3.2809 \(\int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=98 \[ \frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{21 \sqrt{3 x+2}}+\frac{2}{21} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{37}{21} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*Sqrt[2 + 3*x]) - (37*Sqrt[11/3]*EllipticE[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/21 + (2*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[
3/7]*Sqrt[1 - 2*x]], 35/33])/21

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Rubi [A]  time = 0.184923, 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 \[ \frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{21 \sqrt{3 x+2}}+\frac{2}{21} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{37}{21} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*Sqrt[2 + 3*x]) - (37*Sqrt[11/3]*EllipticE[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/21 + (2*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[
3/7]*Sqrt[1 - 2*x]], 35/33])/21

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Rubi in Sympy [A]  time = 18.1579, size = 85, normalized size = 0.87 \[ \frac{2 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{21 \sqrt{3 x + 2}} - \frac{37 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{63} + \frac{22 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{735} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(-2*x + 1)*sqrt(5*x + 3)/(21*sqrt(3*x + 2)) - 37*sqrt(33)*elliptic_e(asin(
sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/63 + 22*sqrt(35)*elliptic_f(asin(sqrt(55)*sqr
t(-2*x + 1)/11), 33/35)/735

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Mathematica [A]  time = 0.163386, size = 92, normalized size = 0.94 \[ \frac{1}{63} \left (\frac{6 \sqrt{1-2 x} \sqrt{5 x+3}}{\sqrt{3 x+2}}-70 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+37 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/Sqrt[2 + 3*x] + 37*Sqrt[2]*EllipticE[ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]], -33/2] - 70*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]], -33/2])/63

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Maple [C]  time = 0.024, size = 159, normalized size = 1.6 \[{\frac{1}{1890\,{x}^{3}+1449\,{x}^{2}-441\,x-378}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 70\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -37\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +60\,{x}^{2}+6\,x-18 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/63*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(70*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))-37*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))+60*x^2+6*x
-18)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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