3.2315 \(\int \frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=115 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)}-\frac{4}{9} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{107}{27} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{41}{27} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-4*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/9 - ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(3*(2 + 3*x
)) - (107*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/27 - (41*Sqrt[7]*ArcTan[Sq
rt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/27

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Rubi [A]  time = 0.239718, antiderivative size = 115, 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 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)}-\frac{4}{9} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{107}{27} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{41}{27} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-4*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/9 - ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(3*(2 + 3*x
)) - (107*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/27 - (41*Sqrt[7]*ArcTan[Sq
rt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/27

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Rubi in Sympy [A]  time = 23.5506, size = 102, normalized size = 0.89 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3 \left (3 x + 2\right )} - \frac{4 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{9} - \frac{107 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{135} - \frac{41 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{27} \]

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)**2,x)

[Out]

-(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(3*(3*x + 2)) - 4*sqrt(-2*x + 1)*sqrt(5*x + 3)/
9 - 107*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/135 - 41*sqrt(7)*atan(sqrt(7)*s
qrt(-2*x + 1)/(7*sqrt(5*x + 3)))/27

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Mathematica [A]  time = 0.143666, size = 107, normalized size = 0.93 \[ \frac{1}{270} \left (-\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} (6 x+11)}{3 x+2}-205 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-107 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(11 + 6*x))/(2 + 3*x) - 205*Sqrt[7]*ArcTan[(-2
0 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] - 107*Sqrt[10]*ArcTan[(1 + 20*x)/(2*
Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/270

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Maple [A]  time = 0.017, size = 146, normalized size = 1.3 \[{\frac{1}{540+810\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 615\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-321\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+410\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -214\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -180\,x\sqrt{-10\,{x}^{2}-x+3}-330\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^2,x)

[Out]

1/270*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(615*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))*x-321*10^(1/2)*arcsin(20/11*x+1/11)*x+410*7^(1/2)*arctan(1/14*
(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-214*10^(1/2)*arcsin(20/11*x+1/11)-180*x*(
-10*x^2-x+3)^(1/2)-330*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)

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Maxima [A]  time = 1.50251, size = 101, normalized size = 0.88 \[ -\frac{107}{270} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{41}{54} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{2}{9} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{7 \, \sqrt{-10 \, x^{2} - x + 3}}{9 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-107/270*sqrt(10)*arcsin(20/11*x + 1/11) + 41/54*sqrt(7)*arcsin(37/11*x/abs(3*x
+ 2) + 20/11/abs(3*x + 2)) - 2/9*sqrt(-10*x^2 - x + 3) - 7/9*sqrt(-10*x^2 - x +
3)/(3*x + 2)

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Fricas [A]  time = 0.229258, size = 153, normalized size = 1.33 \[ \frac{\sqrt{5}{\left (41 \, \sqrt{7} \sqrt{5}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 6 \, \sqrt{5}{\left (6 \, x + 11\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 107 \, \sqrt{2}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{270 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/270*sqrt(5)*(41*sqrt(7)*sqrt(5)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqr
t(5*x + 3)*sqrt(-2*x + 1))) - 6*sqrt(5)*(6*x + 11)*sqrt(5*x + 3)*sqrt(-2*x + 1)
- 107*sqrt(2)*(3*x + 2)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x + 3)*sq
rt(-2*x + 1))))/(3*x + 2)

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

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)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.317401, size = 377, normalized size = 3.28 \[ \frac{41}{540} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{107}{270} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{2}{45} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{154 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{9 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

41/540*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)
))) - 107/270*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)))) - 2/45*sq
rt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 154/9*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))
/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)
*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)