3.2847 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=63 \[ \frac{6 \sqrt{1-2 x} \sqrt{5 x+3}}{7 \sqrt{3 x+2}}-2 \sqrt{\frac{5}{7}} E\left (\sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )|\frac{33}{35}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0933349, antiderivative size = 63, 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 \[ \frac{6 \sqrt{1-2 x} \sqrt{5 x+3}}{7 \sqrt{3 x+2}}-2 \sqrt{\frac{5}{7}} E\left (\sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )|\frac{33}{35}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.58066, size = 56, normalized size = 0.89 \[ \frac{6 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{7 \sqrt{3 x + 2}} - \frac{2 \sqrt{35} E\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*sqrt(-2*x + 1)*sqrt(5*x + 3)/(7*sqrt(3*x + 2)) - 2*sqrt(35)*elliptic_e(asin(sq
rt(55)*sqrt(-2*x + 1)/11), 33/35)/7

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Mathematica [A]  time = 0.112447, size = 62, normalized size = 0.98 \[ \frac{2}{7} \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3}}{\sqrt{3 x+2}}+\sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.027, size = 104, normalized size = 1.7 \[ -{\frac{2}{210\,{x}^{3}+161\,{x}^{2}-49\,x-42}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( \sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ({\frac{\sqrt{11}\sqrt{2}}{11}\sqrt{3+5\,x}},{\frac{i}{2}}\sqrt{11}\sqrt{3}\sqrt{2} \right ) -30\,{x}^{2}-3\,x+9 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/7*(2+3*x)^(1/2)*(1-2*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)*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. \[ \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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