3.2668 \(\int \frac{\sqrt{1-2 x} \sqrt{2+3 x}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{2 \sqrt{1-2 x} \sqrt{3 x+2}}{5 \sqrt{5 x+3}}-\frac{62 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25 \sqrt{33}}+\frac{4}{25} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(5*Sqrt[3 + 5*x]) + (4*Sqrt[33]*EllipticE[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/25 - (62*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/(25*Sqrt[33])

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Rubi [A]  time = 0.186217, antiderivative size = 94, 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 \[ -\frac{2 \sqrt{1-2 x} \sqrt{3 x+2}}{5 \sqrt{5 x+3}}-\frac{62 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25 \sqrt{33}}+\frac{4}{25} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(5*Sqrt[3 + 5*x]) + (4*Sqrt[33]*EllipticE[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/25 - (62*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/(25*Sqrt[33])

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Rubi in Sympy [A]  time = 18.7668, size = 85, normalized size = 0.9 \[ - \frac{2 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{5 \sqrt{5 x + 3}} + \frac{4 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{25} - \frac{62 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)*sqrt(3*x + 2)/(5*sqrt(5*x + 3)) + 4*sqrt(33)*elliptic_e(asin(s
qrt(21)*sqrt(-2*x + 1)/7), 35/33)/25 - 62*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt
(-2*x + 1)/11), 33/35)/875

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Mathematica [A]  time = 0.274968, size = 92, normalized size = 0.98 \[ \frac{1}{25} \left (-\frac{10 \sqrt{1-2 x} \sqrt{3 x+2}}{\sqrt{5 x+3}}+35 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-4 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/Sqrt[3 + 5*x] - 4*Sqrt[2]*EllipticE[ArcSin[Sq
rt[2/11]*Sqrt[3 + 5*x]], -33/2] + 35*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2])/25

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Maple [C]  time = 0.018, size = 159, normalized size = 1.7 \[ -{\frac{1}{750\,{x}^{3}+575\,{x}^{2}-175\,x-150}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +60\,{x}^{2}+10\,x-20 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/25*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*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))-4*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+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. \[ \int \frac{\sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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