3.2637 \(\int \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=191 \[ \frac{2}{45} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac{3}{175} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{1208 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{7875}-\frac{160297 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{141750}-\frac{160297 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{708750}-\frac{5327983 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{708750} \]

[Out]

(-160297*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/141750 - (1208*Sqrt[1 - 2*x]
*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/7875 - (3*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^
(5/2))/175 + (2*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/45 - (5327983*Sqr
t[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/708750 - (160297*Sqrt
[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/708750

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Rubi [A]  time = 0.402518, 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 \[ \frac{2}{45} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac{3}{175} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{1208 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{7875}-\frac{160297 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{141750}-\frac{160297 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{708750}-\frac{5327983 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{708750} \]

Antiderivative was successfully verified.

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

[Out]

(-160297*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/141750 - (1208*Sqrt[1 - 2*x]
*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/7875 - (3*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^
(5/2))/175 + (2*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/45 - (5327983*Sqr
t[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/708750 - (160297*Sqrt
[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/708750

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Rubi in Sympy [A]  time = 38.8297, size = 172, normalized size = 0.9 \[ \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{27} - \frac{41 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{567} - \frac{3284 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{14175} - \frac{153319 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{141750} - \frac{5327983 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2126250} - \frac{160297 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2126250} \]

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)**(1/2),x)

[Out]

2*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*(5*x + 3)**(3/2)/27 - 41*sqrt(-2*x + 1)*(3*x +
2)**(5/2)*sqrt(5*x + 3)/567 - 3284*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3
)/14175 - 153319*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/141750 - 5327983*sqr
t(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/2126250 - 160297*sqrt(3
3)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/2126250

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Mathematica [A]  time = 0.337202, size = 102, normalized size = 0.53 \[ \frac{15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (472500 x^3+821250 x^2+366480 x-133999\right )-5366165 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2126250 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-133999 + 366480*x + 821250*x^2 +
472500*x^3) + 10655966*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 536
6165*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(2126250*Sqrt[2])

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Maple [C]  time = 0.017, size = 179, normalized size = 0.9 \[{\frac{1}{127575000\,{x}^{3}[phone]\,{x}^{2}-29767500\,x-25515000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 425250000\,{x}^{6}+5366165\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -10655966\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}-125240400\,{x}^{3}-317245110\,{x}^{2}-37826610\,x+24119820 \right ) } \]

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)^(1/2),x)

[Out]

1/4252500*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(425250000*x^6+5366165*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))-10655966*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+797269500*x^4-125240400*x^3-317245110*x^2-378
26610*x+24119820)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((15*x^2 + 19*x + 6)*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)**(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. \[ \int{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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