3.2724 \(\int \frac{(1-2 x)^{3/2}}{\sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=98 \[ -\frac{22 \sqrt{1-2 x} \sqrt{3 x+2}}{5 \sqrt{5 x+3}}+\frac{8}{25} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{62}{25} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-22*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(5*Sqrt[3 + 5*x]) + (62*Sqrt[11/3]*EllipticE[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/25 + (8*Sqrt[11/3]*EllipticF[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/25

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Rubi [A]  time = 0.188495, 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 \[ -\frac{22 \sqrt{1-2 x} \sqrt{3 x+2}}{5 \sqrt{5 x+3}}+\frac{8}{25} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{62}{25} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-22*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(5*Sqrt[3 + 5*x]) + (62*Sqrt[11/3]*EllipticE[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/25 + (8*Sqrt[11/3]*EllipticF[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/25

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Rubi in Sympy [A]  time = 17.5005, size = 85, normalized size = 0.87 \[ - \frac{22 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{5 \sqrt{5 x + 3}} + \frac{62 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{75} + \frac{8 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{75} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-22*sqrt(-2*x + 1)*sqrt(3*x + 2)/(5*sqrt(5*x + 3)) + 62*sqrt(33)*elliptic_e(asin
(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/75 + 8*sqrt(33)*elliptic_f(asin(sqrt(21)*sqr
t(-2*x + 1)/7), 35/33)/75

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Mathematica [A]  time = 0.299395, size = 92, normalized size = 0.94 \[ \frac{1}{75} \left (-\frac{330 \sqrt{1-2 x} \sqrt{3 x+2}}{\sqrt{5 x+3}}-70 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-62 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-330*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/Sqrt[3 + 5*x] - 62*Sqrt[2]*EllipticE[ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 70*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[
3 + 5*x]], -33/2])/75

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Maple [C]  time = 0.024, size = 159, normalized size = 1.6 \[{\frac{2}{2250\,{x}^{3}+1725\,{x}^{2}-525\,x-450}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +31\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -990\,{x}^{2}-165\,x+330 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/75*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*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)*Elliptic
E(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))-990*x^2-16
5*x+330)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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