3.2636 \(\int \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=218 \[ \frac{2}{55} \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac{23 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{2475}-\frac{543 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{9625}-\frac{342971 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{866250}-\frac{11346991 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{3898125}-\frac{11346991 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1771875 \sqrt{33}}-\frac{1508889271 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7087500 \sqrt{33}} \]

[Out]

(-11346991*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/3898125 - (342971*Sqrt[1 -
2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/866250 - (543*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(
3 + 5*x)^(5/2))/9625 - (23*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/2475 +
(2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/55 - (1508889271*EllipticE[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(7087500*Sqrt[33]) - (11346991*EllipticF[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1771875*Sqrt[33])

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Rubi [A]  time = 0.481687, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{55} \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac{23 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{2475}-\frac{543 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{9625}-\frac{342971 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{866250}-\frac{11346991 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{3898125}-\frac{11346991 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1771875 \sqrt{33}}-\frac{1508889271 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7087500 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-11346991*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/3898125 - (342971*Sqrt[1 -
2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/866250 - (543*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(
3 + 5*x)^(5/2))/9625 - (23*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/2475 +
(2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/55 - (1508889271*EllipticE[Ar
cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(7087500*Sqrt[33]) - (11346991*EllipticF[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1771875*Sqrt[33])

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Rubi in Sympy [A]  time = 46.3692, size = 201, normalized size = 0.92 \[ \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{33} - \frac{41 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{891} - \frac{4439 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{31185} - \frac{932783 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{1559250} - \frac{21713939 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{7796250} - \frac{1508889271 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{233887500} - \frac{11346991 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{62015625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(-2*x + 1)*(3*x + 2)**(7/2)*(5*x + 3)**(3/2)/33 - 41*sqrt(-2*x + 1)*(3*x +
2)**(7/2)*sqrt(5*x + 3)/891 - 4439*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x + 3
)/31185 - 932783*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/1559250 - 2171393
9*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/7796250 - 1508889271*sqrt(33)*ellip
tic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/233887500 - 11346991*sqrt(35)*elli
ptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/62015625

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Mathematica [A]  time = 0.43942, size = 107, normalized size = 0.49 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (63787500 x^4+156161250 x^3+132234750 x^2+29706255 x-27010769\right )-759987865 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{116943750 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-27010769 + 29706255*x + 13223475
0*x^2 + 156161250*x^3 + 63787500*x^4) + 1508889271*EllipticE[ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]], -33/2] - 759987865*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -3
3/2])/(116943750*Sqrt[2])

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Maple [C]  time = 0.031, size = 184, normalized size = 0.8 \[{\frac{1}{7016625000\,{x}^{3}[phone]\,{x}^{2}-1637212500\,x-1403325000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 57408750000\,{x}^{7}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1508889271\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-59690698650\,{x}^{3}-48677999160\,{x}^{2}[phone]\,x+4861938420 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/233887500*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(57408750000*x^7+184558500
000*x^6+759987865*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))-1508889271*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+73701994500*x^4-59
690698650*x^3-48677999160*x^2+325135590*x+4861938420)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((45*x^3 + 87*x^2 + 56*x + 12)*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((2+3*x)**(5/2)*(3+5*x)**(3/2)*(1-2*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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