3.2884 \(\int \frac{(2+3 x)^{3/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=157 \[ \frac{(3 x+2)^{3/2} (5 x+3)^{3/2}}{\sqrt{1-2 x}}+\frac{9}{5} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{139}{10} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{139}{50} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{4621}{50} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(139*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/10 + (9*Sqrt[1 - 2*x]*Sqrt[2 + 3
*x]*(3 + 5*x)^(3/2))/5 + ((2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (4621
*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/50 + (139*Sqrt[11
/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/50

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Rubi [A]  time = 0.310595, antiderivative size = 157, 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 \[ \frac{(3 x+2)^{3/2} (5 x+3)^{3/2}}{\sqrt{1-2 x}}+\frac{9}{5} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{139}{10} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}+\frac{139}{50} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{4621}{50} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(139*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/10 + (9*Sqrt[1 - 2*x]*Sqrt[2 + 3
*x]*(3 + 5*x)^(3/2))/5 + ((2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (4621
*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/50 + (139*Sqrt[11
/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/50

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Rubi in Sympy [A]  time = 31.5143, size = 139, normalized size = 0.89 \[ \frac{9 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{5} + \frac{139 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{10} + \frac{4621 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{150} + \frac{139 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{150} + \frac{\left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(3/2)/5 + 139*sqrt(-2*x + 1)*sqrt(3*x
+ 2)*sqrt(5*x + 3)/10 + 4621*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7)
, 35/33)/150 + 139*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1
50 + (3*x + 2)**(3/2)*(5*x + 3)**(3/2)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.241802, size = 110, normalized size = 0.7 \[ \frac{-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (30 x^2+106 x-253\right )+4655 \sqrt{2-4 x} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-9242 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{300 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-253 + 106*x + 30*x^2) - 9242*Sqrt[2 - 4*x]*El
lipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 4655*Sqrt[2 - 4*x]*EllipticF[
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(300*Sqrt[1 - 2*x])

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Maple [C]  time = 0.025, size = 169, normalized size = 1.1 \[ -{\frac{1}{9000\,{x}^{3}+6900\,{x}^{2}-2100\,x-1800}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 4655\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -9242\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -13500\,{x}^{4}-64800\,{x}^{3}+48030\,{x}^{2}+125130\,x+45540 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/300*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(4655*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))-9242*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*El
lipticE(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))-1350
0*x^4-64800*x^3+48030*x^2+125130*x+45540)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (15 \, x^{2} + 19 \, x + 6\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(15*x^2 + 19*x + 6)*sqrt(5*x + 3)*sqrt(3*x + 2)/((2*x - 1)*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)*(3+5*x)**(3/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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