3.2906 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=129 \[ \frac{186 \sqrt{1-2 x} \sqrt{5 x+3}}{539 \sqrt{3 x+2}}+\frac{4 \sqrt{5 x+3}}{77 \sqrt{1-2 x} \sqrt{3 x+2}}-\frac{8}{49} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{62}{49} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(4*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) + (186*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(539*Sqrt[2 + 3*x]) - (62*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/49 - (8*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/
33])/49

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Rubi [A]  time = 0.263799, 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 \[ \frac{186 \sqrt{1-2 x} \sqrt{5 x+3}}{539 \sqrt{3 x+2}}+\frac{4 \sqrt{5 x+3}}{77 \sqrt{1-2 x} \sqrt{3 x+2}}-\frac{8}{49} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{62}{49} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) + (186*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(539*Sqrt[2 + 3*x]) - (62*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/49 - (8*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/
33])/49

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Rubi in Sympy [A]  time = 23.8424, size = 114, normalized size = 0.88 \[ \frac{186 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{539 \sqrt{3 x + 2}} - \frac{62 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{539} - \frac{8 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{539} + \frac{4 \sqrt{5 x + 3}}{77 \sqrt{- 2 x + 1} \sqrt{3 x + 2}} \]

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)**(1/2),x)

[Out]

186*sqrt(-2*x + 1)*sqrt(5*x + 3)/(539*sqrt(3*x + 2)) - 62*sqrt(33)*elliptic_e(as
in(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/539 - 8*sqrt(33)*elliptic_f(asin(sqrt(21)*
sqrt(-2*x + 1)/7), 35/33)/539 + 4*sqrt(5*x + 3)/(77*sqrt(-2*x + 1)*sqrt(3*x + 2)
)

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Mathematica [A]  time = 0.267403, size = 122, normalized size = 0.95 \[ \frac{2 \sqrt{3 x+2} \sqrt{5 x+3} (107-186 x)+70 \sqrt{2-4 x} (3 x+2) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+62 \sqrt{2-4 x} (3 x+2) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{539 \sqrt{1-2 x} (3 x+2)} \]

Antiderivative was successfully verified.

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

[Out]

(2*(107 - 186*x)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + 62*Sqrt[2 - 4*x]*(2 + 3*x)*Ellipt
icE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 70*Sqrt[2 - 4*x]*(2 + 3*x)*Ellipt
icF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(539*Sqrt[1 - 2*x]*(2 + 3*x))

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Maple [C]  time = 0.03, size = 159, normalized size = 1.2 \[ -{\frac{2}{16170\,{x}^{3}+12397\,{x}^{2}-3773\,x-3234}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +31\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -930\,{x}^{2}-23\,x+321 \right ) } \]

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)^(1/2),x)

[Out]

-2/539*(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))+31*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellipt
icE(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))-930*x^2-
23*x+321)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-1/((6*x^2 + x - 2)*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)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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