3.2588 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\)

Optimal. Leaf size=151 \[ \frac{4 (5 x+3)^{7/2}}{231 (1-2 x)^{3/2} (3 x+2)^2}+\frac{26 (5 x+3)^{5/2}}{231 \sqrt{1-2 x} (3 x+2)^2}+\frac{65 \sqrt{1-2 x} (5 x+3)^{3/2}}{3234 (3 x+2)^2}+\frac{65 \sqrt{1-2 x} \sqrt{5 x+3}}{1372 (3 x+2)}+\frac{715 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

[Out]

(65*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1372*(2 + 3*x)) + (65*Sqrt[1 - 2*x]*(3 + 5*x)^
(3/2))/(3234*(2 + 3*x)^2) + (26*(3 + 5*x)^(5/2))/(231*Sqrt[1 - 2*x]*(2 + 3*x)^2)
+ (4*(3 + 5*x)^(7/2))/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (715*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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Rubi [A]  time = 0.220137, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{4 (5 x+3)^{7/2}}{231 (1-2 x)^{3/2} (3 x+2)^2}+\frac{26 (5 x+3)^{5/2}}{231 \sqrt{1-2 x} (3 x+2)^2}+\frac{65 \sqrt{1-2 x} (5 x+3)^{3/2}}{3234 (3 x+2)^2}+\frac{65 \sqrt{1-2 x} \sqrt{5 x+3}}{1372 (3 x+2)}+\frac{715 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(65*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1372*(2 + 3*x)) + (65*Sqrt[1 - 2*x]*(3 + 5*x)^
(3/2))/(3234*(2 + 3*x)^2) + (26*(3 + 5*x)^(5/2))/(231*Sqrt[1 - 2*x]*(2 + 3*x)^2)
+ (4*(3 + 5*x)^(7/2))/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (715*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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Rubi in Sympy [A]  time = 17.185, size = 131, normalized size = 0.87 \[ \frac{715 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{9604} + \frac{715 \sqrt{5 x + 3}}{1372 \sqrt{- 2 x + 1}} - \frac{65 \left (5 x + 3\right )^{\frac{3}{2}}}{588 \sqrt{- 2 x + 1} \left (3 x + 2\right )} - \frac{13 \left (5 x + 3\right )^{\frac{5}{2}}}{42 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} + \frac{3 \left (5 x + 3\right )^{\frac{7}{2}}}{14 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

715*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/9604 + 715*sqrt(5*x +
3)/(1372*sqrt(-2*x + 1)) - 65*(5*x + 3)**(3/2)/(588*sqrt(-2*x + 1)*(3*x + 2)) -
13*(5*x + 3)**(5/2)/(42*(-2*x + 1)**(3/2)*(3*x + 2)) + 3*(5*x + 3)**(7/2)/(14*(
-2*x + 1)**(3/2)*(3*x + 2)**2)

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Mathematica [A]  time = 0.11075, size = 85, normalized size = 0.56 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (-10260 x^3-1620 x^2+13627 x+6732\right )}{\left (6 x^2+x-2\right )^2}+2145 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{57624} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(6732 + 13627*x - 1620*x^2 - 10260*x^3))/(-2 +
x + 6*x^2)^2 + 2145*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])
])/57624

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Maple [B]  time = 0.021, size = 257, normalized size = 1.7 \[ -{\frac{1}{57624\, \left ( 2+3\,x \right ) ^{2} \left ( -1+2\,x \right ) ^{2}} \left ( 77220\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+25740\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-49335\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+143640\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-8580\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+22680\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+8580\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -190778\,x\sqrt{-10\,{x}^{2}-x+3}-94248\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/57624*(77220*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+2
5740*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-49335*7^(1/2
)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+143640*x^3*(-10*x^2-x+3
)^(1/2)-8580*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+22680*
x^2*(-10*x^2-x+3)^(1/2)+8580*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)
^(1/2))-190778*x*(-10*x^2-x+3)^(1/2)-94248*(-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 = 1.51346, size = 232, normalized size = 1.54 \[ -\frac{715}{19208} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{475 \, x}{686 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{215}{4116 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{17375 \, x}{2646 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1}{1134 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{1}{36 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{60695}{15876 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-715/19208*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 475/686*x
/sqrt(-10*x^2 - x + 3) - 215/4116/sqrt(-10*x^2 - x + 3) + 17375/2646*x/(-10*x^2
- x + 3)^(3/2) - 1/1134/(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)) + 1/36/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*
x^2 - x + 3)^(3/2)) + 60695/15876/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.224954, size = 147, normalized size = 0.97 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (10260 \, x^{3} + 1620 \, x^{2} - 13627 \, x - 6732\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 2145 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{57624 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/57624*sqrt(7)*(2*sqrt(7)*(10260*x^3 + 1620*x^2 - 13627*x - 6732)*sqrt(5*x + 3
)*sqrt(-2*x + 1) + 2145*(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)**(5/2)/(1-2*x)**(5/2)/(2+3*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.458173, size = 400, normalized size = 2.65 \[ -\frac{143}{38416} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{22 \,{\left (104 \, \sqrt{5}{\left (5 \, x + 3\right )} - 957 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{180075 \,{\left (2 \, x - 1\right )}^{2}} + \frac{11 \,{\left (223 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 80920 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{4802 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-143/38416*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)))) - 22/180075*(104*sqrt(5)*(5*x + 3) - 957*sqrt(5))*sqrt(5*x + 3)*sqrt(-10
*x + 5)/(2*x - 1)^2 + 11/4802*(223*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 + 8092
0*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))/s
qrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2