3.2658 \(\int \frac{\sqrt{1-2 x} (2+3 x)^{5/2}}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=158 \[ \frac{2}{35} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{23}{875} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{859 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{4375}-\frac{314 \sqrt{33} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{21875}-\frac{61151 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{43750} \]

[Out]

(-859*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/4375 - (23*Sqrt[1 - 2*x]*(2 + 3
*x)^(3/2)*Sqrt[3 + 5*x])/875 + (2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/3
5 - (61151*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/43750 -
(314*Sqrt[33]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/21875

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Rubi [A]  time = 0.340862, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{35} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}-\frac{23}{875} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{859 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{4375}-\frac{314 \sqrt{33} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{21875}-\frac{61151 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{43750} \]

Antiderivative was successfully verified.

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

[Out]

(-859*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/4375 - (23*Sqrt[1 - 2*x]*(2 + 3
*x)^(3/2)*Sqrt[3 + 5*x])/875 + (2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/3
5 - (61151*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/43750 -
(314*Sqrt[33]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/21875

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Rubi in Sympy [A]  time = 33.2088, size = 143, normalized size = 0.91 \[ \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{35} - \frac{23 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{875} - \frac{859 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{4375} - \frac{61151 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{131250} - \frac{10362 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{765625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/35 - 23*sqrt(-2*x + 1)*(3*x + 2)
**(3/2)*sqrt(5*x + 3)/875 - 859*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/4375
- 61151*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/131250 - 103
62*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/765625

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Mathematica [A]  time = 0.316911, size = 97, normalized size = 0.61 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (2250 x^2+2655 x-89\right )-30065 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+61151 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{65625 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-89 + 2655*x + 2250*x^2) + 61151*
EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 30065*EllipticF[ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]], -33/2])/(65625*Sqrt[2])

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Maple [C]  time = 0.032, size = 174, normalized size = 1.1 \[{\frac{1}{3937500\,{x}^{3}+3018750\,{x}^{2}-918750\,x-787500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 30065\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -61151\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +2025000\,{x}^{5}+3942000\,{x}^{4}+1279350\,{x}^{3}-1023960\,{x}^{2}-459210\,x+16020 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/131250*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(30065*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))-61151*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2
)*EllipticE(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))+
2025000*x^5+3942000*x^4+1279350*x^3-1023960*x^2-459210*x+16020)/(30*x^3+23*x^2-7
*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{\sqrt{5 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((9*x^2 + 12*x + 4)*sqrt(3*x + 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)**(5/2)*(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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}{\sqrt{5 \, x + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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