3.2312 \(\int (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=116 \[ -\frac{3}{40} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{23}{96} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{253 \sqrt{5 x+3} (1-2 x)^{3/2}}{1920}+\frac{2783 \sqrt{5 x+3} \sqrt{1-2 x}}{6400}+\frac{30613 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]

[Out]

(2783*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6400 + (253*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/19
20 - (23*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/96 - (3*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))
/40 + (30613*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(6400*Sqrt[10])

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Rubi [A]  time = 0.116326, 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 \[ -\frac{3}{40} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{23}{96} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{253 \sqrt{5 x+3} (1-2 x)^{3/2}}{1920}+\frac{2783 \sqrt{5 x+3} \sqrt{1-2 x}}{6400}+\frac{30613 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(2783*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6400 + (253*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/19
20 - (23*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/96 - (3*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))
/40 + (30613*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(6400*Sqrt[10])

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Rubi in Sympy [A]  time = 10.3678, size = 105, normalized size = 0.91 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{40} + \frac{23 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{240} + \frac{253 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{1600} - \frac{2783 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{6400} + \frac{30613 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{64000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(5/2)*(5*x + 3)**(3/2)/40 + 23*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)
/240 + 253*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/1600 - 2783*sqrt(-2*x + 1)*sqrt(5*x +
3)/6400 + 30613*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/64000

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Mathematica [A]  time = 0.0676921, size = 65, normalized size = 0.56 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (-28800 x^3-6880 x^2+23420 x+1959\right )-91839 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{192000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1959 + 23420*x - 6880*x^2 - 28800*x^3) - 91839*
Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/192000

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Maple [A]  time = 0.012, size = 104, normalized size = 0.9 \[{\frac{1}{384000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -576000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-137600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+91839\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +468400\,x\sqrt{-10\,{x}^{2}-x+3}+39180\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-576000*x^3*(-10*x^2-x+3)^(1/2)-137600*x^2
*(-10*x^2-x+3)^(1/2)+91839*10^(1/2)*arcsin(20/11*x+1/11)+468400*x*(-10*x^2-x+3)^
(1/2)+39180*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.47856, size = 95, normalized size = 0.82 \[ \frac{3}{20} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{1}{48} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{253}{320} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{30613}{128000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{253}{6400} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/20*(-10*x^2 - x + 3)^(3/2)*x + 1/48*(-10*x^2 - x + 3)^(3/2) + 253/320*sqrt(-10
*x^2 - x + 3)*x - 30613/128000*sqrt(10)*arcsin(-20/11*x - 1/11) + 253/6400*sqrt(
-10*x^2 - x + 3)

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Fricas [A]  time = 0.218486, size = 90, normalized size = 0.78 \[ -\frac{1}{384000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (28800 \, x^{3} + 6880 \, x^{2} - 23420 \, x - 1959\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 91839 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/384000*sqrt(10)*(2*sqrt(10)*(28800*x^3 + 6880*x^2 - 23420*x - 1959)*sqrt(5*x
+ 3)*sqrt(-2*x + 1) - 91839*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(
-2*x + 1))))

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Sympy [A]  time = 20.4458, size = 316, normalized size = 2.72 \[ \frac{22 \sqrt{5} \left (\begin{cases} \frac{121 \sqrt{2} \left (- \frac{\sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{121} + \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}\right )}{32} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} + \frac{62 \sqrt{5} \left (\begin{cases} \frac{1331 \sqrt{2} \left (- \frac{\sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{1936} - \frac{\sqrt{2} \left (- 10 x + 5\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} + \frac{\operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16}\right )}{8} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} - \frac{12 \sqrt{5} \left (\begin{cases} \frac{14641 \sqrt{2} \left (- \frac{\sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{3872} - \frac{\sqrt{2} \left (- 10 x + 5\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{128}\right )}{16} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

22*sqrt(5)*Piecewise((121*sqrt(2)*(-sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x
+ 3)/121 + asin(sqrt(22)*sqrt(5*x + 3)/11))/32, (x >= -3/5) & (x < 1/2)))/625 +
62*sqrt(5)*Piecewise((1331*sqrt(2)*(-sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5
*x + 3)/1936 - sqrt(2)*(-10*x + 5)**(3/2)*(5*x + 3)**(3/2)/3993 + asin(sqrt(22)*
sqrt(5*x + 3)/11)/16)/8, (x >= -3/5) & (x < 1/2)))/625 - 12*sqrt(5)*Piecewise((1
4641*sqrt(2)*(-sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/3872 - sqrt(2)*
(-10*x + 5)**(3/2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)
*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/1874048 + 5*asin(sqrt(
22)*sqrt(5*x + 3)/11)/128)/16, (x >= -3/5) & (x < 1/2)))/625

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GIAC/XCAS [A]  time = 0.238411, size = 220, normalized size = 1.9 \[ -\frac{1}{320000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{24000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{200} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/320000*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))) + 1/200*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))
)