3.2640 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{3 \sqrt{3 x+2}}+\frac{40}{27} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{8}{27} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{49}{27} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(40*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/27 - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(
3/2))/(3*Sqrt[2 + 3*x]) - (49*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x
]], 35/33])/27 + (8*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]
)/27

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Rubi [A]  time = 0.254076, 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 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{3 \sqrt{3 x+2}}+\frac{40}{27} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{8}{27} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{49}{27} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(40*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/27 - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(
3/2))/(3*Sqrt[2 + 3*x]) - (49*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x
]], 35/33])/27 + (8*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]
)/27

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Rubi in Sympy [A]  time = 24.3653, size = 114, normalized size = 0.88 \[ \frac{40 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{27} - \frac{2 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{3 \sqrt{3 x + 2}} - \frac{49 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{81} + \frac{88 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{945} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

40*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/27 - 2*sqrt(-2*x + 1)*(5*x + 3)**(
3/2)/(3*sqrt(3*x + 2)) - 49*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7),
35/33)/81 + 88*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/945

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Mathematica [A]  time = 0.281544, size = 97, normalized size = 0.75 \[ \frac{1}{81} \left (\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} (15 x+13)}{\sqrt{3 x+2}}-181 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+49 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(13 + 15*x))/Sqrt[2 + 3*x] + 49*Sqrt[2]*Elliptic
E[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 181*Sqrt[2]*EllipticF[ArcSin[Sqrt[2
/11]*Sqrt[3 + 5*x]], -33/2])/81

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Maple [C]  time = 0.024, size = 164, normalized size = 1.3 \[{\frac{1}{2430\,{x}^{3}+1863\,{x}^{2}-567\,x-486}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 181\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -49\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +900\,{x}^{3}+870\,{x}^{2}-192\,x-234 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/81*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(181*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))-49*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellipti
cE(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))+900*x^3+8
70*x^2-192*x-234)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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