3.2783 \(\int \frac{(1-2 x)^{5/2}}{\sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{22 \sqrt{3 x+2} (1-2 x)^{3/2}}{5 \sqrt{5 x+3}}-\frac{388}{225} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1196 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125}+\frac{5594 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125} \]

[Out]

(-22*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x])/(5*Sqrt[3 + 5*x]) - (388*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*Sqrt[3 + 5*x])/225 + (5594*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/1125 + (1196*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*
x]], 35/33])/1125

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Rubi [A]  time = 0.257039, 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 \[ -\frac{22 \sqrt{3 x+2} (1-2 x)^{3/2}}{5 \sqrt{5 x+3}}-\frac{388}{225} \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1196 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125}+\frac{5594 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1125} \]

Antiderivative was successfully verified.

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

[Out]

(-22*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x])/(5*Sqrt[3 + 5*x]) - (388*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*Sqrt[3 + 5*x])/225 + (5594*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/1125 + (1196*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*
x]], 35/33])/1125

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Rubi in Sympy [A]  time = 26.397, size = 114, normalized size = 0.88 \[ - \frac{22 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2}}{5 \sqrt{5 x + 3}} - \frac{388 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{225} + \frac{5594 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3375} + \frac{1196 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-22*(-2*x + 1)**(3/2)*sqrt(3*x + 2)/(5*sqrt(5*x + 3)) - 388*sqrt(-2*x + 1)*sqrt(
3*x + 2)*sqrt(5*x + 3)/225 + 5594*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x +
1)/7), 35/33)/3375 + 1196*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 3
5/33)/3375

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Mathematica [A]  time = 0.462311, size = 97, normalized size = 0.75 \[ \frac{2 \left (\frac{15 \sqrt{1-2 x} \sqrt{3 x+2} (20 x-1077)}{\sqrt{5 x+3}}-7070 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2797 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{3375} \]

Antiderivative was successfully verified.

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

[Out]

(2*((15*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(-1077 + 20*x))/Sqrt[3 + 5*x] - 2797*Sqrt[2]
*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 7070*Sqrt[2]*EllipticF[Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/3375

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Maple [C]  time = 0.026, size = 164, normalized size = 1.3 \[{\frac{2}{101250\,{x}^{3}+77625\,{x}^{2}-23625\,x-20250}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 7070\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +2797\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1800\,{x}^{3}-96630\,{x}^{2}-16755\,x+32310 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3375*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(7070*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))+2797*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*El
lipticE(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))+1800
*x^3-96630*x^2-16755*x+32310)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(-2*x + 1)/((5*x + 3)^(3/2)*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)**(5/2)/(3+5*x)**(3/2)/(2+3*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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