3.2438 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=166 \[ \frac{7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}+\frac{1784635 \sqrt{1-2 x}}{72 \sqrt{5 x+3}}+\frac{7843 \sqrt{1-2 x}}{24 (3 x+2) (5 x+3)^{3/2}}+\frac{77 \sqrt{1-2 x}}{4 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{196735 \sqrt{1-2 x}}{72 (5 x+3)^{3/2}}-\frac{1361195 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]

[Out]

(-196735*Sqrt[1 - 2*x])/(72*(3 + 5*x)^(3/2)) + (7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^
3*(3 + 5*x)^(3/2)) + (77*Sqrt[1 - 2*x])/(4*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (7843*
Sqrt[1 - 2*x])/(24*(2 + 3*x)*(3 + 5*x)^(3/2)) + (1784635*Sqrt[1 - 2*x])/(72*Sqrt
[3 + 5*x]) - (1361195*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(8*Sqrt[7])

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Rubi [A]  time = 0.38767, antiderivative size = 166, 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 \[ \frac{7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}+\frac{1784635 \sqrt{1-2 x}}{72 \sqrt{5 x+3}}+\frac{7843 \sqrt{1-2 x}}{24 (3 x+2) (5 x+3)^{3/2}}+\frac{77 \sqrt{1-2 x}}{4 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{196735 \sqrt{1-2 x}}{72 (5 x+3)^{3/2}}-\frac{1361195 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-196735*Sqrt[1 - 2*x])/(72*(3 + 5*x)^(3/2)) + (7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^
3*(3 + 5*x)^(3/2)) + (77*Sqrt[1 - 2*x])/(4*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (7843*
Sqrt[1 - 2*x])/(24*(2 + 3*x)*(3 + 5*x)^(3/2)) + (1784635*Sqrt[1 - 2*x])/(72*Sqrt
[3 + 5*x]) - (1361195*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(8*Sqrt[7])

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Rubi in Sympy [A]  time = 36.5509, size = 153, normalized size = 0.92 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{9 \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{1784635 \sqrt{- 2 x + 1}}{72 \sqrt{5 x + 3}} - \frac{196735 \sqrt{- 2 x + 1}}{72 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{7843 \sqrt{- 2 x + 1}}{24 \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{77 \sqrt{- 2 x + 1}}{4 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{1361195 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{56} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)/(9*(3*x + 2)**3*(5*x + 3)**(3/2)) + 1784635*sqrt(-2*x + 1)/(
72*sqrt(5*x + 3)) - 196735*sqrt(-2*x + 1)/(72*(5*x + 3)**(3/2)) + 7843*sqrt(-2*x
+ 1)/(24*(3*x + 2)*(5*x + 3)**(3/2)) + 77*sqrt(-2*x + 1)/(4*(3*x + 2)**2*(5*x +
3)**(3/2)) - 1361195*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/56

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Mathematica [A]  time = 0.116622, size = 87, normalized size = 0.52 \[ \frac{\sqrt{1-2 x} \left (80308575 x^4+207031680 x^3+199977747 x^2+85776638 x+13784768\right )}{24 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{1361195 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{16 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(13784768 + 85776638*x + 199977747*x^2 + 207031680*x^3 + 80308575
*x^4))/(24*(2 + 3*x)^3*(3 + 5*x)^(3/2)) - (1361195*ArcTan[(-20 - 37*x)/(2*Sqrt[7
- 14*x]*Sqrt[3 + 5*x])])/(16*Sqrt[7])

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Maple [B]  time = 0.021, size = 298, normalized size = 1.8 \[{\frac{1}{336\, \left ( 2+3\,x \right ) ^{3}} \left ( 2756419875\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+2799688458\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/336*(2756419875*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5
[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+11282
945355*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+1124320050
*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+2898443520*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+2799688458*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))*(1-2*x)^(1/2)/(2+3*x)^3/(-
10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.4937, size = 324, normalized size = 1.95 \[ \frac{1361195}{112} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{1784635 \, x}{36 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1863329}{72 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{149501 \, x}{12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{2401}{243 \,{\left (27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 54 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 8 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{31213}{324 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{1115681}{648 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{13081615}{1944 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1361195/112*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1784635/
36*x/sqrt(-10*x^2 - x + 3) + 1863329/72/sqrt(-10*x^2 - x + 3) + 149501/12*x/(-10
*x^2 - x + 3)^(3/2) + 2401/243/(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)) + 31
213/324/(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)) + 1115681/648/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x +
3)^(3/2)) - 13081615/1944/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.225174, size = 167, normalized size = 1.01 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (80308575 \, x^{4} + 207031680 \, x^{3} + 199977747 \, x^{2} + 85776638 \, x + 13784768\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 4083585 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{336 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/336*sqrt(7)*(2*sqrt(7)*(80308575*x^4 + 207031680*x^3 + 199977747*x^2 + 8577663
8*x + 13784768)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 4083585*(675*x^5 + 2160*x^4 + 276
3*x^3 + 1766*x^2 + 564*x + 72)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sq
rt(-2*x + 1))))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)

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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)**(5/2)/(2+3*x)**4/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.444853, size = 591, normalized size = 3.56 \[ -\frac{11}{48} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{272239}{224} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + 748 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} + \frac{11 \,{\left (63359 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 30251200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 3730664000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{4 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11/48*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 + 272239/224*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)))) + 748*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))) + 11/4*(63359*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 + 302512
00*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 + 3730664000*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