3.2646 \(\int \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=218 \[ \frac{2}{55} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}-\frac{3}{275} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}-\frac{177 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{1925}-\frac{7031 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{11550}-\frac{465127 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{103950}-\frac{465127 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47250 \sqrt{33}}-\frac{30926081 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{94500 \sqrt{33}} \]

[Out]

(-465127*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/103950 - (7031*Sqrt[1 - 2*x]
*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/11550 - (177*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*
x)^(5/2))/1925 - (3*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/275 + (2*Sqrt[1
- 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/55 - (30926081*EllipticE[ArcSin[Sqrt[3/
7]*Sqrt[1 - 2*x]], 35/33])/(94500*Sqrt[33]) - (465127*EllipticF[ArcSin[Sqrt[3/7]
*Sqrt[1 - 2*x]], 35/33])/(47250*Sqrt[33])

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Rubi [A]  time = 0.482675, 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 \[ \frac{2}{55} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}-\frac{3}{275} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}-\frac{177 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{1925}-\frac{7031 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{11550}-\frac{465127 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{103950}-\frac{465127 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47250 \sqrt{33}}-\frac{30926081 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{94500 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-465127*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/103950 - (7031*Sqrt[1 - 2*x]
*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/11550 - (177*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*
x)^(5/2))/1925 - (3*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/275 + (2*Sqrt[1
- 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/55 - (30926081*EllipticE[ArcSin[Sqrt[3/
7]*Sqrt[1 - 2*x]], 35/33])/(94500*Sqrt[33]) - (465127*EllipticF[ArcSin[Sqrt[3/7]
*Sqrt[1 - 2*x]], 35/33])/(47250*Sqrt[33])

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Rubi in Sympy [A]  time = 46.3927, size = 201, normalized size = 0.92 \[ \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{33} - \frac{5 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{99} - \frac{95 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{693} - \frac{6691 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{6930} - \frac{222527 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{51975} - \frac{30926081 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3118500} - \frac{465127 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1653750} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*(5*x + 3)**(5/2)/33 - 5*sqrt(-2*x + 1)*(3*x +
2)**(5/2)*(5*x + 3)**(3/2)/99 - 95*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*(5*x + 3)**(3
/2)/693 - 6691*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/6930 - 222527*sqrt(
-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/51975 - 30926081*sqrt(33)*elliptic_e(asin(
sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/3118500 - 465127*sqrt(35)*elliptic_f(asin(sqr
t(55)*sqrt(-2*x + 1)/11), 33/35)/1653750

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Mathematica [A]  time = 0.375617, size = 107, normalized size = 0.49 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (1417500 x^4+3354750 x^3+2737800 x^2+570555 x-567484\right )-15576890 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1559250 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-567484 + 570555*x + 2737800*x^2
+ 3354750*x^3 + 1417500*x^4) + 30926081*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x
]], -33/2] - 15576890*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(15592
50*Sqrt[2])

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Maple [C]  time = 0.017, size = 184, normalized size = 0.8 \[{\frac{1}{93555000\,{x}^{3}[phone]\,{x}^{2}-21829500\,x-18711000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1275750000\,{x}^{7}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -30926081\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-1295845650\,{x}^{3}-1004184510\,{x}^{2}[phone]\,x+102147120 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3118500*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(1275750000*x^7+3997350000*x
^6+15576890*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))-30926081*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+1442934000*x^4-1295845650*x^
3-1004184510*x^2+16471740*x+102147120)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((75*x^3 + 140*x^2 + 87*x + 18)*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)**(3/2)*(3+5*x)**(5/2)*(1-2*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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