3.2329 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ \frac{407 \sqrt{1-2 x} (5 x+3)^{5/2}}{112 (3 x+2)^3}+\frac{37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{56 (3 x+2)^4}+\frac{3 (1-2 x)^{5/2} (5 x+3)^{5/2}}{35 (3 x+2)^5}-\frac{4477 \sqrt{1-2 x} (5 x+3)^{3/2}}{3136 (3 x+2)^2}-\frac{147741 \sqrt{1-2 x} \sqrt{5 x+3}}{43904 (3 x+2)}-\frac{1625151 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

[Out]

(-147741*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43904*(2 + 3*x)) - (4477*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(3136*(2 + 3*x)^2) + (3*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(35*(2 +
3*x)^5) + (37*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(56*(2 + 3*x)^4) + (407*Sqrt[1 -
2*x]*(3 + 5*x)^(5/2))/(112*(2 + 3*x)^3) - (1625151*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7
]*Sqrt[3 + 5*x])])/(43904*Sqrt[7])

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Rubi [A]  time = 0.270748, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{407 \sqrt{1-2 x} (5 x+3)^{5/2}}{112 (3 x+2)^3}+\frac{37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{56 (3 x+2)^4}+\frac{3 (1-2 x)^{5/2} (5 x+3)^{5/2}}{35 (3 x+2)^5}-\frac{4477 \sqrt{1-2 x} (5 x+3)^{3/2}}{3136 (3 x+2)^2}-\frac{147741 \sqrt{1-2 x} \sqrt{5 x+3}}{43904 (3 x+2)}-\frac{1625151 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^6,x]

[Out]

(-147741*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43904*(2 + 3*x)) - (4477*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(3136*(2 + 3*x)^2) + (3*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(35*(2 +
3*x)^5) + (37*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(56*(2 + 3*x)^4) + (407*Sqrt[1 -
2*x]*(3 + 5*x)^(5/2))/(112*(2 + 3*x)^3) - (1625151*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7
]*Sqrt[3 + 5*x])])/(43904*Sqrt[7])

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Rubi in Sympy [A]  time = 20.1933, size = 165, normalized size = 0.92 \[ - \frac{407 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{5488 \left (3 x + 2\right )^{3}} - \frac{37 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{392 \left (3 x + 2\right )^{4}} + \frac{3 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{35 \left (3 x + 2\right )^{5}} + \frac{4477 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{21952 \left (3 x + 2\right )^{2}} + \frac{147741 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{43904 \left (3 x + 2\right )} - \frac{1625151 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{307328} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)*(3+5*x)**(3/2)/(2+3*x)**6,x)

[Out]

-407*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(5488*(3*x + 2)**3) - 37*(-2*x + 1)**(5/2)*
(5*x + 3)**(3/2)/(392*(3*x + 2)**4) + 3*(-2*x + 1)**(5/2)*(5*x + 3)**(5/2)/(35*(
3*x + 2)**5) + 4477*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(21952*(3*x + 2)**2) + 14774
1*sqrt(-2*x + 1)*sqrt(5*x + 3)/(43904*(3*x + 2)) - 1625151*sqrt(7)*atan(sqrt(7)*
sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/307328

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Mathematica [A]  time = 0.142323, size = 87, normalized size = 0.48 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (57469845 x^4+155783350 x^3+158785356 x^2+71866904 x+12157344\right )}{(3 x+2)^5}-8125755 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{3073280} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^6,x]

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(12157344 + 71866904*x + 158785356*x^2 + 155783
350*x^3 + 57469845*x^4))/(2 + 3*x)^5 - 8125755*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sq
rt[7 - 14*x]*Sqrt[3 + 5*x])])/3073280

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Maple [B]  time = 0.018, size = 298, normalized size = 1.7 \[{\frac{1}{3073280\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1974558465\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+2222994984\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^6,x)

[Out]

1/3073280*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(1974558465*7^(1/2)*arctan(1/14*(37*x+20)*
7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+6581861550*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))*x^4+8775815400*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10
*x^2-x+3)^(1/2))*x^3+804577830*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan
(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+2180966900*x^3*(-10*x^2-x+3)^(1
/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+2222
994984*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/
2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^5

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Maxima [A]  time = 1.51948, size = 306, normalized size = 1.7 \[ \frac{305065}{230496} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{35 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{111 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{392 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{4107 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{5488 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{183039 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{153664 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{2484735}{153664} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{1625151}{614656} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{2189253}{307328} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{724201 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{921984 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^6,x, algorithm="maxima")

[Out]

305065/230496*(-10*x^2 - x + 3)^(3/2) + 3/35*(-10*x^2 - x + 3)^(5/2)/(243*x^5 +
810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 111/392*(-10*x^2 - x + 3)^(5/2)/(81
*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 4107/5488*(-10*x^2 - x + 3)^(5/2)/(27*x^
3 + 54*x^2 + 36*x + 8) + 183039/153664*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4
) + 2484735/153664*sqrt(-10*x^2 - x + 3)*x + 1625151/614656*sqrt(7)*arcsin(37/11
*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2189253/307328*sqrt(-10*x^2 - x + 3) + 7
24201/921984*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 0.225397, size = 167, normalized size = 0.93 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (57469845 \, x^{4} + 155783350 \, x^{3} + 158785356 \, x^{2} + 71866904 \, x + 12157344\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 8125755 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{3073280 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^6,x, algorithm="fricas")

[Out]

1/3073280*sqrt(7)*(2*sqrt(7)*(57469845*x^4 + 155783350*x^3 + 158785356*x^2 + 718
66904*x + 12157344)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 8125755*(243*x^5 + 810*x^4 +
1080*x^3 + 720*x^2 + 240*x + 32)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*
sqrt(-2*x + 1))))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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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((1-2*x)**(3/2)*(3+5*x)**(3/2)/(2+3*x)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.490935, size = 594, normalized size = 3.3 \[ \frac{1625151}{6146560} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{14641 \,{\left (111 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 145040 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 66232320 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 11371136000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 682268160000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{21952 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^6,x, algorithm="giac")

[Out]

1625151/6146560*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)))) - 14641/21952*(111*sqrt(10)*((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)))^9 + 145040*
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)))^7 - 66232320*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 - 11371136000*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 - 682268160000*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))))/(((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)^5