3.2877 \(\int \frac{\sqrt{2+3 x} \sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=85 \[ \frac{\sqrt{3 x+2} \sqrt{5 x+3}}{\sqrt{1-2 x}}+\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}+\sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

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

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Rubi [A]  time = 0.181584, antiderivative size = 85, 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 \[ \frac{\sqrt{3 x+2} \sqrt{5 x+3}}{\sqrt{1-2 x}}+\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{\sqrt{33}}+\sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.2916, size = 76, normalized size = 0.89 \[ \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right ) + \frac{\sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{35} + \frac{\sqrt{3 x + 2} \sqrt{5 x + 3}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.166639, size = 86, normalized size = 1.01 \[ \frac{\sqrt{3 x+2} \sqrt{5 x+3}}{\sqrt{1-2 x}}+\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{\sqrt{2}}-\sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.019, size = 158, normalized size = 1.9 \[ -{\frac{1}{60\,{x}^{3}+46\,{x}^{2}-14\,x-12}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( \sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ({\frac{\sqrt{11}\sqrt{2}}{11}\sqrt{3+5\,x}},{\frac{i}{2}}\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +30\,{x}^{2}+38\,x+12 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*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))+30*x^2+38*x+12
)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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