3.2323 \(\int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=116 \[ -\frac{1}{8} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{11}{32} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{121}{640} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{3993 \sqrt{5 x+3} \sqrt{1-2 x}}{6400}+\frac{43923 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]

[Out]

(3993*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6400 + (121*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/64
0 - (11*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/32 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/8
+ (43923*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(6400*Sqrt[10])

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Rubi [A]  time = 0.107428, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{1}{8} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{11}{32} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{121}{640} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{3993 \sqrt{5 x+3} \sqrt{1-2 x}}{6400}+\frac{43923 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(3993*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6400 + (121*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/64
0 - (11*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/32 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/8
+ (43923*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(6400*Sqrt[10])

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Rubi in Sympy [A]  time = 10.0096, size = 104, normalized size = 0.9 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{20} + \frac{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{200} - \frac{121 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{1600} - \frac{3993 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{6400} + \frac{43923 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{64000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(3/2)*(5*x + 3)**(5/2)/20 + 11*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/200 -
121*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/1600 - 3993*sqrt(-2*x + 1)*sqrt(5*x + 3)/64
00 + 43923*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/64000

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Mathematica [A]  time = 0.0988443, size = 65, normalized size = 0.56 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (-16000 x^3-2400 x^2+11980 x+603\right )-43923 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{64000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(603 + 11980*x - 2400*x^2 - 16000*x^3) - 43923*S
qrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/64000

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Maple [A]  time = 0.006, size = 104, normalized size = 0.9 \[{\frac{1}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}}+{\frac{11}{200} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}\sqrt{1-2\,x}}-{\frac{121}{1600} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{3993}{6400}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{43923\,\sqrt{10}}{128000}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(1-2*x)^(3/2)*(3+5*x)^(5/2)+11/200*(3+5*x)^(5/2)*(1-2*x)^(1/2)-121/1600*(3+
5*x)^(3/2)*(1-2*x)^(1/2)-3993/6400*(1-2*x)^(1/2)*(3+5*x)^(1/2)+43923/128000*((1-
2*x)*(3+5*x))^(1/2)/(3+5*x)^(1/2)/(1-2*x)^(1/2)*10^(1/2)*arcsin(20/11*x+1/11)

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Maxima [A]  time = 1.507, size = 95, normalized size = 0.82 \[ \frac{1}{4} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{1}{80} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{363}{320} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{43923}{128000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{363}{6400} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(-10*x^2 - x + 3)^(3/2)*x + 1/80*(-10*x^2 - x + 3)^(3/2) + 363/320*sqrt(-10*
x^2 - x + 3)*x - 43923/128000*sqrt(10)*arcsin(-20/11*x - 1/11) + 363/6400*sqrt(-
10*x^2 - x + 3)

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Fricas [A]  time = 0.217772, size = 90, normalized size = 0.78 \[ -\frac{1}{128000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (16000 \, x^{3} + 2400 \, x^{2} - 11980 \, x - 603\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 43923 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/128000*sqrt(10)*(2*sqrt(10)*(16000*x^3 + 2400*x^2 - 11980*x - 603)*sqrt(5*x +
3)*sqrt(-2*x + 1) - 43923*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-
2*x + 1))))

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Sympy [A]  time = 17.827, size = 269, normalized size = 2.32 \[ \begin{cases} - \frac{25 i \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{\sqrt{10 x - 5}} + \frac{275 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{4 \sqrt{10 x - 5}} - \frac{1573 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{32 \sqrt{10 x - 5}} - \frac{1331 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{640 \sqrt{10 x - 5}} + \frac{43923 i \sqrt{x + \frac{3}{5}}}{6400 \sqrt{10 x - 5}} - \frac{43923 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{64000} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{43923 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{64000} + \frac{25 \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{\sqrt{- 10 x + 5}} - \frac{275 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{4 \sqrt{- 10 x + 5}} + \frac{1573 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{32 \sqrt{- 10 x + 5}} + \frac{1331 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{640 \sqrt{- 10 x + 5}} - \frac{43923 \sqrt{x + \frac{3}{5}}}{6400 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-25*I*(x + 3/5)**(9/2)/sqrt(10*x - 5) + 275*I*(x + 3/5)**(7/2)/(4*sqr
t(10*x - 5)) - 1573*I*(x + 3/5)**(5/2)/(32*sqrt(10*x - 5)) - 1331*I*(x + 3/5)**(
3/2)/(640*sqrt(10*x - 5)) + 43923*I*sqrt(x + 3/5)/(6400*sqrt(10*x - 5)) - 43923*
sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/64000, 10*Abs(x + 3/5)/11 > 1), (43
923*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/64000 + 25*(x + 3/5)**(9/2)/sqrt(-
10*x + 5) - 275*(x + 3/5)**(7/2)/(4*sqrt(-10*x + 5)) + 1573*(x + 3/5)**(5/2)/(32
*sqrt(-10*x + 5)) + 1331*(x + 3/5)**(3/2)/(640*sqrt(-10*x + 5)) - 43923*sqrt(x +
3/5)/(6400*sqrt(-10*x + 5)), True))

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GIAC/XCAS [A]  time = 0.243228, size = 220, normalized size = 1.9 \[ -\frac{1}{192000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{24000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{3}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/192000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(
5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) -
1/24000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3)*sqrt(-10*x + 5)
- 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 3/400*sqrt(5)*(2*(20*x + 1)
*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))
)