3.2725 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{136 \sqrt{1-2 x} \sqrt{3 x+2}}{3 \sqrt{5 x+3}}+\frac{14 \sqrt{1-2 x}}{3 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4}{5} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{136}{5} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

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

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Rubi [A]  time = 0.261014, 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 \[ -\frac{136 \sqrt{1-2 x} \sqrt{3 x+2}}{3 \sqrt{5 x+3}}+\frac{14 \sqrt{1-2 x}}{3 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4}{5} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{136}{5} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.6862, size = 114, normalized size = 0.88 \[ - \frac{136 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{3 \sqrt{5 x + 3}} + \frac{14 \sqrt{- 2 x + 1}}{3 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{136 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{15} + \frac{44 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{175} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-136*sqrt(-2*x + 1)*sqrt(3*x + 2)/(3*sqrt(5*x + 3)) + 14*sqrt(-2*x + 1)/(3*sqrt(
3*x + 2)*sqrt(5*x + 3)) + 136*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7
), 35/33)/15 + 44*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/1
75

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Mathematica [A]  time = 0.247226, size = 131, normalized size = 1.02 \[ \frac{70 \sqrt{2} \left (15 x^2+19 x+6\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-136 \sqrt{2} \left (15 x^2+19 x+6\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} (68 x+43)}{15 (3 x+2) (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

(-30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(43 + 68*x) - 136*Sqrt[2]*(6 + 19
*x + 15*x^2)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 70*Sqrt[2]*(6
+ 19*x + 15*x^2)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(15*(2 + 3*
x)*(3 + 5*x))

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Maple [C]  time = 0.029, size = 159, normalized size = 1.2 \[ -{\frac{2}{450\,{x}^{3}+345\,{x}^{2}-105\,x-90}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -68\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +2040\,{x}^{2}+270\,x-645 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(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))-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+
270*x-645)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((-2*x + 1)^(3/2)/((15*x^2 + 19*x + 6)*sqrt(5*x + 3)*sqrt(3*x + 2)), 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*x)**(3/2)/(2+3*x)**(3/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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