3.2915 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{23180 \sqrt{1-2 x} \sqrt{3 x+2}}{5929 \sqrt{5 x+3}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{124}{539} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{4636}{539} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

4/(77*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + (186*Sqrt[1 - 2*x])/(539*Sqrt
[2 + 3*x]*Sqrt[3 + 5*x]) - (23180*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(5929*Sqrt[3 + 5*
x]) + (4636*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/539 +
(124*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/539

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Rubi [A]  time = 0.348393, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{23180 \sqrt{1-2 x} \sqrt{3 x+2}}{5929 \sqrt{5 x+3}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{124}{539} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{4636}{539} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

4/(77*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + (186*Sqrt[1 - 2*x])/(539*Sqrt
[2 + 3*x]*Sqrt[3 + 5*x]) - (23180*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(5929*Sqrt[3 + 5*
x]) + (4636*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/539 +
(124*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/539

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Rubi in Sympy [A]  time = 32.7121, size = 143, normalized size = 0.89 \[ - \frac{23180 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{5929 \sqrt{5 x + 3}} + \frac{186 \sqrt{- 2 x + 1}}{539 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{4636 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5929} + \frac{124 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5929} + \frac{4}{77 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-23180*sqrt(-2*x + 1)*sqrt(3*x + 2)/(5929*sqrt(5*x + 3)) + 186*sqrt(-2*x + 1)/(5
39*sqrt(3*x + 2)*sqrt(5*x + 3)) + 4636*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2
*x + 1)/7), 35/33)/5929 + 124*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7
), 35/33)/5929 + 4/(77*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.266626, size = 98, normalized size = 0.61 \[ \frac{2 \left (\frac{69540 x^2+9544 x-22003}{\sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}+\sqrt{2} \left (1295 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2318 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{5929} \]

Antiderivative was successfully verified.

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

[Out]

(2*((-22003 + 9544*x + 69540*x^2)/(Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) +
Sqrt[2]*(-2318*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 1295*Ellipti
cF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/5929

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Maple [C]  time = 0.033, size = 159, normalized size = 1. \[ -{\frac{2}{177870\,{x}^{3}+136367\,{x}^{2}-41503\,x-35574} \left ( 1295\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2318\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +69540\,{x}^{2}+9544\,x-22003 \right ) \sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5929*(1295*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*1
1^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))-2318*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))+69540*x^2+9544*x-22003)*(3+5*x)^(1/2)*(2+3*x)
^(1/2)*(1-2*x)^(1/2)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-1/((30*x^3 + 23*x^2 - 7*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/(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. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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