3.2577 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx\)

Optimal. Leaf size=79 \[ \frac{2 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{2 \sqrt{5 x+3}}{49 \sqrt{1-2 x}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

[Out]

(-2*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]) + (2*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2))
- (2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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Rubi [A]  time = 0.124907, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{2 \sqrt{5 x+3}}{49 \sqrt{1-2 x}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]) + (2*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2))
- (2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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Rubi in Sympy [A]  time = 10.8325, size = 73, normalized size = 0.92 \[ - \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{343} - \frac{2 \sqrt{5 x + 3}}{49 \sqrt{- 2 x + 1}} + \frac{2 \left (5 x + 3\right )^{\frac{3}{2}}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

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),x)

[Out]

-2*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/343 - 2*sqrt(5*x + 3)/
(49*sqrt(-2*x + 1)) + 2*(5*x + 3)**(3/2)/(21*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.120111, size = 65, normalized size = 0.82 \[ \frac{2 \sqrt{5 x+3} (41 x+18)}{147 (1-2 x)^{3/2}}-\frac{\tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3 + 5*x]*(18 + 41*x))/(147*(1 - 2*x)^(3/2)) - ArcTan[(-20 - 37*x)/(2*Sqr
t[7 - 14*x]*Sqrt[3 + 5*x])]/(49*Sqrt[7])

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Maple [B]  time = 0.02, size = 154, normalized size = 2. \[{\frac{1}{1029\, \left ( -1+2\,x \right ) ^{2}} \left ( 12\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-12\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+3\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +574\,x\sqrt{-10\,{x}^{2}-x+3}+252\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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),x)

[Out]

1/1029*(12*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-12*7^(
1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+3*7^(1/2)*arctan(1/14*
(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+574*x*(-10*x^2-x+3)^(1/2)+252*(-10*x^2-x+
3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50844, size = 140, normalized size = 1.77 \[ \frac{1}{343} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{205 \, x}{147 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{125 \, x^{2}}{6 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{37}{588 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1385 \, x}{84 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{67}{28 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/343*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 205/147*x/sqrt
(-10*x^2 - x + 3) + 125/6*x^2/(-10*x^2 - x + 3)^(3/2) - 37/588/sqrt(-10*x^2 - x
+ 3) + 1385/84*x/(-10*x^2 - x + 3)^(3/2) + 67/28/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.226159, size = 107, normalized size = 1.35 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (41 \, x + 18\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 3 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1029 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1029*sqrt(7)*(2*sqrt(7)*(41*x + 18)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 3*(4*x^2 -
4*x + 1)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(4*x^2
- 4*x + 1)

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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),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.265058, size = 153, normalized size = 1.94 \[ \frac{1}{3430} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{2 \,{\left (41 \, \sqrt{5}{\left (5 \, x + 3\right )} - 33 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{3675 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3430*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)
))) + 2/3675*(41*sqrt(5)*(5*x + 3) - 33*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(
2*x - 1)^2