3.2589 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=173 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^3}+\frac{15755 \sqrt{5 x+3}}{86436 \sqrt{1-2 x}}-\frac{2365 \sqrt{5 x+3}}{8232 \sqrt{1-2 x} (3 x+2)}-\frac{187 \sqrt{5 x+3}}{588 \sqrt{1-2 x} (3 x+2)^2}+\frac{32 \sqrt{5 x+3}}{441 \sqrt{1-2 x} (3 x+2)^3}-\frac{2585 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

[Out]

(15755*Sqrt[3 + 5*x])/(86436*Sqrt[1 - 2*x]) + (32*Sqrt[3 + 5*x])/(441*Sqrt[1 - 2
*x]*(2 + 3*x)^3) - (187*Sqrt[3 + 5*x])/(588*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (2365*S
qrt[3 + 5*x])/(8232*Sqrt[1 - 2*x]*(2 + 3*x)) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x
)^(3/2)*(2 + 3*x)^3) - (2585*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(192
08*Sqrt[7])

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Rubi [A]  time = 0.40268, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^3}+\frac{15755 \sqrt{5 x+3}}{86436 \sqrt{1-2 x}}-\frac{2365 \sqrt{5 x+3}}{8232 \sqrt{1-2 x} (3 x+2)}-\frac{187 \sqrt{5 x+3}}{588 \sqrt{1-2 x} (3 x+2)^2}+\frac{32 \sqrt{5 x+3}}{441 \sqrt{1-2 x} (3 x+2)^3}-\frac{2585 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(15755*Sqrt[3 + 5*x])/(86436*Sqrt[1 - 2*x]) + (32*Sqrt[3 + 5*x])/(441*Sqrt[1 - 2
*x]*(2 + 3*x)^3) - (187*Sqrt[3 + 5*x])/(588*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (2365*S
qrt[3 + 5*x])/(8232*Sqrt[1 - 2*x]*(2 + 3*x)) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x
)^(3/2)*(2 + 3*x)^3) - (2585*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(192
08*Sqrt[7])

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Rubi in Sympy [A]  time = 36.459, size = 160, normalized size = 0.92 \[ - \frac{2585 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{134456} + \frac{15755 \sqrt{5 x + 3}}{86436 \sqrt{- 2 x + 1}} - \frac{2365 \sqrt{5 x + 3}}{8232 \sqrt{- 2 x + 1} \left (3 x + 2\right )} - \frac{187 \sqrt{5 x + 3}}{588 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} + \frac{32 \sqrt{5 x + 3}}{441 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2585*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/134456 + 15755*sqrt
(5*x + 3)/(86436*sqrt(-2*x + 1)) - 2365*sqrt(5*x + 3)/(8232*sqrt(-2*x + 1)*(3*x
+ 2)) - 187*sqrt(5*x + 3)/(588*sqrt(-2*x + 1)*(3*x + 2)**2) + 32*sqrt(5*x + 3)/(
441*sqrt(-2*x + 1)*(3*x + 2)**3) + 11*(5*x + 3)**(3/2)/(21*(-2*x + 1)**(3/2)*(3*
x + 2)**3)

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Mathematica [A]  time = 0.155882, size = 87, normalized size = 0.5 \[ \frac{\frac{14 \sqrt{5 x+3} \left (-567180 x^4-552780 x^3+169221 x^2+304730 x+75888\right )}{(1-2 x)^{3/2} (3 x+2)^3}-7755 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{806736} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[3 + 5*x]*(75888 + 304730*x + 169221*x^2 - 552780*x^3 - 567180*x^4))/((
1 - 2*x)^(3/2)*(2 + 3*x)^3) - 7755*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]
*Sqrt[3 + 5*x])])/806736

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Maple [B]  time = 0.023, size = 305, normalized size = 1.8 \[{\frac{1}{806736\, \left ( 2+3\,x \right ) ^{3} \left ( -1+2\,x \right ) ^{2}} \left ( 837540\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}+837540\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-348975\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-7940520\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-449790\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-7738920\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+31020\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+2369094\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+62040\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +4266220\,x\sqrt{-10\,{x}^{2}-x+3}+1062432\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^4,x)

[Out]

1/806736*(837540*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+
837540*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4-348975*7^(
1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-7940520*x^4*(-10*x^2
-x+3)^(1/2)-449790*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^
2-7738920*x^3*(-10*x^2-x+3)^(1/2)+31020*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-
10*x^2-x+3)^(1/2))*x+2369094*x^2*(-10*x^2-x+3)^(1/2)+62040*7^(1/2)*arctan(1/14*(
37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+4266220*x*(-10*x^2-x+3)^(1/2)+1062432*(-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 = 1.51144, size = 324, normalized size = 1.87 \[ \frac{2585}{268912} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{78775 \, x}{86436 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{11755}{172872 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{17875 \, x}{12348 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1}{1701 \,{\left (27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 54 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 8 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{239}{15876 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{4997}{31752 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{901885}{666792 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2585/268912*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 78775/86
436*x/sqrt(-10*x^2 - x + 3) + 11755/172872/sqrt(-10*x^2 - x + 3) + 17875/12348*x
/(-10*x^2 - x + 3)^(3/2) - 1/1701/(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)) +
239/15876/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(-10*x^2 - x + 3)^(3/2)*x + 4*(-1
0*x^2 - x + 3)^(3/2)) - 4997/31752/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x
+ 3)^(3/2)) + 901885/666792/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.223124, size = 167, normalized size = 0.97 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (567180 \, x^{4} + 552780 \, x^{3} - 169221 \, x^{2} - 304730 \, x - 75888\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 7755 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{806736 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/806736*sqrt(7)*(2*sqrt(7)*(567180*x^4 + 552780*x^3 - 169221*x^2 - 304730*x -
75888)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 7755*(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)**(5/2)/(1-2*x)**(5/2)/(2+3*x)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.566207, size = 482, normalized size = 2.79 \[ \frac{517}{537824} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{88 \,{\left (151 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1023 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1260525 \,{\left (2 \, x - 1\right )}^{2}} - \frac{11 \,{\left (3629 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 2900800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 755384000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{67228 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

517/537824*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)))) - 88/1260525*(151*sqrt(5)*(5*x + 3) - 1023*sqrt(5))*sqrt(5*x + 3)*sqrt(-
10*x + 5)/(2*x - 1)^2 - 11/67228*(3629*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 +
2900800*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 + 755384000*sqrt(10)*((sqrt(2)*sq
rt(-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