3.2365 \(\int \frac{(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{2 (1-2 x)^{5/2}}{165 (5 x+3)^{3/2}}-\frac{38 (1-2 x)^{3/2}}{165 \sqrt{5 x+3}}-\frac{38}{275} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{19}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-2*(1 - 2*x)^(5/2))/(165*(3 + 5*x)^(3/2)) - (38*(1 - 2*x)^(3/2))/(165*Sqrt[3 +
5*x]) - (38*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/275 - (19*Sqrt[2/5]*ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]])/25

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Rubi [A]  time = 0.0958403, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{2 (1-2 x)^{5/2}}{165 (5 x+3)^{3/2}}-\frac{38 (1-2 x)^{3/2}}{165 \sqrt{5 x+3}}-\frac{38}{275} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{19}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(5/2))/(165*(3 + 5*x)^(3/2)) - (38*(1 - 2*x)^(3/2))/(165*Sqrt[3 +
5*x]) - (38*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/275 - (19*Sqrt[2/5]*ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]])/25

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Rubi in Sympy [A]  time = 9.14851, size = 87, normalized size = 0.91 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{165 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{38 \left (- 2 x + 1\right )^{\frac{3}{2}}}{165 \sqrt{5 x + 3}} - \frac{38 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{275} - \frac{19 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(5/2)/(165*(5*x + 3)**(3/2)) - 38*(-2*x + 1)**(3/2)/(165*sqrt(5*x
+ 3)) - 38*sqrt(-2*x + 1)*sqrt(5*x + 3)/275 - 19*sqrt(10)*asin(sqrt(22)*sqrt(5*
x + 3)/11)/125

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Mathematica [A]  time = 0.139098, size = 62, normalized size = 0.65 \[ \frac{19}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-\frac{2 \sqrt{1-2 x} \left (45 x^2+145 x+73\right )}{75 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(73 + 145*x + 45*x^2))/(75*(3 + 5*x)^(3/2)) + (19*Sqrt[2/5]*Ar
cSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/25

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Maple [A]  time = 0.016, size = 113, normalized size = 1.2 \[ -{\frac{1}{750} \left ( 1425\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+1710\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+900\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+513\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +2900\,x\sqrt{-10\,{x}^{2}-x+3}+1460\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/750*(1425*10^(1/2)*arcsin(20/11*x+1/11)*x^2+1710*10^(1/2)*arcsin(20/11*x+1/11
)*x+900*x^2*(-10*x^2-x+3)^(1/2)+513*10^(1/2)*arcsin(20/11*x+1/11)+2900*x*(-10*x^
2-x+3)^(1/2)+1460*(-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 [A]  time = 1.49668, size = 161, normalized size = 1.68 \[ -\frac{19}{250} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{75 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{25 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{11 \, \sqrt{-10 \, x^{2} - x + 3}}{375 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{283 \, \sqrt{-10 \, x^{2} - x + 3}}{375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-19/250*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1/75*(-10*x^2 - x + 3)^(3/2)/(1
25*x^3 + 225*x^2 + 135*x + 27) + 3/25*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9
) - 11/375*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 283/375*sqrt(-10*x^2 - x
+ 3)/(5*x + 3)

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Fricas [A]  time = 0.220018, size = 122, normalized size = 1.27 \[ -\frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (45 \, x^{2} + 145 \, x + 73\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 57 \, \sqrt{2}{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/750*sqrt(5)*(4*sqrt(5)*(45*x^2 + 145*x + 73)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 5
7*sqrt(2)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(5)*sqrt(2)*(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((1-2*x)**(3/2)*(2+3*x)/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278669, size = 220, normalized size = 2.29 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{30000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{6}{625} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{19}{125} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{61 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{2500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{183 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{1875 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 6/625
*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 19/125*sqrt(10)*arcsin(1/11*sqrt(22)*sq
rt(5*x + 3)) - 61/2500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x +
3) + 1/1875*(183*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*s
qrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3