time = 0.140498, antiderivative size = 81, 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 \[ \frac{2 \sqrt{\frac{5}{7}} \sqrt{-5 x-3} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{3 x+2}\right )|\frac{2}{35}\right )}{3 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{7 \sqrt{3 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.1759, size = 94, normalized size = 1.16 \[ \frac{2 \sqrt{5} \sqrt{- 15 x - 9} \sqrt{- 2 x + 1} E\left (\operatorname{asin}{\left (\sqrt{5} \sqrt{3 x + 2} \right )}\middle | \frac{2}{35}\right )}{21 \sqrt{- \frac{6 x}{7} + \frac{3}{7}} \sqrt{5 x + 3}} - \frac{2 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{7 \sqrt{3 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(5)*sqrt(-15*x - 9)*sqrt(-2*x + 1)*elliptic_e(asin(sqrt(5)*sqrt(3*x + 2)),
2/35)/(21*sqrt(-6*x/7 + 3/7)*sqrt(5*x + 3)) - 2*sqrt(-2*x + 1)*sqrt(5*x + 3)/(7
*sqrt(3*x + 2))

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Mathematica [C]  time = 0.149255, size = 70, normalized size = 0.86 \[ \frac{-6 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-2 i \sqrt{33} (3 x+2) E\left (i \sinh ^{-1}\left (\sqrt{15 x+9}\right )|-\frac{2}{33}\right )}{63 x+42} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] - (2*I)*Sqrt[33]*(2 + 3*x)*Ellipti
cE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(42 + 63*x)

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Maple [C]  time = 0.026, size = 159, normalized size = 2. \[ -{\frac{1}{630\,{x}^{3}+483\,{x}^{2}-147\,x-126}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +60\,{x}^{2}+6\,x-18 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/21*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(35*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)*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))+60*x^2+6*x
-18)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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