3.2659 \(\int \frac{\sqrt{1-2 x} (2+3 x)^{3/2}}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=127 \[ \frac{2}{25} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{9}{125} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{17}{625} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{146}{625} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-9*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/125 + (2*Sqrt[1 - 2*x]*(2 + 3*x)^
(3/2)*Sqrt[3 + 5*x])/25 - (146*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/625 - (17*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/625

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Rubi [A]  time = 0.262722, antiderivative size = 127, 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 \[ \frac{2}{25} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{9}{125} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{17}{625} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{146}{625} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-9*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/125 + (2*Sqrt[1 - 2*x]*(2 + 3*x)^
(3/2)*Sqrt[3 + 5*x])/25 - (146*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/625 - (17*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/625

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Rubi in Sympy [A]  time = 26.2838, size = 114, normalized size = 0.9 \[ \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{25} - \frac{9 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{125} - \frac{146 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{625} - \frac{17 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/25 - 9*sqrt(-2*x + 1)*sqrt(3*x +
2)*sqrt(5*x + 3)/125 - 146*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7),
35/33)/625 - 17*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/187
5

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Mathematica [A]  time = 0.277499, size = 97, normalized size = 0.76 \[ \frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} (30 x+11)-105 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+292 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1250} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(11 + 30*x) + 292*Sqrt[2]*Elliptic
E[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 105*Sqrt[2]*EllipticF[ArcSin[Sqrt[2
/11]*Sqrt[3 + 5*x]], -33/2])/1250

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Maple [C]  time = 0.016, size = 169, normalized size = 1.3 \[{\frac{1}{37500\,{x}^{3}+28750\,{x}^{2}-8750\,x-7500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 105\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -292\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +9000\,{x}^{4}+10200\,{x}^{3}+430\,{x}^{2}-2570\,x-660 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1250*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(105*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))-292*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Elli
pticE(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))+9000*x
^4+10200*x^3+430*x^2-2570*x-660)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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