3.2579 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\)

Optimal. Leaf size=144 \[ \frac{5 \sqrt{5 x+3}}{42 \sqrt{1-2 x}}-\frac{5 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)}-\frac{3 \sqrt{5 x+3}}{14 \sqrt{1-2 x} (3 x+2)^2}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac{5 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]

[Out]

(5*Sqrt[3 + 5*x])/(42*Sqrt[1 - 2*x]) + (11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2
+ 3*x)^2) - (3*Sqrt[3 + 5*x])/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (5*Sqrt[3 + 5*x]
)/(28*Sqrt[1 - 2*x]*(2 + 3*x)) - (5*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])
])/(28*Sqrt[7])

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Rubi [A]  time = 0.328194, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{5 \sqrt{5 x+3}}{42 \sqrt{1-2 x}}-\frac{5 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)}-\frac{3 \sqrt{5 x+3}}{14 \sqrt{1-2 x} (3 x+2)^2}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac{5 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(5*Sqrt[3 + 5*x])/(42*Sqrt[1 - 2*x]) + (11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2
+ 3*x)^2) - (3*Sqrt[3 + 5*x])/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (5*Sqrt[3 + 5*x]
)/(28*Sqrt[1 - 2*x]*(2 + 3*x)) - (5*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])
])/(28*Sqrt[7])

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Rubi in Sympy [A]  time = 28.6151, size = 133, normalized size = 0.92 \[ - \frac{5 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{196} + \frac{5 \sqrt{5 x + 3}}{42 \sqrt{- 2 x + 1}} - \frac{5 \sqrt{5 x + 3}}{28 \sqrt{- 2 x + 1} \left (3 x + 2\right )} - \frac{3 \sqrt{5 x + 3}}{14 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} + \frac{11 \sqrt{5 x + 3}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/196 + 5*sqrt(5*x + 3)/
(42*sqrt(-2*x + 1)) - 5*sqrt(5*x + 3)/(28*sqrt(-2*x + 1)*(3*x + 2)) - 3*sqrt(5*x
+ 3)/(14*sqrt(-2*x + 1)*(3*x + 2)**2) + 11*sqrt(5*x + 3)/(21*(-2*x + 1)**(3/2)*
(3*x + 2)**2)

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Mathematica [A]  time = 0.100992, size = 85, normalized size = 0.59 \[ \frac{\sqrt{1-2 x} \sqrt{5 x+3} \left (-180 x^3-60 x^2+91 x+36\right )}{84 \left (6 x^2+x-2\right )^2}-\frac{5 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{56 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(36 + 91*x - 60*x^2 - 180*x^3))/(84*(-2 + x + 6*x^2
)^2) - (5*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(56*Sqrt[7])

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Maple [B]  time = 0.022, size = 257, normalized size = 1.8 \[{\frac{1}{1176\, \left ( 2+3\,x \right ) ^{2} \left ( -1+2\,x \right ) ^{2}} \left ( 540\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+180\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-345\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-2520\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-60\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-840\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+60\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1274\,x\sqrt{-10\,{x}^{2}-x+3}+504\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1176*(540*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+180*7
^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-345*7^(1/2)*arctan
(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-2520*x^3*(-10*x^2-x+3)^(1/2)-60
*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-840*x^2*(-10*x^2-x
+3)^(1/2)+60*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1274*x*(
-10*x^2-x+3)^(1/2)+504*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^
2/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50885, size = 232, normalized size = 1.61 \[ \frac{5}{392} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{25 \, x}{42 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{5}{84 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{125 \, x}{126 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{1}{378 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{43}{756 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{205}{252 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/392*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 25/42*x/sqrt(-
10*x^2 - x + 3) + 5/84/sqrt(-10*x^2 - x + 3) + 125/126*x/(-10*x^2 - x + 3)^(3/2)
+ 1/378/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(-10*x^2 - x + 3)^(3/2)*x + 4*(-10*
x^2 - x + 3)^(3/2)) - 43/756/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^
(3/2)) + 205/252/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.226914, size = 147, normalized size = 1.02 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (180 \, x^{3} + 60 \, x^{2} - 91 \, x - 36\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 15 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1176 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1176*sqrt(7)*(2*sqrt(7)*(180*x^3 + 60*x^2 - 91*x - 36)*sqrt(5*x + 3)*sqrt(-2*
x + 1) - 15*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)
/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.430934, size = 400, normalized size = 2.78 \[ \frac{1}{784} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{8 \,{\left (157 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1056 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{180075 \,{\left (2 \, x - 1\right )}^{2}} - \frac{33 \,{\left (83 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 41720 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{4802 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/784*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))
)) - 8/180075*(157*sqrt(5)*(5*x + 3) - 1056*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x +
5)/(2*x - 1)^2 - 33/4802*(83*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)))^3 + 41720*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)^2