3.2314 \(\int \frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{2+3 x} \, dx\)

Optimal. Leaf size=106 \[ \frac{1}{6} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{107}{180} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{4091 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{540 \sqrt{10}}+\frac{14}{27} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/180 + ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/6 + (409
1*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(540*Sqrt[10]) + (14*Sqrt[7]*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/27

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Rubi [A]  time = 0.237023, 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 \[ \frac{1}{6} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{107}{180} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{4091 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{540 \sqrt{10}}+\frac{14}{27} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/180 + ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/6 + (409
1*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(540*Sqrt[10]) + (14*Sqrt[7]*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/27

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Rubi in Sympy [A]  time = 23.4516, size = 97, normalized size = 0.92 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{6} + \frac{107 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{180} + \frac{4091 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{5400} + \frac{14 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(3/2)*sqrt(5*x + 3)/6 + 107*sqrt(-2*x + 1)*sqrt(5*x + 3)/180 + 4091*
sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/5400 + 14*sqrt(7)*atan(sqrt(7)*sqrt(-2*
x + 1)/(7*sqrt(5*x + 3)))/27

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Mathematica [A]  time = 0.157991, size = 100, normalized size = 0.94 \[ \frac{60 \sqrt{1-2 x} \sqrt{5 x+3} (137-60 x)+2800 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+4091 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{10800} \]

Antiderivative was successfully verified.

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

[Out]

(60*(137 - 60*x)*Sqrt[1 - 2*x]*Sqrt[3 + 5*x] + 2800*Sqrt[7]*ArcTan[(-20 - 37*x)/
(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 4091*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 -
2*x]*Sqrt[30 + 50*x])])/10800

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Maple [A]  time = 0.013, size = 98, normalized size = 0.9 \[ -{\frac{1}{10800}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2800\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -4091\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +3600\,x\sqrt{-10\,{x}^{2}-x+3}-8220\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10800*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2800*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))-4091*10^(1/2)*arcsin(20/11*x+1/11)+3600*x*(-10*x^2-x+3)^(1
/2)-8220*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50075, size = 93, normalized size = 0.88 \[ -\frac{1}{3} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{4091}{10800} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{7}{27} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{137}{180} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(-10*x^2 - x + 3)*x + 4091/10800*sqrt(10)*arcsin(20/11*x + 1/11) - 7/27
*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 137/180*sqrt(-10*x^
2 - x + 3)

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Fricas [A]  time = 0.230471, size = 122, normalized size = 1.15 \[ -\frac{1}{10800} \, \sqrt{10}{\left (6 \, \sqrt{10}{\left (60 \, x - 137\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 280 \, \sqrt{10} \sqrt{7} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 4091 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10800*sqrt(10)*(6*sqrt(10)*(60*x - 137)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 280*sq
rt(10)*sqrt(7)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) -
4091*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. \[ \int \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3 x + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.282784, size = 234, normalized size = 2.21 \[ -\frac{7}{270} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1}{900} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 173 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{4091}{10800} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/270*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*
sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)
))) - 1/900*(12*sqrt(5)*(5*x + 3) - 173*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) +
4091/10800*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))