3.2855 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{680 \sqrt{1-2 x} \sqrt{3 x+2}}{77 \sqrt{5 x+3}}+\frac{6 \sqrt{1-2 x}}{7 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4}{7} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{136}{7} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(6*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (680*Sqrt[1 - 2*x]*Sqrt[2 +
3*x])/(77*Sqrt[3 + 5*x]) + (136*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2
*x]], 35/33])/7 + (4*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/7

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Rubi [A]  time = 0.266163, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{680 \sqrt{1-2 x} \sqrt{3 x+2}}{77 \sqrt{5 x+3}}+\frac{6 \sqrt{1-2 x}}{7 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4}{7} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{136}{7} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(6*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (680*Sqrt[1 - 2*x]*Sqrt[2 +
3*x])/(77*Sqrt[3 + 5*x]) + (136*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2
*x]], 35/33])/7 + (4*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/7

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Rubi in Sympy [A]  time = 25.4708, size = 114, normalized size = 0.88 \[ - \frac{680 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{77 \sqrt{5 x + 3}} + \frac{6 \sqrt{- 2 x + 1}}{7 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{136 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{77} + \frac{12 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{245} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-680*sqrt(-2*x + 1)*sqrt(3*x + 2)/(77*sqrt(5*x + 3)) + 6*sqrt(-2*x + 1)/(7*sqrt(
3*x + 2)*sqrt(5*x + 3)) + 136*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7
), 35/33)/77 + 12*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/2
45

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Mathematica [A]  time = 0.173814, size = 130, normalized size = 1.01 \[ -\frac{2 \left (-35 \sqrt{2} \left (15 x^2+19 x+6\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+68 \sqrt{2} \left (15 x^2+19 x+6\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+\sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} (1020 x+647)\right )}{77 (3 x+2) (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(647 + 1020*x) + 68*Sqrt[2]*(6 +
19*x + 15*x^2)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 35*Sqrt[2]*(
6 + 19*x + 15*x^2)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(77*(2 +
3*x)*(3 + 5*x))

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Maple [C]  time = 0.032, size = 159, normalized size = 1.2 \[ -{\frac{2}{2310\,{x}^{3}+1771\,{x}^{2}-539\,x-462}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -68\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +2040\,{x}^{2}+274\,x-647 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/77*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*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))-68*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellipti
cE(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))+2040*x^2+
274*x-647)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (15 \, x^{2} + 19 \, x + 6\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((15*x^2 + 19*x + 6)*sqrt(5*x + 3)*sqrt(3*x + 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(1/(2+3*x)**(3/2)/(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. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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