3.2629 \(\int \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=160 \[ \frac{2}{35} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}-\frac{27}{875} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{823 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{2625}-\frac{823 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13125}-\frac{55019 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{26250} \]

[Out]

(-823*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2625 - (27*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*(3 + 5*x)^(3/2))/875 + (2*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))
/35 - (55019*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/26250
- (823*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/13125

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Rubi [A]  time = 0.329839, antiderivative size = 160, 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 \[ \frac{2}{35} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}-\frac{27}{875} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{823 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{2625}-\frac{823 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13125}-\frac{55019 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{26250} \]

Antiderivative was successfully verified.

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

[Out]

(-823*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2625 - (27*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*(3 + 5*x)^(3/2))/875 + (2*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))
/35 - (55019*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/26250
- (823*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/13125

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Rubi in Sympy [A]  time = 31.3051, size = 143, normalized size = 0.89 \[ \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{21} - \frac{37 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{525} - \frac{796 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{2625} - \frac{55019 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{78750} - \frac{823 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{39375} \]

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)**(5/2)*sqrt(5*x + 3)/21 - 37*sqrt(-2*x + 1)*(3*x + 2)
**(3/2)*sqrt(5*x + 3)/525 - 796*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/2625
- 55019*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/78750 - 823*
sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/39375

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Mathematica [A]  time = 0.303513, size = 97, normalized size = 0.61 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (2250 x^2+2445 x-166\right )-27860 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+55019 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{39375 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-166 + 2445*x + 2250*x^2) + 55019
*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 27860*EllipticF[ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]], -33/2])/(39375*Sqrt[2])

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Maple [C]  time = 0.015, size = 174, normalized size = 1.1 \[{\frac{1}{2362500\,{x}^{3}+1811250\,{x}^{2}-551250\,x-472500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 27860\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -55019\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +2025000\,{x}^{5}+3753000\,{x}^{4}+1065150\,{x}^{3}-1032990\,{x}^{2}-405240\,x+29880 \right ) } \]

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]

1/78750*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(27860*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*1
1^(1/2)*3^(1/2)*2^(1/2))-55019*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))+2
025000*x^5+3753000*x^4+1065150*x^3-1032990*x^2-405240*x+29880)/(30*x^3+23*x^2-7*
x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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