3.2715 \(\int \frac{(1-2 x)^{3/2}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=98 \[ -\frac{4}{45} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{202}{225} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{272}{225} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-4*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/45 + (272*Sqrt[11/3]*EllipticE[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/225 - (202*Sqrt[11/3]*EllipticF[ArcSin[Sq
rt[3/7]*Sqrt[1 - 2*x]], 35/33])/225

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Rubi [A]  time = 0.189681, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{4}{45} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{202}{225} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{272}{225} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-4*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/45 + (272*Sqrt[11/3]*EllipticE[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/225 - (202*Sqrt[11/3]*EllipticF[ArcSin[Sq
rt[3/7]*Sqrt[1 - 2*x]], 35/33])/225

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Rubi in Sympy [A]  time = 16.9958, size = 85, normalized size = 0.87 \[ - \frac{4 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{45} + \frac{272 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{675} - \frac{2222 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{7875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/45 + 272*sqrt(33)*elliptic_e(asin(
sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/675 - 2222*sqrt(35)*elliptic_f(asin(sqrt(55)*
sqrt(-2*x + 1)/11), 33/35)/7875

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Mathematica [A]  time = 0.165156, size = 92, normalized size = 0.94 \[ \frac{1}{675} \left (-60 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+3605 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-272 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-60*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] - 272*Sqrt[2]*EllipticE[ArcSin[Sq
rt[2/11]*Sqrt[3 + 5*x]], -33/2] + 3605*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[
3 + 5*x]], -33/2])/675

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Maple [C]  time = 0.019, size = 164, normalized size = 1.7 \[ -{\frac{1}{20250\,{x}^{3}+15525\,{x}^{2}-4725\,x-4050}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 3605\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -272\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1800\,{x}^{3}+1380\,{x}^{2}-420\,x-360 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/675*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(3605*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))-272*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ell
ipticE(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+1380*x^2-420*x-360)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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