3.2852 \(\int \frac{(2+3 x)^{3/2}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=98 \[ -\frac{2 \sqrt{1-2 x} \sqrt{3 x+2}}{55 \sqrt{5 x+3}}-\frac{4}{25} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{31}{25} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(55*Sqrt[3 + 5*x]) - (31*Sqrt[3/11]*EllipticE[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/25 - (4*Sqrt[3/11]*EllipticF[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/25

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Rubi [A]  time = 0.189557, 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 \[ -\frac{2 \sqrt{1-2 x} \sqrt{3 x+2}}{55 \sqrt{5 x+3}}-\frac{4}{25} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{31}{25} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(55*Sqrt[3 + 5*x]) - (31*Sqrt[3/11]*EllipticE[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/25 - (4*Sqrt[3/11]*EllipticF[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/25

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Rubi in Sympy [A]  time = 18.3756, size = 87, normalized size = 0.89 \[ - \frac{2 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{55 \sqrt{5 x + 3}} - \frac{31 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{275} - \frac{12 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)*sqrt(3*x + 2)/(55*sqrt(5*x + 3)) - 31*sqrt(33)*elliptic_e(asin
(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/275 - 12*sqrt(35)*elliptic_f(asin(sqrt(55)*s
qrt(-2*x + 1)/11), 33/35)/875

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Mathematica [A]  time = 0.188827, size = 92, normalized size = 0.94 \[ \frac{1}{275} \left (-\frac{10 \sqrt{1-2 x} \sqrt{3 x+2}}{\sqrt{5 x+3}}+35 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+31 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/Sqrt[3 + 5*x] + 31*Sqrt[2]*EllipticE[ArcSin[S
qrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 35*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2])/275

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Maple [C]  time = 0.026, size = 159, normalized size = 1.6 \[ -{\frac{1}{8250\,{x}^{3}+6325\,{x}^{2}-1925\,x-1650}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +31\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +60\,{x}^{2}+10\,x-20 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/275*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*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))+60*x^2+1
0*x-20)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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