3.2410 \(\int \frac{(1-2 x)^{5/2} (2+3 x)}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=116 \[ -\frac{3}{40} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{49 \sqrt{5 x+3} (1-2 x)^{5/2}}{1200}+\frac{539 \sqrt{5 x+3} (1-2 x)^{3/2}}{4800}+\frac{5929 \sqrt{5 x+3} \sqrt{1-2 x}}{16000}+\frac{65219 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{16000 \sqrt{10}} \]

[Out]

(5929*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/16000 + (539*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/4
800 + (49*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/1200 - (3*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x]
)/40 + (65219*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(16000*Sqrt[10])

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Rubi [A]  time = 0.119928, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{3}{40} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{49 \sqrt{5 x+3} (1-2 x)^{5/2}}{1200}+\frac{539 \sqrt{5 x+3} (1-2 x)^{3/2}}{4800}+\frac{5929 \sqrt{5 x+3} \sqrt{1-2 x}}{16000}+\frac{65219 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{16000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(5929*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/16000 + (539*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/4
800 + (49*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/1200 - (3*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x]
)/40 + (65219*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(16000*Sqrt[10])

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Rubi in Sympy [A]  time = 10.3337, size = 105, normalized size = 0.91 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{40} + \frac{49 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{1200} + \frac{539 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{4800} + \frac{5929 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{16000} + \frac{65219 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{160000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(7/2)*sqrt(5*x + 3)/40 + 49*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/1200
+ 539*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/4800 + 5929*sqrt(-2*x + 1)*sqrt(5*x + 3)/1
6000 + 65219*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/160000

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Mathematica [A]  time = 0.0795555, size = 65, normalized size = 0.56 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (28800 x^3-35360 x^2+2980 x+21537\right )-195657 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{480000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(21537 + 2980*x - 35360*x^2 + 28800*x^3) - 19565
7*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/480000

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Maple [A]  time = 0.013, size = 104, normalized size = 0.9 \[{\frac{1}{960000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 576000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-707200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+195657\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +59600\,x\sqrt{-10\,{x}^{2}-x+3}+430740\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/960000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(576000*x^3*(-10*x^2-x+3)^(1/2)-707200*x^2*
(-10*x^2-x+3)^(1/2)+195657*10^(1/2)*arcsin(20/11*x+1/11)+59600*x*(-10*x^2-x+3)^(
1/2)+430740*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.51056, size = 101, normalized size = 0.87 \[ \frac{3}{5} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} - \frac{221}{300} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{149}{2400} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{65219}{320000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{7179}{16000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/5*sqrt(-10*x^2 - x + 3)*x^3 - 221/300*sqrt(-10*x^2 - x + 3)*x^2 + 149/2400*sqr
t(-10*x^2 - x + 3)*x - 65219/320000*sqrt(10)*arcsin(-20/11*x - 1/11) + 7179/1600
0*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.225762, size = 90, normalized size = 0.78 \[ \frac{1}{960000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (28800 \, x^{3} - 35360 \, x^{2} + 2980 \, x + 21537\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 195657 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/960000*sqrt(10)*(2*sqrt(10)*(28800*x^3 - 35360*x^2 + 2980*x + 21537)*sqrt(5*x
+ 3)*sqrt(-2*x + 1) + 195657*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt
(-2*x + 1))))

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Sympy [A]  time = 81.2204, size = 296, normalized size = 2.55 \[ - \frac{7 \sqrt{2} \left (\begin{cases} \frac{1331 \sqrt{5} \left (\frac{5 \sqrt{5} \left (- 2 x + 1\right )^{\frac{3}{2}} \left (10 x + 6\right )^{\frac{3}{2}}}{7986} + \frac{3 \sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{1936} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{16}\right )}{625} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{2} + \frac{3 \sqrt{2} \left (\begin{cases} \frac{14641 \sqrt{5} \left (\frac{5 \sqrt{5} \left (- 2 x + 1\right )^{\frac{3}{2}} \left (10 x + 6\right )^{\frac{3}{2}}}{3993} + \frac{7 \sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (20 x + 1\right )}{3872} + \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6} \left (12100 x - 2000 \left (- 2 x + 1\right )^{3} + 6600 \left (- 2 x + 1\right )^{2} - 4719\right )}{1874048} - \frac{\sqrt{5} \sqrt{- 2 x + 1} \sqrt{10 x + 6}}{22} + \frac{35 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{128}\right )}{3125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*sqrt(2)*Piecewise((1331*sqrt(5)*(5*sqrt(5)*(-2*x + 1)**(3/2)*(10*x + 6)**(3/2
)/7986 + 3*sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(20*x + 1)/1936 - sqrt(5)*sqrt(
-2*x + 1)*sqrt(10*x + 6)/22 + 5*asin(sqrt(55)*sqrt(-2*x + 1)/11)/16)/625, (x <=
1/2) & (x > -3/5)))/2 + 3*sqrt(2)*Piecewise((14641*sqrt(5)*(5*sqrt(5)*(-2*x + 1)
**(3/2)*(10*x + 6)**(3/2)/3993 + 7*sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(20*x +
1)/3872 + sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)*(12100*x - 2000*(-2*x + 1)**3 +
6600*(-2*x + 1)**2 - 4719)/1874048 - sqrt(5)*sqrt(-2*x + 1)*sqrt(10*x + 6)/22 +
35*asin(sqrt(55)*sqrt(-2*x + 1)/11)/128)/3125, (x <= 1/2) & (x > -3/5)))/2

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GIAC/XCAS [A]  time = 0.248846, size = 274, normalized size = 2.36 \[ \frac{1}{800000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 119\right )}{\left (5 \, x + 3\right )} + 6163\right )}{\left (5 \, x + 3\right )} - 66189\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 184305 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{30000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 59\right )}{\left (5 \, x + 3\right )} + 1293\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 4785 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{25} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/800000*sqrt(5)*(2*(4*(8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt
(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))
- 1/30000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x +
5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 1/400*sqrt(5)*(2*(20*x
- 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x
+ 3))) + 1/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(
5*x + 3)*sqrt(-10*x + 5))