3.2712 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^{5/2}}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=191 \[ \frac{2}{45} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{178 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{4725}+\frac{403 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{118125}-\frac{87476 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{590625}-\frac{104663 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2953125}-\frac{6515539 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5906250} \]

[Out]

(-87476*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/590625 + (403*Sqrt[1 - 2*x]*(
2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/118125 + (178*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3
+ 5*x])/4725 + (2*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/45 - (6515539*
Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5906250 - (104663*
Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2953125

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Rubi [A]  time = 0.406636, 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 \[ \frac{2}{45} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{178 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{4725}+\frac{403 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{118125}-\frac{87476 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{590625}-\frac{104663 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2953125}-\frac{6515539 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5906250} \]

Antiderivative was successfully verified.

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

[Out]

(-87476*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/590625 + (403*Sqrt[1 - 2*x]*(
2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/118125 + (178*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3
+ 5*x])/4725 + (2*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/45 - (6515539*
Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5906250 - (104663*
Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2953125

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Rubi in Sympy [A]  time = 39.4106, size = 172, normalized size = 0.9 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{45} + \frac{178 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{4725} + \frac{403 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{118125} - \frac{87476 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{590625} - \frac{6515539 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{17718750} - \frac{1151293 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{103359375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(3/2)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/45 + 178*sqrt(-2*x + 1)*(3*x
+ 2)**(5/2)*sqrt(5*x + 3)/4725 + 403*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x +
3)/118125 - 87476*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/590625 - 6515539*sq
rt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/17718750 - 1151293*sqr
t(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/103359375

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Mathematica [A]  time = 0.356656, size = 105, normalized size = 0.55 \[ \frac{6515539 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (472500 x^3+193500 x^2-378045 x-110554\right )+612332 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{8859375 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(6515539*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]*(-110554 - 378045*x + 193500*x^2 + 472500*x^3) + 61
2332*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(8859375*Sqrt[2])

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Maple [C]  time = 0.017, size = 179, normalized size = 0.9 \[{\frac{1}{531562500\,{x}^{3}[phone]\,{x}^{2}-124031250\,x-106312500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -425250000\,{x}^{6}+3061660\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -6515539\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -500175000\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-91264440\,x-19899720 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/17718750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-425250000*x^6+3061660*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))-6515539*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))-500175000*x^5+305950500*x^4+486034650*x^3+31722810*x^2-9126
4440*x-19899720)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{\sqrt{5 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(18*x^3 + 15*x^2 - 4*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((1-2*x)**(3/2)*(2+3*x)**(5/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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