3.2263 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=115 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{3 (3 x+2)}+\frac{10}{9} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{41}{27} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{107 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{27 \sqrt{7}} \]

[Out]

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

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Rubi [A]  time = 0.236141, 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 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{3 (3 x+2)}+\frac{10}{9} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{41}{27} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{107 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{27 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.3148, size = 100, normalized size = 0.87 \[ \frac{10 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{9} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{3 \left (3 x + 2\right )} + \frac{41 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{54} + \frac{107 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{189} \]

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)**2,x)

[Out]

10*sqrt(-2*x + 1)*sqrt(5*x + 3)/9 - sqrt(-2*x + 1)*(5*x + 3)**(3/2)/(3*(3*x + 2)
) + 41*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/54 + 107*sqrt(7)*atan(sqrt(7)*sq
rt(-2*x + 1)/(7*sqrt(5*x + 3)))/189

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Mathematica [A]  time = 0.226645, size = 107, normalized size = 0.93 \[ \frac{1}{756} \left (\frac{84 \sqrt{1-2 x} \sqrt{5 x+3} (15 x+11)}{3 x+2}+214 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+287 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((84*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(11 + 15*x))/(2 + 3*x) + 214*Sqrt[7]*ArcTan[(-2
0 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 287*Sqrt[10]*ArcTan[(1 + 20*x)/(2*
Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/756

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Maple [A]  time = 0.016, size = 146, normalized size = 1.3 \[ -{\frac{1}{1512+2268\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 642\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-861\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+428\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -574\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1260\,x\sqrt{-10\,{x}^{2}-x+3}-924\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^2,x)

[Out]

-1/756*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(642*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-
10*x^2-x+3)^(1/2))*x-861*10^(1/2)*arcsin(20/11*x+1/11)*x+428*7^(1/2)*arctan(1/14
*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-574*10^(1/2)*arcsin(20/11*x+1/11)-1260*x
*(-10*x^2-x+3)^(1/2)-924*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)

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Maxima [A]  time = 1.51733, size = 101, normalized size = 0.88 \[ \frac{41}{108} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{107}{378} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{5}{9} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{\sqrt{-10 \, x^{2} - x + 3}}{9 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

41/108*sqrt(10)*arcsin(20/11*x + 1/11) - 107/378*sqrt(7)*arcsin(37/11*x/abs(3*x
+ 2) + 20/11/abs(3*x + 2)) + 5/9*sqrt(-10*x^2 - x + 3) + 1/9*sqrt(-10*x^2 - x +
3)/(3*x + 2)

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Fricas [A]  time = 0.233769, size = 161, normalized size = 1.4 \[ \frac{\sqrt{7} \sqrt{2}{\left (6 \, \sqrt{7} \sqrt{2}{\left (15 \, x + 11\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 41 \, \sqrt{7} \sqrt{5}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 107 \, \sqrt{2}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{756 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/756*sqrt(7)*sqrt(2)*(6*sqrt(7)*sqrt(2)*(15*x + 11)*sqrt(5*x + 3)*sqrt(-2*x + 1
) + 41*sqrt(7)*sqrt(5)*(3*x + 2)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*
x + 3)*sqrt(-2*x + 1))) - 107*sqrt(2)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)/
(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(3*x + 2)

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

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)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.325645, size = 377, normalized size = 3.28 \[ -\frac{107}{3780} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{41}{108} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1}{9} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{22 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{9 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-107/3780*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)))) + 41/108*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)))) + 1/9*s
qrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 22/9*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)