3.2262 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx\)

Optimal. Leaf size=106 \[ \frac{1}{6} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{41}{72} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{793 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{216 \sqrt{10}}-\frac{2}{27} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-41*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/72 + (Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/6 + (793*
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(216*Sqrt[10]) - (2*Sqrt[7]*ArcTan[Sqrt[1 - 2*
x]/(Sqrt[7]*Sqrt[3 + 5*x])])/27

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Rubi [A]  time = 0.23443, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{1}{6} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{41}{72} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{793 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{216 \sqrt{10}}-\frac{2}{27} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-41*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/72 + (Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/6 + (793*
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(216*Sqrt[10]) - (2*Sqrt[7]*ArcTan[Sqrt[1 - 2*
x]/(Sqrt[7]*Sqrt[3 + 5*x])])/27

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Rubi in Sympy [A]  time = 22.689, size = 97, normalized size = 0.92 \[ \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{6} - \frac{41 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{72} + \frac{793 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2160} - \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-2*x + 1)*(5*x + 3)**(3/2)/6 - 41*sqrt(-2*x + 1)*sqrt(5*x + 3)/72 + 793*sqr
t(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/2160 - 2*sqrt(7)*atan(sqrt(7)*sqrt(-2*x +
1)/(7*sqrt(5*x + 3)))/27

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Mathematica [A]  time = 0.141606, size = 100, normalized size = 0.94 \[ \frac{300 \sqrt{1-2 x} \sqrt{5 x+3} (12 x-1)-160 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+793 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{4320} \]

Antiderivative was successfully verified.

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

[Out]

(300*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-1 + 12*x) - 160*Sqrt[7]*ArcTan[(-20 - 37*x)/(
2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 793*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*
x]*Sqrt[30 + 50*x])])/4320

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Maple [A]  time = 0.013, size = 98, normalized size = 0.9 \[{\frac{1}{4320}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 160\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +793\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +3600\,x\sqrt{-10\,{x}^{2}-x+3}-300\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4320*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(160*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-
10*x^2-x+3)^(1/2))+793*10^(1/2)*arcsin(20/11*x+1/11)+3600*x*(-10*x^2-x+3)^(1/2)-
300*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.48795, size = 93, normalized size = 0.88 \[ \frac{5}{6} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{793}{4320} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1}{27} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{5}{72} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/6*sqrt(-10*x^2 - x + 3)*x + 793/4320*sqrt(10)*arcsin(20/11*x + 1/11) + 1/27*sq
rt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 5/72*sqrt(-10*x^2 - x
+ 3)

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Fricas [A]  time = 0.231911, size = 122, normalized size = 1.15 \[ \frac{1}{4320} \, \sqrt{10}{\left (30 \, \sqrt{10}{\left (12 \, x - 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 16 \, \sqrt{10} \sqrt{7} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 793 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4320*sqrt(10)*(30*sqrt(10)*(12*x - 1)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 16*sqrt(1
0)*sqrt(7)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 793
*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{3 x + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.294094, size = 234, normalized size = 2.21 \[ \frac{1}{270} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1}{360} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 41 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{793}{4320} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/270*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))
)) + 1/360*(12*sqrt(5)*(5*x + 3) - 41*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 7
93/4320*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))