3.2807 \(\int \frac{\sqrt{2+3 x} (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{1}{5} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{67}{45} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{67}{225} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{4451}{450} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-67*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/45 - (Sqrt[1 - 2*x]*Sqrt[2 + 3*x
]*(3 + 5*x)^(3/2))/5 - (4451*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/450 - (67*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/225

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Rubi [A]  time = 0.257354, 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 \[ -\frac{1}{5} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{67}{45} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{67}{225} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{4451}{450} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-67*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/45 - (Sqrt[1 - 2*x]*Sqrt[2 + 3*x
]*(3 + 5*x)^(3/2))/5 - (4451*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/450 - (67*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/225

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Rubi in Sympy [A]  time = 25.4065, size = 114, normalized size = 0.88 \[ - \frac{\sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{5} - \frac{67 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{45} - \frac{4451 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1350} - \frac{737 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{7875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(3/2)/5 - 67*sqrt(-2*x + 1)*sqrt(3*x +
2)*sqrt(5*x + 3)/45 - 4451*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7),
35/33)/1350 - 737*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/7
875

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Mathematica [A]  time = 0.239582, size = 95, normalized size = 0.74 \[ \frac{4451 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} (45 x+94)+448 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{675 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(4451*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*(3*Sqrt[2 - 4*x]*Sq
rt[2 + 3*x]*Sqrt[3 + 5*x]*(94 + 45*x) + 448*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]], -33/2]))/(675*Sqrt[2])

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Maple [C]  time = 0.016, size = 169, normalized size = 1.3 \[{\frac{1}{40500\,{x}^{3}+31050\,{x}^{2}-9450\,x-8100}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 2240\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4451\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -40500\,{x}^{4}-115650\,{x}^{3}-55410\,{x}^{2}+27840\,x+16920 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1350*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(2240*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))-4451*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*El
lipticE(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))-4050
0*x^4-115650*x^3-55410*x^2+27840*x+16920)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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