3.2303 \(\int \frac{\sqrt{1-2 x} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{12 (1-2 x)^{3/2}}{275 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2}}{825 (5 x+3)^{3/2}}+\frac{3}{55} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{3 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(3/2))/(825*(3 + 5*x)^(3/2)) - (12*(1 - 2*x)^(3/2))/(275*Sqrt[3 +
5*x]) + (3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/55 + (3*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]
)/(5*Sqrt[10])

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Rubi [A]  time = 0.117089, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{12 (1-2 x)^{3/2}}{275 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2}}{825 (5 x+3)^{3/2}}+\frac{3}{55} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{3 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(3/2))/(825*(3 + 5*x)^(3/2)) - (12*(1 - 2*x)^(3/2))/(275*Sqrt[3 +
5*x]) + (3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/55 + (3*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]
)/(5*Sqrt[10])

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Rubi in Sympy [A]  time = 10.6196, size = 85, normalized size = 0.9 \[ - \frac{12 \left (- 2 x + 1\right )^{\frac{3}{2}}}{275 \sqrt{5 x + 3}} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{825 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{3 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{55} + \frac{3 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{50} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-12*(-2*x + 1)**(3/2)/(275*sqrt(5*x + 3)) - 2*(-2*x + 1)**(3/2)/(825*(5*x + 3)**
(3/2)) + 3*sqrt(-2*x + 1)*sqrt(5*x + 3)/55 + 3*sqrt(10)*asin(sqrt(22)*sqrt(5*x +
3)/11)/50

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Mathematica [A]  time = 0.148167, size = 60, normalized size = 0.64 \[ \frac{\sqrt{1-2 x} \left (297 x^2+278 x+59\right )}{165 (5 x+3)^{3/2}}-\frac{3 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(59 + 278*x + 297*x^2))/(165*(3 + 5*x)^(3/2)) - (3*ArcSin[Sqrt[5/
11]*Sqrt[1 - 2*x]])/(5*Sqrt[10])

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Maple [A]  time = 0.016, size = 113, normalized size = 1.2 \[{\frac{1}{3300} \left ( 2475\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+2970\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+5940\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+891\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +5560\,x\sqrt{-10\,{x}^{2}-x+3}+1180\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3300*(2475*10^(1/2)*arcsin(20/11*x+1/11)*x^2+2970*10^(1/2)*arcsin(20/11*x+1/11
)*x+5940*x^2*(-10*x^2-x+3)^(1/2)+891*10^(1/2)*arcsin(20/11*x+1/11)+5560*x*(-10*x
^2-x+3)^(1/2)+1180*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x
)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Fricas [A]  time = 0.223124, size = 113, normalized size = 1.2 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (297 \, x^{2} + 278 \, x + 59\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 99 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{3300 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3300*sqrt(10)*(2*sqrt(10)*(297*x^2 + 278*x + 59)*sqrt(5*x + 3)*sqrt(-2*x + 1)
+ 99*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*
x + 1))))/(25*x^2 + 30*x + 9)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.287414, size = 220, normalized size = 2.34 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{330000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{9}{625} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{3}{50} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{131 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{27500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{393 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{20625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/330000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 9/62
5*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 3/50*sqrt(10)*arcsin(1/11*sqrt(22)*sqr
t(5*x + 3)) - 131/27500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x +
3) + 1/20625*(393*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4
*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3