3.2529 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx\)

Optimal. Leaf size=122 \[ \frac{2 (5 x+3)^{5/2}}{7 \sqrt{1-2 x} (3 x+2)^2}+\frac{5 \sqrt{1-2 x} (5 x+3)^{3/2}}{98 (3 x+2)^2}+\frac{165 \sqrt{1-2 x} \sqrt{5 x+3}}{1372 (3 x+2)}+\frac{1815 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

[Out]

(165*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1372*(2 + 3*x)) + (5*Sqrt[1 - 2*x]*(3 + 5*x)^
(3/2))/(98*(2 + 3*x)^2) + (2*(3 + 5*x)^(5/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (1
815*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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Rubi [A]  time = 0.171623, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2 (5 x+3)^{5/2}}{7 \sqrt{1-2 x} (3 x+2)^2}+\frac{5 \sqrt{1-2 x} (5 x+3)^{3/2}}{98 (3 x+2)^2}+\frac{165 \sqrt{1-2 x} \sqrt{5 x+3}}{1372 (3 x+2)}+\frac{1815 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(165*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1372*(2 + 3*x)) + (5*Sqrt[1 - 2*x]*(3 + 5*x)^
(3/2))/(98*(2 + 3*x)^2) + (2*(3 + 5*x)^(5/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (1
815*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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Rubi in Sympy [A]  time = 13.696, size = 107, normalized size = 0.88 \[ \frac{165 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1372 \left (3 x + 2\right )} + \frac{1815 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9604} + \frac{55 \left (5 x + 3\right )^{\frac{3}{2}}}{98 \sqrt{- 2 x + 1} \left (3 x + 2\right )} - \frac{\left (5 x + 3\right )^{\frac{5}{2}}}{14 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

165*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1372*(3*x + 2)) + 1815*sqrt(7)*atan(sqrt(7)*sq
rt(-2*x + 1)/(7*sqrt(5*x + 3)))/9604 + 55*(5*x + 3)**(3/2)/(98*sqrt(-2*x + 1)*(3
*x + 2)) - (5*x + 3)**(5/2)/(14*sqrt(-2*x + 1)*(3*x + 2)**2)

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Mathematica [A]  time = 0.119916, size = 77, normalized size = 0.63 \[ \frac{\frac{14 \sqrt{5 x+3} \left (8110 x^2+11525 x+4068\right )}{\sqrt{1-2 x} (3 x+2)^2}+1815 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{19208} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[3 + 5*x]*(4068 + 11525*x + 8110*x^2))/(Sqrt[1 - 2*x]*(2 + 3*x)^2) + 18
15*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/19208

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Maple [B]  time = 0.02, size = 209, normalized size = 1.7 \[ -{\frac{1}{19208\, \left ( 2+3\,x \right ) ^{2} \left ( -1+2\,x \right ) } \left ( 32670\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+27225\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-7260\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+113540\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-7260\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +161350\,x\sqrt{-10\,{x}^{2}-x+3}+56952\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/19208*(32670*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+2
7225*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-7260*7^(1/2)
*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+113540*x^2*(-10*x^2-x+3)^(
1/2)-7260*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+161350*x*(-
10*x^2-x+3)^(1/2)+56952*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
^2/(-1+2*x)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.53416, size = 193, normalized size = 1.58 \[ -\frac{1815}{19208} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{20275 \, x}{6174 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{83665}{37044 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1}{378 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{125}{1764 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1815/19208*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 20275/61
74*x/sqrt(-10*x^2 - x + 3) + 83665/37044/sqrt(-10*x^2 - x + 3) + 1/378/(9*sqrt(-
10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) - 12
5/1764/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.234898, size = 127, normalized size = 1.04 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (8110 \, x^{2} + 11525 \, x + 4068\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1815 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{19208 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/19208*sqrt(7)*(2*sqrt(7)*(8110*x^2 + 11525*x + 4068)*sqrt(5*x + 3)*sqrt(-2*x
+ 1) + 1815*(18*x^3 + 15*x^2 - 4*x - 4)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*
x + 3)*sqrt(-2*x + 1))))/(18*x^3 + 15*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)**(5/2)/(1-2*x)**(3/2)/(2+3*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.399426, size = 381, normalized size = 3.12 \[ -\frac{363}{38416} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{242 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1715 \,{\left (2 \, x - 1\right )}} + \frac{121 \,{\left (\sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 360 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{98 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-363/38416*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)))) - 242/1715*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 121/98*(sqr
t(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 360*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(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