3.2335 \(\int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=138 \[ -\frac{1}{10} (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{11}{32} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{121}{128} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{1331 \sqrt{5 x+3} (1-2 x)^{3/2}}{2560}+\frac{43923 \sqrt{5 x+3} \sqrt{1-2 x}}{25600}+\frac{483153 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25600 \sqrt{10}} \]

[Out]

(43923*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25600 + (1331*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])
/2560 - (121*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/128 - (11*(1 - 2*x)^(5/2)*(3 + 5*x)^
(3/2))/32 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/10 + (483153*ArcSin[Sqrt[2/11]*Sqr
t[3 + 5*x]])/(25600*Sqrt[10])

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Rubi [A]  time = 0.131163, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{1}{10} (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{11}{32} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{121}{128} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{1331 \sqrt{5 x+3} (1-2 x)^{3/2}}{2560}+\frac{43923 \sqrt{5 x+3} \sqrt{1-2 x}}{25600}+\frac{483153 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25600 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(43923*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25600 + (1331*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])
/2560 - (121*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/128 - (11*(1 - 2*x)^(5/2)*(3 + 5*x)^
(3/2))/32 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/10 + (483153*ArcSin[Sqrt[2/11]*Sqr
t[3 + 5*x]])/(25600*Sqrt[10])

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Rubi in Sympy [A]  time = 11.7412, size = 124, normalized size = 0.9 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{25} + \frac{33 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{7}{2}}}{1000} - \frac{121 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{4000} - \frac{1331 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{6400} - \frac{43923 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{25600} + \frac{483153 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{256000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(3/2)*(5*x + 3)**(7/2)/25 + 33*sqrt(-2*x + 1)*(5*x + 3)**(7/2)/1000
- 121*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/4000 - 1331*sqrt(-2*x + 1)*(5*x + 3)**(3/2
)/6400 - 43923*sqrt(-2*x + 1)*sqrt(5*x + 3)/25600 + 483153*sqrt(10)*asin(sqrt(22
)*sqrt(5*x + 3)/11)/256000

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Mathematica [A]  time = 0.107415, size = 70, normalized size = 0.51 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (256000 x^4+227200 x^3-124640 x^2-147140 x+16407\right )-483153 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{256000} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(16407 - 147140*x - 124640*x^2 + 227200*x^3 + 2
56000*x^4) - 483153*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/256000

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Maple [A]  time = 0.007, size = 120, normalized size = 0.9 \[{\frac{1}{25} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{{\frac{7}{2}}}}+{\frac{33}{1000} \left ( 3+5\,x \right ) ^{{\frac{7}{2}}}\sqrt{1-2\,x}}-{\frac{121}{4000} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}\sqrt{1-2\,x}}-{\frac{1331}{6400} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{43923}{25600}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{483153\,\sqrt{10}}{512000}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25*(1-2*x)^(3/2)*(3+5*x)^(7/2)+33/1000*(3+5*x)^(7/2)*(1-2*x)^(1/2)-121/4000*(3
+5*x)^(5/2)*(1-2*x)^(1/2)-1331/6400*(3+5*x)^(3/2)*(1-2*x)^(1/2)-43923/25600*(1-2
*x)^(1/2)*(3+5*x)^(1/2)+483153/512000*((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 = 1.50077, size = 113, normalized size = 0.82 \[ -\frac{1}{10} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{11}{16} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{11}{320} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3993}{1280} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{483153}{512000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{3993}{25600} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*(-10*x^2 - x + 3)^(5/2) + 11/16*(-10*x^2 - x + 3)^(3/2)*x + 11/320*(-10*x^
2 - x + 3)^(3/2) + 3993/1280*sqrt(-10*x^2 - x + 3)*x - 483153/512000*sqrt(10)*ar
csin(-20/11*x - 1/11) + 3993/25600*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.217055, size = 97, normalized size = 0.7 \[ -\frac{1}{512000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (256000 \, x^{4} + 227200 \, x^{3} - 124640 \, x^{2} - 147140 \, x + 16407\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 483153 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/512000*sqrt(10)*(2*sqrt(10)*(256000*x^4 + 227200*x^3 - 124640*x^2 - 147140*x
+ 16407)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 483153*arctan(1/20*sqrt(10)*(20*x + 1)/(
sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 147.195, size = 311, normalized size = 2.25 \[ \begin{cases} - \frac{100 i \left (x + \frac{3}{5}\right )^{\frac{11}{2}}}{\sqrt{10 x - 5}} + \frac{1045 i \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{4 \sqrt{10 x - 5}} - \frac{2783 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{16 \sqrt{10 x - 5}} - \frac{1331 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{640 \sqrt{10 x - 5}} - \frac{14641 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{2560 \sqrt{10 x - 5}} + \frac{483153 i \sqrt{x + \frac{3}{5}}}{25600 \sqrt{10 x - 5}} - \frac{483153 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{256000} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{483153 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{256000} + \frac{100 \left (x + \frac{3}{5}\right )^{\frac{11}{2}}}{\sqrt{- 10 x + 5}} - \frac{1045 \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{4 \sqrt{- 10 x + 5}} + \frac{2783 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{16 \sqrt{- 10 x + 5}} + \frac{1331 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{640 \sqrt{- 10 x + 5}} + \frac{14641 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{2560 \sqrt{- 10 x + 5}} - \frac{483153 \sqrt{x + \frac{3}{5}}}{25600 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-100*I*(x + 3/5)**(11/2)/sqrt(10*x - 5) + 1045*I*(x + 3/5)**(9/2)/(4*
sqrt(10*x - 5)) - 2783*I*(x + 3/5)**(7/2)/(16*sqrt(10*x - 5)) - 1331*I*(x + 3/5)
**(5/2)/(640*sqrt(10*x - 5)) - 14641*I*(x + 3/5)**(3/2)/(2560*sqrt(10*x - 5)) +
483153*I*sqrt(x + 3/5)/(25600*sqrt(10*x - 5)) - 483153*sqrt(10)*I*acosh(sqrt(110
)*sqrt(x + 3/5)/11)/256000, 10*Abs(x + 3/5)/11 > 1), (483153*sqrt(10)*asin(sqrt(
110)*sqrt(x + 3/5)/11)/256000 + 100*(x + 3/5)**(11/2)/sqrt(-10*x + 5) - 1045*(x
+ 3/5)**(9/2)/(4*sqrt(-10*x + 5)) + 2783*(x + 3/5)**(7/2)/(16*sqrt(-10*x + 5)) +
1331*(x + 3/5)**(5/2)/(640*sqrt(-10*x + 5)) + 14641*(x + 3/5)**(3/2)/(2560*sqrt
(-10*x + 5)) - 483153*sqrt(x + 3/5)/(25600*sqrt(-10*x + 5)), True))

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GIAC/XCAS [A]  time = 0.249129, size = 317, normalized size = 2.3 \[ -\frac{1}{3840000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{7}{384000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{2000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3840000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 13640
5)*(5*x + 3) + 60555)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11
*sqrt(22)*sqrt(5*x + 3))) - 7/384000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 21
79)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*
sqrt(22)*sqrt(5*x + 3))) + 1/2000*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))) + 9
/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)))