3.2647 \(\int \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=191 \[ \frac{2}{45} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}-\frac{31}{945} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{223}{945} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{29357 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{17010}-\frac{29357 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{85050}-\frac{488149 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{42525} \]

[Out]

(-29357*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/17010 - (223*Sqrt[1 - 2*x]*Sq
rt[2 + 3*x]*(3 + 5*x)^(3/2))/945 - (31*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/
2))/945 + (2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/45 - (488149*Sqrt[11/3
]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/42525 - (29357*Sqrt[11/3]*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/85050

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Rubi [A]  time = 0.406946, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{45} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}-\frac{31}{945} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{223}{945} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{29357 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{17010}-\frac{29357 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{85050}-\frac{488149 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{42525} \]

Antiderivative was successfully verified.

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

[Out]

(-29357*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/17010 - (223*Sqrt[1 - 2*x]*Sq
rt[2 + 3*x]*(3 + 5*x)^(3/2))/945 - (31*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/
2))/945 + (2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/45 - (488149*Sqrt[11/3
]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/42525 - (29357*Sqrt[11/3]*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/85050

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Rubi in Sympy [A]  time = 38.8456, size = 172, normalized size = 0.9 \[ \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{27} - \frac{5 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{63} - \frac{208 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{945} - \frac{29357 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{17010} - \frac{488149 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{127575} - \frac{322927 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{2976750} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*(5*x + 3)**(5/2)/27 - 5*sqrt(-2*x + 1)*(3*x +
2)**(3/2)*(5*x + 3)**(3/2)/63 - 208*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(3/2
)/945 - 29357*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/17010 - 488149*sqrt(33)
*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/127575 - 322927*sqrt(35)*ell
iptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/2976750

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Mathematica [A]  time = 0.328572, size = 102, normalized size = 0.53 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (94500 x^3+156150 x^2+65250 x-26009\right )-983815 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+1952596 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{255150 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-26009 + 65250*x + 156150*x^2 + 9
4500*x^3) + 1952596*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 983815*
EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(255150*Sqrt[2])

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Maple [C]  time = 0.017, size = 179, normalized size = 0.9 \[{\frac{1}{15309000\,{x}^{3}[phone]\,{x}^{2}-3572100\,x-3061800}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 85050000\,{x}^{6}+983815\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1952596\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-28187100\,{x}^{3}-59755710\,{x}^{2}-6283110\,x+4681620 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/510300*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(85050000*x^6+983815*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))-1952596*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))[phone]*x^5+146623500*x^4-28187100*x^3-59755710*x^2-6283110*x+
4681620)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^2 + 30*x + 9)*sqrt(5*x + 3)*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)**(5/2)*(1-2*x)**(1/2)*(2+3*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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