3.2853 \(\int \frac{\sqrt{2+3 x}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac{2 \sqrt{\frac{7}{5}} \sqrt{-5 x-3} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{3 x+2}\right )|\frac{2}{35}\right )}{11 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x} \sqrt{3 x+2}}{11 \sqrt{5 x+3}} \]

[Out]

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

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Rubi [A]  time = 0.140416, 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 \[ \frac{2 \sqrt{\frac{7}{5}} \sqrt{-5 x-3} E\left (\sin ^{-1}\left (\sqrt{5} \sqrt{3 x+2}\right )|\frac{2}{35}\right )}{11 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x} \sqrt{3 x+2}}{11 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.0439, size = 94, normalized size = 1.16 \[ \frac{2 \sqrt{5} \sqrt{- 15 x - 9} \sqrt{- 2 x + 1} E\left (\operatorname{asin}{\left (\sqrt{5} \sqrt{3 x + 2} \right )}\middle | \frac{2}{35}\right )}{55 \sqrt{- \frac{6 x}{7} + \frac{3}{7}} \sqrt{5 x + 3}} - \frac{2 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{11 \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**(1/2)/(3+5*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)/(55*sqrt(-6*x/7 + 3/7)*sqrt(5*x + 3)) - 2*sqrt(-2*x + 1)*sqrt(3*x + 2)/(1
1*sqrt(5*x + 3))

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Mathematica [C]  time = 0.129661, size = 61, normalized size = 0.75 \[ \frac{2}{55} \left (-\frac{5 \sqrt{1-2 x} \sqrt{3 x+2}}{\sqrt{5 x+3}}-i \sqrt{33} E\left (i \sinh ^{-1}\left (\sqrt{15 x+9}\right )|-\frac{2}{33}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*((-5*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/Sqrt[3 + 5*x] - I*Sqrt[33]*EllipticE[I*ArcS
inh[Sqrt[9 + 15*x]], -2/33]))/55

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Maple [C]  time = 0.024, size = 159, normalized size = 2. \[{\frac{1}{1650\,{x}^{3}+1265\,{x}^{2}-385\,x-330}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 2\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -60\,{x}^{2}-10\,x+20 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/55*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(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))-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))-60*x^2-10*x
+20)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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