3.2854 \(\int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=63 \[ 2 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{10 \sqrt{1-2 x} \sqrt{3 x+2}}{11 \sqrt{5 x+3}} \]

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) + 2*Sqrt[3/11]*EllipticE[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]

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Rubi [A]  time = 0.0950023, 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 \[ 2 \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{10 \sqrt{1-2 x} \sqrt{3 x+2}}{11 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) + 2*Sqrt[3/11]*EllipticE[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]

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Rubi in Sympy [A]  time = 9.44459, size = 56, normalized size = 0.89 \[ - \frac{10 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{11 \sqrt{5 x + 3}} + \frac{2 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*sqrt(-2*x + 1)*sqrt(3*x + 2)/(11*sqrt(5*x + 3)) + 2*sqrt(33)*elliptic_e(asin
(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/11

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Mathematica [A]  time = 0.114814, size = 106, normalized size = 1.68 \[ -\frac{2 \left (5 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\sqrt{2} (5 x+3) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+\sqrt{2} (5 x+3) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{55 x+33} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(5*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + Sqrt[2]*(3 + 5*x)*EllipticE[A
rcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - Sqrt[2]*(3 + 5*x)*EllipticF[ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]], -33/2]))/(33 + 55*x)

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Maple [C]  time = 0.028, size = 158, normalized size = 2.5 \[ -{\frac{2}{330\,{x}^{3}+253\,{x}^{2}-77\,x-66}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( \sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ({\frac{\sqrt{11}\sqrt{2}}{11}\sqrt{3+5\,x}},{\frac{i}{2}}\sqrt{11}\sqrt{3}\sqrt{2} \right ) -\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ({\frac{\sqrt{11}\sqrt{2}}{11}\sqrt{3+5\,x}},{\frac{i}{2}}\sqrt{11}\sqrt{3}\sqrt{2} \right ) +30\,{x}^{2}+5\,x-10 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/11*(3+5*x)^(1/2)*(1-2*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)*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^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/1
1*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. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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