3.2683 \(\int (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=191 \[ \frac{2}{45} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2}+\frac{62 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}{1575}-\frac{347 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{39375}-\frac{84134 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{354375}-\frac{84134 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1771875}-\frac{5684677 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3543750} \]

[Out]

(-84134*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/354375 - (347*Sqrt[1 - 2*x]*S
qrt[2 + 3*x]*(3 + 5*x)^(3/2))/39375 + (62*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x
)^(3/2))/1575 + (2*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/45 - (568467
7*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3543750 - (84134
*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1771875

_______________________________________________________________________________________

Rubi [A]  time = 0.389632, 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 \[ \frac{2}{45} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2}+\frac{62 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}{1575}-\frac{347 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{39375}-\frac{84134 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{354375}-\frac{84134 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1771875}-\frac{5684677 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3543750} \]

Antiderivative was successfully verified.

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

[Out]

(-84134*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/354375 - (347*Sqrt[1 - 2*x]*S
qrt[2 + 3*x]*(3 + 5*x)^(3/2))/39375 + (62*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x
)^(3/2))/1575 + (2*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/45 - (568467
7*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3543750 - (84134
*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1771875

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 38.8296, size = 172, normalized size = 0.9 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{27} - \frac{107 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{945} - \frac{2384 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}}{7875} + \frac{167227 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{354375} - \frac{5684677 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{10631250} - \frac{84134 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{5315625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(3/2)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/27 - 107*(-2*x + 1)**(3/2)*(3
*x + 2)**(3/2)*sqrt(5*x + 3)/945 - 2384*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*x
+ 3)/7875 + 167227*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/354375 - 5684677*
sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/10631250 - 84134*sqr
t(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/5315625

_______________________________________________________________________________________

Mathematica [A]  time = 0.322789, size = 105, normalized size = 0.55 \[ \frac{5684677 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (472500 x^3+153000 x^2-359685 x-84697\right )+581651 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{5315625 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(5684677*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*(3*Sqrt[2 - 4*x]
*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-84697 - 359685*x + 153000*x^2 + 472500*x^3) + 581
651*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(5315625*Sqrt[2])

_______________________________________________________________________________________

Maple [C]  time = 0.017, size = 179, normalized size = 0.9 \[ -{\frac{1}{318937500\,{x}^{3}[phone]\,{x}^{2}-74418750\,x-63787500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 425250000\,{x}^{6}+5684677\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2908255\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}-317371500\,{x}^{4}-441589950\,{x}^{3}-10447080\,{x}^{2}[phone]\,x+15245460 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10631250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(425250000*x^6+5684677*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))-2908255*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))[phone]*x^5-317371500*x^4-441589950*x^3-10447080*x^2+8252
9670*x+15245460)/(30*x^3+23*x^2-7*x-6)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (6 \, x^{2} + x - 2\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(6*x^2 + x - 2)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1), 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((1-2*x)**(3/2)*(2+3*x)**(3/2)*(3+5*x)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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