3.2580 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=173 \[ \frac{465 \sqrt{5 x+3}}{9604 \sqrt{1-2 x}}-\frac{85 \sqrt{5 x+3}}{2744 \sqrt{1-2 x} (3 x+2)}-\frac{23 \sqrt{5 x+3}}{196 \sqrt{1-2 x} (3 x+2)^2}-\frac{32 \sqrt{5 x+3}}{147 \sqrt{1-2 x} (3 x+2)^3}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac{9395 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

[Out]

(465*Sqrt[3 + 5*x])/(9604*Sqrt[1 - 2*x]) + (11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2
)*(2 + 3*x)^3) - (32*Sqrt[3 + 5*x])/(147*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (23*Sqrt[3
+ 5*x])/(196*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (85*Sqrt[3 + 5*x])/(2744*Sqrt[1 - 2*x
]*(2 + 3*x)) - (9395*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[
7])

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Rubi [A]  time = 0.404115, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{465 \sqrt{5 x+3}}{9604 \sqrt{1-2 x}}-\frac{85 \sqrt{5 x+3}}{2744 \sqrt{1-2 x} (3 x+2)}-\frac{23 \sqrt{5 x+3}}{196 \sqrt{1-2 x} (3 x+2)^2}-\frac{32 \sqrt{5 x+3}}{147 \sqrt{1-2 x} (3 x+2)^3}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac{9395 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 5*x)^(3/2)/((1 - 2*x)^(5/2)*(2 + 3*x)^4),x]

[Out]

(465*Sqrt[3 + 5*x])/(9604*Sqrt[1 - 2*x]) + (11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2
)*(2 + 3*x)^3) - (32*Sqrt[3 + 5*x])/(147*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (23*Sqrt[3
+ 5*x])/(196*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (85*Sqrt[3 + 5*x])/(2744*Sqrt[1 - 2*x
]*(2 + 3*x)) - (9395*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[
7])

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Rubi in Sympy [A]  time = 35.966, size = 160, normalized size = 0.92 \[ - \frac{9395 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{134456} + \frac{465 \sqrt{5 x + 3}}{9604 \sqrt{- 2 x + 1}} - \frac{85 \sqrt{5 x + 3}}{2744 \sqrt{- 2 x + 1} \left (3 x + 2\right )} - \frac{23 \sqrt{5 x + 3}}{196 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} - \frac{32 \sqrt{5 x + 3}}{147 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} + \frac{11 \sqrt{5 x + 3}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}} \]

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)**4,x)

[Out]

-9395*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/134456 + 465*sqrt(5
*x + 3)/(9604*sqrt(-2*x + 1)) - 85*sqrt(5*x + 3)/(2744*sqrt(-2*x + 1)*(3*x + 2))
- 23*sqrt(5*x + 3)/(196*sqrt(-2*x + 1)*(3*x + 2)**2) - 32*sqrt(5*x + 3)/(147*sq
rt(-2*x + 1)*(3*x + 2)**3) + 11*sqrt(5*x + 3)/(21*(-2*x + 1)**(3/2)*(3*x + 2)**3
)

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Mathematica [A]  time = 0.158621, size = 87, normalized size = 0.5 \[ \frac{\frac{14 \sqrt{5 x+3} \left (-150660 x^4-193860 x^3+17127 x^2+80510 x+19296\right )}{(1-2 x)^{3/2} (3 x+2)^3}-28185 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{806736} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 5*x)^(3/2)/((1 - 2*x)^(5/2)*(2 + 3*x)^4),x]

[Out]

((14*Sqrt[3 + 5*x]*(19296 + 80510*x + 17127*x^2 - 193860*x^3 - 150660*x^4))/((1
- 2*x)^(3/2)*(2 + 3*x)^3) - 28185*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*
Sqrt[3 + 5*x])])/806736

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Maple [B]  time = 0.022, size = 305, normalized size = 1.8 \[{\frac{1}{806736\, \left ( 2+3\,x \right ) ^{3} \left ( -1+2\,x \right ) ^{2}} \left ( 3043980\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}+3043980\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-1268325\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-2109240\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-1634730\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-2714040\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+112740\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+239778\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+225480\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1127140\,x\sqrt{-10\,{x}^{2}-x+3}+270144\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^4,x)

[Out]

1/806736*(3043980*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5
+3043980*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4-1268325*
7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-2109240*x^4*(-10*
x^2-x+3)^(1/2)-1634730*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)*x^2-2714040*x^3*(-10*x^2-x+3)^(1/2)+112740*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/
2)/(-10*x^2-x+3)^(1/2))*x+239778*x^2*(-10*x^2-x+3)^(1/2)+225480*7^(1/2)*arctan(1
/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1127140*x*(-10*x^2-x+3)^(1/2)+270144*
(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3/(-1+2*x)^2/(-10*x^2-x
+3)^(1/2)

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Maxima [A]  time = 1.51164, size = 324, normalized size = 1.87 \[ \frac{9395}{268912} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{2325 \, x}{9604 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{5395}{57624 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{7625 \, x}{12348 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{1}{567 \,{\left (27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 54 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 8 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{169}{5292 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{1987}{10584 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{2165}{222264 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)/((3*x + 2)^4*(-2*x + 1)^(5/2)),x, algorithm="maxima")

[Out]

9395/268912*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2325/960
4*x/sqrt(-10*x^2 - x + 3) + 5395/57624/sqrt(-10*x^2 - x + 3) + 7625/12348*x/(-10
*x^2 - x + 3)^(3/2) + 1/567/(27*(-10*x^2 - x + 3)^(3/2)*x^3 + 54*(-10*x^2 - x +
3)^(3/2)*x^2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^2 - x + 3)^(3/2)) - 169/5
292/(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)) + 1987/10584/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(
3/2)) + 2165/222264/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.226505, size = 167, normalized size = 0.97 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (150660 \, x^{4} + 193860 \, x^{3} - 17127 \, x^{2} - 80510 \, x - 19296\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 28185 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{806736 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)/((3*x + 2)^4*(-2*x + 1)^(5/2)),x, algorithm="fricas")

[Out]

-1/806736*sqrt(7)*(2*sqrt(7)*(150660*x^4 + 193860*x^3 - 17127*x^2 - 80510*x - 19
296)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 28185*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 +
4*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(108*
x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)

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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)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.558812, size = 482, normalized size = 2.79 \[ \frac{1879}{537824} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{8 \,{\left (512 \, \sqrt{5}{\left (5 \, x + 3\right )} - 3201 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1260525 \,{\left (2 \, x - 1\right )}^{2}} - \frac{99 \,{\left (727 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 548800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 20776000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{67228 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)/((3*x + 2)^4*(-2*x + 1)^(5/2)),x, algorithm="giac")

[Out]

1879/537824*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqr
t(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqr
t(22)))) - 8/1260525*(512*sqrt(5)*(5*x + 3) - 3201*sqrt(5))*sqrt(5*x + 3)*sqrt(-
10*x + 5)/(2*x - 1)^2 - 99/67228*(727*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)))^5 + 5
48800*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 + 20776000*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)^3