3.2639 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{\sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=129 \[ \frac{2}{15} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{41}{135} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{41}{675} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{974}{675} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-41*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/135 + (2*Sqrt[1 - 2*x]*Sqrt[2 +
3*x]*(3 + 5*x)^(3/2))/15 - (974*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2
*x]], 35/33])/675 - (41*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/675

_______________________________________________________________________________________

Rubi [A]  time = 0.256972, 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 \[ \frac{2}{15} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{41}{135} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{41}{675} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{974}{675} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-41*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/135 + (2*Sqrt[1 - 2*x]*Sqrt[2 +
3*x]*(3 + 5*x)^(3/2))/15 - (974*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2
*x]], 35/33])/675 - (41*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/675

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.6568, size = 114, normalized size = 0.88 \[ \frac{2 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{15} - \frac{41 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{135} - \frac{974 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2025} - \frac{451 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{23625} \]

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)**(1/2),x)

[Out]

2*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(3/2)/15 - 41*sqrt(-2*x + 1)*sqrt(3*x
+ 2)*sqrt(5*x + 3)/135 - 974*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7)
, 35/33)/2025 - 451*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)
/23625

_______________________________________________________________________________________

Mathematica [A]  time = 0.25003, size = 92, normalized size = 0.71 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} (90 x+13)-595 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+1948 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2025 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(13 + 90*x) + 1948*EllipticE[ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 595*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]], -33/2])/(2025*Sqrt[2])

_______________________________________________________________________________________

Maple [C]  time = 0.016, size = 169, normalized size = 1.3 \[{\frac{1}{121500\,{x}^{3}+93150\,{x}^{2}-28350\,x-24300}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 595\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1948\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +81000\,{x}^{4}+73800\,{x}^{3}-9930\,{x}^{2}-18930\,x-2340 \right ) } \]

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)^(1/2),x)

[Out]

1/4050*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(595*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))-1948*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ell
ipticE(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))+81000
*x^4+73800*x^3-9930*x^2-18930*x-2340)/(30*x^3+23*x^2-7*x-6)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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