3.2465 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)} \, dx\)

Optimal. Leaf size=108 \[ -\frac{5}{12} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{455}{144} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{3035}{432} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{27 \sqrt{7}} \]

[Out]

(-455*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/144 - (5*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/12 +
(3035*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/432 + (2*ArcTan[Sqrt[1 - 2*x]/
(Sqrt[7]*Sqrt[3 + 5*x])])/(27*Sqrt[7])

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Rubi [A]  time = 0.236484, antiderivative size = 108, 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 \[ -\frac{5}{12} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{455}{144} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{3035}{432} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{27 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-455*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/144 - (5*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/12 +
(3035*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/432 + (2*ArcTan[Sqrt[1 - 2*x]/
(Sqrt[7]*Sqrt[3 + 5*x])])/(27*Sqrt[7])

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Rubi in Sympy [A]  time = 22.5449, size = 99, normalized size = 0.92 \[ - \frac{5 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{12} - \frac{455 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{144} + \frac{3035 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{864} + \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{189} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/12 - 455*sqrt(-2*x + 1)*sqrt(5*x + 3)/144 + 3
035*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/864 + 2*sqrt(7)*atan(sqrt(7)*sqrt(-
2*x + 1)/(7*sqrt(5*x + 3)))/189

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Mathematica [A]  time = 0.17759, size = 100, normalized size = 0.93 \[ \frac{-420 \sqrt{1-2 x} \sqrt{5 x+3} (60 x+127)+64 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+21245 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{12096} \]

Antiderivative was successfully verified.

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

[Out]

(-420*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(127 + 60*x) + 64*Sqrt[7]*ArcTan[(-20 - 37*x)/
(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 21245*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 -
2*x]*Sqrt[30 + 50*x])])/12096

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Maple [A]  time = 0.019, size = 98, normalized size = 0.9 \[ -{\frac{1}{12096}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 64\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -21245\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +25200\,x\sqrt{-10\,{x}^{2}-x+3}+53340\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12096*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(64*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))-21245*10^(1/2)*arcsin(20/11*x+1/11)+25200*x*(-10*x^2-x+3)^(1
/2)+53340*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50766, size = 93, normalized size = 0.86 \[ -\frac{25}{12} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{3035}{1728} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{1}{189} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{635}{144} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/12*sqrt(-10*x^2 - x + 3)*x + 3035/1728*sqrt(10)*arcsin(20/11*x + 1/11) - 1/1
89*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 635/144*sqrt(-10*
x^2 - x + 3)

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Fricas [A]  time = 0.228761, size = 138, normalized size = 1.28 \[ -\frac{1}{12096} \, \sqrt{7} \sqrt{2}{\left (30 \, \sqrt{7} \sqrt{2}{\left (60 \, x + 127\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 3035 \, \sqrt{7} \sqrt{5} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 32 \, \sqrt{2} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12096*sqrt(7)*sqrt(2)*(30*sqrt(7)*sqrt(2)*(60*x + 127)*sqrt(5*x + 3)*sqrt(-2*
x + 1) - 3035*sqrt(7)*sqrt(5)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x +
3)*sqrt(-2*x + 1))) + 32*sqrt(2)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)
*sqrt(-2*x + 1))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.288813, size = 234, normalized size = 2.17 \[ -\frac{1}{1890} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1}{144} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 91 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{3035}{1728} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1890*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
)))) - 1/144*(12*sqrt(5)*(5*x + 3) + 91*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) +
3035/1728*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))))