3.2578 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^2} \, dx\)

Optimal. Leaf size=122 \[ \frac{4 (5 x+3)^{5/2}}{231 (1-2 x)^{3/2} (3 x+2)}+\frac{190 (5 x+3)^{3/2}}{1617 \sqrt{1-2 x} (3 x+2)}+\frac{95 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 (3 x+2)}+\frac{95 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]

[Out]

(95*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3773*(2 + 3*x)) + (190*(3 + 5*x)^(3/2))/(1617*
Sqrt[1 - 2*x]*(2 + 3*x)) + (4*(3 + 5*x)^(5/2))/(231*(1 - 2*x)^(3/2)*(2 + 3*x)) +
(95*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[7])

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Rubi [A]  time = 0.172, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{4 (5 x+3)^{5/2}}{231 (1-2 x)^{3/2} (3 x+2)}+\frac{190 (5 x+3)^{3/2}}{1617 \sqrt{1-2 x} (3 x+2)}+\frac{95 \sqrt{1-2 x} \sqrt{5 x+3}}{3773 (3 x+2)}+\frac{95 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(95*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3773*(2 + 3*x)) + (190*(3 + 5*x)^(3/2))/(1617*
Sqrt[1 - 2*x]*(2 + 3*x)) + (4*(3 + 5*x)^(5/2))/(231*(1 - 2*x)^(3/2)*(2 + 3*x)) +
(95*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[7])

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Rubi in Sympy [A]  time = 13.3028, size = 99, normalized size = 0.81 \[ \frac{95 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2401} + \frac{95 \sqrt{5 x + 3}}{343 \sqrt{- 2 x + 1}} - \frac{95 \left (5 x + 3\right )^{\frac{3}{2}}}{147 \left (- 2 x + 1\right )^{\frac{3}{2}}} + \frac{3 \left (5 x + 3\right )^{\frac{5}{2}}}{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )} \]

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)**2,x)

[Out]

95*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/2401 + 95*sqrt(5*x + 3
)/(343*sqrt(-2*x + 1)) - 95*(5*x + 3)**(3/2)/(147*(-2*x + 1)**(3/2)) + 3*(5*x +
3)**(5/2)/(7*(-2*x + 1)**(3/2)*(3*x + 2))

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Mathematica [A]  time = 0.0925864, size = 77, normalized size = 0.63 \[ \frac{\sqrt{5 x+3} \left (-660 x^2+310 x+549\right )}{1029 (1-2 x)^{3/2} (3 x+2)}+\frac{95 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{686 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[3 + 5*x]*(549 + 310*x - 660*x^2))/(1029*(1 - 2*x)^(3/2)*(2 + 3*x)) + (95*A
rcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(686*Sqrt[7])

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Maple [B]  time = 0.021, size = 209, normalized size = 1.7 \[ -{\frac{1}{ \left ( 28812+43218\,x \right ) \left ( -1+2\,x \right ) ^{2}} \left ( 3420\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-1140\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-1425\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+9240\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+570\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -4340\,x\sqrt{-10\,{x}^{2}-x+3}-7686\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^2,x)

[Out]

-1/14406*(3420*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-11
40*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-1425*7^(1/2)*a
rctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+9240*x^2*(-10*x^2-x+3)^(1/2)
+570*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-4340*x*(-10*x^2-
x+3)^(1/2)-7686*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)/(-1+2*x
)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.51639, size = 163, normalized size = 1.34 \[ -\frac{95}{4802} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{550 \, x}{1029 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{20}{1029 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1825 \, x}{441 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{1}{189 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{3250}{1323 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-95/4802*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 550/1029*x/
sqrt(-10*x^2 - x + 3) - 20/1029/sqrt(-10*x^2 - x + 3) + 1825/441*x/(-10*x^2 - x
+ 3)^(3/2) + 1/189/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) + 3
250/1323/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.223447, size = 127, normalized size = 1.04 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (660 \, x^{2} - 310 \, x - 549\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 285 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{14406 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/14406*sqrt(7)*(2*sqrt(7)*(660*x^2 - 310*x - 549)*sqrt(5*x + 3)*sqrt(-2*x + 1)
+ 285*(12*x^3 - 4*x^2 - 5*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)
*sqrt(-2*x + 1))))/(12*x^3 - 4*x^2 - 5*x + 2)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.330088, size = 313, normalized size = 2.57 \[ -\frac{19}{9604} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{66 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{343 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} - \frac{2 \,{\left (116 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1023 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{25725 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-19/9604*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(2
2)))) + 66/343*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) - 2/25725*(116*sqrt(5)*(5*x + 3) - 1023*sqrt(5))*sqrt(5*x + 3)*sqrt(-1
0*x + 5)/(2*x - 1)^2